I have, too slowly, been coming to the realisation that
what I understand in mathematics is far ahead of what I can
write up in a polished form and that much of my contribution to the
progress of mathematics comes from my lectures. Hopefully, putting as many
of these lectures on the web as I can manage will help
mathematics move along as quickly as possible. (This is not a complete list.
There are some years where I seem to have no records at all, and
others for which I have records only of colloquium and invited
conference talks.)
Talks of Arun Ram in 2021

Specializations of Macdonald polynomials,
Solvable Lattice Models Seminar,
Stanford University, 12 October 2021.
Abstract:
Macdonald often included in his talks the
"specialization square" for symmetric Macdonald polynomials,
which has monomial symmetric functions on the top edge,
elementary symmetric functions on the right edge,
HallLittlewood polynomials on the left edge, Schur functions
on the diagonal and Jack polynomials in the upper right corner.
This talk will explore the analogous "specialization
square" for nonsymmetric Macdonald polynomials. The primary
tools are the monomial expansion formulas, the Eexpansion
formula, and the creation formula.

The quantum group is a fake,
Working seminar, University of Melbourne, 30 August 2021.
Abstract:
This talk was an attempt to organize the results of Drinfeld,
proving that the quantum group is isomorphic to the nonquantum group
and that this isomorphism (up to the twist that corrects the Rmatrix
to be symmetric) takes the trivial associator to the KZ associator
and the Rmatrix to the exponential of the Casimir. The focus was on
the sl_{2} case where, hopfully,
everything can be written out explicitly.

Principal specializatons of Macdonald polynomials,
Workshop on Macdonald polynomials,
Indian Institute of Science, Bangalore, 30 June 2021.
YouTube
Abstract:
The principal specialization of a Schur function
(or Weyl character) has a striking factorization as a product,
and specializing the parameter gives the Weyl dimension formula.
In type GL_n this gives the hook formula for the number of
column strict tableaux. The nonspecialized formula is often called
the quantum dimension. Amazingly the Macdonald polynomials have
similar formulas for their principal specializations, which might
be thought of as elliptic Weyl dimension formulas.

Five (q,t) analogues of Kostka numbers,
Workshop on Macdonald polynomials,
Indian Institute of Science, Bangalore, 28 June 2021.
YouTube
Abstract:
The Kostka numbers arise as the coefficients in the monomial expansion
of Schur functions. In the Macdonald polynomial case I know 5 different analogues
of these numbers coming from
 The monomial expansion of the P_{λ}
 The expansion of e_{μ} in terms of the P_{λ}
 The expansion of g_{μ} in terms of the P_{λ}
 The expansion of P_{μ}(q,qt)
in terms of the P_{λ}(q,t)
 The expansion of J_{μ} in terms of the big
Schurs S_{λ}
In the first three cases I know formulas for these as
weighted sums of column strict tableaux.
I do not understand the relationships between these
5 different analogues of Kostka numbers.

The (q,t) Weyl character formula and the
(q,t) BosonFermion correspondence,
Workshop on Macdonald polynomials,
Indian Institute of Science, Bangalore, 21 June 2021.
YouTube
Abstract:
One of the favorite formulas for the Schur function is as a quotient
of two determinants. I will explain the corresponding formula for Macdonald
polynomials. The bosonfermion correspondence is the correspondence between
symmetric functions and skew symmetric functions which is given by multiplying
by the Vandermonde determinant. I will explain the (q,t) version of this correspondence
and how the Weyl character formula fits into this correspondence.

Level 0 modules of affine Lie algebras,
Solvable Lattice Models Seminar,
Stanford University, 16 June 2021.
YouTube
Abstract:
The favourite Rmatrices and transfer matrices
(which give the 6 vertex model) arise from the evaluation
representation of the standard representation of the quantum group
of sl(n). This is a level 0 representation of quantum affine sl(n)
and the lattice models are based on tensor powers of this representation.
With Finn McGlade and Yaping Yang we have written a survey about the
classification of these modules (by dominant weights for
extremal weight modules and by Drinfeld polynomials for
finite dimensional modules), their characters (which are qWhittaker
functions) and their crystals. I will try to sketch how I think this
category (of level 0 modules) is the controlling structure for
vertex operators, Fock spaces (Kyoto path model) and the Algebraic Bethe ansatz.

Representations of affine Hecke algebras IV:
Where do Hecke algebras come from?,
Representation Theory seminar,
University of Melbourne, 15 June 2021.
Abstract:
This will be a talk about the origins of the game:
 double centralisers and correspondence between
some representations of A and representations of the centralizer
 Convolution, Inducing from the trivial representation,
double coset algebras,
and correspondence between some representations of the group and
the representations of the double coset algebra
 Generators and relations for padic groups, cosets and double cosets
 Computation of the IwahoriHecke algebras and correspondence
between some representations of the padic groups
and representations of the Hecke algebra.

Representations of affine Hecke algebras III: Standard modules,
Representation Theory seminar,
University of Melbourne, 8 June 2021.
Abstract:
Most standard modules are built by induction.
I'll describe principal series modules and the basic theorems about
their simplicity and their composition factors. Then I'll define
tempered and square integrable modules, and explain how the
standard modules corespond to generalized Springer fibers and
the square integrable modules correspond to cuspidal nilpotent elements.
This provides an indexing of irreducible representations of
affine Hecke algebras by DeligneLanglands parameters.

Representations of affine Hecke algebras II: Central characters, weight spaces
and intertwiners,
Representation Theory seminar,
University of Melbourne, 1 June 2021.
Abstract:
I will try to explain how the representation theory of
the affine Hecke algebra is coded by local regions in a
hyperplane arrangement, sometimes called the Shi arrangement.
I'll introduce central characters and explain how to study
irreducible representations of affine Hecke algebras
by weight spaces and intertwiners. I'll explain the indexing of
calibrated simple Hmodules and the type GL_{n} indexing
of simple Hmodules by multisegments.

Representations of affine Hecke algebras I: Isogeny and lifting
representations of cyclotomic Hecke algebras,
Representation Theory seminar,
University of Melbourne, 25 May 2021.
Abstract:
In this first talk, I'll define the isogeneous affine
Hecke algebras and explain how to compare their representations.
Then I'll explain how to get the representations of finite Hecke algebras
of the finite complex reflection groups G(r,p,n) from the affine Hecke algebra
of type GL_{n}.

Decomposition numbers for standard objects in categories O,
Representation Theory seminar,
University of Melbourne, 11 May 2021.
Abstract:
The Verma modules are indexed by their highest weight.
They have a simple quotient and so the irreducibles are indexed by their
highest weight. The composition factors of a Verma module must all lie
in the same orbit of the Weyl group. In the affine case the orbits take three
different shapes depending on whether it is positive level, negative level, or
critical level. In each case there is a different family of KazhdanLusztig
type
polynomials that describes the multiplicity of the irreducible in the layers
of the Jantzen filtration of the Verma module. I'll try to explain what these
affine Weyl group orbits and KazhdanLusztig polynomials are.

Open boundary Hecke and TemperleyLieb algebras,
Mathematical Physics seminar,
University of Melbourne, 4 May 2021.
Abstract:
In 2005 Vladimir Rittenberg explained me the idea of two boundary
spin chains. In my discussions with him, I outlined how I would like
to think about this question, in terms of the affine Hecke algebra of
a padic symplectic group. Then one can use the JucysMurphy type elements
and the description of the center that one gets from BernsteinLusztig 1980,
and the classification of irreducible representations
that one gets from KazhdanLusztig 1987. In joint work with
Zajj Daugherty we have, over several years, been working out the details
for this sketch. One part of our work has been to explain the conversion
between this method to the classification of irreducible open boundary
TemperleyLieb representations that was done by De GierNichols in 2007.

The universal Pieri rule and Pieri/difference operator duality,
Macdonald polynomial reading seminar,
University of Melbourne, 21 March 2021.
Abstract:
This talk was a survey of material in Chapter 6 Section 6 of
Macdonald's book Symmetric functions and Hall polynomials.
I explained how Macdonald's Pieri rule for Macdonald polynomial is
given by evaluations of the coefficients of the expansion of
Macdonald's difference operator and then argued that this (along
with symmetry) is a forecasting of "duality" on the double affine
Hecke algebra. This is done by introducing the double affine Weyl group,
where there is an obvious symmetry between the Xs and the Ys, and then
look at its polynomial representation.

The elliptic hook length formula,
Macdonald polynomial reading seminar,
University of Melbourne, 14 March 2021.
Abstract:
This talk was a survey of material in Chapter 6 Section 6 of
Macdonald's book Symmetric functions and Hall polynomials.
I explained how the Weyl dimension formula (for GL_{n}) corresponds
to a specialization of Schur functions, how the principal specialization
of a Schur function is the quantum dimension, and argued that perhaps
the pricipal specialization of the Macdonald polynomial can be considered
as an elliptic dimesion. Then I gave a combinatorial/pictorial explanation
of Macdonald's proof of the hook formula for this elliptic dimension.
Talks of Arun Ram in 2020

Examples in affine combinatorial representation theory,
Minicourse at
DMRT2020, Discussion Meeting on Representation Theory 2020
Indian Institute of Science, Bengaluru, 1012 December 2020.
Although the theory applies to all quantum affine algebras we shall
focus on some illustrative examples in type GL_n. The goal is
to highlight some amazing connections between combinatorics,
representations, mathematical physics and probability.
Lecture 1: Examples of Macdonald polynomials
Abstract:
We'll study examples of GL_{n} Macdonald polynomials.
The basic tool is the affine Weyl group. This lecture is
motivated by the papers of HaglundHaimanLoehr 2006 and Lenart 2008.
An amazing connection is between tableau formulas for Macdonald
polynomials and sequences of elements in the affine Weyl group.
Lecture 2: Examples of level zero extremal weight modules
Abstract:
We'll construct the standard and simple level 0 modules
corresponding to skew shape Young diagrams (for the quantum affine
algebra of type GL_{n}). This lecture is motivated by the papers of
Drinfeld 1986, Cherednik 1987 and NazarovTarasov 1998.
An amazing result is that the irreducible modules bases
indexed by tableaux and that the characters of the
standard modules are specialisations of Macdonald polynomials.
Lecture 3: Examples connecting to probability and conformal field theory
Abstract:
V = ℂ^{n}[t,t^{1}] and its
relation to Rmatrices, transfer matrices, and Macdonald polynomials.
This lecture is motivated by the papers of TakhtajanFaddeev 1979,
KashiwaraMiwaStern 1995 and BorodinWheeler 2018.
An amazing result is that an eigenvector of the transfer matrix
is related to the stationary distribution of the ASEP
(asymmetric exclusion process)
and has coefficients which are specialisations of Macdonald polynomials.

Extremal weight modules, global Weyl modules and local Weyl modules
,
Special session in Representation Theory,
Australian Mathematical Society Meeting, 810 December 2020.
Abstract:
This talk will be a brief review of the indexing and structure of level 0 standard modules for affine Lie algebras. Basically, the extremal weight modules are shaped like infinite tubes, and the local Weyl modules have the shape of the torus obtain by gluing the ends of the tube together. The Heisenberg subalgebra inside the affine Lie algebra moves vectors up and down the tube.

Periodic permutations and Macdonald polynomials,
Combinatorics seminar, Univ. of Southern California, 11 November 2020.
Abstract:
HaglundHaimanLoehr gave formulas for Macdonald
polynomials as sums over nonattacking fillings (of boxes) counted
with statistics (arm, leg, maj and coinv). The goal of this work is to
try to make the HaglundHaimanLoehr statistics “fall out" of an
analysis of reduced words for periodic permutations.
At the root of this relationship is the intertwiner construction
of the (nonsymmetric) Macdonald polynomials.

Flags, crystals, and orthogonal polynomials,
Colloquium, University of Talca, 15 October 2020.
Abstract:
When nonmathematicians ask me what research I do, I say symmetry. From symmetry of polyhedra, to symmetries of universes like spheres and tori, now we are fascinated by "paths" of symmetries, and this is the source of loop groups. It turns out that loop groups capture amazing geometry, combinatorics, and representation theory. I will endeavour to explain what the integrable
representations for loop groups look like (paraboloids, mountains, craters, and tubes)
and how these shapes are a reflection of the corresponding geometry (of an infinite dimensional flag variety).
And then, miraculously, the characters of these modules turn out to be (specialised) Macdonald polynomials!

Curves of symmetries and categories of modules,
Colloquium, Macquarie University, 9 October 2020.
Abstract:
When nonmathematicians ask me what research I do, I say symmetry.
From symmetry of polyhedra, to symmetries of universes like spheres
and tori, now were are fascinated by "paths" of symmetries, and this is
the source of loop groups. It turns out that loop groups capture
amazing geometry, combinatorics, and extremely structured
tensor categories. I will endeavour to explain what the integrable modules
for loop groups look like: paraboloids, mountains, valleys, and
tubes.

Integrable modules for affine Lie algebras,
Algebra seminar, University of Georgia Athens, 6 October 2020.
YouTube
Abstract:
These modules naturally divide themselves into three categories:
positive level, negative level and level 0.
The positive level modules are highest weight,
the negative level ones are lowest weight, and the level 0 ones are neither.
But all three classes of modules have some nice character formulas,
a good crystal theory in the sense of KashiwaraLusztigLittelmann,
and BorelWeilBott type geometric constructions.
The geometric constructions use, respectively, the thin affine flag variety
(for positive level), the thick affine flag variety (for negative level),
and the semiinfinite flag variety (for level 0).

Presenting your research: Six points for giving talks,
Univ. of Melbourne Mathematics and Statistics MSc Masterclasses,
University of Melbourne, 9 September 2020.
Abstract:
This talk covered: Audience, Preparation, Cutting material,
Time management, Presentation tools, Explain it, "Talk machine",
differences between talks, developing confidence, finding the right
conference, Finding funding opportunities, and technology for
online presentations.

Teaching Math in The Next Life,
University of Western Sydney Abend Seminars
, 30 July 2020.
Abstract:
For many years I've been thinking about how to teach mathematics
with honesty and inspiration.
This has resulted in ideas like "Reality teaching", "Proof machine",
"Marking apocalypse", and "Just do it".
And then a virus came, and the new life began, online, on Zoom.
This will be a talk about the adventures of the past life and
the preparations for the next.

Nonsymmetric Macdonald polynomials: Part I
and Part II,
Séminaire Corteel,
UC Berkeley, 12 and 19 May 2020.
Abstract:
Following on from the statement that the Key polynomials
and Demazure atoms are specialisations of nonsymmetric Macdonald polynomials,
I will explain how I like to construct the nonsymmetric Macdonald polynomials.
Depending on which "flavor" of the construction is used,
one obtains the HHL formula or the alcove walk formula.
For this talk I will stick to type A (more precisely, type GLn).

Presenting your research: Six points for giving talks,
Univ. of Melbourne Mathematics and Statistics MSc Masterclasses,
University of Melbourne, 23 April 2020.
Abstract:
This talk covered: Audience, Preparation, Cutting material,
Time management, Presentation tools, Explain it, "Talk machine",
differences between talks, developing confidence, finding the right
conference, Finding funding opportunities, and technology for
online presentations.

Formulas for Macdonald polynomials, Representation Theory seminar,
University of Melbourne, 27 March 2020.
Abstract:
I will review/compare and contrast some of the formulas for
nonysmmetric (and symmetric) Macdonald polynomials including the
HaglundHaimanLoehr formula, the RamYip formula, and the recent
formulas of de GierCantiniWheeler and CorteelWilliamsMandelshtam.
One result I'd like to highlight provides the specialisations of the
RamYip formula for q and t taking values 0 or infinity.
I may also make some comments about the Macdonald polynomials
for type (C^{∨}, C), which are called Koornwinder polynomials,
and Macdonald polynomials for other classical (unitary, orthogonal
and symplectic) types.

Mendelssohn Salon 1828; Elliptic functions  Kosmos  Beethoven Sonata
in A flat Major Opus 110,
written and
performed by Michael Leslie (RichardStraussKonservatorium, Munich)
and Arun Ram (University of Melbourne),
at Tempo Rubato,
Brunswick, Victoria Australia, 12 March 2020.
Abstract:
Berlin 1828: The salons of Abraham and Lea Mendelssohn were the foci for some of the greatest scientific and musical minds of the day. Alexander von Humboldt, famous for his scientific findings from his voyage to the wild Americas, had set up a hut for measuring the earth's magnetic field in the Mendelssohn garden. In the previous year Beethoven had died leaving a controversial musical legacy, while Abel and Jacobi created a mathematical revolution between 1827 and 1829 with the development of elliptic functions. The discussion and debate this evening will be music and mathematics: the sounds of Beethoven, the elliptic orbits of the planets, and the thrill of the Kosmos. This talk aims to recreate the passion and exuberance which were the hallmark of those legendary gatherings.

Level 0 and the Bethe ansatz, Pure mathematics seminar,
University of Melbourne, 21 February 2020.
Abstract:
First I will review the mechanics for finding eigenvalues and eigenvectors
of the Murphy elements in the group algebra of the symmetric group
as a model method for finding the eigenvalues and eigenvectors of the transfer
matrices that appear in the Algebraic Bethe ansatz.
Second I will review the connection between transfer matrices
and pseudoquasitriangular Hopf (psqtH) algebras. Third I will visit the 3 favourite
families of pqtHalgebras and hint at their relation to affine Lie algebras.
Finally I will explain, via an illustrative (9dimensional)
example for sl_{3}, the construction of the eigenvalues and eigenvectors
in a level 0 representation.

Introductory preseminar for Level 0 and the Bethe ansatz,
Introductory seminar for the Pure mathematics seminar, University of Melbourne, 21 February 2020.
Abstract:
This was an introductory preseminar for the Pure Mathematics seminar
that same afternoon.

Level 0 and the Bethe ansatz, invited speaker
at
Talking Across Fields, a conference in Honor of Persi Diaconis, Stanford University, 31 Jan.2 February 2020.
Abstract:
This is about the relation between representations of
affine Lie algebras, the energy (Hamiltonian) for the Bethe ansatz
and the combinatorics of Macdonald polynomials. Although I probably
won't get to talk about the shuffles in the background (sorry Persi), I
probably will talk about Young tableaux, quantum groups, and
Heisenberg spin chains.

Level 0 and the Bethe ansatz, Mathematical Physics seminar,
University of Melbourne, 21 January 2020.
Abstract:
First I will review the mechanics for finding eigenvalues and
eigenvectors of the Murphy elements in the group algebra of the symmetric
group as a model method for finding the eigenvalues and eigenvectors of
transfer matrices. Second I will review the connection between transfer
matrices and pseudoquasitriangular Hopf (psqtH) algebras. Third I will
visit the 3 favourite families of pqtHalgebras and explain the special
case that gives the classical XXX, XXZ, and XYZ Hamiltonians. Finally
I will explain, via an illustrative (9dimensional)
example for sl_{3}, the construction of the eigenvalues
and eigenvectors.
Talks of Arun Ram in 2019

Tits buildings and the structure of Chevalley groups,
Representation Theory seminar,
University of Melbourne, 15 October 2019.
Abstract:
This was an introductory talk covering philosophy motivating the definition
of a Tits building,
how to picture a building in cases where it is a Coxeter complex,
where it is the building of a finite Chevalley group,
and in cases where it is an affine building. We ended with the connection
between certain additive root subgroups and the corresponding pictorial
point of view that motivates the MoyPrasad filtrations for groups over
local fields.

The être of Michèle Vergne,
a short talk about Michèle Vergne for the
Women in Maths Day
at the University of Melbourne, 29 August 2019.

Presenting your research: 6 points to think about for giving talks,
MasterClass program for Master Students,
University of Melbourne, 27 August 2019.

Maybe I could be a mathematician: A story of growing up alongside
vinyl, CD, MP3 and YouTubeRed,
A lecture performance presented by
National Science Week
and the Faculty of Science at University of
Melbourne at the
Bendigo Discovery Science and Technology Centre, 22 August 2019.
Abstract:
In this unique presentation Professor Arun Ram tells a series of stories,
interweaving mathematics and music. Humorous, educational, personal,
often all at once, these collected stories illuminate the remarkable
journey of an enquiring mind who became a mathematician.
A mathematical mixtape from a lifelong road trip.
 The Music of Mathematics,
Castlemaine Primary School,
Castlemaind Victoria Australia, 23 August 2019.
Abstract:
This was a presentation for school students, including an excerpt from
the Glass Bead Game on the emotion of
mathematics leading to Maria Callas and Alexander Grothendieck,
and excerpts from Maybe I could be a Mathematician,
(45 minutes, 9:009:45, 56 students in grade 5 and 6;
This visit was covered by the
Castlemaine Mail).
 The Music of Mathematics,
Girton Grammar School,
Bendigo Victoria Australia, 22 August 2019.
Abstract:
This was a presentation for school students, including an excerpt from
the Glass Bead Game on the emotion of
mathematics leading to Maria Callas and Alexander Grothendieck,
an excerpt from the Mendelssohn Salon,
and an excerpt from Maybe I could be a Nathematician,
a juggling analysis with Anthony Mays, a poem of Sofia Kovalevsky and
Carl Gauss with Anita Ponsiang and Anthony Mays, a portrait of
Persi Diaconis and a card trick performed by Anthony Mays.
(75 minutes, 9:3010:45, with students from Girton Grammar as well
students from Holy Rosary Primary School, White Hills Primary School,
Marist College, Cursoe Secondary College, a total of 150 students for 75min,
from grades 5,6,7,8, 9; This visit was covered by the
Bendigo advertiser)

The musical sensation of mathematics,
A lecture performance at
ExtraSensory,
presented by
National Science Week
and the Royal Society of Victoria,
at Parliament of Victoria,hMelbourne 10 August 2019.
Abstract:
It's the pivotal scene of your favourite movie, the soundtrack builds,
the tension rises. You experience a wave of sound and roller coaster of
emotions. Music makes us feel things, but how could this possibly
relate to mathematics?
With a collage of musical excerpts and mathematical ideas,
join us for an emotional ride where music evokes the drama and
excitement of major mathematical advances. Hear Bach's music illuminate
Euler's circle formula and Maria Callas' incomparable voice capture
the expanse of Alexander Grothendieck's highways of geometry.

The triumvirate of affine flag varieties,
Invited speaker at the conference
Flags, Galleries and Reflections at University of Sydney,
58 August 2019.
Abstract:
This will be a survey talk about the positive level (thin) affine flag variety,
the negative level affine flag variety (thick) affine flag variety,
and the level 0 (semiinfinite) affine flag variety.
The first part will be a discussion of what I know (gallery models for points)
and the second part will be a discussion of things I don't know
but would like to know
(a universal BorelWeilBott theorem covering all three cases simultaneously
and the Ktheory of Springer fibres as DAHAmodules).

Mendelssohn Salon 1828; Elliptic functions  Kosmos  Beethoven Sonata
in A flat Major Opus 110,
written by Michael Leslie (RichardStraussKonservatorium, Munich)
and Arun Ram (University of Melbourne), performed by Joshua Hooke and
Arun Ram (University of Melbourne),
Federation Hall, Southbank, Melbourne, 4 August July 2019.
Abstract:
Berlin 1828: The salons of Abraham and Lea Mendelssohn were the foci for some of the greatest scientific and musical minds of the day. Alexander von Humboldt, famous for his scientific findings from his voyage to the wild Americas, had set up a hut for measuring the earth's magnetic field in the Mendelssohn garden. In the previous year Beethoven had died leaving a controversial musical legacy, while Abel and Jacobi created a mathematical revolution between 1827 and 1829 with the development of elliptic functions. The discussion and debate this evening will be music and mathematics: the sounds of Beethoven, the elliptic orbits of the planets, and the thrill of the Kosmos. This talk aims to recreate the passion and exuberance which were the hallmark of those legendary gatherings.

Mendelssohn Salon 1828; Elliptic functions  Kosmos  Beethoven Sonata
in A flat Major Opus 110,
written by Michael Leslie (RichardStraussKonservatorium, Munich)
and Arun Ram (University of Melbourne), performed by Joshua Hooke and
Arun Ram (University of Melbourne),
The Women's College, University of Sydney, 6 August 2019.
Abstract:
Berlin 1828: The salons of Abraham and Lea Mendelssohn were the foci for some of the greatest scientific and musical minds of the day. Alexander von Humboldt, famous for his scientific findings from his voyage to the wild Americas, had set up a hut for measuring the earth's magnetic field in the Mendelssohn garden. In the previous year Beethoven had died leaving a controversial musical legacy, while Abel and Jacobi created a mathematical revolution between 1827 and 1829 with the development of elliptic functions. The discussion and debate this evening will be music and mathematics: the sounds of Beethoven, the elliptic orbits of the planets, and the thrill of the Kosmos. This talk aims to recreate the passion and exuberance which were the hallmark of those legendary gatherings.

Mendelssohn Salon 1828; Elliptic functions  Kosmos  Beethoven Sonata
in A flat Major Opus 110,
written by Michael Leslie (RichardStraussKonservatorium, Munich)
and Arun Ram (University of Melbourne), performed by Joshua Hooke and
Arun Ram (University of Melbourne),
Wonthaggi Baptist Church, 31 July 2019.
Abstract:
Berlin 1828: The salons of Abraham and Lea Mendelssohn were the foci for some of the greatest scientific and musical minds of the day. Alexander von Humboldt, famous for his scientific findings from his voyage to the wild Americas, had set up a hut for measuring the earth's magnetic field in the Mendelssohn garden. In the previous year Beethoven had died leaving a controversial musical legacy, while Abel and Jacobi created a mathematical revolution between 1827 and 1829 with the development of elliptic functions. The discussion and debate this evening will be music and mathematics: the sounds of Beethoven, the elliptic orbits of the planets, and the thrill of the Kosmos. This talk aims to recreate the passion and exuberance which were the hallmark of those legendary gatherings.
 The Music of Mathematics,
Wonthaggi Secondary College,
Wonthaggi Gippsland Victoria Australia, 31 July 2019.
Abstract:
This was a presentation for school students, including an excpert from
the Glass Beadn Game on the emotion of
mathematics leading to Maria Callas and Alexander Grothendieck,
an excerpt from the Mendelssohn Salon,
and an excerpt from Maybe I could be a Nathematician.
(13:1514:00 for about 160200 students in years 7,8 9, this school visit
was covered by WIN News Goppsland)
 The Music of Mathematics,
Mary McKillop Catholic Regional College,
Leongatha Gippsland Victoria Australia, 31 July 2019.
Abstract:
This was a presentation for school students, including an excpert from
the Glass Beadn Game on the emotion of
mathematics leading to Maria Callas and Alexander Grothendieck,
an excerpt from the Mendelssohn Salon,
and an excerpt from Maybe I could be a Nathematician.
(9:009:45 for about 84 students in Year 10).
 The Music of Mathematics,
Chairo Christian College,
Leongatha Gippsland Victoria Australia, 31 July 2019.
Abstract:
This was a presentation for school students, including an excpert from
the Glass Beadn Game on the emotion of
mathematics leading to Maria Callas and Alexander Grothendieck,
an excerpt from the Mendelssohn Salon,
and an excerpt from Maybe I could be a Nathematician.
(12:0012:45 for about 54 students years 710).

Combinatorics in level 0,
Representation Theory seminar, University of Melbourne 19 February 2019.
Abstract:
I gave an introduction to level 0 representations of the affine Lie algebra
with lots of pictures:
including their indexing, and their character theory via crystals (via
LS paths following NaitoSagaki).

A point of view on Conformal Field Theory,
invited lecture at the conference
Subfactors in Sydney,
University of New South Wales, 7 February 2019.
Abstract:
I will give a proposal for studying conformal field theory
by combinatorics of crystals (in the sense of Kashiwara and Lusztig),
more precisely, via crystals of level 0 integrable representations
of affine Lie algebras.

The door is ajar, Concluding speaker,
Strengthening Engagement and Achievement in Mathematics and Science (SEAMS)
Senior Residential Camp, 2123 January 2019, Melbourne University.
Abstract:
The door is ajar for the next generation of Mathematicians and Scientists.
Talks of Arun Ram in 2018

Two boundary TemperleyLieb and the exotic nilcone,
speaker at Geometric and Categorical Representation Theory,
MATRIX, Creswick, Australia, 20 December 2019, 14 December 2019
Abstract:
This talk is based on work with Zajj Daugherty, Iva Halacheva and Arik Wilbert.
We describe the representations of the two boundary TemperleyLieb algebra
combinatorially, diagrammatically and geometrically.
The diagrammatic approach follows de GierNichols and GreenMartinParker.
The geometric approach follows S. Kato and uses the exotic nilcone
to construct representations of the two boundary Hecke algebra which
are then identified as representations of the two boundary TemperleyLieb
algebra. The combinatorial approach uses the theory of multisegments and
standard Young tableaux.

Schubert calculus on semiinfinite flag varieties,
Representation Theory Seminar, University of Melbourne, 9 October 2018
Abstract:
I will endeavour to explain the main facets of the way that I think about Schubert calculus, focusing on the case of the semiinfinite flag variety. I will review the definition of the semiinfinite flag variety, the Schubert classes, the action of polynomials, the moment graph description, the pushpull operators and the PieriChevalley formula.
None of this is my work except, perhaps, a certain point of view on the subject.

Representations of two boundary Hecke and TemperleyLieb algebras
(TBHA and TBTL),
Representation Theory Seminar, University of Melbourne, 14 August 2018
Abstract:
I will discuss classifications, constructions and combinatorics of
irreducible and standard modules of TBHA and TBTL.
The TBTA is the affine Hecke algebra of type C with arbitrary
"unequal" parameters. The TBTL is a quotient of the TBHA by
local idempotents (for rank 2 sub root systems).
The TBTL has been of interest in statistical mechanics:
Heisenberg spin chains with boundaries (de GierNichols).
The geometry construction of TBHAmodules (Kato) for
unequal parameters is via the exotic nilpotent cone.

Maybe I could be a mathematician: A story of growing up alongside
vinyl, CD, MP3 and YouTubeRed,
A lecture performance presented by
The Institute for Enquiring Minds,
RMIT Cinema Theatre Building 80, Melbourne, 31 July 2018
Abstract:
In this unique presentation Professor Arun Ram tells a series of stories,
interweaving mathematics and music. Humorous, educational, personal,
often all at once, these collected stories illuminate the remarkable
journey of an enquiring mind who became a mathematician.
A mathematical mixtape from a lifelong road trip.

The door is ajar, Concluding speaker,
Strengthening Engagement and Achievement in Mathematics and Science (SEAMS)
Senior Residential Camp, 35 July 2018, Melbourne University.
Abstract:
The door is ajar for the next generation of Mathematicians and Scientists.

Mendelssohn Salon 1828; Elliptic functions  Kosmos  Beethoven Sonata
in A flat Major Opus 110,
written and
performed by Michael Leslie (RichardStraussKonservatorium, Munich)
and Arun Ram (University of Melbourne),
at the Fergg Salon, Munich, Germany, 16 March 2018.
Abstract:
Berlin 1828: The salons of Abraham and Lea Mendelssohn were the foci for some of the greatest scientific and musical minds of the day. Alexander von Humboldt, famous for his scientific findings from his voyage to the wild Americas, had set up a hut for measuring the earth's magnetic field in the Mendelssohn garden. In the previous year Beethoven had died leaving a controversial musical legacy, while Abel and Jacobi created a mathematical revolution between 1827 and 1829 with the development of elliptic functions. The discussion and debate this evening will be music and mathematics: the sounds of Beethoven, the elliptic orbits of the planets, and the thrill of the Kosmos. This talk aims to recreate the passion and exuberance which were the hallmark of those legendary gatherings.

Mendelssohn Salon 1828; Elliptic functions  Kosmos  Beethoven Sonata
in A flat Major Opus 110,
written and
performed by Michael Leslie (RichardStraussKonservatorium, Munich)
and Arun Ram (University of Melbourne),
at the ClemannZimansky Salon, Basel, Switzerland, 11 March 2018.
Abstract:
Berlin 1828: The salons of Abraham and Lea Mendelssohn were the foci for some of the greatest scientific and musical minds of the day. Alexander von Humboldt, famous for his scientific findings from his voyage to the wild Americas, had set up a hut for measuring the earth's magnetic field in the Mendelssohn garden. In the previous year Beethoven had died leaving a controversial musical legacy, while Abel and Jacobi created a mathematical revolution between 1827 and 1829 with the development of elliptic functions. The discussion and debate this evening will be music and mathematics: the sounds of Beethoven, the elliptic orbits of the planets, and the thrill of the Kosmos. This talk aims to recreate the passion and exuberance which were the hallmark of those legendary gatherings.

Flag varieties and Schubert Calculus,
4 lectures at the NZMRI Summer school,
Nelson New Zealand, 713 January 2018.
Lecture 1,
Lecture 2,
Lecture 3,
Lecture 4.
Abstract:
Schubert calculus is the study of the structure of flag varieties.
An example of a flag variety is the set of kdimensional subspaces
of a vector space (when k=1, this gives us projective space).
The structure of flag varieties is intimately connected to the
combinatorics of symmetric functions, and Schubert calculus is the
mechanics for doing computations in flag varieties with polynomials.
This course will be a survey of how we play this game.
Talks of Arun Ram in 2017

Are there Symmetric group crystals?
, invited speaker at the conference
Representation Theory of Symmetric Groups and Related Algebras,
Institute for Mathematical Sciences, National University of Singpore, 1120 Dec 2017.
(video on YouTube)
(pdf slides)
(talk notes short version)
(talk notes with diagrams)
(Towards B⊗B⊗B)
(B⊗B⊗B computation)
(badnews proof and computations)
Abstract:
Several recent conversations about the Kronecker problem
(positive combinatorial expressions for the decomposition of tensor products
of irreducible symmetric group representation in characteristic 0)
made me feel that it would be useful to think about the possibility of a
theory of crystals for symmetric group representations (a graph based
categorification of the character ring of the symmetric group).
Initial explorations
have been fascinating. I will give a summary of what I have learned about
what such a theory would need to look like.

Combinatorics of level 0 representations,
invited speaker at the conference Future Directions in Representation Theory, University of Sydney, 58 December 2017.
Abstract:
Recent work of KatoNaitoSagakiFeiginMakyedonskyi has provided an
improved understanding of the combinatorics of integrable level 0 representations of
the affine Lie algebra. In particular, there is a connection to the Schubert calculus of the
semiinfinite flag variety from which KatoNaitoSagaki prove an analogue of the
PieriChevalley formula for the semiinfinite flag variety (line bundle multiplied with a
Schubert class). The connection to crystals in this new formula (analogous to that in the
PieriChevalley formula of PittieRam) provides a geometric interpretation of (a part of)
the path model which was used by RamYip to give a formula for Macdonald polynomials.

Combinatorics of level 0 representations,
Algebra and Topology seminar, Australian National University, 28 November 2017.
Abstract:
Recent work of Kato and KatoNaitoSagaki has provided an improved understanding
of the combinatorics of integrable level 0 representations of the affine Lie algebra. In particular, there
is a connection to the Schubert calculus of the semiinfinite flag variety from which KatoNaitoSagaki prove
an analogue of the PieriChevalley formula for the semiinfinite flag variety (line bundle multiplied with a Schubert class).
The connection to crystals in this new formula (analogous to that in the PieriChevalley formula of PittieRam)
provides a geometric interpretation of (a part of) the path model which was used by RamYip
to give a formula for Macdonald polynomials.

Combinatorics of semiinfinite flag varieties,
invited speaker at the conference International Festival in Schubert Calculus, SunYat Sen University, Guangzhou China, 511 November 2017.
See http://math.sysu.edu.cn/Files/ArticleFiles/20171103/02a464080f1d4d73a49224afa953f412.pdf
Abstract:
I will discuss the path model/alcove walk model for the semiinfinite flag
variety. This model has connections to the path model formula for Macdonald polynomials
and to the Littelmann path model for representations of affine Lie algebras at level 0.

TOYKAMP,
short presentation for the School Review, School of Mathematics and Statistics,
University of Melbourne, 1 November 2017.
Abstract:
TOYKAMP is a pure math playgroup that meets weekly on Thursday afternoons.
We do weaving (Fine Springer Fibres to make Higgs Bundles) and
box building (black boxes to put the fibres and bundles in to make Modules).
The participants are usually local, but we have some Commuting Operators
who come in to run the black box modules and navigate Spectral Curves with
them. Our systems are often Integrable into a larger mathematics department
(if the bundle is sufficiently ample). Everyone is welcome.

Fusion following KazhdanLusztig,
contributed talk at the conference
Tensor Categories and Field Theory, University of Melbourne 59 June 2017.
Abstract:
In recent discussions with S. Kanade, K. Kawasetsu and D.
Ridout we have been
trying to understand what fusion products are and how to compute them,
following work of
Gaberdiel, Kausch, Nahm, Huang, Lepowsky, Zhang, Li, Tsuchiya, Miyamoto
and many others.
I will outline the rigorous approach of KazhdanLusztig from Tensor
structures arising from affine Lie algebras I, J. Amer. Math. Soc. 1993.

Mendelssohn Salon 1828; Elliptic functions  Kosmos  Beethoven Sonata
in A flat Major Opus 110,
written and
performed by Michael Leslie (RichardStraussKonservatorium, Munich)
and Arun Ram (University of Melbourne)
Wyselaskie Auditorium, Parkville, Melbourne, Australia, 20 July 2017.
Abstract:
Berlin 1828: The salons of Abraham and Lea Mendelssohn were the foci for some of the greatest scientific and musical minds of the day. Alexander von Humboldt, famous for his scientific findings from his voyage to the wild Americas, had set up a hut for measuring the earth's magnetic field in the Mendelssohn garden. In the previous year Beethoven had died leaving a controversial musical legacy, while Abel and Jacobi created a mathematical revolution between 1827 and 1829 with the development of elliptic functions. The discussion and debate this evening will be music and mathematics: the sounds of Beethoven, the elliptic orbits of the planets, and the thrill of the Kosmos. This talk aims to recreate the passion and exuberance which were the hallmark of those legendary gatherings.

Level 0 representations and Macdonald polynomials,
Mathematical Physics Seminar, University of Melbourne, 16 May 2017.
Abstract:
I will describe the passage from extremal weight modules, to global Weyl modules and then to local Weyl modules (for affine algebras and currents). Then I will explain how characters of these modules are (a specialisation) of Macdonald polynomials. If time permits I will outline why these level 0 modules are fundamental to some conformal field theory constructions (Fock spaces, KhnizhnikZamolodchikov, Rmatrices, and energy functions).

Rmatrices, elliptic stable envelopes, Schubert calculus,
Working seminar, University of Melbourne 10 April 2017.
Abstract:
This talk provided some summary of the recent paper
by G. Felder, R. Rimanyi and A. Varchenko
entitled
Elliptic dynamical quantum groups and equivariant elliptic cohomology

Niveaux de la forêt cristal,
Colloquium, University of Reims, 21 March 2017.
Abstract:
La forêt de cristal affine est mystérieuse, traversée par de nombreux chemins d'un homme petit (Littelmann). Les chemins montent aux niveaux positifs et descendent aux niveaux négatifs. À chaque endroit de la forêt on découvre les traces des groupes des chats de mauvaise humeur (KacMoody groups), mais les groupes des chats de mauvaise humeur sont toujours cachés dans l'ombre. Caché ... et voilà! (Kashiwara) "tien le saké!" (Tanisaki) ... Les japonais habitent aussi les tunnels au niveau zéro: cas d'eaux (Kato) japonaises, les tunnels sont d'ici au semiinfini.

A Fock space plan,
Invited speaker, Workshop on Asymptotic Representation Theory,
part of the trimester on Combinatorics and Interactions,
Institut Henri Poincaré,
Paris 23 February 2017.
Abstract:
This talk has two goals:
(a) to explain the construction of 'abstract Fock space' following the recent
paper of LaniniRamSobaje arXiv1612.03120,
(b) to outline the relationship of Fock space to symmetrisation/antisymmetrization,
BosonFermion correspondence, Brownian motion to Whittaker process, and
geometric Langlands.

Really cool numbers and Research methodology through Berlioz,
Special lecture, University of Melbourne 18 January 2017.
Abstract:
This was a lecture for High School students (Year 9/10) attending the
"Melbourne Summer School 2017", a program run by the
MGSE (Melbourne Graduate School of Education) specifically for the
Master of Teaching (Secondary) Internship program (this
is different to the "VCE Summer School (VCESS)).
Many thanks to interns Andres Alzate and Andrew Jacobs who
organised and invited me to give this lecture.
The goal was to provide the students a "University experience" in a
lecture theatre and discussion of what a professional mathematician does.
Talks of Arun Ram in 2016

What is fusion of representations,
Working seminar, University of Melbourne 4 October 2016.
Abstract:
This will be a survey of KazhdanLusztig's paper "Tensor Structures on Affine Lie algebras I", J. Amer. Math. Soc. 1993, where the notion of fusion of representations (as used in representations of affine Lie algebras and conformal field theory) is put on rigorous mathematical footing in 3 different points of view.

Vertex algebras, Associative algebras and Poisson algebras,
Working seminar, University of Melbourne 24 August 2016.
Abstract:
This talk is a summary of the definitions of Vertex algebras,
associative filtered algebras and Poisson algebras, and the Zhu functors
between them. The content was mostly taken from Arakawa's paper
"Rationality of Walgebras: principal nilpotent cases" arXiv:1211.7124.

Alcove walks and the Peterson isomorphism,
Banff conference on Whittaker Functions: Number Theory, Geometry and Physics (16w5039), BIRS Banff Canada
Banff conference on Whittaker Functions: Number Theory, Geometry and Physics (16w5039), BIRS Banff Canada 2429 July 2016.
Abstract:
I will describe a combinatorial (alcove walk) labelling of the points of
the moduli space of curves (of genus 0) in the flag variety. The idea is
that this geometric labelling can explain the sometimes magical "quantum
to affine" phenomena relating quantum cohomology of the flag variety to
the cohomology of the affine Grassmannian. This is joint work with Liz
Milicevic.

A combinatorial gadget for decompositions numbers for quantum groups at roots of unity
, invited speaker at the conference
Representation Theory and Physics, University of Leeds, 1822 July 2016.
Abstract:
This is joint work with Martina Lanini and Paul Sobaje in which we produce a
generalization of the qFock space (as used, for example, by Ariki,
LascouxLeclercThibon, Hayashi, MisraMiwa, KashiwaraMiwaStern) to all
Lie types. This gadget captures the decomposition numbers of standard modules
for representations of quantum groups at roots of unity in the same way that
the usual qFock space does for type A. In classical type, via SchurWeyl
duality, it will also see the decomposition numbers of affine BMW algebras
in the same way that the usual qFock space does for the affine Hecke
algebras of type A.

The door is ajar, Concluding speaker,
Strengthening Engagement and Achievement in Mathematics and Science (SEAMS)
Senior Residential Camp, 29 June  1 July 2016, Melbourne University.
Abstract:
The door is ajar for the next generation of Mathematicians and Scientists.

Combinatorics of affine Springer fibres,
Algebra seminar, Sydney University 17 June 2016.
Abstract:
This talk will be a survey of the relation between affine Springer fibres and
representations of the double affine Hecke algebra. I will likely focus on
a favourite example of the elliptic homogeneous case where I can draw a nice
picture illustrating how the affine Springer fibre is decomposed into
cells indexed by connected components of complement of a hyperplane
arrangement called the Shi arrangement (the same one that appears in the
Ktheory version of the ChevalleyShephardTodd theorem for reflection groups).
These regions then correspond to a Macdonald polynomial basis of the
corresponding representation of the double affine Hecke algebra.

Combinatorics of representations of affine Lie algebras,
Algebra seminar, Sydney University 16 June 2016.
Abstract:
This will be a survey of my current understanding of the combinatorial
representation theory of affine Lie algebras. For category O at negative
level, Verma modules have finite composition series with decomposition numbers
determined by KazhdanLusztig polynomials. The structure of affine Weyl group
orbits controls the pretty patterns. For category O at positive level,
Verma modules have infinite compositions with decomposition
numbers given by inverse KazhdanLusztig polynomials, and at critical level,
the patterns correspond to the periodic KazhdanLusztig polynomials.
I’ll also discuss parabolic category O. Finite dimensional modules (which
are level 0) are indexed by Drinfeld polynomials and then there are various
collections of smooth representations where our combinatorial understanding
has increased greatly in recent years.

Combinatorics of the loop Grassmannian,
Algebra seminar, Sydney University 14 June 2016.
Abstract:
I will explain what the loop Grassmannian and
the affine flag variety are and how to label their points.
This labelling is a refinement of the labelling of crystal bases
by Littelmann paths. I’ll show the picture which summarises the connection to
the affine Hecke algebra and the spherical affine Hecke algebra.
I’ll give a summary of the relationship between MirkovicVilonen cycles
and the crystal bases and explain how this is reflected in the path model
indexing.

The BRST Complex and quantised Hamiltonian reduction,
Working seminar, University of Melbourne, 16 May 2016.
Abstract:
This will be a description of the KacRoanWakimoto construction of the
BRST complex. This process, often called quantised Hamiltonian
reduction produces a chain complex generated by charged free
fermions, neutral free fermions, currents. The cohomology of this
complex produces vertex algebras called Walgebras (whose
representations correspond to minimal models from conformal field
theory).

Does BRST reduction produce branes?,
MUMS seminar, University of Melbourne, 13 May 2016.
Abstract:
This talk will be a story about a recent
experience in my role as a professional mathematician. The
experience drew me into a new field with new collaborators and new
project to work on. I will describe how this came about, what the
process was, and how one learns and expands into new subjects and
directions, in this case, String Theory.

Parking functions, the Shi arrangement and Macdonald polynomials,
Philadelphia Area Combinatorics and Algebraic Geometry seminar, Drexel University 3 May 2016.
Abstract:
A parking function is a sequence (b_{1}, ..., b_{n})
of positive integers which,
when rearranged in increasing order (a_{1} ≤ a_{2} ≤ ...
≤ a_{n}), is such that a_{i}≤ i.
I will first convert parking functions to elements of the affine Weyl group
which correspond to regions of the Shi hyperplane arrangement and bases of a
module for the rational Cherednik algebra (or double affine Hecke algebra).
As explained, for example, in papers of VaragnoloVasserot and OblomkovYun,
this module can be realized as the cohomology (or Ktheory) of an affine
Springer fiber. These bases are closely connected to Macdonald polynomials.
GoreskyKotwitzMacpherson explain how to chop up the affine Springer fiber
into tractable pieces indexed by the (generalised) parking functions
(paving by Hessenbergs). I'll start by drawing the pictures and then explain
how to read the connections off the picture.

The Geometric Peterson isomorphism,
University of Ottawa, Workshop on Equivariant generalized Schubert calculus and its applications, 30 April 2016.
Abstract:
I will describe a combinatorial (alcove walk) labelling of the points of
the moduli space of curves (of genus 0) in the flag variety. The idea is
that this geometric labelling can explain the sometimes magical "quantum
to affine" phenomena relating quantum cohomology of the flag variety to
the cohomology of the affine Grassmannian. This is joint work with Liz
Milicevic.

A combinatorial gadget for decompositions numbers for quantum groups at roots of unity,
CUNY Representation Theory seminar, CUNY Graduate Center, New York, 29 April 2016.
Abstract:
This is joint work with Martina Lanini and Paul Sobaje in which
we produce a generalization of the qFock space (as used, for example,
by Ariki, LascouxLeclercThibon, Hayashi, MisraMiwa, KashiwaraMiwaStern)
to all Lie types. This gadget captures the decomposition numbers of
standard modules for representations of quantum groups at roots of unity in
the same way that the usual qFock space does for type A. In classcal type,
via SchurWeyl duality, it will also see the decomposition numbers of affine
BMW algebras in the same way that the usual qFock space does for affine
Hecke algebras of type A.

Double affine Hecke algebras,
Working seminar, University of Melbourne 8 April 2016.
Abstract:
This will an introductory talk about the double affine Hecke algebra and
the algebraic and geometric construction of its "polynomial representation".

Hitchin Fibers, Higgs bundles and Springer fibers,
Working seminar, University of Melbourne 8 April 2016.
Abstract:
This will be a summary of the section 6 of the paper of OblomkovYun
arxiv:1407.5685 which explains the relationship between affine Springer
fibers and homogeneous Hitchin fibres. This relationship is a key part
of the setup of Ngo in his proof of Langlands “Fundamental Lemma”.

Affine Springer fibers,
Summer working seminar, University of Melbourne 12 January 2016.
Abstract:
In a paper of GoreskyKottwitzMacpherson they explain how
to view many affine Springer fibers as towers with a Hessenberg foundation.
I will hopefully explain how they use the MoyPrasad filtration to do this.

Cohomology of flag varieties,
Summer working seminar, University of Melbourne 12 January 2016.
Abstract:
In a recent paper of OblomkovYun they view the cohomology of the
affine flag variety as a module for a Hecke algebra. I will survey their paper.

Curves in flag varieties,
Summer working seminar, University of Melbourne 5 January 2016.
Abstract:
In his 1997 unpublished notes setting up his description of the quantum
cohomology of flag varieties, D. Peterson described a beautiful correspondence
between curves in flag varieties and points of the affine flag variety. I
will try to give an interpretation (and some examples) of
this correspondence in terms of alcove walks.
Talks of Arun Ram in 2015

Picturing representation rings,
Talk at the conference Geometric and categorical representation theory, Mooloolaba Queensland, 17 December 2015.
Abstract:
The goal is to provide pictures, yes PICTURES, for navigating
the Grothendieck ring of category O for the affine Lie algebras. The representations
are divided into levels (by the action of the central element) and blocks (by
the dot action of the affine Weyl group) and are connected together by
KazhdanLusztig polynomials for the affine Weyl group (via the affine Hecke
algebra). The negative level representations with integral highest weight correspond
(by a KhnizhnikZamolodchikov functor) to representations of quantum groups
at a root of unity, and this part of the Grothendieck ring appears, in type A, in a different
incarnation, as a Fock space of partitions (boxes in a corner). We explain how
the Fock space generalises to all affine KacMoody Lie algebras. This is joint work
with Martina Lanini and Paul Sobaje.

Parking functions and the Shi arrangement,
Summer working seminar, University of Melbourne 7 December 2015.
Abstract:
A parking function is a sequence (b_{1}, ..., b_{n})
of positive integers which,
when rearranged in increasing order (a_{1} ≤ a_{2} ≤ ...
≤ a_{n}), is such that a_{i}≤ i.
Persi has raised the question of random walks on parking functions,
and for computing statistics for various possible distributions on parking
functions. In order to provide some tools for solving this, I want to
convert parking functions to elements of the affine Weyl group which
correspond to regions of the Shi hyperplane arrangement
and bases of a module for the rational Cherednik algebra (or double
affine Hecke algebra). As explained, for example, by a recent paper of
OblomkovYun, this module can be realized as the cohomology of an
affine Springer fiber. I'll start by drawing all the pictures
and then explaining how to read all the connections off the picture.

Picturing the flag variety,
Talk at the Victorian Algebra Conference 2015, University of Western Sydney, Paramatta, 30 November 2015.
Abstract:
What is the flag variety? A lattice? A projective space? The VeblenYoung theorem? Cosets? Subspaces? A building? A simplicial complex? A generalised generalised quadrangle? An incidence geometry? A Chevalley group? Borel subalgebras? Borel subgroups? The LUP factorisation? The Bruhat decomposition? The Schubert calculus? A polarization? A twisted Chevalley group? An isotropic subsapce? An ovoid? A finite simple group? Perhaps the right answer is the title of Wassily Kandinsky’s book, POINT AND LINE TO PLANE.

Some projects to work on: Fusion, Elliptic, glass beads TQFTs, moduli spaces, Hecke algebras, etc,
Summer working seminar, University of Melbourne 29 October 2015.
Abstract:
This goal here was to produce a list of possible projects for Vacation scholars,
Masters students, PhD students, for postdocs, for research careers and
for grant proposals. A problem was only considered suitable for the list if
it could be naturally shaped for any one of these purposes.

Unitary, Orthogonal and Symplectic groups,
Summer working seminar, University of Melbourne 27 October 2015.
Abstract:
This talk was an effort to unify approaches to these groups:
(1) exponentiation from the Lie algebra, (2) as twisted Chevalley groups of
invariants of general linear groups with respect to an involution,
(3) as ordinary Chevalley groups coming from a root system,
(4) as given by generators and relations as found in Section 10 of
BruhatTits paper "Groupes reductifs sur un corp local I".

The glass bead game,
Public Lecture as part of the
BrisScience series at The EDGE, Brisbane, 7 July 2015.
Abstract:
This talk will take a virtual tour of the toy store with our friends
Maria Callas, Alexander Grothendieck and Hermann Hesse.
There are pleasant games with glass beads, athletic
games skiing the moguls, and violent games where
everything gets smashed. There are crystals and hurricanes
and, of course, a few polynomials. The point of the talk is
to tell some stories related to current research in symmetry.
 Views from Castalia, Guest lecture at the AMSI Winter School, University of Queensland, 7 July 2015.
Abstract: There are three mountains (the preprojective variety, the quiver variety
and the loop Grassmannian). These three mountains feed the spring at their base
(the quantum group). At the spring live three nymphs, very similar, but different
(the semicanonical basis, the canonical basis and the MV basis). Each of these
is nourished by the special nutrients of the corresponding source (the preprojective
variety, the quiver variety and the loop Grassmannian). But the shadow of the
muses is the same (the MV polytope). And, as we shall see, there is a Glass Bead
Game in this story as well.

Theta functions as matrix coefficients for Heisenberg representations,
Number theory seminar, University of Melbourne, 16 March 2015.
Abstract:
This talk is the third of a series of working seminar talks with the
aim of reading the first few sections of Mumford's Tata Lectures on Theta III.

Heisenberg groups, abelian varieties and theta functions,
Number theory seminar, University of Melbourne 9 March 2015.
Abstract:
This talk is the second of a series of working seminar talks with the
aim of reading the first few sections of Mumford's Tata Lectures on Theta III.

Two boundary Hecke and TemperleyLieb algebras,
Pure Math Seminar, University of Melbourne 6 March 2015.
Abstract:
This talk will be a survey of recent work with Z. Daugherty.
I will introduce the two pole braid group, the two pole Hecke algebra,
and the two pole TemperleyLieb algebra. I will explain the connection
to the affine Hecke algebra of type C and the indexing of irreducible
calibrated representations. Then I will explain how to get actions on
tensor space from Rmatrices and give descriptions of the irreducible
representations which appear this way. As a consequence one obtains
some understanding of the answers to some questions and conjectures
of de Gier and Nichols. A preprint with the first part of this story
is now available at
http://researchers.ms.unimelb.edu.au/~aram@unimelb/preprints.html

The glass bead game,
Talks for the Melbourne Universty Mathematics Students Society MUMS,
University of Melbourne, 6 March 2015.
Abstract:
This talk will take a virtual tour of the toy store with our friends
Maria Callas, Alexander Grothendieck and Hermann Hesse.
There are pleasant games with glass beads, athletic
games skiing the moguls, and violent games where
everything gets smashed. There are crystals and hurricanes
and, of course, a few polynomials. The point of the talk is
to tell some stories related to the topic of my research.

Heisenberg groups and representations,
Number theory seminar, University of Melbourne 2 March 2015.
Abstract:
This talk is the first of a series of working seminar talks with the
aim of reading the first few sections of Mumford's Tata Lectures on Theta III.
Talks of Arun Ram in 2014

The Peterson isomorphism between quantum cohomology and affine Schubert calculus for flag varieties,
Moduli spaces seminar, University of Melbourne 28 November 2014.
Abstract:
Lam and Shimozono, in the paper arXiv:0705.1386, say "We show that the quantum cohomology of a flag variety is a quotient of the homology of the loop Grassmannian after localization and describe the map explicitly on the level of Schubert classes. As a consequence, all threepoint genus zero GromovWitten invariants of G/P are identified with homology Schubert structure constants of the loop Grassmannian, establishing the equivalence of the quantum and homology affine Schubert calculi. This is an unpublished result stated by Dale Peterson in 1997." Allen Knutson talks about the Peterson isomorphism at
https://plus.google.com/+AllenKnutson/posts/Dz6eTTtG3ML
When I looked at Peterson notes (which I have had typed and put on the resources link from my web page), I was amazed to see an explicit bijection between the moduli spaces, essentially the same as that used in the form of "quasimap spaces" by Braverman and coauthors. This renders the "mysterious Peterson isomorphism" completely unmysterious. I will see what I can do in the talk to present this bijection between the moduli spaces.

Heisenberg groups and Theta functions,
Three talks in the Number Theory seminar, University of Melbourne 7, 19 and 24 November 2014.
Abstract:
These talks were a survey of Chapter 1 of Igusa's book on Theta function.
We introduced the Heisenberg group, the Schrodinger representation,
discussed the symplectic group Sp_{2n}(R), its maximal compact subgroup,
the Siegel upper half space, the integral symplectic group, the normalizer
of the Heisenberg group in Aut(L^{2}(R)), its maximal compact subgroup,
the favourite discrete subgroup arising as normalizer of a lattice, and
the expression of periodic functions under a lattice
in terms of theta functions of characteristic m,m^{*} and modulus τ.

Two boundary Hecke algebras and SchurWeyl duality,
Talk at the Representation Theory conference/retreat at the ANU coastal campus
at Kioloa, organised by Scott Morrison and Tony Licata, 12 November 2014.
Abstract:
This was a summary of the forthcoming paper with Zajj Daugherty.

Affine and degenerate affine BMW algebras,
Talk at the
Workshop on Diagram Algebras, University of Stuttgart, Germany, 812 September 2014.
Abstract:
This talk will be a survey of recent and forthcoming papers with
Zajj Daugherty and Rahbar Virk. In this work the affine and degenerate affine
BMW algebras are completely parallel: with parallel presentations, parallel structure
and parallel representation theory. These algebras act on tensor space and are in
SchurWeyl duality with orthogonal and symplectic groups and quantum groups.
The centers of these algebras are described by symmetric functions with a cancellation
property as in the description of the cohomology and Ktheory of symplectic and
orthogonal Grassmannians. The representation theory of these algebras can be described
by multisegments and the decomposition numbers are given by KazhdanLusztig polynomials.

The glass bead game,
Public lecture at ICMS
as part of the workshop
Algebraic Lie theory and representation theory, International Centre for Mathematical Sciences,
Edinburgh, Scotland, 3 September 2014.
Abstract:
This talk will take a virtual tour of the toy store with our friends
Maria Callas, Alexander Grothendieck and Hermann Hesse.
There are pleasant games with glass beads, athletic
games skiing the moguls, and violent games where
everything gets smashed. There are crystals and hurricanes
and, of course, a few polynomials. The point of the talk is
to tell some stories related to the topic of the concurrent ICMS
research workshop on Algebraic Lie theory.

Generalized Fock space for decomposition numbers for quantum groups at roots of unity,
Invited talk at the ICM 2014 Satellite Conference on
Representation Theory and Related Topics,
69 August 2014, EXCO, Deagu, South Korea
Abstract:
In this work we provide a generalization of the Type A Fock space. This provides a combinatorially constructed "vector space" with a bar involution such that the barinvariant basis captures the decomposition numbers of Weyl modules for quantum group (of a finite dimensional complex semisimple Lie algebra) at a root of unity. This is joint work with Martina Lanini and Paul Sobaje.

Double affine Artin groups and the modular group (or, more to the point,
Did you know there are bugs on your donut?),
Algebra seminar at the University of North Texas, Denton, June 16, 2014.
Abstract:
The double affine Artin group is used to construct the double affine
Hecke algebra which is used to construct Macdonald polynomials. In
this work we analyse the topological construction of the double affine
Artin group to uncover the underlying elliptic curves and the action
of the modular group. This is joint work with Z. Daugherty and S. Griffeth.
(Note: Double affine Hecke algebras are undegenerated symplectic
reflection algebras and Macdonald polynomials are generalized spherical
and Whittaker functions for padic groups).

Lyndon words and the loop Grassmannian,
Workshop on
Categorification and geometric representation theory, at the
Thematic semester: New Directions in Lie Theory
June 913, 2014.
Abstract:
In the representation theory of KLR algebras, the cuspidal modules are
indexed by the good Lyndon word giving the maximal weight, and the simple
modules are heads of the standard modules (shuffle products of cuspidal
modules). This is a categorification of the relation between the PBW
basis and the canonical basis of the quantum group. In this work we analyse
how a similar phenomenon appears inside the loop Grassmannian. The key
which makes it work is that the generators of the group are exponentials
of the generators of the Lie algebra, and the combinatorics of the Lie
algebra (which produces the Lyndon words) is reflected in the SteinbergTits
relations for the Chevalley group. This enables one to use Lyndon words to
control the combinatorics of expressions in the Chevalley group. This is
joint work with A. Ghitza and S. Kannan.

Are 3pole braids elliptic?,
Colloquium, Dartmouth College, 26 April 2013.
Abstract:
The goal of this work, partly joint with Zajj Daugherty and Stephen Griffeth,
is to understand the source and implications
of the action of the modular group on the double affine braid group.
The example of the double affine braid group of type C_n has a pleasant presentation
as braids with n strands and three poles, and the modular group moves the poles around.
To gain understanding of our objects it is enlightening to draw lots of fun pictures of braids.
PRETEND THERE IS A PICTURE OF A 3POLE BRAID HERE.

Are 3pole braids elliptic?,
Workshop on
Combinatorial Representation Theory, at the
Thematic semester: New Directions in Lie Theory
April 2125, 2014.
Abstract:
The goal of this work, partly joint with Zajj Daugherty and Stephen Griffeth,
is to understand the source and implications
of the action of the modular group on the double affine braid group.
The example of the double affine braid group of type C_n has a pleasant presentation
as braids with n strands and three poles, and the modular group moves the poles around.
To gain understanding of our objects it is enlightening to draw lots of fun pictures of braids.
PRETEND THERE IS A PICTURE OF A 3POLE BRAID HERE.
The modular group action on the double affine braid group
should be of great importance in the study of Macdonald polynomials but
has not yet become a familiar object in the Macdonald polynomial theory.
In his PhD thesis, van der Lek gave three presentations of the double affine
braid group, and these were followed by the presentations of Cherednik,
IonSahi and Haiman. In this work, we compare these presentations, analyze
their relationship to the family of abelian varieties that appear in the
character theory of affine Lie algebras, and do some explicit examples.
 Artin groups and affine Artin groups related to Coxeter groups,
Talk,
Talk preparation,
Deformation retract,
Notes from an old seminar on 14.10.2013,
Coxeter groups seminar, University of Melbourne, 11 April 2014.
Abstract:
This was a survey of the results of the first three chapters of van der Lek's
PhD thesis, focusing on the construction of the configuration
spaces which have as fundamental group the Artin group and the
affine Artin group corresponding to a Coxeter group W.
 Introduction to Elliptic functions,
Talk 1,
Talk 2,
Talk 3,
Number theory seminar, University of Melbourne, 7,14 and 21 March 2014.
Abstract:
This was a leisurely 3talk introduction to elliptic functions and modular forms:
the analogy to trigonometry, theta functions, the Weierstrass Pfunction,
Eisenstein series, elliptic curves, the moduli space of elliptic curves, abelian varieties, and the moduli space of elliptic curves. The punchline was that elliptic functions are global sections of line bundles on elliptic curves and modular forms are global sections of line bundles on the moduli space of elliptic curves.

Proof Machine,
MUMS Seminar, University of Melbourne, 7 March 2014.
Abstract:
There is one tool that has saved my mathematical confidence (and career) more times than any other, my Proof Machine. In this talk I will discuss the theory behind how (and why) it works, and power it up and show how it works on a few examples. With a little practice wielding the hammer anyone can use the Proof Machine to build rock solid proofs, both within mathematics and also outside mathematics.

Proof Machine,
Department Seminar, University of South Australia, 14 February 2014.
Abstract:
There is one tool that has saved my mathematical confidence (and career) more times than any other, my Proof Machine. In this talk I will discuss the theory behind how (and why) it works, and power it up and show how it works on a few examples. With a little practice wielding the hammer anyone can use the Proof Machine to build rock solid proofs, both within mathematics and also outside mathematics.
Talks of Arun Ram in 2013

Fock space and representations of quantum groups,
Taipei Conference on Representation Theory IV, Insitute
of Mathematics, Academica Sinica, 2023 December 2013.
Abstract:
The qdeformed Fock space has several incarnations:
Hayashi's formulation via spin and oscillator representations, or
via semiinfinite wedges, or by a quotient of a polynomial representation
as in LeclercThibon. Misra and Miwa made it a representation of
quantum affine sl(l) and KashiwaraMiwaStern provided an intimate relationship
with the affine Hecke algebra of type A. Ariki used it in his study of the representations
of cyclotomic Hecke algebras at roots of unity, and it has played an important role
in much further work, notably the works of Uglov and LeclercThibon. All this is
mostly for type A. In this work, joint with Martina Lanini and Paul Sobaje, we use affine Hecke
algebras to generalise this construction to all Lie types, in such a way that we
produce a good model for the (graded) Grothendieck group of finite dimensional
(and category O) representations of quantum groups at roots of unity. Our construction
makes clear the relationship to the periodic module introduced in Lusztig's work on
Jantzen's generic decomposition patterns, the inverse KazhdanLusztig polynomials
for affine Weyl groups, the Steinberg tensor product theorem, and the KazhdanLusztig
conjectures for affine Lie algebras at negative integer level (as proved in the work of KashiwaraTanisaki).
The result is a combinatorial model for working with the decomposition numbers
for quantum groups at roots of unity in general type, completely analogous to the Fock space
construction ubiquitous to the type A case.
 What is a Schubert polynomial?, Pure Math Seminar, University of Queensland, 26 November 2013.
Abstract:
A Schubert polynomial is what comes out of the black box called
"cohomology" when you put a Schubert variety in. As technology gets more advanced
generalised cohomologies (say Tequivariant elliptic cohomology)
produce new Schubert polynomials, ... or do they?
It seems that the connecting cable for the actual Schubert varieties doesn't
quite fit the socket in the new cohomologies and the old workaround of
replacing the input by a BottSamelson variety no longer produces the right
answer. But the new cohomologies definitely have some killer apps so it's hard
to resist them.
 Views from Castalia,
Insitute for Basic Sciences Symposium
Representation Theory and Algebraic Structures, Seoul Korea, 7 August 2013.
Abstract:
The title of this talk comes from the novel "The Glass Bead Game" by
Hermann Hesse. The setting of the story is a serene location in the
mountains called Castalia.
There are three mountains (the preprojective variety, the quiver variety
and the loop Grassmannian). These three mountains feed the spring at their
base (the quantum group). At the spring live three nymphs, very similar, but
different (the semicanonical basis, the canonical basis and the MV basis).
Each of these is nourished by the special nutrients of the corresponding
source (the preprojective variety, the quiver variety and the loop
Grassmannian). But the shadow of the muses is the same (the MV polytope).
And, as we shall see, there is a Glass Bead Game in this story as well.
 Views from Castalia, Algebra seminar, University of Sydney, 12 July, 2013.
Abstract: There are three mountains (the preprojective variety, the quiver variety
and the loop Grassmannian). These three mountains feed the spring at their base
(the quantum group). At the spring live three nymphs, very similar, but different
(the semicanonical basis, the canonical basis and the MV basis). Each of these
is nourished by the special nutrients of the corresponding source (the preprojective
variety, the quiver variety and the loop Grassmannian). But the shadow of the
muses is the same (the MV polytope). And, as we shall see, there is a Glass Bead
Game in this story as well.
 Alcove walks and MirkovicVilonen cycles,
Number theory seminar, University of Melbourne, 10 July 2013.
Abstract:
In this talk I gave some background on the relation between characters of
irreducible representations of the Langlands dual group and MirkovicVilonen
cycles in the loop Grassmanian. The method of GaussentLittelmann provides a
construction of the MVcycles using the combinatorics of alcove walks. I
illustrated this method with the computation of the 27 MVcycles corresponding
to the irreducible representation of PGL_3(ℂ) with highest weight
2α_{1}+2α_{2}.
 Combinatorics, representations,
homgeneous spaces and elliptic cohomology,
Plenary lecture at PRIMA 2013 Pacific Rim Mathematical Association Congress,
2428 June 2013, Xuhui (downtown) campus of Shanghai Jiao Tong University.
Abstract:
This talk will give an elementary survey of recent developments in combinatorial
representation theory, where input from stable homotopy theory and algebraic topology enters.
The theory of pcompact groups founded by Dwyer and Wilkerson gives rise to homogeneous
spaces that generalize the combinatorics of Lie theory, and the framework of generalized cohomology
theories (elliptic cohomology and cobordism) provides new insight into classical intersection theory
computations for Grassmannians and flag varieties. In both cases, the study of possible generalizations of the Weyl
character formula, and their relation to AtiyahBott localization formulas provides useful motivation and insight.
The combinatorial aspect provides powerful elementary models for all of these objects.
 Combinatorics, Representations
and elliptic cohomology,
Colloquium at Stanford University, Stanford, California USA, 9 May 2013.
Abstract:
This talk will summarize generalizations of the Weyl character formula
to spherical functions and Whittaker functions (for groups over local fields),
to Macdonald polynomials (generalizations of AskeyWilson polynomials) and
to affine Lie algebras (using theta functions on (very special) abelian varieties).
This leads to the combinatorial study of the equivariant elliptic cohomology
(and equivariant cobordism) of flag varieties.
 A probabilistic interpretation of the Macdonald polynomials,
talk in the RTGC seminar at University of California, Berkeley, 8 May 2013.
Abstract:
P. Hanlon studied a random walk on partitions that has the Jack polynomials as
eigenvectors. This random walk arises from a Markov chain on permutations by
"lumping". In this work (joint with Persi Diaconis) we generalise this process to
a much more vigorous walk which has eigenvectors, the Macdonald polynomials.
 Combinatorial representation theory and elliptic cohomology,
Colloquium at University of Oregon, Eugene, Oregon USA, 6 May 2013.
Abstract:
On passing through a pedestrian underpass near the train station in Bonn one might notice
graffiti declaring passion between CRT and EC and wonder and speculate about what kind
of relationship this is. Actually this graffiti was put there late one night by
a mathematician visiting the Hausdorff Institute of Mathematics. Indeed this is a soap
opera of the best sort with Combinatorial Representation Theory and Elliptic Cohomology
being an uncomfortable pair from different echelons of society that, as a rule, never mix.
Of course the relationship is manipulated by family (generalised cohomology theories),
some friends (the Schubert classes) that were once true have now become elusive, and there
are other, somewhat uncomfortable, acquaintances (abelian varieties and theta functions) who
cannot seem to keep their nose out of what we think is not their business. And who is it
that is spying and scheming for their own purposes? (shh...the affine Lie algebras).
 Bringing together Demazure, Macdonald and Whittaker,
talk at the research seminar of the ICERM
special program in Spring 2013,
Brown University, Providence, 30 April 2013.
Abstract:
I will draw my favourite "Langlands" picture and explain how this specializes to the
Weyl character formula and how it generalizes to Macdonald polynomials. Using the
affine Hecke algebra to unify the perspective clarifies the role of Demazure operators and
provides a definition of Macdonald Whittaker functions and motivates some tantalizing
positivity conjectures on (q,t)weight multiplicities.
 Generalized Schubert calculus,
talk at special session on "???" at the Southwestern section
American Mathematical Society Meeting, Boulder, Colorado USA
1415 April 2013,
Abstract:
We show how to study the generalized cohomology (cobordism) of flag varieties using
two models, the Borel model and the moment graph model. We study the differences between
the Schubert classes and the BottSamelson classes. We will illustrate the results by
showing some rank 2 computations which generalize various earlier computations of
GriffethRam (in equivariant Ktheory) and of CalmesPetrovZainoulline (in nonequivariant
cobordism).
 Elliptic and Cobordism Schubert calculus,
talk at Whittaker functions, Schubert
calculus and crystals, a conference at the ICERM special program in Spring 2013,
ICERM, 48 March, 2013,
Brown University, Providence, 4 March 2013.
Abstract:
In this work we explain combinatorial methods for studying the equivariant generalised
cohomology of flag varieties via the Borel model and the moment graph model. We will
discuss BGGDemazure operators and explain why BottSamelson classes cannot be the same
as Schubert classes. As examples, we compute some Schubert products in equivariant
cobordism. This is a report on joint work with Nora Ganter.
In the actual talk there was not time for doing examples, instead I announced a conjectural
definition of Schubert classes and gave a conjectural formula for the class of a divisor
Schubert class.
 Schubert calculus,
Tutorial on Schubert calculus for the ICERM special program in Spring 2013,
ICERM, Brown University, Providence,
1 March 2013.
Abstract:
This talk was an introduction to the Schubert calculus using the example of
type C_{2}. We reviewed the Borel model and the moment graph model in the
context of equivariant cohomology. We defined BGG operators and BottSamelson
classes, and defined Schubert classes in terms of the moment graph picture using
the support condition and the degree condition. We computed these in the cases
of type C_{2}.
 Alcove path models and Macdonald polynomials, Tutorial on Macdonald polynomials for the ICERM special program in Spring 2013,
ICERM, Brown University, Providence,
25 February 2013.
Abstract:
This talk was a description of the double affine Hecke algebra, Macdonald
polynomials and the path model formulas for Macdonald polynomials from the paper
of RamYip, A combinatorial formulas for Macdonald polynomials.
 padic integrals to sums over
crystals, Tutorial on Crystals for the ICERM special program in Spring 2013,
ICERM, Brown University, Providence,
18 February 2013.
Abstract:
This talk was a description of the main results, and the extended example in the paper
of ParkinsonRamSchwer Combinatorics in affine flag varieties.
I described, in detail, the indexing
of points in the affine flag variety by (Littelmann) paths with additional labels.
The focus was on the application of these tools
to the padic integrals that arise in the local representation theory of the
Langlands program.
Talks of Arun Ram in 2012
 Modular forms, Hecke algebras and G_{2} flags, Number Theory seminar, University of Melbourne, 18 December 2012.
Abstract:
In this talk, we reviewed the relationship between modular forms and spaces of intertwining operators
between induced representations and the associated Hecke algebra actions.
The second part of the talk was focused on providing representatives for the (G(Q), K) double cosets
in G(A), where A is the field of finite adeles, in terms of partial flag varieties.
We used G_{2} as the primary example, decribing null flags in the 7dimensional
representation. Everything followed Gross and the PhD thesis of Pollack.
 The Octonions and G_{2}, Number Theory seminar, University of Melbourne, 7 November 2012.
Abstract:
This talk was an attempt to begin to understand the construction of the integral form of G_{2} for which G_{2}(ℝ) is compact. This group is fundamental to the study of
the algebraic modular forms for the G_{2}, as defined by Gross. Preliminary work on
decomposing the space of algebraic modular forms as a Hecke module for this case has i
been done by Pollack and LanskyPollack. The group is constructed as the automorphism group
of a maximal order in the octonions.
 Clifford Theory and Hecke algebras, Groups and
Combinatorics Seminar, University of Western Australia, 23 October 2012.
Abstract:
The usual Clifford theory describes the irreducible representations of group G in terms of those of a normal subgroup. Generalizing, Clifford theory constructs the irreducible representations
of semidirect product rings and invariant rings. In this work with Z. Daugherty we use Clifford theory to index the irreducible representations of two pole Hecke algebras and relate this indexing to a labeling coming from statistical mechanics (following work of de Gier and Nichols) and to a geometric labeling (coming from Ktheory of Steinberg varieties following KazhdanLusztig). Despite the mathsphysics and geometric motivations for the project, in the talk I shall assume only that the audience is familiar with the notions of groups, rings, and modules.
 Soergel bimodules and Rouquier complexes, AlgebraGeometryTopology seminar, University of Melbourne, 19 October 2012.
Abstract:
Soergel bimodules are the primary tools in the recently announced proof by EliasWilliamson of the positivity of the coefficients of KazhdanLusztig polynomials for Coxeter groups. I will define these and explore some examples. Time permitting, I will explain how these are used in Schubert calculus.
 Flags, Cohomology and Positivity, MUMS (Melbourne University Mathematics Society) lecture, University of Melbourne 03 August, 2012.
Abstract: Flags are fundamental in linear algebra. I'll explain how they relate to lower triangular, upper triangular decompositions
of matrices, and provide a method for representing collections
of flags in equivariant cohomology. Then we'll do a few little computations
and explain a couple little questions for which we'd like to understand
the answers better.
 Combinatorial Representation Theory: Minicourse of five lectures: Representation Theory, Reflection groups and Groups of Lie type,
Representations of the Symmetric group from KLR algebras,
The Weyl character formula and Geometric Langlands,
Crystals from paths and MV polytopes,
and
Crystals from KLR and preprojective algebras, XXII Brazilian Algebra meeting,
1520 July 2012.
Abstract: This would be an introduction to modern methods in combinatorial representation theory. The primary tool is the affine Hecke algebra, which comes in many forms: as the group algebra of the symmetric group, as a quiver Hecke algebra, as a convolution algebra of functions on a padic group. From this we can derive the structure and combinatorics of the cohomology of flag varieties, projective spaces and Grassmannians, and the detailed "crystal" structure of the representations of complex simple Lie algebras. I will give an introductory tour to these techniques in the interplay between combinatorics, representation theory and geometry.
 FImodules: a new approach to stability for S_n representations,
Seminar, University of Melbourne, 22 June 2012.
Abstract: This talk will be a summary of a recent paper of Church, Ellenberg and Farb,
arXiv:1204.4533
FImodules: a new approach to stability for S_n representations.
 Schubert calculus for the affine Grassmannian, Seminar, University of Melbourne, 11 May 2012.
Abstract: This talk will be a summary of a recent paper of Lam and Shimozono.
arXiv:1105.2170, kDouble Schur functions and
equivariant (co)homology of the affine Grassmannian.
 Views from Castalia, Colloquium, University of Southern California, 27 April 2012.
Abstract: There are three mountains (the preprojective variety, the quiver variety
and the loop Grassmannian). These three mountains feed the spring at their base
(the quantum group). At the spring live three nymphs, very similar, but different
(the semicanonical basis, the canonical basis and the MV basis). Each of these
is nourished by the special nutrients of the corresponding source (the preprojective
variety, the quiver variety and the loop Grassmannian). But the shadow of the
muses is the same (the MV polytope). And, as we shall see, there is a Glass Bead
Game in this story as well.
 Elliptic Schubert Calculus, Colloquium, University of California, Los Angeles, 26 April 2012.
Abstract: Traditional Schubert calculus is the combinatorial study of intersections
of Schubert varieties inside the flag variety. This is usually done by computing
cup products in cohomology or equivariant cohomology. This talk is a summary of our
study of how to extend the story to other equivariant cohomology theories: Ktheory,
elliptic cohomology and cobordism.
The talk is intended to be a general audience survey: to give a feel for what flag varieties "look like",
what cohomology theories do for us, and the combinatorial structure (reflection group symmetry)
that makes the game go. The subject is fascinating in the confluence of different parts of mathematics:
Lie groups, loop groups, symmetries of regular polytopes, and algebraic topology.
This talk is based on ongoing joint work with Nora Ganter.
 Generalized equivariant cohomology of flag varieties, Geometry/Physics seminar, Northwestern University, 24 April 2012.
Abstract: I will review some of the KacPeterson paper on affine Lie algebras and modular forms in order to set up a framework for the equivariant elliptic cohomology version of Schubert polynomials. I will also discuss some parts of an analogous story for equivariant cobordism Schubert polynomials and compare and contrast four cases: cohomology, Ktheory, elliptic cohomology and cobordism. This is work in progress with Nora Ganter.
 Views of Castalia, Geometry/Physics seminar, Northwestern University, 24 April 2012.
Abstract: There are three mountains (the preprojective variety, the quiver variety and the loop Grassmannian). These three mountains feed the spring at their base (the quantum group). At the spring live three nymphs, very similar, but different (the semicanonical basis, the canonical basis and the MV basis). Each of these is nourished by the special nutrients of the corresponding source (the preprojective variety, the quiver variety and the loop Grassmannian). At three particular moments of each day (dawn, noon, dusk) the shadow of the muses on the valley below is the same (the MV polytope). And, as we shall see, there is a Glass Bead Game in this story as well.
 Views from Castalia, Special Geometry seminar, University of Texas at Austin, 23 April 2012.
Abstract: There are three mountains (the preprojective variety, the quiver variety
and the loop Grassmannian). These three mountains feed the spring at their base
(the quantum group). At the spring live three nymphs, very similar, but different
(the semicanonical basis, the canonical basis and the MV basis). Each of these
is nourished by the special nutrients of the corresponding source (the preprojective
variety, the quiver variety and the loop Grassmannian). But the shadow of the
muses is the same (the MV polytope). And, as we shall see, there is a Glass Bead
Game in this story as well.
 Generalized equivariant cohomology of flag varieties, Infinite dimensional algebra seminar, MIT, 20 April 2012.
Abstract: I will review some of the KacPeterson paper on affine Lie algebras and modular forms in order to set up a framework for the equivariant elliptic cohomology version of Schubert polynomials. I will also discuss some parts of an analogous story for equivariant cobordism Schubert polynomials and compare and contrast four cases: cohomology, Ktheory, elliptic cohomology and cobordism. This is work in progress with Nora Ganter.
 Schubert Calculus I, II and III, Seminar, University of Melbourne, 9, 11 and 13 April 2012.
Abstract: Schubert calculus I  Examples of the Borel picture for generalised cohomologies of flag varieties, Schubert calculus II  Generalised cohomology of the BottSamelson map, Schubert calculus III  BGG/Demazure operators for generalized cohomology theories: These talks are intended as a (hopefully) leisurely tour of the generalised Schubert calculus that we are setting up in our join work with Nora Ganter. In each case I will try to indicate how to work with the appropriate rings and maps, both in terms of generators and relations, and via moment graphs. Traditional Schubert calculus is the combinatorial study of intersections of Schubert varieties inside the flag variety. This is usually done by computing
cup products in cohomology or equivariant cohomology. This talk is a summary of our study of how to extend the story to other equivariant cohomology theories: Ktheory, elliptic cohomology and cobordism.
 Combinatorics and growth in Chevalley groups and their representations, Connecting finite and infinite mathematics through symmetry, 13 February 2012, University of Wollongong, 2 February 2012.
Abstract: I will endeavour to illustrate the ‘path model’, a powerful tool for labeling points in Chevalley groups in a way which is consistent with all finite dimensional representations (‘finite’ quotients of the group) at once. The main tools are to have a set of generators and relations which interacts in a controllable way with the reflection group geometry of the group (coming from automorphisms) and a powerful combinatorics for manipulating computations using these generators and relations.
 Views from 20 years trekking on the LS path, CMIIMSc Mathematics Colloquium, in celebration of the 80 birthday of CS Seshadri, 2327 January 2012, Chennai Mathematical Institute, Chennai, 23 January 2012.
Abstract: My travels on the LS (LakshmibaiSeshadri) path have brought many personal realizations and its vistas have allowed me to look out over many beautiful structures. As best I can, I shall give a brief summary (1 hour instead of 20 years) including: Kashiwara crystals, combinatorial aspects of Schubert calculus, affine Hecke algebras and structure of Chevalley groups.
Talks of Arun Ram in 2011
 RTT algebras and Algebraic Bethe ansatz, working seminar at University of Melbourne, 6 December 2011.
Abstract: This is a review of RTT algebras (as found in Drinfeld's ICM paper in section 10),
spectral subalgebras (following Drinfeld's ICM paper section 11), lattice models (following deGier's thesis chapter 5),
the source of the Yangian (following Drinfeld and ChariPressley) and a formulation of the algebraic Bethe ansatz
(following Takhtajan and Faddeev, Kulish and Reshitikhin, Kirillov and Reshetikhin, and Nazarov and Tarasov). If time
permits I will formulate the Lmatrices in the Drinfeld double context (following ReshetikhinTakhtajanFaddeev).
 Schubert Calculus, working seminar at University of Melbourne, 28 October 2011.
Abstract: In connection with my work with Nora on the Elliptic Schubert
Calculus I have found that I need to read two recent papers
on Complex Cobordism and Schubert calculus:
CalmesPetrovZainoulline arXiv:0905.1341
and KriticheckoKrishna arXiv:1104.1089
Viewing this talk as a working seminar, I will try to present some survey
of the contents of these papers from my perspective.
 An introduction to the KnizhnikZamolodchikov equations, working seminar at University of Melbourne, 19 September 2011.
Abstract: The KnizhnikZamolodchikov connection provides a remarkable passage between representation theory, differential equations and conformal field theory. I will try to give a brief introduction to this fascinating correspondence.
 Elliptic Schubert calculus, invited talk at Perspectives in Algebraic Lie Theory, at the Isaac Newton Institute, Cambridge, 1216 September 2011.
Abstract: BernsteinGelfandGelfand and Demazure operators
(also called divided difference operators) are the foundation of the
Schubert calculus, the study of the cohomology and Ktheory of the
flag variety in terms of the natural basis coming from the Schubert varieties.
In this work, joint with Nora Ganter, we define elliptic (double) Schubert polynomials
by using rings of theta functions. The elliptic Schubert polynomials are combinatorial realisations of the
classes of Schubert varieties in equivariant elliptic cohomology. We set up the BGG calculus
and the corresponding moment graph perspective for elliptic cohomology of
flag varieties and make the connection to the representation
theory of affine Lie algebras (following KacPeterson, Grojnowski, Ando).
 Elliptic Schubert calculus, invited talk at Algebraic Cycles and the Geometry of Group Orbits, A
Conference on the occasion of the 60th birthday of Peter O'Sullivan, at the Australian National University, Canberra, 24 September 2011.
Abstract: BernsteinGelfandGelfand and Demazure operators
(also called divided difference operators) are the foundation of the
Schubert calculus, the study of the cohomology and Ktheory of the
flag variety in terms of the natural basis coming from the Schubert varieties.
In this work, joint with Nora Ganter, we produce an analogous study of
the elliptic cohomology of the flag varieties, by using rings of theta functions
and the appropriate elliptic cohomology versions of the Thom isomorphism and
localisation at Tfixed points. Additionally, we make the connection to the representation
theory of affine Lie algebras (following KacPeterson, Grojnowski, Ando).
 Cohomology of Grassmannians and isotropic Grassmanians, working seminar talk at University of Melbourne, 2 June 2011.
Abstract: The classical example of the Schubert calculus is the
case of the Grassmannian where the Schubert classes can be represented
as Schur functions, and the problem is then solved with representation theory.
The case of the isotropic Grassmanians (orthogonal and symplectic groups)
was treated by Pragacz, making contact with the Schur Qfunctions that arise in the
projective representation theory of the symmetric group (and Nora's recent work).
I recently understood how this generalises to pcompact groups, what the generalizations
of the Schur Qfunctions are and note, following Morris, an interesting connection to
the evaluation of characters of the symmetric groups at rregular conjugacy classes
(the type of phenomena that Jamie is noting ought to happen in the pcompact setting).
A future goal here is to get the whole picture clear also in Ktheory and Tequivariant Ktheory.
 Affine Weyl group, Heisenberg groups and classifying complex reflection groups, working seminar talk at University of Melbourne, 26 May 2011.
Abstract: I will try to outline the generalised Cartan matrix approach to the
affine Weyl group, as found in Kac's book on Infinite dimensional Lie algebras,
and explain its relation to the Heisenberg group, and an attempt to extend it
so that it might have a chance of providing a new way
to approach the classification of complex reflection groups. This is some
meld of current work with Nora on elliptic stuff, and what Don Taylor and I came
up with during his visit in April.
 Heisenberg groups, abelian varieties and theta functions, working seminar talk at University of Melbourne, 12 May 2011.
 Rank 1 reductive Lie groups, in perspective, working seminar talk at University of Melbourne, 5 May 2011.
Abstract: I will discuss the Lie groups SU(2), Spin(3),
Sp(1), SL(2), PGL(2) their relationships, and their adjoint representations
in the context of the other reductive Lie groups of Lie types
A,B, C and D. I'll derive the adjoint representation in terms of
Hamiltonians=Quaternions, Pauli matrices and Chevalley generators.
 The BorelWeilBott Theorem, working seminar talk at University of Melbourne, 29 April 2011.
Abstract: The lecture will essentially be a version of a lecture I gave at the
WinterSchool on The interaction of Geometry and Combinatorics in Representation Theory
at the Hausdorff Institute in Bonn in January.
Understanding this is, hopefully, a path to a full understanding of
the Weyl character formula, cohomology and Ktheory for compact and pcompact groups,
the elliptic cohomology of flag varieties, theta functions,
Looijenga line bundles, and modular forms from abelian varieties.
 Polytopes, shuffles, quivers and flags, seminar talk at University of Melbourne, 8 March 2011.
Abstract: There are two geometries that show remarkable similarities: that of quiver varieties and that of affine flag varieties. By work of BravermanGaitsgory and GaussentLittelmann and KashiwaraSaito and KamnitzerBaumann one sees the crystals, in the sense of Kashiwara, coming from both quivers and flags. In the picture of LeclercGeissSchroer one sees how elements of the shuffle algebra come from quiver varieties. In joint work with A. Ghitza and S. Kannan we are seeing shuffle elements coming from affine flag varieties.
 A probabilistic interpretation of Macdonald polynomials , invited speaker at the Combinatorial Representation Theory day at the Leibniz Universität Hannover, 18 February, 2011.
Abstract: P. Hanlon studied a random walk on partitions that has the Jack polynomials as eigenvectors. This random walk arises from a Markov chain on permutations by "lumping". In this work we generalise this process to a much more vigorous walk which has eigenvectors, the Macdonald polynomials.
 Polytopes, shuffles, quivers and flags, invited speaker at the La troisième du séminaire de combinatoire énumérative et analytique at the Institut Henri Poincaré, Paris, 3 February, 2011.
Abstract: There are two geometries that show remarkable similarities: that of quiver varieties and that of affine flag varieties. By work of BravermanGaitsgory and GaussentLittelmann and KashiwaraSaito and KamnitzerBaumann one sees the crystals, in the sense of Kashiwara, coming from both quivers and flags. In the picture of LeclercGeissSchroer one sees how elements of the shuffle algebra come from quiver varieties. In joint work with A. Ghitza and S. Kannan we are seeing shuffle elements coming from affine flag varieties. Following my recent joint work Ghitza and S. Kannan, I will explain the purely combinatorial approach for seeing the moment polytopes and the shuffle elements.
 Combinatorics of the flag variety: Minicourse of three lectures: Chevalley groups and Hecke algebras,
Cohomology, and The BorelWeilBott theorem, Hausdorff Insitute of Mathematics, special Trimester on "The interaction of geometry and combinatorics in Representation Theory", Winterschool, 1014 January 2011.
Abstract: This was a review of the cohomology and Ktheory of G/B,
following a readign of the paper of Lam, Schilling and Shimozono, arXiv 0901.1506.
Talks of Arun Ram in 2010
 On the cohomology of G/B, Working seminar, University of Melbourne, December 15, 2010.
Abstract: This was a review of the cohomology and Ktheory of G/B,
following a readign of the paper of Lam, Schilling and Shimozono, arXiv 0901.1506.
 Elliptic cohomology and Weyl character formulas, Invited speaker at the IGA/AMSI Workshop
"Dirac Operators in Geometry, Topology, Representation Theory, and Physics" at University of Adelaide October 1822, 2010.
Abstract: In this work, joint with Nora Ganter, we establish
an elliptic cohomology version of the AtiyahSegalLefschetz
fixed point formula and apply it to the flag variety of a compact
Lie group. We make contact with the work of Looijenga on
Root systems and Elliptic Curves and the work of Kac and Peterson
on Affine Lie algebras and Modular Forms and
obtain Weyl characters for the loop group as push forwards in elliptic cohomology.

The Glass Bead Game, Colloquium,
University of Queensland, Brisbane, 11 October 2010.
Abstract: This title is taken from the novel of Hermann Hesse. In joint work
with A. Kleshchev, we were amused to discover a glass bead game for constructing
representations of quiver Hecke algebras (algebras recently defined by KhovanovLauda and
Rouquier whose representation theory categorifies quantum groups of KacMoody Lie algebras).
In fact, the glass bead game is tantalizingly simple, and may soon be marketed in your local toy store.
I will explain how this game works, and some of the fascinating numerology that appears in the
scoring of the plays.
 What are KLRBMW algebras?,Invited speaker at the International Conference on the NonCommutative Rings and Combinatorial Representation Theory" at Pondicherry University September 23, 2010.
Abstract: In 2008, KhovanovLauda and Rouquier defined a family of diagram algebras whose representations have the property that their
characters are elements of the quantum group. The characters of the simple modules of the KLR algebras are the canonical basis elements of the quantum group. In the type A case, the KLR algebras are a graded version of the affine Hecke algebra. This talk will be a survey,
with the question of the definition of the KLR BirmanWenzlMurakami and Brauer algebras, as the motivation.
 Towards elliptic Chevalley groups and flag varieties,Seminar at Chennai Mathematical Institute, India, September 1, 2010.
Abstract: Assuming that the double affine Hecke algebra is
a shadow of a double loop group or an "elliptic" Chevalley group we
learn about the structure of the elliptic flag variety. Though we are not yet
ready to make a proper definition we can see many properties
of the object which should be the analogue of the flag variety
for the elliptic case. This talk is based on
joint work with Martha Yip on the combinatorics of the double
affine Hecke algebra and with Nora Ganter on obtaining
Weyl character formulas from elliptic cohomology using an
elliptic cohomology analogue of the AtiyahSegal Lefschetz fixed
point formula and localization.
 Musings towards elliptic buildings,Invited speaker at the ICM Satellite conference "Buildings, Finite Geometries and Groups" at the Indian Statistical Institute, Bangalore, India, during August 29  31, 2010.
Abstract: Assuming that the double affine Hecke algebra is
a shadow of a double loop group or an "elliptic" Chevalley group we
learn about the structure of the elliptic building. Though we are not yet
ready to make a Tits style definition we can see many properties
of the object which should be the analogue of the Tits building
for the double loop and the elliptic cases. This talk is based on
joint work with Martha Yip on the combinatorics of the double
affine Hecke algebra and with Nora Ganter on obtaining
Weyl character formulas from elliptic cohomology.
 Symmetry and identities, Melbourne University Mathematics
Society (MUMS) seminar, University of Melbourne, 20 August 2010.
Abstract: I will explain some relationship between
classical partition identities of Euler, Gauss and Jacobi
are related to the symmetries of rigid polyhedra and the
way that they fill up space.
 On affine BMW algebras,Invited speaker at the International Conference on Representation Theory, Xian China August 9 August 14, 2010.
 The Glass Bead Game, Colloquium, University of Adelaide, 25 June 2010.
Abstract: This title is taken from the novel of Hermann Hesse. In joint work
with A. Kleshchev, we were amused to discover a glass bead game for constructing
representations of quiver Hecke algebras (algebras recently defined by KhovanovLauda and
Rouquier whose representation theory categorifies quantum groups of KacMoody Lie algebras).
In fact, the glass bead game is tantalizingly simple, and may soon be marketed in your local toy store.
I will explain how this game works, and some of the fascinating numerology that appears in the
scoring of the plays.
 Affine BMW algebras,
Pure Mathematics Seminar,
University of Adelaide, 25 June 2010.
Abstract: I will describe a family of algebras of tangles (which give rise to link invariants
following the methods of ReshetikhinTuraev and Jones) and describe some aspects of their
structure and their representation theory. The main goal will be to explain how to use
universal Verma modules for the symplectic group to compute the representation theory
of affine BMW (BirmanMurakamiWenzl) algebras.
 Combinatorics and Spherical functions, invited talk at the BIRS Workshop 10w5096, Whittaker Functions, Crystal Bases, and Quantum Groups, Banff Canada
June 611, 2010.
 What is a line bundle?,
Informal Seminar,
University of Melbourne, 4 June 2010.
 Three examples when the space of global sections of line bundles
is interesting (BorelWeilBott, modular forms, and toric varieties from polytopes),
Informal Seminar,
University of Melbourne, 2 June 2010.
 Lecture 1: Quantum groups and Lyndon words, Lecture 2: Graded quiver algerbas and their representations, Lecture 3: Indexings of canonical bases: Lyndon words, MV polytopes and the path model, Invited speaker at the 64th Séminaire Lotharingien de Combinatoire, Program, Institut Camille Jordan  Bâtiment Braconnier, Lyon, Sunday, March 28th, 2010 (evening) to Wednesday March 31st, 2010.
Talks of Arun Ram in 2009

The Glass Bead Game,
Short talk to the Vacation Scholars, University of Melbourne, 18 December 2009.
This was a short presentation of the bead game in recent work with A. Kleshchev on
homogeneous representations of Quiver Hecke algebras.

Moment maps on flag varieties and piecewise linear functions,
Seminar/Introduction, University of Melbourne, 26 November 2009.
We discuss the theorem of BorelBottWeil and the Weyl character formula via localization.

Introduction to equivariant cohomology,
Seminar/Introduction, University of Melbourne, 19 November 2009.
We will introduce foundational material on BrionVergne lattice point counting via the JeffreyKirwan localization formula.

Introduction to categories,
Seminar/Introduction, University of Melbourne, 12 November 2009.
We introduce the notions of chain complexes, categories, natural transformations, totalization, homotopy and derived categories.
 Universal Verma modules and Translation, at the
53rd Annual meeting of the Australian Mathematical Society, Special session in Algebra and Number Theory,
University of South Australia, Adelaide 28 Spet. 1 October, 2009.
Abstract: We will introduce a framework for studying the combinatorics
of translation functors in a "universally integral" framework and explain
a unified perspective on GabberJoseph's approach to the KazhdanLusztig conjectures,
Kleshchev and Brundan's approach to modular branching rules, and
the MisraMiwa Fock space. This talk is based on joint work with Peter Tingley.
 Universal Verma modules and Translation, at the
International workshop on combinatorial and geometric approach to representation
theory,
Seoul National University, Korea, 2124 September, 2009.
YouTube
Abstract: We will introduce a framework for studying the combinatorics
of translation functors in a "universally integral" framework and explain
a unified perspective on GabberJoseph's approach to the KazhdanLusztig conjectures,
Kleshchev and Brundan's approach to modular branching rules, and
the MisraMiwa Fock space. This talk is based on joint work with Peter Tingley.
 Why I care about pcompact groups, Reading seminar, University of Melbourne, 21 August 2009.
Abstract: A survey of symmetric functions, Schur functions, Weyl characters, the BorelWeilBott theorem, the cohomology and Ktheory of flag varieties, the classification of pcompact groups and the ClarkEwing formula.
 Poles, strings, braids and lattices, Colloquium, La Trobe University, 1 May 2009.
Abstract: The double affine braid group has important applications
to Macdonald polynomials, group representations, mathematical
physics and combinatorics. The classical type double affine braid
groups have nice pictorial presentations which exhibit the
tantalizing symmetries at play. In this talk I'll draw some of these
pictures and explain their role in topology, harmonic analysis,
combinatorics and the study of symmetry.
 Lyndon Bases, "blackboard seminar", University of Melbourne, 31 March 2009.
Abstract: I will define Lyndon words and good Lyndon words
and explain how we associate standard and simple quiver Hecke
algebra modules to these words. I will not assume any memory
of last week's talk.
 Quiver Hecke alagebras, "blackboard seminar", University of Melbourne, 24 March 2009.
Abstract: Quiver Hecke algebras were recently defined by KhovanovLauda
and, independently, by Rouquier. The importance of these algebras
is that the category of graded modules for the quiver Hecke algebras is a
categorfication of the DrinfeldJimbo quantum group. I will give a survey
of this exciting new subject, perhaps highlighting some of our recent results
joint with Kleshchev.
 A path model formula for Macdonald polynomials,Séminaire sur les Algèbres Enveloppantes et Théorie des Représentations, Paris Jussieu, 6 March 2009.
Abstract: The path model of Littelmann provides a combinatorial
formula for Weyl characters. In this talk we shall explain the generalization
of the Littelmann formula to Macdonald polynomials.
 A path model formula for Macdonald polynomials, Seminar Algebra and Topologie, University of Basel, 20 February 2009.
Abstract: The path model of Littelmann provides a combinatorial
formula for Weyl characters. In this talk we shall explain the generalization
of the Littelmann formula to Macdonald polynomials.
 Two boundary Hecke algebras and tantalizer algebras,
Algebra seminar, Maxwell Institute for the mathematical sciences, University of Edinburgh, 17 February 2009.
Abstract: The double affine Hecke algebra (DAHA) of type C has special properties (6 parameters!) and distinguished quotients. The Macdonald polynomials for this Hecke algebra are the Koornwinder polynomials and the
AskeyWilson polynomials. One interesting quotient of the DAHA is the two boundary TemperleyLieb algebra.
The 2 boundary TemperleyLieb algebra points the way to a family of centralizer algebras which includes the 2
boundary BMW (BirmanMurakamiWenzl) algebras. This talk will be a medley of vignettes around double affine
type C braid groups and quotient algebras.
 Two boundary Hecke algebras and tantalizer algebras, Algebra seminar at Cambridge University, 28 January 2009.
Abstract: The double affine Hecke algebra (DAHA) of type C has special properties (6 parameters!) and distinguished quotients. The Macdonald polynomials for this Hecke algebra are the Koornwinder polynomials and the
AskeyWilson polynomials. One interesting quotient of the DAHA is the two boundary TemperleyLieb algebra.
The 2 boundary TemperleyLieb algebra points the way to a family of centralizer algebras which includes the 2
boundary BMW (BirmanMurakamiWenzl) algebras. This talk will be a medley of vignettes around double affine
type C braid groups and quotient algebras.
 Symmetry, Polynomials and quantisation Lecture1 Lecture2 Lecture3 Lecture4,
Minicourse of four lectures at the program Algebraic Lie Theory, Isaac Newton Institute, 1223 January 2009.
Abstract: These talks will provide a pictorial approach to Weyl groups, braid groups and their Hecke algebras. With the pictures in hand, we can use them to study orthogonal polynomials, representations of braid groups, solutions of difference and differential equations, integrable systems and the quantisations that produce them.
Talks of Arun Ram in 2008
 Beads on runners, Invited talk at the special session Group actions and Representation Theory at the 7th AustraliaNew Zealand Mathematics Convention, Christchurch, New Zealand, 812 December 2008.
Abstract: KhovanovLauda algebras are a family of algebras whose representation theory provides a categorification of quantum groups. In this work we classify and construct homogeneous representations of these algebras. The construction generalises the construction of irreducible representations of the symmetric groups and the notions of partitions, skew shapes, and abaci.
 The mysteries of symmetry, Colloquium, Australian National University, 20 November 2008.
Abstract: In recent joint work with Martha Yip we gave a combinatorial formula for Macdonald polynomials. The formula is a weighted sum of paths and the construction of the paths is completely elementary. The mystery is that these paths are describing subtle information about fancier objects: loop groups, integrable hierarchies of differential equations, representation theory and cohomology theories. I will try to formulate some of my speculations about how these objects are related. The underlying symmetry is certainly touching many parts of modern mathematics and it is all the more amazing that the elementary combinatorics of paths has something deep to say about it all.
 Beads on runners, Colloquium, Monash University, 6 November 2008.
Abstract: We think of beads on runners like an abacus, or
like one of those games for toddlers where the children slide
the beads on the runners (these games are sometimes found in
waiting rooms of the offices of pediatricians). In joint work with
A. Kleshchev we have shown this is a perfect model for representations
of KhovanovLauda algebras, the recently discovered algebras
whose representations categorify quantum groups. I shall explain the
bead and runner model and how to have your toddlers compute
representations of KhovanovLauda algebras while waiting
for the doctor at the medical centre. The model generalizes partitions
and their classical connection to the symmetric group. At the
end of the talk I will explain how these algebras are related
to Lie algebras and quantum groups and why they are considered
a great new advance in the art of "categorification".
 Short lecture at the University of Melbourne/BHP Billiton School Mathematics competition, 11 October 2008.
Abstract: This was a 10 minute talk to school students  maths competition winners. I told them that I went into mathematics for the lifestyle and pointed out the existence of a coffee shop/restaurant on the lakefront in Lugano on Lago Como in Swizerland. Then we looked at the wonderful Bratelli diagram on Tom Halverson's web page, and finally I told them that Persi Diaconis has a knack for finding uses of pure maths in other arenas and will be visiting Melbourne in 2010.
 A combinatorial formula for Macdonald polynomials, Victorian Algebra Conference, RMIT Melbourne, 23 October 2008.
 Generalising Pascal's triangle, Melbourne University Mathematics and Statistics Society (MUMS), lunchtime seminar, 12 September 2008.
 Two boundary Hecke algebras and tantalizer algebras, Invited speaker at the International conference on
Combinatorics and Representation Theory, Graduate School of Mathematics,
Nagoya University, 15 September 2008.
Abstract: The double affine Hecke algebra (DAHA) of type C has special properties (6 parameters!) and distinguished quotients. One interesting quotient is the two boundary TemperleyLieb algebra. The 2 boundary TemperleyLieb algebra points the way to a family of centralizer algebras which includes the 2 boundary BMW (BirmanMurakamiWenzl) algebras. This talk will survey this family of algebras.

Introduction to Buildings,
AlgebraGeometryTopology Discussion session, University of Melbourne, 21 August 2008.
Abstract: This is a brief introduction to buildings in hopes of strengthening the analogy to the curve complex and FenchelNielsen coordinates.

M_{g,n} and Fock space,
AlgebraGeometryTopology Processing seminar, University of Melbourne, 18 August 2008.
Abstract: In his January lectures at MSRI, Okounkov outlined how to use
partition combinatorics and Fock space to give formulas for the
coefficients of the M_{g,n} volume polynomial. I shall try to explain this
combinatorics and summarize the results of OkounkovPandharipande.

M_{g,n}, counting and recursions
(pdf file),
AlgebraGeometryTopology Processing seminar, University of Melbourne, 4 August 2008.
Abstract: I will attempt an elementarification/bigpicturification of the
topic of Paul Norbury's talk last week.

Introduction to the Path Model,
Seminar/Introduction, University of Melbourne, 2 July 2008.
About the path model:
I mean the model of P. Littelmann which generalises
Dyck paths to give combinatorial models for representations
of compact Lie groups. I am interested in using it to
determine the polytope whose integer points describe
the places where LittlewoodRichardson coefficients
(also called ClebshGordon or tensor product coefficients)
are nonzero.

Type C Hecke and TemperleyLieb algebras,
Seminar/Introduction, University of Melbourne, 27 June 2008.

A combinatorial formula for Macdonald polynomials,
Stanford Combinatorics and Geometry seminar,
Stanford University, April 30, 2008.
Abstract:
We will explain a common generalization of Littelmann's formula
for Weyl characters and Schwer's formula for spherical functions
for a padic group. These formulas hold for arbitrary
Lie type.

Symmetric functions d'après Macdonald,
Keynote presentation at the 4th annual
Graduate Student
Combinatorics Conference, UC Davis, April 1213, 2008.
Abstract:
This talk will be a road map to Macdonald's classic book
on Symmetric functions, highlighting the combinatorics, representation
theory and geometry coded by symmetric function identities.

Path Models
(pdf
notes), invited talk at
Topics in Combinatorial Representation Theory,
MSRI, Berkeley, March 1721, 2008.
Abstract:
Recent years have seen big developments in the theory and applications
of path models. The new applications are in understanding the combinatorics
of the affine Hecke algebra, spherical functions, and the geometry of points in affine flag
varieties.
This talk will survey some of these recent results.

Tantalizer algebras,
Colloquium
University of Utah, March 6, 2008.
Abstract:
Abstract: Tantalizer is short for tensor power centralizer.
These algebras often come as algebras of diagrams or of tangles,
and so working with them requires drawing lots of pictures. Their
structure and representation theory contains and immense amount of
information about the representation theory of groups and quantum groups
of types GL, SO, Sp, and they can be used to construct corresponding link
polynomials and 3manifold invariants. This talk will be a survey of some
recent developments in tantalizer algebras.

Path Models,
Representation Theory seminar,
University of Utah, March 7, 2008.
Abstract: This talk will be a survey of applications
of path models: The Weyl character formula, Schubert calculus, spherical
functions, normal forms in Chevalley groups, and indexing of points in
affine flag varieties and MirkovicVilonen cycles.

Tantalizers, invited talk at
University of California Lie Theory Workshop,
a conference in honor of Georgia Benkart,
University of California, San Diego, February 1618, 2008.
Abstract:
A tensor power centralizer algebra (tantalizer) is the algebra of
commuting operators for a Lie group or quantum group action on tensor space.
The favourite examples are the group algebra of the symmetric
group and the Brauer algebra. This talk will survey some recent work
on tantalizers: giving definitions and
recent results for affine and graded BMW algebras and some two boundary
tantalizers.

Minicourse: Combinatorics of Lie Type, three lectures at the
Introductory Workshop on Combinatorial Representation Theory
at MSRI, January 2225, 2008.
Talks of Arun Ram in 2007
 Combinatorial Representation Theory 20082018,
Colloquium,
University of Minnesota, November 15, 2007.
Abstract: The 1997 survey article of BarceloRam entitled Combinatorial Representation Theory “defined” the field and set out its structure. In 2007 this field is thriving and vibrant. In Spring 2008 there will be a full semester program at MSRI entitled Combinatorial Representation Theory. Where is the field now? What has happened in the interim 19972007? More importantly, what will happen in Combinatorial Representation Theory in 20082018?
 Generalizing partitions and standard tableaux,
combinatorics seminar, University of Minnesota, November 15, 2007.
Abstract: The irreducible representations of the symmetric group are indexed by partitions and bases of these representations are indexed by standard tableaux. The representation theory of the affine Hecke algebras provides a generalization of partitions and standard tableaux. I will explain these combinatorial indexings and how they arise.
 Two row partitions and the TemperleyLieb algebra, Combinatorics seminar, University of Wisconsin, Madison, October 15, 2007.
Abstract: Following a good idea of V. Rittenberg, two boundary diagram algebras are getting more and more attention, with two boundary TemperleyLieb algebras being a fundamental example. This talk will begin to answer the question: Which Type C affine Hecke algebra representations are two boundary TemperleyLieb representations and what is a good combinatorial set for indexing these representations?
 Boundary diagram algebras,
Representation theory seminar, University of Wisconsin, Madison,
October 12, 2007.
This talk was a repeat of a talk given at University of Koln on
19 November 2005.
 Centers of tantalizers, Representation theory seminar, University of Wisconsin, Madison, September 14, 2007.
Abstract: Many diagram algebras arise as tantalizers. The SchurWeyl duality makes it possible to steal most of the center of the tantalizer from the corresponding dual object in the duality. I will outline this process and explain how combinatorial results pop out of the picture. This talk is based on joint work with Zajj Daugherty and Rahbar Virk.

Today I feel like a mathematician  personality, music and geometry, The 21st Behrend Memorial Lecture, a public lecture at the University of Melbourne, August 21, 2007.
Abstract: What does it feel like to be a mathematician? Who are the people who discovered and proved the Weil conjectures (one of the great human achievements of the 20th century)? Are they artists, musicians, or scientists? So, what does it feel like to be a mathematician, really?
 Combinatorics in affine flag varieties, 6 July 2007; invited talk
at GL07, Geometry and Lie Theory, a
conference in honor of Gus Lehrer's 60th birthday,
University of Sydney, July 26 and July 913, 2007.
Abstract: This talk is about the combinatorics of indexing points in
affine flag varieties. It is possible to make choices so that the points
are indexed by a refinement of Littelmann's path model in such a way that
the Schubert cell and the MirkovicVilonen slice are easily read off the
"path" indexing of the point. From this, the relations for the affine Hecke
algebra can be derived, both in the IwahoriMatsumoto and in the Bernstein
generators. If time permits I will discuss the action of the "root
operators" on points, and/or the relation to the Kamnitzer and
BaumannGaussent indexings of MirkovicVilonen cycles.
 What is a Weyl group?, Summer Representation Theory seminar, University of WisconsinMadison, 14 June 2007.
 Level l Fock spaces and the polynomial representation of
Cherednik's double affine Hecke algebra,American Institute of Mathematics workshop: Arithmetic harmonic analysis on character and quiver varieties,
American Institute of Mathematics, Palo Alto, June 48, 2007.

Introduction to Buildings and Combinatorial Representation Theory
(pdf file),
American Institute of Mathematics workshop on Buildings and Combinatorial Representation Theory, Palo Alto, March 26, 2007.
 Introduction to moment maps on flag varieties, Lie Theory seminar, University of Wisconsin, Madison, 21 March, 2007.
Talks of Arun Ram in 2006
 Plenary lecture at the 2006 Fall
American Mathematical Society Southeastern Section Meeting, University of Arkansas,
Fayetteville, Arkansas, 34 November 2006.
 Lecture at the workshop Modern Math: An Introduction to 200708 Programs at MSRI, at the Society for the Advancement of Chicanos and Native Americans in Science , National Conference, Tampa, Florida, October 2425, 2006.
 Combinatorial Hopf algebras: An outsider's survey,
Minicourse at the conference Hopf algebras, Combinatorics and Quantum field Theory,
MaxPlanck Institute for Mathematics in the Sciences, Leipzig,
Germany, 2528 September 2006.
 Path models and Chevalley groups, Oberwolfach meeting on
Finite groups and representation theory, March 2531, 2006.
 Representations and translation, Special lecture in Quantum groups
course, KdV Institut, Amsterdam, March 22, 2006.
 Row reduction and loop groups, Lie Theory seminar,
University of Wisconsin, Madison, February 28, 2006.

The Schur Hopf algebra, Combinatorics seminar,
University of Wisconsin, Madison, February 27, 2006.
 Alcove walks and reductive groups over local fields, Lie group and Representation Theory seminar,
University of Maryland, February 3, 2006.
In this talk I presented the precise combinatorial construction
of the generalized MV cycles by labeled alcove walks.
 Examples of groups: Lecture 1 
Reflection groups and braid groups,
Lecture 2 
Matrix groups and Lie groups,
Special minicourse,
University of Rome "La Sapienza",
January 2324, 2006.
These lectures were for advanced undergraduates in mathematics
in Italy: introductory lectures on braid groups, reflection groups,
matrix groups and Lie groups.
 Hecke algebras and spherical functions,
Harmonic analysis seminar, University of Rome "La Sapienza",
January 18, 2006.
Talks of Arun Ram in 2005

Boundary diagram algebras
(pdf file),
Seminar on Transformation groups & mathematical physics,
a joint seminar of the Universities of Koln, Hamburg, Bochum,
Bremen and Darmstadt, University of Koln, November 19, 2005.
This talk is about diagram algebras which come from the twoboundary
braid group (braids with two poles). This is a generalization of
recent work (from statistical mechanics) on twoboundary TemperleyLieb
algebras. The generalized setting naturally includes two boundary Hecke
algebras and twoboundary BMW algebras. These algebras are like affine
Hecke algebras (of type A) and affine BMW algebras except with two poles.
 pcompact groups,
Oberseminar Geometrie, University of Fribourg,
Switzerland, October 26, 2005
This talk was the first time I realised that applying Milnor's
construction of the classifying space to the
pcompact groups of ClarkEwing gives
the space where path models
(such as the Littelmann path models) live.
 Random walks, spherical functions and representations,
Colloquium, University of Fribourg,
Switzerland, October 25, 2005.
 pcompact groups,
Algebra seminar, University of Rome "La Sapienza",
October 20, 2005.
This talk was my first attempt to learn something about pcompact groups.
 Diagram algebras as tantalizers,
Colloquium, University of Rome "Tor Vergata",
October 19, 2005.
This talk was the first time I realized and defined the
graded version of the group algebra of the
affine braid group which has, as quotients,
the graded BMW algebras (also called cyclotomic NazarovWenzl algebras),
and the graded Hecke algebras.
 Representations of affine Hecke algebras,
Algebra seminar, University of Rome "La Sapienza",
October 13, 2005.
This talk was a survey on the representations of affine Hecke algebras.
 Alcove walks and Iwahori cosets Algebra seminar, University of Rome "La Sapienza",
October 6, 2005.
This talk was where I first worked out the generalization of
the MV cycles to G/I, in the example SL_2. In other words,
coset representatives for the cosets in U^+vI\cap IwI,
coset representatives for the cosets in U^+vI\cap IwI,
where I is an Iwahori, and v and w are elements of the affine
Weyl group.
 Picturing Hecke algebras and loop groups,
Algebra seminar, University of Rome "La Sapienza",
September 29, 2005.
This talk was an attempt to explain the alcove walk method of looking at
affine Hecke algebras and loop groups.
 Random walks, spherical functions and representations,
Colloquium, University of Stuttgart, July 4, 2005.

qcrystals ,
Invited speaker in the special session in honor of Adriano Garsia at
the conference Formal Power Series and Algebraic Combinatorics 2005,
June 2425, 2005.
 Commuting elements in diagram algebras, Algebra seminar, University of Wuppertal, June 7, 2005.
 Verma crystals, Algebra seminar, University of Lyon 1, May 27, 2005.
 Random walks, spherical functions and representations, Colloquium, University of Lyon 1, May 26, 2005.
 Walks, crystals and polytopes, Algebra seminar, University of Caen, May 24, 2005.
 Random walks, spherical functions and representations, Colloquium, University of Freiburg, May 13, 2005.
 Murphy elements in diagram algebras, Plenary speaker at the conference Cellular and diagram algebras and
their applications in mathematics and physics,
University of Leicester, England, April 310, 2005.
 Combinatorial Representation Theory, Colloquium, MaxPlanckInstitut fur Mathematik, March 24, 2005.

Combiantorial Representation theory IICrystals,
Talk 2 of a lecture series at University of Zaragoza, Spain,
February 24, 2005.
 Combinatorial Representation theory ITowers and Centralizers,
Talk 1 of a lecture series at University of Zaragoza, Spain,
February 22, 2005.
 Representations of Reflection groups, Seminar, Bernoulli Centre, EPFL, Lausanne,
January 19, 2005.
Talks of Arun Ram in 2004
Talks of Arun Ram in 2003
Talks of Arun Ram in 2002

Plenary speaker for the 46th Annual conference of
the Australian Mathematical Society, Newcastle, Australia,
September 30October 3, 2002.
 Plenary speaker for the Quantum groups day at the
XXIV International Colloquium on Group Theortical
Methods in Physics, Paris, France, July 1520, 2002.

Geometry seminar at University of Wuppertal,
Wuppertal, Germany, July 2, 2002.
 Invited speaker at the conference on Computational Lie
theory, at the Centre Recherches Mathematiques of the University
of Montreal, May 27June 7, 2002.
 Colloquium, University of Massachusetts, Amherst,
March 28, 2002.
 Plenary speaker, two one hour lectures, at
the MidAtlantic Algebra Conference, Wake Forest University,
March 1617, 2002.
Talks of Arun Ram in 2001
 February 1, 2001, Symmetric functions seminar, Isaac Newton
Institute, Cambridge, England.
 February 16, 2001, Colloquium, University of Southampton, England.
 February 21, 2001, Algebra seminar, Cambridge University, England.
 February 26, 2001, Speaker, Isaac Newton Insitute Colloquium for
General Scientific Audience.
 March 8, 2001, Colloquium, City University, London, England.
 March 21, 2001, Colloquium, University of Birmingham, England.
 March 2531, 2001, Invited speaker at the conference,
``Representations of Finite groups'' Oberwolfach, Germany.
 April 30, 2001, Algebra seminar, University of Copenhagen, Denmark.
 May 2, 2001, Algebra semianr, University of Aarhus, Denmark.
 May 11, 2001, Colloquium, University of Warwick, England.
 May 21, 2001, Algebra seminar, University of Leicester.
 May 23, 2001, Algebra seminar, University of Glasgow, Scotland.
 May 31, 2001, Seminaire Chevalley, Institut Henri Poincare, Paris, France.
 May 31, 2001, Colloquium, Ecole Normale Superieur, Paris, France.
 June 25, 2001, Invited plenary speaker at ``The Heritage
of Schur's 1901 dissertation: a conference in honor of J.A. Green''.
 June 12, 2001, Algebra seminar, University of Oxford.
 June 12, 2001, Representation theory seminar, University of Oxford.
 June 18, 2001, Speaker, Symmetric functions seminar, Isaac Newton Institute,
Cambridge, England.
 October 67, 2001, Invited speaker at the conference
``Midwest Lie algebras and Related Topics'' conference,
DePaul University.
 October 2226, 2001, Invited speaker at the conference
``Combinatorial and Geometric Representation Theory'',
Seoul, Korea.
 December 716, 2001, Invited speaker at the conference
``Algebra and Geometry'', University of Hyderabad, India.
 December 1622, 2001, Invited speaker at the special year
on ``Recent results and conjectures on Hilbert functions'',
IIT Bombay, India.
Talks of Arun Ram in 2000
 December 1, 2000, Colloquium, University of WisconsinMilwaukee.
 November 2223, 2000, Series of two talks, algebra seminar,
Instituto de Matematica, UNAM, Morelia,
 November 21, 2000, Colloquium, Instituto de Matematica, UNAM, Morelia,
Mexico.
 October 1620, 2000, Invited speaker at ``Algebres de Hecke affines
et groupes reductifs'', LuminyMarseille, France.
 Algebra seminar, University of Sydney, Sydney Australia, August 11, 2000.
 Colloquium, Macquarie University, Sydney Australia, August 7, 2000.
 Algebra seminar, Mathmatisches Institut B, Universität Stuttgart,
June 8, 2000.
 Algebra seminar, University of Strasbourg, France, June 7, 2000.
 BaselFreiburgStrasbourg joint Algebraic Groups Seminar, May 30, 2000.
 Combinatorics seminar, University of Michigan, Ann Arbor, March 31, 2000.
 Algebra seminar, University of Oregon, Eugene, March 7, 2000.
 Colloquium, University of Oregon, Eugene, March 6, 2000.
Talks of Arun Ram in 1999
 Invited speaker at the KIAS Lie Theory Conference at the
Korea Institute for Advanced Study (KIAS), Seoul, Korea, October 58, 1999.
 Invited speaker at the conference Quantum groups and knot theory,
at L'Institut de Recherche Mathématique Avancée, Strasbourg, France,
September 2729, 1999.
 Invited lecturer at Seoul National University Math Camp, Chunan, Korea, June 21, 1999.
 Invited lecture series (Minicourse on Hecke algebra representations) at the Korea Institute for Advanced Study
(KIAS), Seoul, Korea, June 1517, 1999.

Tableaux, hyperplanes and representations, plenary talk at the
11th Conference on Formal Power Series and Algebraic
Combinatorics, Barcelona, June 711, 1999.
 Tableaux, hyperplanes and representations,
Colloquium, Center for Communications Research, Princeton, May 18, 1999.
 Affine braids, quantum groups, and Jantzen filtrations,
Invited speaker at the special session on
Representations of Lie algebras
at the American Math. Society meeting, Buffalo, NY, April 2425, 1999.
 Lie Theory seminar, MIT, April 21, 1999.
 Combinatorics seminar, MIT, April 19, 1999.
 Algebra and Geometry seminar, Stanford University, April 8, 1999.
 Lie group seminar, Rutgers University, March 5, 1999.
 Tableaux, hyperplanes and constructing representations,
a lecture series at the Institute of Advanced Study, February 1999.
Talks of Arun Ram in 1998
Talks of Arun Ram in 1997
Talks of Arun Ram in 1996
Talks of Arun Ram in 1995
Talks of Arun Ram in 1994
Talks of Arun Ram in 1993
Talks of Arun Ram in 1992
Talks of Arun Ram in 1991
Talks of Arun Ram in 1990
Talks of Arun Ram in 1989