 |
MAST90030 |
Semester II 2025 |
Lecturer: Arun Ram, 174 Peter Hall Building, email: aram@unimelb.edu.au
Time and Location:
- Lecture:
- Wednesday 10:00-11:00 Evan Williams Theatre, Peter Hall Building
- Thursday 10:00-11:00 Evan Williams Theatre, Peter Hall Building
- Friday 9:00-10:00 Evan Williams Theatre, Peter Hall Building
- Consultation/Problem/ProofandWriting/ResearchExploration/Tutorial sessions:
- Wednesday 11:00-12:00 in Russell Love
- Thursday 9:00-10:00 in Evan Williams
- Friday 11:00-12:00 in Russell Love
I am often available for additional questions/discussions after class on Friday.
I am available by appointment. I rarely come in to the University before 12
as I have other responsibilities in the morning until 12. If you email me
suggesting some weekday afternoon times for an appointment that work for you
then I can choose one of them that also works for me. If you email me and
don't suggest some times that work for you,
then I will respond by asking you to suggest some
weekday afternoon times that work for you.
Some of my thoughts about teaching have been written down in the Lecture script:
Teaching Math in the Next Life
I am very interested in listening to and discussing any
thoughts/reaction/improvements that you have in relation to this content.
The student representatives are
??? ??? email: ???@student.unimelb.edu.au
??? ??? email: ???@student.unimelb.edu.au
The handbook entry for this course is at https://handbook.unimelb.edu.au/2025/subjects/mast90030/print
Main Topics
- (1) Partitions and symmetric functions
- (2) Posets, lattices and generating functions
- (3) The symmetric group and GLn(𝔽q)
- (4) Catalan, Modified Hall-Littlewood and Modified Macdonald polynomials
- (5) Moment graphs and Kazdhan-Lusztig polynomials
- (6) Flag varieites, Schubert varieties and Springer fibers
Announcements
- No books, notes, calculators, tablets, ipads, phones, etc at the exam.
- Prof. Ram reads email but generally does not respond by email. Usually these are collated and reponses to email queries are provided in the first few minutes of lectures. That way all students can benefit from the answer to the query.
- Compared to the other resources on this page Lecture Capture is not an efficient way to absorb material for this course. If you want to get a hang on the naterial covered in a particular lecture, you should make a handwritten copy of the Handwritten Lecture Notes for that lecture which are posted below, and ask any questions that come up as you make your handwritten copy in the Proof and Solution writing sessions.
Assessment
Final exam
The problem sheets. The final exam will be constructed by ???
Assignments
To be determined
Tutorial sheets
To be determined
Resources
Recommended are:
- The notes on this page
Another favourite resource for some of the the material we will cover is
Richard Stanley, Enumberative combinatorics, Volumes I and II, ????, 20??.
Another favourite resource for some of the material we will cover is
I.G. Macdonald, Symmetric functions and Hall polynomials, Second edition, Oxford University Press, 1995.
Notes written by Arun Ram
| Resources in preparation |
In class lectures
Week 1: Partitions, the exponential, and the symmetric group
- 28 July 2025 Lecture 1: Partitions and Bratelli diagrams
Draft Hand written Lecture Notes
- 30 July 2025 Lecture 2: The binomial theorem and the exponential
Draft Hand written Lecture Notes
- 1 August 2025 Lecture 3: The symmetric group
Draft Hand written Lecture Notes
Week 2: Posets, lattices and rank generating functions
- 4 August 2025 Lecture 4: Posets and lattices
Draft Hand written Lecture Notes
- 6 August 2025 Lecture 5: The subspace lattice 𝔾(𝔽n)
Draft Hand written Lecture Notes
- 8 August 2025 Lecture 6: The subset lattice 𝕊(n), and simple refelctions in
Sn
Draft Hand written Lecture Notes
Week 3: Fundamental symmetric functions
- 11 August 2025 Lecture 7: Generating functions and monomial formulas
Draft Hand written Lecture Notes
- 13 August 2025 Lecture 8: Power sums and the Cauchy-Macdonald kernel
Draft Hand written Lecture Notes
- 15 August 2025 Lecture 9: Binomial theorems
Draft Hand written Lecture Notes
Week 4: Crystals
- 18 August 2025 Lecture 10: The tensor category of crystals
Draft Hand written Lecture Notes
- 20 August 2025 Lecture 11: The irreducible crystals B(λ)
Draft Hand written Lecture Notes
- 22 August 2025 Lecture 12: Schur functions and the Weyl character formulas
Week 5: Littlewood-Richardson rule, combinatorial R-matrix, RSK, Cauchy identities, Pieri rules, and characters
- 25 August 2025 Lecture 13: Products and restrictions and the
Littlwood-Richardson rule
- 27 August 2025 Lecture 14: The comibnatorial R-matrix, RSK and
Cauchy identities
- 29 August 2025 Lecture 15: Pieri tules and characters of the symmetric group
and the Hecke algebra
Week 6: q-t-Catalan
- 1 September 2025 Lecture 16: q-t-Catalan by Dyck paths
- 3 September 2025 Lecture 17: ∇en and diagonal coinvariants
- 5 September 2025 Lecture 18: Modified Macdonald polynomials and Garsia-Haiman modules
Week 7: G/B
- 8 September 2025 Lecture 19: Generator and relations for GLn(𝔽)
- 10 September 2025 Lecture 20: The Bruhat decomposition and the Poincare polynomial
- 12 September 2025 Lecture 21: Schubert varieties, and Grassmannians
Week 8: Moment graphs and Kazhdan-Lusztig polynomials
- 15 September 2025 Lecture 22: HT(G/B)
- 17 September 2025 Lecture 23: Sheaves on moment graphs
- 19 September 2025 Lecture 24: Kazhdan-Lusztig polynomials
Week 9: Springer fibers
- 22 September 2025 Lecture 25: Cells in Springer fibers
- 24 September 2025 Lecture 26: Modified Hall-Littlewood polynomials
- 26 September 2025 Lecture 27: AFL Grand Final Eve holiday
Non-teaching week 29 September - 3 October
Week 10: Review and Student talks
- 6 October 2025 Lecture 28: Review and Student Talks
- 8 October 2025 Lecture 29: Review and Student Talks
- 10 October 2025 Lecture 30: Review and Student Talks
Week 11: Review and Student Talks
- 13 October 2025 Lecture 31: Review and Student Talks
- 15 October 2025 Lecture 32: Review and Student Talks
- 17 October 2025 Lecture 33: Review and Student Talks
Week 12: Reivew and Student Talks
- 20 October 2025 Lecture 34: Review and Student Talks
- 22 October 2025 Lecture 35: Review and Student Talks
- 24 October 2025 Lecture 36: Review and Student Talks