## TOYKAMP

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.

 Ting Xue Omar Foda Yaping Yang Kari Vilonen Arun Ram Michael Wheeler Paul Norbury

## Representation Theory seminar 2017 University of Melbourne School of Mathematics and Statistics

 Ting Xue Lecture Notes 17 August 2017 Omar Foda Lecture Notes 21 September 2017 Yaping Yang Lecture Notes 10 August 2017 Kari Vilonen Lecture Notes 7 September 2017 Arun Ram Lecture Notes 31 August 2017 Michael Wheeler Lecture Notes 28 September 2017 Paul Norbury Lecture Notes 24 August 2017

From the Lecture notes:

• DAHA: The BLACK BOX theorem  $\begin{array}{ccc}{H}^{\mathrm{DAHA}}& \text{acts on}& K\left({𝔛}_{\gamma }\right)\\ {H}^{\mathrm{trigDAHA}}& \text{acts on}& {H}^{*}\left({𝔛}_{\gamma }\right)\\ {H}_{1,c}& \text{acts on}& \mathrm{gr}{H}^{*}\left({𝔛}_{\gamma }\right)\end{array}$
• Affine Springer Fibres inside affine flag varieties  ${𝔛}_{\gamma ,ℙ}=\left\{gℙ\in {\mathrm{Fl}}_{ℙ} | \mathrm{Ad}\left({g}^{-1}\right)\gamma \in \mathrm{Lie}ℙ\right\}$
• Moduli space of Higgs bundles  ${ℳ}_{G,ℒ}=\left\{\left(ℱ,ℒ\right) | ℱ \text{a principal} G\text{-bundle on} X, \varphi \in {H}^{0}\left(X,\mathrm{ad}\left(\mathrm{𝔤\mathrm{\right)\otimes ℒ\right)\right\}}}$
• Hitchin map  $\begin{array}{ccc}{ℳ}_{G,ℒ}& ⟶& {𝒜}_{G,ℒ}\\ {ℳ}_{a}& ⟼& a\end{array}$
• Spectral Curves  ${ℳ}_{a}\cong \stackrel{‾}{\mathrm{Pic}}\left({C}_{a}\right)=\mathrm{Pic}\left({C}_{a}\right){×}^{\prod _{C\to Z}{\mathrm{Pic}}_{C}\left({C}_{a}\right)}\prod _{C\to Z}\stackrel{‾}{\mathrm{Pic}}\left({C}_{a}\right)$
• the WEAVING theorem  $\prod _{C\to x}{\stackrel{‾}{\mathrm{Pic}}}_{C}\left({C}_{a}\right)={𝔛}_{{a}_{x}}$

## Representation Theory seminar 2017 University of Melbourne School of Mathematics and Statistics

 Ting Xue Lecture Notes 17 August 2017 Omar Foda Lecture Notes 21 September 2017 Yaping Yang Lecture Notes 10 August 2017 Kari Vilonen Lecture Notes 7 September 2017 Arun Ram Lecture Notes 31 August 2017 Michael Wheeler Lecture Notes 28 September 2017 Paul Norbury Lecture Notes 24 August 2017

## If you want to join our TOYKAMP please come to Room 107 on Thursday afternoon at 3:30 (when Yaping Yang is in town).

The point is
that affine Springer fibres
are weaving together to form the
moduli space of Higgs bundles
which is gonna be
Pic     of a spectral curve.

The
Cohomology is a Black Box
containing the affine Springer fibers
and making a DAHA module

and that DAHA module
has got commuting operators acting on it
and in important cases
it corresponds
to an integrable system.

That's an integrable system for which
the Energies
are eigenvalues of the
Macdonald polynomials
and it provides some kind of Fourier analysis
imbedded in the Langlands correspondence