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.

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
$\begin{array}{ccc}{H}^{\mathrm{DAHA}}& \text{acts on}& K\left({\U0001d51b}_{\gamma}\right)\\ {H}^{\mathrm{trigDAHA}}& \text{acts on}& {H}^{*}\left({\U0001d51b}_{\gamma}\right)\\ {H}_{1,c}& \text{acts on}& \mathrm{gr}{H}^{*}\left({\U0001d51b}_{\gamma}\right)\end{array}$ - Affine Springer Fibres inside affine flag varieties
${\U0001d51b}_{\gamma ,\mathbb{P}}=\{g\mathbb{P}\in {\mathrm{Fl}}_{\mathbb{P}}\hspace{0.17em}|\hspace{0.17em}\mathrm{Ad}\left({g}^{-1}\right)\gamma \in \mathrm{Lie}\mathbb{P}\}$ - Moduli space of Higgs bundles
${\mathcal{M}}_{G,\mathcal{L}}=\left\{\right(\mathcal{F},\mathcal{L}\left)\hspace{0.17em}\right|\hspace{0.17em}\mathcal{F}\hspace{0.17em}\text{a principal}\hspace{0.17em}G\text{-bundle on}\hspace{0.17em}X,\hspace{0.17em}\varphi \in {H}^{0}(X,\mathrm{ad}(\mathrm{\U0001d524\mathrm{)\otimes \mathcal{L})\}}}$ - Hitchin map
$\begin{array}{ccc}{\mathcal{M}}_{G,\mathcal{L}}& \u27f6& {\mathcal{A}}_{G,\mathcal{L}}\\ {\mathcal{M}}_{a}& \u27fc& a\end{array}$ - Spectral Curves
${\mathcal{M}}_{a}\cong \stackrel{\u203e}{\mathrm{Pic}}\left({C}_{a}\right)={\scriptscriptstyle \mathrm{Pic}\left({C}_{a}\right)}{{\scriptscriptstyle \times}}^{{\displaystyle \prod _{C\to Z}}{\mathrm{Pic}}_{C}\left({C}_{a}\right)}{\displaystyle \prod _{C\to Z}}{\scriptscriptstyle \stackrel{\u203e}{\mathrm{Pic}}\left({C}_{a}\right)}$ - the WEAVING theorem
$\prod _{C\to x}}{\stackrel{\u203e}{\mathrm{Pic}}}_{C}\left({C}_{a}\right)={\U0001d51b}_{{a}_{x}$

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 |

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

This was the plan for a School Pure Mathematics Research presentation for the School Review 1 November 2017.