Title: Applications of algebraic microlocal analysis in symplectic geometry and representation theory
Location: BA6183, Bahen Center, 40 St. George St.
Speaker: James Mracek
Speaker URL: http://blog.math.toronto.edu/GraduateBlog/files/2017/01/ut-thesis-002.pdf
Institution: University of Toronto

PhD Advisors: Lisa Jeffrey – Clifton Cunningham

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This thesis investigates applications of microlocal geometry in both representation theory and symplectic geometry. Accordingly, there are two bodies of work contained herein.

The first part of this thesis investigates a conjectural geometrization of local Arthur packets. These packets of representations of a p-adic group were invented by Arthur for the purpose of classifying the automorphic discrete spectrum of special orthogonal and symplectic groups. While their existence has been established, an explicit construction of Arthur packets remains difficult. In the case of real groups, Adams, Barbasch, and Vogan showed how one can use a geometrization of the local Langlands correspondence to construct packets of equivariant D-modules that satisfy similar endoscopic transfer properties as the ones defining Arthur packets. We classify the contents of these “microlocal” packets in the analogue of these varieties for p-adic groups, under certain restrictions, for a plethora of split classical groups.

The goal of the second part of this thesis is to find a way to make sense of the Duistermaat-Heckman function for a Hamiltonian action of a compact torus on an infinite dimensional symplectic manifold. We show that the Duistermaat-Heckman theorem can be understood in the language of hyperfunction theory, then apply this generalization to study the Hamiltonian T×S1 action on ΩSU(2). The essential reason for introducing hyperfunction theory is that the local contribution to the Duistermaat-Heckman polynomial near the image of a fixed point is a Green's function for an infinite order differential equation. Since infinite order differential operators do not act on Schwarz distributions, we are forced to use this more general theory.

https://seminars.math.toronto.edu/seminars/list/events.py/process?action...

Speaker: Clifton Cunningham

Title: Arthur packets packets for p-adic groups and vanishing cycles of perverse sheaves

Abstract: This talk will explain how to adapt the approach developed by Adams-Barbasch-Vogan to Arthur packets from Real groups to p-adic groups, and will illustrate this adaptation with several examples. We will also sketch a proof showing that Arthur packets are p-adic ABV packets for unipotent representations of p-adic special orthogonal groups.

Joint with: Bin Xu, Ahmed Moussaoui, Andrew Fiori, James Mracek

https://www-math.umd.edu/research/seminars/lie-groups-and-rep-theory-sem...

Speaker: Lassina Dembélé, Max Planck Institut für Mathematik

Abstract: In this talk, we discuss the existence of certain lifts of Hilbert modular forms. We then use those lifts to provide theoretical evidence for a conjecture of Dick Gross on the modularity of abelian varieties not of GL_type.
(This is joint work with Clifton Cunningham.)

Location: MS 337 and https://zoom.us/j/344804314

https://zoom.us/j/344804314

Update on the paper.
Sketch of the proof of Conjecture 1.
Cuspidal support decomposition, revisited.

Speaker: Kiran Kedlaya, University of California, San Diego (http://www.kskedlaya.org/)

Topic: Perfectoid spaces

Location: from UBC and on https://bluejeans.com/211681494/browser

More details on joining the lecture: http://www.pims.math.ca/scientific-event/170220-rakk

Notes from Prof. Kedlaya (and also part of the 2017 Arizona Winter School):

- https://cloud.sagemath.com/projects/7da27120-4e06-4340-bbcb-70054d89ac67...
- Supplemented with background from Jared Weinstein's lecture notes: http://swc.math.arizona.edu/aws/2017/2017WeinsteinNotes.pdf

This lecture series is part of the PIMS Focus Group on Representations in Arithmetic.

Speaker: Kiran Kedlaya, University of California, San Diego (http://www.kskedlaya.org/)

Topic: Adic spaces

Location: from UBC and on https://bluejeans.com/211681494/browser

More details on joining the lecture: http://www.pims.math.ca/scientific-event/170220-rakk

Notes from Prof. Kedlaya (and also part of the 2017 Arizona Winter School):

- https://cloud.sagemath.com/projects/7da27120-4e06-4340-bbcb-70054d89ac67...
- Supplemented with background from Jared Weinstein's lecture notes: http://swc.math.arizona.edu/aws/2017/2017WeinsteinNotes.pdf

This lecture series is part of the PIMS Focus Group on Representations in Arithmetic.

Speakers: Ahmed, Bin and Clifton.

Topic: Review Arthur's main local result in the endoscopic classification, adapted to pure inner twists of quasi-split classical p-adic groups (Conjecture 9.4.2 in Arthur's book). Also review Kaletha's notion of rigid inner twists. We need this to state Voganish Conjecture 1, properly.

Location: Bin and Clifton at the University of Calgary, Room MS 337, Ahmed in Paris, and on https://zoom.us/j/525655184

http://www.pims.math.ca/scientific-event/170209-ntsgkft

Speaker: Kam-Fai Tam

Location: University of British Columbia, Room ESB 4127

Part I: Representations of reductive groups over local fields
In this talk, we describe the representations of certain reductive groups over local fields and the representations of Weil groups. Then we review the class field theory for local fields and the Langlands correspondence for these reductive groups. The talk will be a brief overview of the representation theory of reductive groups over local fields, largely based on examples of low-rank groups.

Part II: Endoscopic classification of essentially tame supercuspidal representations for quasi-split classical groups
Continue from the last talk, we specify our reductive group \mathbf{G} to be a quasi-split classical group (special orthogonal, symplectic, unitary) over a p-adic field of odd residual characteristic. We describe the endoscopic classification, proved by Arthur and Mok, of certain supercuspidal representations and their L-packets of \mathbf{G}, under some regularity and tameness conditions. These representations can be described by inducing types constructed by Bushnell-Kutzko, Stevens, or Yu, and they correspond to Langlands parameters related to characters of elliptic maximal tori of \mathbf{G}.
(This work is partly joint with Corinne Blondel.)