## Memorial Service for Yuri Cher

The Department of Mathematics will hold a Memorial Service for our late graduate student, Yuri Cher, on Tuesday, March 28, 2017 at 10:30 a.m. in the Math lounge.

A remembrance book will be available in the main office the week of March 20.

## Departmental PhD Thesis Exam – Yiannis Loizides

Everyone is welcome to attend.  Refreshments will be served in the Math Lounge before the exam.

Tuesday, April 25, 2017
2:10 p.m.
BA6183

PhD Candidate:  Yiannis Loizides
Supervisor:  Eckhard Meinrenken
Thesis title:  Norm-square localization for Hamiltonian LG-spaces

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Abstract:

In this thesis we prove norm-square localization formulas for two invariants of Hamiltonian loop group spaces: twisted Duistermaat-Heckman measures and a K-theoretic `quantization’.  The terms in the formulas are indexed by the components of the critical set of the norm-square of the moment map.  These results are analogous to results proved by Paradan in the case of Hamiltonian G-spaces.  An important application of the norm-square localization formula is to prove that the multiplicity of the fundamental level k representation in the quantization is a quasi-polynomial function of k.  This is closely related to the [Q,R]=0 theorem of Alekseev-Meinrenken-Woodward for Hamiltonian loop group spaces.

A copy of the thesis can be found here: YLThesis

## Departmental PhD Thesis Exam – Jack Klys

Everyone is welcome to attend.  Refreshments will be served in the Math Lounge before the exam.

Thursday, February 23, 2017
2:10 p.m.
BA6183

PhD Candidate:  Jack Klys
Supervisor:  Jacob Tsimerman
Thesis title: Statistics of class groups and related topics

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Abstract:

We prove several results concerning class groups of number fields and function fields.

Firstly we compute all the moments of the $p$-torsion in the first step of a filtration of the class group defined by Gerth \cite{gerthprank} for cyclic number fields of degree $p$, unconditionally for $p=3$ and under GRH in general. We show that it satisfies a distribution which Gerth conjectured as an extension of the Cohen-Lenstra-Martinet conjectures. In the $p=3$ case this gives the distribution of the $3$-torsion of the class group modulo the Galois invariant part. We follow the strategy used by Fouvry and Kl$\ddot{\text{u}}$ners in their proof of the distribution of the $4$-torsion in quadratic fields \cite{fk1}.

Secondly, we compute all the moments of a normalization of the function which counts unramified $H_{8}$-extensions of quadratic number fields, where $H_{8}$ is the quaternion group of order $8$, and show that the values of this function determine a constant distribution. Furthermore we propose a similar modification to the non-abelian Cohen-Lenstra heuristics for unramified $G$-extensions of quadratic fields for several other 2-groups $G$, which we conjecture will give finite moments which determine a distribution. These are all cases in which the unnormalized average is known or conjectured to be infinite. Our method additionally can be used to determine the asymptotics of the unnormalized counting function, which we also do for unramified $H_{8}$-extensions. This part of the thesis is joint work with Brandon Alberts.

Thirdly we present several new examples of reflection principles which apply to both class groups of number fields and picard groups of of curves over $\mathbb{P}^{1}/\mathbb{F}_{p}$. This proves a conjecture of Lemmermeyer \cite{franz1} about equality of 2-rank in subfields of $A_{4}$, up to a constant not depending on the discriminant in the number field case, and exactly in the function field case.  More generally we prove similar relations for subfields of a Galois extension with group $G$ for the cases when $G$ is $S_{3}$, $S_{4}$, $A_{4}$, $D_{2l}$ and $\mathbb{Z}/l\mathbb{Z}\rtimes\mathbb{Z}/r\mathbb{Z}$.  The method of proof uses sheaf cohomology on 1-dimensional schemes, which reduces to Galois module computations.

A copy of the thesis can be found here: klysthesis

## Departmental PhD Thesis Exam – James Mracek

Everyone is welcome to attend.  Refreshments will be served in the Math Lounge before the exam.

Tuesday, April 25th, 2017
11:10 a.m.
BA6183

PhD Candidate:  James Mracek
Supervisor:  Lisa Jeffrey/ Clifton Cunningham
Thesis title:  Applications of algebraic microlocal analysis in symplectic geometry and representation theory

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Abstract:

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 $T\times S^1$ action on $\Omega SU(2)$. 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.

A copy of the thesis can be found here: ut-thesis (002)

## Departmental PhD Thesis Exam – Jeremy Lane

Everyone is welcome to attend.  Refreshments will be served in the Math Lounge before the exam.

Friday, February 10, 2017
10:10 a.m.
BA1210

PhD Candidate:  Jeremy Lane
Supervisor:  Yael Karshon
Thesis title:  On the topology of collective integrable systems

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Abstract:

This thesis studies the topological properties of momentum maps of a large family of completely integrable systems called collective completely integrable systems.

The first result concerns the topological monodromy of a collective completely integrable system on a product of three two-spheres. This system is called a “Heisenberg spin-chain” for its connection with a quantum integrable system of the same name.

The remainder of the thesis is concerned with collective systems that are “constructed by Thimm’s trick” with the “action coordinates of Guillemin and Sternberg.”  We observe that the open dense subset of a symplectic manifold where such systems define a Hamiltonian torus action is connected. This observation was absent from the literature until this point. We also prove that if a system is constructed in this manner from a chain of Lie subalgebras then the image is given by an explicit set of inequalities called branching inequalities.

When the symplectic manifold in question is a multiplicity free Hamiltonian $U(n)$-manifold, the construction of Thimm’s trick with the action coordinates of Guillemin and Sternberg yields a completely integrable Hamiltonian torus action on a connected open dense subset. If the momentum map is proper, then we are able to prove lower bounds for the Gromov width of the symplectic manifold  from the classification of the connected open dense subset as a non-compact symplectic toric manifold. Convex multiplicity free manifolds of compact Lie groups have been classified by (Knop, 2010) and (Losev, 2009) and, accordingly, our lower bounds are given in terms of the combinatorics of the classifying data: the momentum set and a lattice. This result is the first estimate for the Gromov width of general multiplicity free manifolds of a nonabelian group.

This result  relies crucially on connectedness of the open dense subset and the explicit description of the momentum map image. The proof is a generalization of methods used to prove lower bounds for the Gromov width of $U(n+1)$ coadjoint orbits (Pabiniak, 2014)  which are an example of multiplicity free $U(n)$-manifolds.

A copy of the thesis can be found here: ut-thesis-revised-March-9

## Departmental PhD Thesis Exam – Asif Zaman

Everyone is welcome to attend.  Refreshments will be served in the Math Lounge before the exam.

Wednesday, February 1, 2017
3:10 p.m.
BA6183

PhD Candidate:  Asif Zaman
Supervisor:  John Friedlander
Thesis title:  Analytic estimates for the Chebotarev Density Theorem and their applications

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Abstract:

In this thesis, we study the distribution of prime ideals within the Chebotarev Density Theorem. The theorem states that the Artin symbols attached to prime ideals are equidistributed within the Galois group of a given Galois extension.

We exhibit field-uniform unconditional bounds with explicit constants for the least prime ideal in the Chebotarev Density Theorem, that is, the prime ideal of least norm with a specified Artin symbol. Moreover, we provide a new upper bound for the number of prime ideals with a specified Artin symbol which is valid for a wide range and sharp, short of precluding a putative Siegel zero. To achieve these results, we establish explicit statistical information on the zeros of Hecke L-functions and the Dedekind zeta function. Our methods were inspired by works of Linnik, Heath-Brown, and Maynard in the classical case and the papers of Lagarias–Odlyzko, Lagarias–Montgomery–Odlyzko, and Weiss in the Chebotarev setting.

We include applications for primes represented by certain binary quadratic forms, congruences of coefficients for modular forms, and the group structure of elliptic curves reduced modulo a prime. In particular, we establish the best known unconditional upper bounds for the least prime represented by a positive definite primitive binary quadratic form and for the Lang–Trotter conjectures on elliptic curves.

A copy of the thesis can be found here: thesis_Zaman_v1

## Departmental PhD Thesis Exam – Jerrod Smith

Everyone is welcome to attend.  Refreshments will be served in the Math Lounge before the exam.

Wednesday, February 1, 2017
2:10 p.m.
BA6183

PhD Candidate:  Jerrod Smith
Supervisor:  Fiona Murnaghan
Thesis title:  Construction of relative discrete series representations for $p$-adic $\mathbf{GL}_n$

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Abstract:

Let $F$ be a nonarchimedean local field of characteristic zero and odd residual characteristic.  Let $G$ be the $F$-points of a connected reductive group defined over $F$ and let $\theta$ be an $F$-rational involution of $G$.  Define $H$ to be the closed subgroup of $\theta$-fixed points in $G$.
The quotient variety $H \backslash G$ is a $p$-adic symmetric space.  A fundamental problem in the harmonic analysis on $H \backslash G$ is to understand the irreducible subrepresentations of the right-regular representation of $G$ acting on the space $L^2(H\backslash G)$ of complex-valued square integrable functions on $H\backslash G$.  The irreducible subrepresentations of $L^2(H\backslash G)$ are called relative discrete series representations.

In this thesis, we give an explicit construction of relative discrete series representations for three $p$-adic symmetric spaces, all of which are quotients of the general linear group.  We consider $\mathbf{GL}_n(F) \times \mathbf{GL}_n(F) \backslash \mathbf{GL}_{2n}(F)$, $\mathbf{GL}_n(F) \backslash \mathbf{GL}_n(E)$, and $\mathbf {U}_{E/F} \backslash \mathbf{GL}_{2n}(E)$, where $E$ is a quadratic Galois extension of $F$ and $\mathbf {U}_{E/F}$ is a quasi-split unitary group.  All of the representations that we construct are parabolically induced from $\theta$-stable parabolic subgroups admitting a certain type of Levi subgroup.  In particular, we give a sufficient condition for the relative discrete series representations that we construct to be non-relatively supercuspidal.  Finally, in an appendix, we describe all of the relative discrete series of $\mathbf{GL}_{n-1}(F)\times \mathbf{GL}_{1}(F) \backslash \mathbf{GL}_{n}(F)$.

A copy of the thesis can be found here: thesis_jmsmith-27-01-2017

## Funeral service for Yuri Cher

Monday, October 24, 2016

York Cemetery and Funeral Centre
160 Beecroft Rd, North York
ON M2N 5Z5

Visitation: 11:00 a.m.-12:00 p.m.

Service: 12:00 – 1:00 p.m.

Burial: 1:00 p.m.

## MSc supervised project presentation – Yuguang Bai

Everyone is welcome to attend.

Thursday, September 22, 2016
4:10 p.m.
BA6180

MSc Candidate:  Yuguang Bai
Supervisor:  Pierre Milman
Project title: Illustrative Proof of Hirzebruch-Riemann-Roch Theorem for Algebraic Curves

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Abstract:

I will be going over something I did for my Master’s project this past summer. Namely, the idea for the proof of the Hirzebruch-Riemann-Roch Theorem for Algebraic Curves, which shows how the topological genus is equal to an algebraic invariant, called the arithmetic genus.

The talk will not be rigorous and should be accessible for new graduate students. Knowledge of undergraduate topology and algebra recommended.