*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:

A copy of the thesis can be found here:

Exam PhD
*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

Exam PhD
*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:

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Abstract:

A copy of the thesis can be found here:

Exam PhD
*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: jeremy-lane-thesis-draft-february-7

Exam PhD
*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

Exam PhD
*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

Exam PhD
**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.

*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.

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

Thursday, September 8, 2016

11:10 a.m.

BA6183

PhD Candidate: Louis-Philippe Thibault

Supervisor: Ragnar Buchweitz

Thesis title: Tensor product of preprojective algebras and preprojective structure on skew-group algebras

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Abstract:

We investigate properties of finite subgroups $G<SL(n, k)$ for which the skew-group algebra $k[x_1,\ldots, x_n]\#G$ does not have a grading structure of (higher) preprojective algebra. Namely, we prove that if a finite subgroup $G<SL(n, k)$ is conjugate to a finite subgroup of $SL(n_1, k)\times SL(n_2, k)$, for some $n_1, n_2\geq 1$, $n_1+n_2 =n$, then the skew-group algebra $R\#G$ is not Morita equivalent to a (higher) preprojective algebra. Motivated by this question, we study preprojective algebras over Koszul algebras. We give a quiver construction for the preprojective algebra over a basic Koszul $n$-representation-infinite algebra. Moreover, we show that such algebras are derivation-quotient algebras whose relations are given by a superpotential. The main problem is also related to the preprojective algebra structure on the tensor product $\Pi:=\Pi_1\otimes_k \Pi_2$ of two Koszul preprojective algebras. We prove that a superpotential in $\Pi$ is given by the shuffle product of superpotentials in $\Pi_1$ and $\Pi_2$. Finally, we prove that if $\Pi$ has a grading structure such that it is $n$-Calabi-Yau of Gorentstein parameter $1$, then its degree $0$ component is the tensor product of a Calabi-Yau algebra and a higher representation-infinite algebra. This implies that it is infinite-dimensional, which means in particular that $\Pi$ is not a preprojective algebra.

A copy of the thesis can be found here: ut-thesis-Thibault

Exam PhD