## Departmental PhD Thesis Exam – Parker Glynn-Adey

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

Monday, May 2, 2016
2:10 p.m.
BA6183

Supervisor:  Rina Rotman
Thesis title: Width, Ricci Curvature, and Bisecting Surfaces

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Abstract:

In this thesis we studied width-volume inequalities, bisecting surfaces in three spheres, and the planar case of Larry Guth’s sponge problem.  Our main result is a width-volume inequality for conformally non-negatively Ricci curved manifolds.  We obtain several estimates on the size of minimal hypersurfaces in such manifolds.  Concerning geometric subdivision and 3-spheres, we give a positive answer to a question of Papasoglu.  Regarding the sponge problem, we show that any open bounded Jordan measurable set in the plane of small area admits an expanding embedding in to a strip of unit height.  We also prove that a generalization of the planar sponge problem is NP-complete. This thesis is partially based on joint work with Ye. Liokumovich [G-ALiokumovich2014] and Z. Zhu [G-AZhu2015]

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

## Departmental PhD Thesis Title – Alex Weekes

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

Friday, May 6, 2016
11:10 a.m.
BA6183

PhD Candidate:  Alex Weekes
Supervisor:  Joel Kamnitzer
Thesis title:  Highest weights for truncated shifted Yangians

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Abstract:

Truncated shifted Yangians are a family of algebras which are conjectured to quantize slices to Schubert varieties in the affine Grassmannian. In this thesis we study the highest weight theory of these algebras, and explore connections with Nakajima quiver varieties and their cohomology. We give a conjectural parametrization of the set of highest weights in terms of product monomial crystals, which are related to Nakajima’s monomial crystal. In type A we prove this conjecture.

Our main tool in describing the set of highest weights is the B–algebra, which is a non-commutative generalization of the notion of torus fixed-point subscheme. We give a conjectural presentation for this algebra based on calculations using Yangians, and show how this presentation admits a natural geometric interpretation in terms of the equivariant cohomology of quiver varieties. We conjecture that this gives an explicit presentation for the equivariant cohomology ring of the Nakajima quiver variety of a finite ADE quiver, and show that this conjecture could be deduced from a special case. We give a proof of this conjecture in type A.

This work can be thought of in the context of symplectic duality. In our case, slices to Schubert varieties in the affine Grassmannian are expected to be symplectic dual to Nakajima quiver varieties. The relationship between B–algebras and equivariant cohomology is part of a general conjecture of Nakajima for symplectic dual varieties. These ideas represent a first approximation to expected connections between the category $\mathcal O$’s for a symplectic dual pair of varieties.

A copy of the thesis can be found here: Alex Weekes – thesis

## 2016 F. V. ATKINSON TEACHING AWARD – Call for nominations

The Department of Mathematics invites nominations for the

F. V. ATKINSON TEACHING AWARD

This prize honors outstanding teaching by post-doctoral fellows and other junior research faculty not on the tenure track. It is named in memory of former chair Frederick Atkinson, “friend to many, scholar, enthusiastic teacher and gifted researcher” [Angelo Mingarelli, Math. Nachr. 278, 2005].

Inspiring accounts of his work and life can be found on:

http://www-history.mcs.st-and.ac.uk/Biographies/Atkinson.html

Previous winners are:

• Marina Chugunova (2010),
• Magdalena Czubak, Brian Smithling (2011),
• Steven Rayan (2012),
• Oded Yacobi, Jesse Gell-Redman (2013),
• Prashant Athavale, David Penneys and Alex Rennet (2014),
• Nicholas Hoell, Geoffrey Scott (2015)

Nominations can be made by individuals or groups, and should be sent by e-mail or in writing to:

Ed Bierstone, bierston@math.utoronto.ca
cc to Mary Pugh, ugchair@math.utoronto.ca
and Donna Birch, dbirch@math.utoronto.ca

no later than:

FRIDAY, APRIL 29, 2016

• your reasons for the nomination; (what is special about the nominee’s classes ?)
• your relevant interactions with the nominee; (e.g., as current/former student, supervisor, or colleague)
• the course and year on which your assessment is based. (Courses may be at any level but should have taken place on the St. George Campus.)

Below you will find a list of eligible postdoctoral fellows.  Please let us know if you find any errors or omissions.  If your favourite instructors do not appear on this list, they may be eligible instead for the DeLury Award (for graduate students), or the Arts & Science Outstanding Teaching Award.

We are looking forward to hearing from you!  AtkinsonAward2016

ELIGIBLE POSTDOCS (2016, in alphabetical order)

• Belotto da Silva, Andre
• Cho, Sungmun
• Garcia Fritz, Natalia
• Garcia Martinez, Luis
• Garcia-Raboso, Alberto
• Gazeau, Maxime
• George, William
• Izosimov, Anton
• Kolpakov, Aleksandr
• Le, Daniel
• Leung, Kin Kwan
• Lin, Chen-Yun
• Manin, Fedor
• Mouquin, Victor
• Ojeda-Aristizabal, Diana
• Peng, Yinhe
• Qing, Yulan
• Shen, Xin
• Song, Yanli
• Wang, Qingyun

## Departmental PhD Thesis Exam – Kyle Thompson

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

Thursday, April 21, 2016
2:10 p.m.
BA6183

PhD Candidate:  Kyle Thompson
Supervisor:  Bob Jerrard
Thesis title:  Dynamics of Superconducting Interfaces

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Abstract:

We look for solutions to a two-component system of nonlinear wave equations with the properties that one component has an interface and the other is exponentially small except near the interface of the first component. The second component can be identified with a superconducting current confined to an interface. A formal analysis suggests that for suitable initial data, the energy of solutions concentrate about a codimension one timelike surface $\Gamma$ whose dynamics are coupled in a highly nonlinear way to the phase of the superconducting current.

We provide a rigorous verification of the predictions these formal arguments make for solutions with an equivariant symmetry in two dimensions subject to a non-degeneracy condition.

A copy of the thesis can be found here:  KThompson_Thesis

## Mathematics Graduate Career Event – May 12, 2016

Have you ever wondered what you can do with an advanced degree in mathematics?  Well, you’re in luck!  The Mathematics Graduate Students Association (MGSA ) is hosting a panel discussion with U of T mathematics alumni who are working in the exciting fields of data science, consulting, education, finance and many other interesting fields. Please come to this panel to discover:

1. career options available to advanced degree holders in mathematics,

2. what skills you can cultivate for a specific career,

3. what kinds of mathematics are used industry-specific careers,

and much more.

http://www.math.utoronto.ca/mgsa/career-2016-05.php
Chiara Moraglia (Actuarial Analyst, Mercer)
Joseph Geraci (Chief Science Officer, NetraMark)
Patrick Walls (Mathematics Instructor, UBC)
Eric Hart (Data Scientist, Angoss Software)
Xiao Liu (Global Analytics and Financial Engineering, Scotiabank)
Mario Morfin (Co-founder MOAI-Solutions Inc.)

Registration deadline:  April 25, 2016. See poster here: MathCareerPoster2016

## Departmental PhD Thesis Exam – Tyler Wilson

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

Tuesday, March 8, 2016
10:10 a.m.
BA1240

PhD Candidate:  Tyler Wilson
Supervisor:  Mary Pugh, Francis Dawson
Thesis title:  Stabilization, Extension and Unification of The Lattice Boltzmann Method Using Information Theory

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Abstract:

A novel Lattice Boltzmann method is derived using the Principle of Minimum Discrimination Information (MinxEnt) via the minimization of Kullback-Leibler Divergence (KLD). Approximations of this method yields the single relaxation time (SRT-LBM), two relaxation time (TRT-LBM), multiple relaxation time (MRT-LBM) Lattice Boltzmann Methods as well as Entropic Lattice Boltzmann Method (ELBM) and Ehrenfest Step LBM (EF-LBM).  Specifi cally it is shown that these methods can be understood as approximations of a method for constrained KLD minimization. By carrying out the actual single step Newton-Raphson minimization (MinxEnt-LBM) a more accurate and stable Lattice Boltzmann Method can be implemented. To demonstrate this, 2D Poiseulle flow, 1D shock tube and lid-driven cavity flow simulations are carried out and compared to SRT-LBM, TRT-LBM MRT-LBM and EF-LBM.

A copy of the thesis can be found here: twilson-thesis

## Departmental PhD Thesis Exam – Greg Fournodavlos

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

Monday, March 21, 2016
1:10 p.m.
BA6183

PhD Candidate: Greg Fournodavlos
Supervisor:  Spyros Alexakis
Thesis title: Stability of singularities in geometric evolutionary PDE

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Abstract:

We study questions of stability of two types of singularities encountered in geometric evolutionary PDE, one in Ricci flow and the other in the context of the Einstein field equations in vacuum.

In the first part of the thesis we introduce certain spherically symmetric singular Ricci solitons and study their stability under the Ricci flow from a dynamical PDE point of view.   The solitons in question exist for all dimensions $n+1\ge 3$, and all have a  point singularity where the curvature blows up; their evolution under the Ricci flow is in sharp contrast to the evolution of their smooth counterparts. In particular, the family of diffeomorphisms associated with the Ricci flow “pushes away” from the singularity causing the  evolving soliton  to open up immediately  becoming an incomplete (but non-singular) metric.  We study the local-in time stability of this dynamical evolution, under spherically symmetric perturbations of the singular initial metric.  We prove a local well-posedness result for the Ricci flow in suitably weighted Sobolev spaces, which in particular implies  that the “opening up” of the singularity persists for the perturbations as well.

The second problem we study concerns the backwards-in-time stability of the Schwarzschild singularity from a dynamical PDE point of view. More precisely, considering a spacelike hypersurface $\Sigma_0$ in the interior of the black hole region, tangent to the singular hypersurface $\{r=0\}$ at a single sphere, we study the problem of perturbing the Schwarzschild data on $\Sigma_0$ and solving the Einstein vacuum equations backwards in time.   We obtain a local backwards well-posedness result for small perturbations lying in certain weighted Sobolev spaces. The perturbed spacetimes all have a singularity at a “collapsed” sphere on $\Sigma_0$, where the leading asymptotics of the curvature and the metric match those of their Schwarzschild counterparts to a suitably high order.  As in the Schwarzschild backward evolution, the pinched initial hypersurface $\Sigma_0$ “opens up” instantly, becoming a regular spacelike (cylindrical) hypersurface.  This result thus yields classes of examples of non-symmetric vacuum spacetimes, evolving forward-in-time from regular initial data, which form a Schwarzschild type singularity at a collapsed sphere. We rely on a precise asymptotic analysis of the Schwarzschild geometry near the singularity which turns out to be at the threshold that our energy methods can handle.

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

## Departmental PhD Thesis Exam – Payman Eskandari

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

Thursday, May 5, 2016
11:10 a.m.
BA6183

PhD Candidate:  Payman Eskandari
Supervisor:  Kumar Murty
Thesis title:   Algebraic Cycles, Fundamental Group of a Punctured Curve, and Applications in Arithmetic

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Abstract:

The results of this thesis can be divided into two parts, geometric and arithmetic. Let $X$ be a smooth projective curve over $\mathbb{C}$, and $e,\infty\in X(\mathbb{C})$ be distinct points. Let $L_n$ be the mixed Hodge structure of functions on $\pi_1(X-\{\infty\},e)$ given by iterated integrals of length $\leq n$ (as defined by Hain). In the geometric part, inspired by a work of Darmon, Rotger, and Sols, we express the mixed Hodge extension $\mathbb{E}^\infty_{n,e}$ given by the weight filtration on $\displaystyle{\frac{L_n}{L_{n-2}}}$ in terms of certain null-homologous algebraic cycles on $X^{2n-1}$. These cycles are constructed using the diagonal embeddings of $X^{n-1}$ into $X^n$.

The arithmetic part of the thesis gives some number-theoretic applications of the geometric part. We assume that $X=X_0\otimes_K\mathbb{C}$ and $e,\infty\in X_0(K)$, where $K$ is a subfield of $\mathbb{C}$ and $X_0$ is a projective curve over $K$. Let $Jac$ be the Jacobian of $X_0$. We use the extension $\mathbb{E}^\infty_{n,e}$ to associate to each $Z\in CH_{n-1}(X_0^{2n-2})$ a point $P_Z\in Jac(K)$, which can be described analytically in terms of iterated integrals. The proof of $K$-rationality of $P_Z$ uses that the algebraic cycles constructed in the first part are defined over $K$. Assuming a certain plausible hypothesis on the Hodge filtration on $L_n$ holds, we show that an algebraic cycle $Z\in CH_{n-1}(X_0^{2n-2})$ for which $P_Z$ is torsion, gives rise to relations between periods of $L_2$. Interestingly, these relations are non-trivial even when one takes $Z$ to be the diagonal in $X_0^2$. In the elliptic curve case, we show unconditionally that a certain relation between periods of $L_2$ (which is induced by the diagonal in $X_0^2$) exists if and only if $e-\infty$ is torsion.

The geometric result of the thesis in $n=2$ case, and the fact that one can associate to $\mathbb{E}^\infty_{2,e}$ a family of points in $Jac(K)$, are due to Darmon, Rotger, and Sols. Our contribution is in generalizing the picture to higher weights.

A copy of the thesis can be found here:  Eskandari_thesis

## Departmental PhD Thesis Exam – Ali Mousavidehshikh

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

Wednesday, March 30, 2016
11:10 a.m.
BA1220

PhD Candidate: Ali Mousavidehshikh
Supervisor:  Ragnar Buchweitz
Thesis title: Constructing endomorphism rings of large finite global dimension

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Abstract:

For a numerical semigroup $\mathcal{H}$ with generators $\alpha_{1},\alpha_{2},…,\alpha_{s}$, let $R$ be
the subring of the ring  of formal power series $k[[t]]$ (where $k$ is a field of characteristic zero)
with generators $t^{\alpha_{1}},t^{\alpha_{2}},…,t^{\alpha_{s}}$. More precisely,
\begin{eqnarray*}
R:=k[[t^{\alpha_{1}},t^{\alpha_{2}},…,t^{\alpha_{s}}]]=\left\lbrace \sum_{i\geq 0}a_{i}t^{i}:~a_{i}\in k,~i\in \mathcal{H}\right\rbrace
\end{eqnarray*}
For $R\neq k[[t]]$, we construct ascending chains of rings $R=R_{1}\subsetneq R_{2}\subsetneq …\subsetneq R_{l}=k[[t]]$, and we then consider $E=\text{End}_{R_{1}}\left(\oplus_{i=1}^{l}R_{i}\right)$. Our arguments show that the global dimension of $E$ depends on $R_{1}$ and the way we construct our ascending chain. This leads to an investigation of two types of constructions for our ascending chain, which we call the “greedy” and “lazy” constructions. In the “greedy” construction we choose $R_{i+1}$ as the endomorphism ring of the radical of $R_{i}$. In the “lazy” (or, as Iyama calls it, saturated) construction we choose $R_{i+1}$ so that $\dim_{k}(R_{i+1}/R_{i})=1$ and the conductor of $R_{i+1}$ is strictly larger than that of $R_{i}$.  We introduce a special family of rings $\lbrace R_{1}^{i}:i\in \mathbb{N}\rbrace$, thinking of each as the beginning of an ascending chain and we let $\lbrace E^{i}:i\in \mathbb{N}\rbrace$ be the set of corresponding endomorphism rings.

This thesis consists of three main results. Firstly, if for each $i$ our chain is constructed via the “lazy” construction, then $\lbrace \text{gl.dim}(E^{i}):i\in \mathbb{N}\rbrace$ is an unbounded set. Secondly, under some additional assumptions on the set $\lbrace R_{1}^{i}:i\in \mathbb{N}\rbrace$ we compute the precise values in the set $\lbrace \text{gl.dim}(E^{i}):i\in \mathbb{N}\rbrace$. Thirdly, if for each $i$ the chain is constructed via the “greedy” construction, then $\text{gl.dim}(E^{i})=2$ for all $i$.

A copy of the thesis can be found here: Mousavidehshikh_Ali_201611_PhD_thesis pdf

## Departmental PhD Thesis Exam – Cui Cui (Amanda) Luo

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

Tuesday, November 10, 2015
2:00 p.m.
BA7256

PhD Candidate: Cui Cui (Amanda) Luo
Supervisor:  Luis Seco
Thesis title: Stochastic Correlation and Portfolio Optimization by Multivariate Garch

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Abstract:

Modeling time varying volatility and correlation in financial time series is an important element in pricing, risk management and portfolio management. The main goal of this thesis is to investigate the performance of multivariate GARCH model in stochastic volatility and correlation forecast and apply theses techniques to develop a new model to enhance the dynamic portfolio performance in several context, including hedge fund portfolio construction.

First, we examine the performance of various univariate GARCH models and regime-switching stochastic volatility models in crude oil market. Then these univariate models discussed are extended to multivariate settings and the empirical evaluation provides evidence on the use of the orthogonal GARCH in correlation forecasting and risk management.

The recent financial turbulence exposed and raised serious concerns about the optimal portfolio selection problem in hedge funds. The dynamic portfolio constructions performance of a broad set of a multivariate stochastic volatility models is examined. It provides further evidence on the use of the orthogonal GARCH in dynamic portfolio constructions and risk management.

Further in this work, a new portfolio optimization model is proposed in order to improve the dynamic portfolio performance. We enhance the safety-first model with standard deviation constraint and derive an analytic formula by filtering the returns with GH skewed t distribution and OGARCH. It is found that the proposed model outperforms the classic mean-variance model and mean-CVAR model during financial crisis period for a fund of hedge fund.

A copy of the thesis can be found here: thesis