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.
We look forward to having you join us.
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.
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
Everyone is welcome to attend. Refreshments will be served in the Math Lounge before the exam.
Wednesday, August 3, 2016
2:10 p.m.
BA6183
PhD Candidate: Ivan Livinskyi
Supervisor: Steve Kudla
Thesis title: On the integrals of the Kudla-Millson theta series
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Abstract:
The Kudla-Millson theta series $\theta_{km}$ of a pseudoeuclidean space $V$ of signature $(p,q)$ and lattice $L$ is a differential form on the symmetric space $\mathcal D$ attached to the pseudoorthogonal group $\mathrm{O}(p,q)$ that transforms like a genus $n$ Siegel modular form of weight $(p+q)/2$. Any integral of $\theta_{km}$ inherits the modular transformation law and becomes a nonholomorphic Siegel modular form. A special case of such integral is the well-known Zagier Eisenstein series $\mathcal{F}(\tau)$ of weight $3/2$ as showed by Funke.
We show that for $n=1$ and $p=1$ the integral of $\theta_{km}$ along a geodesic path coincides with the Zwegers theta function $\widehat{\Theta}_{a,b}$. We construct a higher-dimensional generalization of Zwegers theta functions as integrals of $\theta_{km}$ over geodesic simplices for $n\geq 2$.
If $\Gamma$ is a discrete group of isometries of $V$ that preserve the lattice $L$ and act trivially on the cosets $L^\ast/L$, then the fundamental region $\Gamma\backslash \mathcal D$ is an arithmetic locally symmetric space. We prove that the integral of $\theta_{km}$ over $\Gamma\backslash \mathcal D$ converges and compute it in some cases. In particular, we extend the results of Kudla to the cases $p=1$, and $q$ odd.
A copy of the thesis can be found here: Livinsky_Thesis
Everyone is welcome to attend. Refreshments will be served in the Math Lounge before the exam.
Wednesday, July 20, 2016
11:10 a.m.
BA6183
PhD Candidate: Jeremy Voltz
Supervisor: Kostya Khanin
Thesis title: Two results on Asymptotic Behaviour of Random Walks in Random Environment
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Abstract:
In the first chapter of this thesis, we consider a model of directed polymer in $1+1$ dimensions in a product-type random environment $\omega(t,x) = b(t) F(x)$, where the fields $F$ and $b$ are i.i.d., with $F(x)$ continuous, symmetric and bounded, and $b(t) = \pm 1$ with probability $1/2$. Thus $\omega$ can be viewed as the field $F$ oscillating in time. We consider directed last-passage percolation through this random field; namely, we investigate the behavior of the length $n$ polymer path with maximal action, where the action of a path is simply the sum of the environment variables it moves through.
We prove a law of large numbers for the maximal action of the path from the origin to a fixed endpoint $(n, \lfloor \alpha n \rfloor)$, and investigate the limiting shape function $a(\alpha)$. We prove that this shape function is non-linear, and has a corner at $\alpha = 0$, thus indicating that this model does not belong to the KPZ universality class. We conjecture that this shape function has a linear piece near $\alpha = 0$.
With probability tending to $1$, the maximizing path with free endpoint will localize on an edge with $F$ values far from each other. Under an assumption on the arrival time to this localization site, we prove that the path endpoint and the centered action of the path, both rescaled by $n^{-2/3}$, converge jointly to a universal law, given by the maximizer and value of a functional on a Poisson point process.
In the second chapter, we consider a class of multidimensional random walks in random environment, where the environment is of the type $p_0 + \gamma \xi$, with $p_0$ a deterministic, homogeneous environment with underlying drift, and $\xi$ an i.i.d. random perturbation. Such environments were considered by Sabot in \cite{Sabot2004}, who finds a third-order expansion in the perturbation for the non-null velocity (which is guaranteed to exist by Sznitman and Zerner’s LLN \cite{Sznitman1999}). We prove that this velocity is an analytic function of the perturbation, by applying perturbation theory techniques to the Markov operator for a certain Markov chain in the space of environments.
A copy of the thesis can be found here: ut-thesis-Voltz
Congratulations to the 2016 awards winners and graduate scholarships recipients.
F. V. Atkinson Teaching Award for Postdoctoral Fellows
Daniel B. DeLury Teaching Assistant Award
Inaugural Ida Bulat Teaching Award for Graduate Students
CI Teaching Excellence Award
Vivekananda Graduate Scholarship for international students
Canadian Mathematical Society Graduate Scholarship
Coxeter Graduate Scholarship
International Graduate Student Scholarship
Margaret Isobel Elliott Graduate Scholarship
We thank the faculty members, undergraduate students and course instructors who took the time to submit their nominations for the various awards.
Congratulations once again to André, Anton, Tracey, Beatriz, Nikita, Asif, Payman, Tyler, Peter, Huan, Zhifei, Anne, Chia-Cheng, and Shuangjian!
We are extremely happy to announce this year’s winners of the Frederick V. Atkinson Teaching Awards for Post Doctoral Fellows:
The prize honors outstanding teaching by post-doctoral fellows and other junior research faculty not on the tenure track.
The awards recipients will receive a monetary award and a certificate during our awards/graduation reception scheduled for Thursday, May 26 at 3:10 p.m. in the Math lounge.
Congratulations to André and Anton!
Everyone is welcome to attend. Refreshments will be served in the Math Lounge before the exam.
Tuesday, June 28, 2016
3:10 p.m.
BA6183
PhD Candidate: Andrew Stewart
Co-Supervisors: Balint Virag
Thesis title: On the scaling limit of the range of a random walk bridge on regular trees
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Abstract:
Consider a nearest neighbour random walk $X_n$ on the $d$-regular tree ${\Bbb T}_d$, where $d\geq 2$, conditioned on $X_n = X_0$. This is known as the random walk bridge. We derive Gaussian-like tail bounds for the return probabilities of the random walk bridge on the scale of $n^{1/2}$. This contrasts with the case of the unconditioned random walk, where Gaussian-like tail bounds exists on the scale of $n$.
We introduce the notion of the infinite bridge, which is known to arise as the distributional limit of the random walk bridge. We also establish some preliminary facts about the infinite bridge.
By showing that the Brownian Continuum Random Tree (BCRT) is characterized by its random self-similarity property, we prove that the range of the random walk bridge converges in distribution to the BCRT when rescaled by $Cn^{-1/2}$ for an appropriate constant $C$
A draft of the thesis can be found here: andrew-stewart-thesis