## Departmental PhD Thesis Exam – Vincent Gelinas

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

Monday, June 18,  2018
11:10 a.m.
BA6183

PhD Candidate:  Vincent Gelinas
Co-Supervisors:   Joel Kamnitzer, Colin Ingalls
Thesis title:  Contributions to the Stable Derived Categories of Gorenstein Rings

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Abstract:

The stable derived category ${\rm D}_{sg}(R)$ of a Gorenstein ring $R$ is defined as the Verdier quotient of the bounded derived category ${\rm D}^b(\modsf R)$ by the thick subcategory of perfect complexes, that is, those with finite projective resolutions, and was introduced by Ragnar-Olaf Buchweitz as a homological measure of the singularities of $R$. This thesis contributes to its study, centered around representation theoretic, homological and Koszul duality aspects.

In Part I, we first complete (over $\C$) the classification of homogeneous complete intersection isolated singularities $R$ for which the graded stable derived category ${\rm D}^{\Z}_{sg}(R)$ (respectively, ${\rm D}^b(\coh X)$ for $X = \proj R$) contains a tilting object. This is done by proving the existence of a full strong exceptional collection of vector bundles on a $2n$-dimensional smooth complete intersection of two quadrics $X = V(Q_1, Q_2) \subseteq \mathbb{P}^{2n+2}$, building on work of Kuznetsov. We then use recent results of Buchweitz-Iyama-Yamaura to classify the indecomposable objects in ${\rm D}_{sg}^{\Z}(R_Y)$ and the Betti tables of their complete resolutions, over $R_Y$ the homogeneous coordinate rings of $4$ points on $\mathbb{P}^1$ and $4$ points on $\mathbb{P}^2$ in general position.

In Part II, for $R$ a Koszul Gorenstein algebra, we study a natural pair of full subcategories whose intersection $\mathcal{H}^{\mathsf{lin}}(R) \subseteq {\rm D}_{sg}^{\Z}(R)$ consists of modules with eventually linear projective resolutions. We prove that such a pair forms a bounded t-structure if and only if $R$ is absolutely Koszul in the sense of Herzog-Iyengar, in which case there is an equivalence of triangulated categories ${\rm D}^b(\mathcal{H}^{\mathsf{lin}}(R)) \cong {\rm D}_{sg}^{\Z}(R)$. We then relate the heart to modules over the Koszul dual algebra $R^!$. As first application, we extend the Bernstein-Gel’fand-Gel’fand correspondence beyond the case of exterior and symmetric algebras, or more generally complete intersections of quadrics and homogeneous Clifford algebras, to any pair of Koszul dual algebras $(R, R^!)$ with $R$ absolutely Koszul Gorenstein. In particular the correspondence holds for the coordinate ring of elliptic normal curves of degree $\geq 4$ and for the anticanonical model of del Pezzo surfaces of degree $\geq 4$. We then relate our results to conjectures of Bondal and Minamoto on the graded coherence of Artin-Schelter regular algebras and higher preprojective algebras; we characterise when these conjectures hold in a restricted setting, and give counterexamples to both in all dimension $\geq 4$.

A copy of the thesis can be found here:  thesis

Over the last few years, the generosity of faculty, alumni and friends of the Department have allowed us to create a number of significant scholarships to support graduate students. This year’s winners are listed below.

1) Ida Bulat Memorial Graduate Fellowship:

Lennart Döppenschmitt (student of Marco Gualtieri)

2) Vivekananda Graduate Scholarship for International students:

Debanjana Kundu (student of Kumar Murty)

Saied Sorkhou (student of Joe Repka)

Mateusz Olechnowicz (student of Jacob Tsimerman, Patrick Ingram)

Abhishek Oswal (student of Jacob Tsimerman)

6) Margaret Isobel Elliott Graduate Scholarship:

Keegan Da Silva Barbosa (student of Stevo Todorcevic)

7) Irving Kaplansky Scholarship:

Jamal Kawach (student of Stevo Todorcevic)

Congratulations to all!

## Panel discussion: What can you do with a PhD in math anyway?

A career panel for graduate students in mathematics

When: April 20, 2018
Where: BA6183
Time: 3:00-4:00 p.m.

Panelists

Alex Bloemendal:
Alex  is a computational scientist at the Broad Institute of MIT and Harvard and at the Analytic and Translational Genetics Unit of Massachusetts General Hospital. As a member of Broad institute member Ben Neale’s lab, Alex leads a group in developing new methods to analyze genetic data, harnessing its unprecedented scope and scale to discover the genetic causes of disease. He also co-founded and directs the Models, Inference & Algorithms initiative at the Broad, bridging computational biology, mathematical theory, and machine learning. Alex is an institute scientist at the Broad.  Alex was previously a research scientist in the Program for Evolutionary Dynamics and a Simons Fellow in the Department of Mathematics at Harvard University. His research in probability theory and random matrices focused on questions of signal and noise in high-dimensional data; he proved an open conjecture with wide-reaching applications for fields including population genetics. He also earned a teaching award for an advanced course on probability.  Alex received an Hon. B.Sc., M.Sc., and Ph.D. in mathematics from the University of Toronto

Aaron Chow:
Aaron is a Senior Information Security Consultant – Security Engineering at CIBC. He graduated from our doctoral program in 2014.

Dorian Goldman:
Dorian develops mathematical models using modern methods in machine learning and statistics for Conde Nast. He’s also an Adjunct Professor of data science at Columbia University, where he’s teaching a course on using data science in industry which has received overwhelmingly positive reviews. He completed his MSc degree in mathematics at UofT and his PhDs at the Courant Institute (NYU) and UPMC (Paris VI) and worked full time as a research-only fellow and instructor of mathematics at DPMMS at the University of Cambridge. He worked in Germany, France, England and the USA over the past several years while completing his degrees and gained considerable experience in variational methods, differential equations and applied analysis. He transitioned into data science and machine learning three years ago, and became very passionate about the mathematical sophistication and significant impact that the field has.

Diana Ojeda:
Diana got her PhD in set theory at Cornell University and was a postdoc at U of T from 2014 to 2017.  She now works as a SoC Engineer at Intel, developing modelling and analysis tools for FPGAs.

Ben Schachter:
Benjamin Schachter is a Consultant at the Boston Consulting Group, based in the Toronto office. He joined BCG full time in January 2018, after previously working at BCG as a summer Consultant in 2016. Ben has primarily worked in the technology, media, and telecommunications (TMT) practice area.  Ben completed his PhD in mathematics at the University of Toronto in 2017; his research focused on optimal transport and the calculus of variations.  Ben also holds an MSc in mathematics from the University of Western Ontario and an MA and BA (hons.), both in economics, from the University of Toronto.

## Yuri Cher – Posthumous Degree Award Ceremony

The Department of Mathematics will be holding a posthumous degree award ceremony for

Yuri Cher

on Tuesday, March 28, 2017
at
10:30 a.m.

in

BA6183
40 St. George St.

~
Everyone is welcome to attend.  Refreshments will be served in the Graduate Lounge.

Yuri Cher memorial invitation

## 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.

## 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

*****

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.

## Departmental PhD Thesis Exam – Louis-Philippe Thibault

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

********

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

## Departmental PhD Thesis Exam – Ivan Livinskyi

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

****
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