Departmental PhD Thesis Exam – Benjamin Briggs

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

Friday, May 18, 2018
2:10 p.m.

PhD Candidate:  Benjamin Briggs
Co-Supervisors:   Joel Kamnitzer, Srikanth Iyengar
Thesis title:  Local Commutative Algebra and Hochschild Cohomology Through the
Lens of Koszul Duality



This thesis splits into two halves, the connecting theme being Koszul duality. The first part concerns local commutative algebra. Koszul duality here manifests in the homotopy Lie algebra. In the second part, which is joint work with Vincent G\’elinas, we study Hochschild cohomology and its characteristic action on the derived category.

We begin by defining the homotopy Lie algebra $\pi^*(\phi)$ of a local homomorphism $\phi$ (or of a ring) in terms of minimal models, slightly generalising a classical theorem of Avramov. Then, starting with work of F\'{e}lix and Halperin, we introduce a notion of Lusternik-Schnirelmann category for local homomorphisms (and rings). In fact, to $\phi$ we associate a sequence $\cat_{0}(\phi)\geq \cat_1(\phi)\geq \cat_2(\phi)\geq \cdots$ each $\cat_i(\phi)$ being either a natural number or infinity. We prove that these numbers characterise weakly regular, complete intersection, and (generalised) Golod homomorphisms. We present examples which demonstrate how they can uncover interesting information about a homomorphism. We give methods for computing these numbers, and in particular prove a positive characteristic version of F\'{e}lix and Halperin’s Mapping Theorem.

A motivating interest in L.S. category is that finiteness of $\cat_2(\phi)$ implies the existence of certain six-term exact sequences of homotopy Lie algebras, following classical work of Avramov. We introduce a variation $\pic(\phi)$ of the homotopy Lie algebra which enjoys long exact sequences in all situations, and construct a comparison $\pic(\phi)\to \pi^*(\phi)$ which is often an isomorphism.
This has various consequences; for instance, we use it to characterise quasi-complete intersection homomorphisms entirely in terms of the homotopy Lie algebra.

In the second part of this thesis we introduce a notion of $A_\infty$ centre for minimal $A_\infty$ algebras. If $A$ is an augmented algebra over a field $k$ we show that the image of the natural homomorphism $\chi_k:\HH(A,A)\to {\rm Ext}^*_A(k,k)$ is exactly the $A_\infty$ centre of $A$, generalising a theorem of Buchweitz, Green, Snashall and Solberg from the case of a Koszul algebra. This is deduced as a consequence of a much wider enrichment of the entire characteristic action $\chi:\HH(A,A)\to {\sf Z}(D(A))$. We give a number of representation theoretic applications.

A copy of the thesis can be found here:  ThesisBriggs

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

A career panel for graduate students in mathematics

Graduate Career Poster

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

Reception and networking to follow in the graduate lounge.



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.

Departmental PhD Thesis Exam – Anup Dixit

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

Wednesday, March 21, 2018
2:10 p.m.

PhD Candidate:  Anup Dixit
Supervisor:   Kumar Murty
Thesis title:  The Lindelof class of L-functions


Meromorphic functions, called L-functions, play a vital role in number theory.  In 1989, Selberg defined a class of L-functions that serves as an axiomatic model for L-functions arising from geometry and arithmetic. Even though the Selberg class successfully captures many characteristics common to most L-functions, it fails to be closed under addition. This creates obstructions, in particular, not allowing us to interpolate between L-functions. To overcome this limitation, V. K. Murty defined a general class of L-functions based on their growth rather than functional equation and Euler product. This class, which is called the Lindelof class of L-functions, is endowed with the structure of a ring.

In this thesis, we study further properties of this class, specifically, its ring structure and topological structure. We also study the zero distribution and the a-value distribution of elements in this class and prove certain uniqueness results, showing that distinct elements cannot share complex values and L-functions in this class cannot share two distinct values with any other meromorphic function. We also establish the value distribution theory for this class with respect to the universality property, which states that every holomorphic function is approximated infinitely often by vertical shifts of an L-function. In this context, we precisely formulate and give some evidence towards the Linnik-Ibragimov conjecture.

A copy of the thesis can be found here: Anup-Dixit-Thesis

Departmental PhD Thesis Exam – Fulgencio Lopez

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

Friday, March 16 , 2018
2:30 p.m.
Fields Institute – 210

PhD Candidate:  Fulgencio Lopez
Supervisor:   Stevo Todorcevic
Thesis title: Construction schemes and their applications


We study capturing construction schemes, a new combinatorial tool introduced by Todorcevíc to build uncountable structures. It consists of a ranked family of finite sets that provides a framework to do recursive constructions of uncountable objects by working with finite amal-gamations of finite isomorphic substructures, the uncountable substructures of the final object can be further study using capturing.

In this Thesis we study the consistency of capturing construction schemes, and related defi-
nitions, we prove results of consistency, and give several applications of this tool both to infinite combinatorics and Banach space theory. For example, we show weaker forms of capturing, such as n-capturing, form a strict hierarchy which is related to the m-Knaster Hierarchy.  We also show how capturing construction schemes can be used in constructing Suslin trees and Haus-dorff gaps of a special kind in an intuitive manner. And give some applications to the theory of nonseparable Banach spaces.

A copy of the thesis: Thesis Fulgencio Lopez

Departmental PhD Thesis Exam – Yuan Yuan Zheng

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

Friday, March 16 , 2018
1:30 p.m.
Fields Institute – 210

PhD Candidate:  Yuan Yuan Zheng
Supervisor:   Stevo Todorcevic
Thesis title:  Parametrizing topological Ramsey spaces


We prove a general theorem indicating that essentially all infinite-dimensional Ramsey-type theorems proven using topological Ramsey space theory can be parametrized by products of infinitely many perfect sets. This theorem has applications in several known spaces, showing that certain ultrafilters are preserved under both side-by-side and iterated Sacks forcing. In particular, the well-known result of `selective ultrafilters on the natural numbers are preserved under Sacks forcing’ is extended to the corresponding ultrafilters on richer structures. We also characterize ultrafilters in topological Ramsey spaces in an abstract
setting. The technique of combinatorial forcing is crucial in the proof of the general parametrization theorem, and ultra-Ramsey theory plays an important role in the applications.

A copy of the thesis can be found here: Yuan Yuan thesis

Departmental PhD Thesis Exam – Nan Wu

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

Thursday, March 15 , 2018
5:10 p.m.

PhD Candidate:  Nan Wu
Co-Supervisors:  Alex Nabutovsky, Hau-tieng Wu
Thesis title:  Differential Geometry Approach for Unsupervised Machine Learning Algorithms


Since its introduction in 2000, locally linear embedding (LLE) algorithm has been widely applied in data science.

In this thesis, we provide an asymptotical analysis of LLE under the manifold setup. First, by study the regularized barycentric problem, we derive the corresponding kernel function of LLE. Second, we show that when the point cloud is sampled from a general closed manifold, asymptotically LLE algorithm does not always recover the Laplace-Beltrami operator, and the result may depend on the non-uniform sampling. We demonstrate that a careful choosing of the regularization is necessary to ensure the recovery of the Laplace-Beltrami operator. A comparison with the other commonly applied nonlinear algorithms, particularly the diffusion map, is provided. Moreover, we discuss the relationship between two common nearest neighbor search schemes and the relationship of LLE with the locally linear regression. At last, we consider the case when the point cloud is sampled from a manifold with boundary.

We show that if the regularization is chosen correctly, LLE algorithm asymptotically recovers a linear second order differential operator with “free” boundary condition. Such operator coincides with Laplace-Beltrami operator in the interior of the manifold. We further modify LLE algorithm to the Dirichlet Graph Laplacian algorithm which can be used to recover the Laplace-Beltrami operator of the manifold with Dirichlet boundary condition.

A copy of the thesis can be found here: Nan Wu’s thesis

Departmental PhD Thesis Exam – Alexander Mangerel

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

Wednesday, February 14, 2018
2:10 p.m.

PhD Candidate:  Alexander Mangerel
Supervisor:  John Friedlander
Thesis title: Topics in Multiplicative and Probabilistic Number Theory


A heuristic in analytic number theory stipulates that sets of positive integers cannot simultaneously be additively and multiplicatively structured. The practical verification of this heuristic is the source of a great number of difficult problems. An example of this is the well-known Hardy-Littlewood tuples conjecture, which asserts that, infinitely often, one should be able to find additive patterns of fairly general shape in the primes. Conjectures of this type are also at least morally equivalent to the expectation that a multiplicative function, unless it has a special form, behaves randomly on additively structured sets.

In this thesis, we consider several problems involving the behaviour of multiplicative functions interacting with additively structured sets. Two main topics are studied: i) the estimation of \emph{mean values} of multiplicative functions, i.e., the limiting average behaviour of partial sums of multiplicative functions along an interval whose length tends to infinity; and ii) the estimation of \emph{correlations} of multiplicative functions, i.e., the behaviour of simultaneous values of multiplicative functions at arguments that are additively related. A number of applications of the study of these topics are also addressed.

First, we prove quantitative versions of mean value theorems due to Wirsing and Hal\'{a}sz for multiplicative functions that often take values outside of the unit disc. This has a broad realm of applications. In particular we are able to extend a further theorem of Hal\'{a}sz, proving local limit theorems for vectors of certain types of additive functions. We thus confirm a probabilistic heuristic in the \emph{small deviation} regime and beyond for the functions in question.

In a different direction, we consider the collection of periodic, completely multiplicative functions, also known as Dirichlet characters. Upper bounds for the maximum size of the partial sums of these functions on intervals of positive integers is connected with the class number problem in algebraic number theory, and with I.M. Vinogradov’s conjecture on the distribution of quadratic non-residues. By refining a quantitative mean value theorem for multiplicative functions, we significantly improve the existing upper bounds on the maximum size of partial sums of odd order Dirichlet characters, both unconditionally and assuming the Generalized Riemann Hypothesis. We also show that our conditional results are best possible unconditionally, up to a bounded power of $\log\log\log\log q$.

Regarding correlations, we prove a quantitative version of the bivariate Erd\H{o}s-Kac theorem. That is, we show that the joint distribution of pairs of values of certain additive functions is asymptotically an uncorrelated bivariate Gaussian, and find a quantitative error term in this approximation. We use this probabilistic result to prove a theorem on the joint distribution of certain natural variants of the M\”{o}bius function at additively-related integers as a partial result in the direction of Chowla’s conjecture on two-point correlations of the M\”{o}bius function. We also apply our result to understanding the set of pairs of consecutive integers with the same number of divisors.

A major theme in the thesis relates to how a multiplicative function can be rigidly characterized globally by certain local properties. As a first example, we show that a completely multiplicative function that only takes finitely many values, vanishes at only finitely many primes and whose partial sums are uniformly bounded, must be a non-principal Dirichlet character. This solves a 60-year-old open problem of N.G. Chudakov. We also solve a folklore conjecture due to Elliott, Ruzsa and others on the gaps between consecutive values of a unimodular completely multiplicative function, showing that these gaps cannot be uniformly large. This is a corollary of several stronger results that are proved regarding the distribution of consecutive values of multiplicative functions. For instance, we classify the set of all unimodular completely multiplicative functions $f$ such that $\{f(n)\}_n$ is dense in $\mb{T}$ and for which the sequence of pairs $(f(n),f(n+1))$ is dense in $\mb{T}^2$. In so doing, we resolve a conjecture of K\'{a}tai.

Finally, we make some progress on some natural variants of Chowla’s conjecture on sign patterns of the Liouville function. In particular, we prove that certain natural collections of multiplicative functions $f: \mb{N} \ra \{-1,+1\}$ are such that the tuples of values they produce on \emph{almost all} 3- and 4-term arithmetic progressions equidistribute among all sign patterns of length 3 and 4, respectively. Some of the aforementioned results are joint work with O. Klurman, or with Y. Lamzouri.

A copy of the thesis can be found here: APMangerPhDThesisFeb13

Departmental PhD Thesis Exam – Ali Feizmohammadi

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

Monday, January 29, 2018
2:10 p.m.

PhD Candidate:  Ali Feizmohammadi
Co-Supervisors:  Spyros Alexakis, Adrian Nachman
Thesis title: Unique Reconstruction of a Potential from the Dirichlet to Neumann Map in Locally CTA Geometries


Let $(M^3,g)$ be a compact smooth Riemannian manifold with smooth boundary and suppose that $U$ is an open set in $M$ such that $g|_U$ is Euclidean. Let $\Gamma= U \cap \partial M$ be connected and suppose that $U$ is the convex hull of $\Gamma$. We will study the uniqueness of an unknown potential for the Schr\”{o}dinger operator $ -\triangle_g + q $ from the associated Dirichlet to Neumann map, $\Lambda_q$. Indeed, we will prove that if the potential $q$ is a priori explicitly known in $U^c$ then one can uniquely reconstruct $q$ from $\Lambda_q$. We will also give a reconstruction algorithm for the potential. More generally we will also discuss the cases where $\Gamma$ is not connected or $g|_U$ is conformally transversally anisotropic and derive the analogous result.

A copy of the thesis can be found here:  Ali Feiz’s Dissertation

Departmental PhD Thesis Exam – Nikita Nikolaev

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

Monday, November 13, 2017
3:10 p.m.

PhD Candidate:  Nikita Nikolaev
Supervisor:  Marco Gualtieri
Thesis title:  Abelianisation of Logarithmic Connections



This thesis studies an equivalence between meromorphic connections of higher rank and abelian connections. Given a complex curve $X$ and a spectral cover $\pi : \Sigma \to X$, we construct a functor $\pi^\textup{ab} : \mathsf{Conn}_X \to \mathsf{Conn}_\Sigma$, called the \textit{abelianisation functor}, from some category of connections on $X$ with logarithmic singularities to some category of abelian connections on $\Sigma$, and we prove that $\pi^\textup{ab}$ is an equivalence of categories.  At the level of the corresponding moduli spaces $\mathbb{M}_X, \mathbb{M}_\Sigma$, which are known to be holomorphic symplectic varieties, this equivalence recovers a symplectomorphism constructed by Gaiotto, Moore, Neitzke in their work on Spectral Networks (2013).  Moreover, the moduli space $\mathbb{M}_\Sigma$ is a torsor for an algebraic torus, so in fact $\pi^\textup{ab}$ provides a Darboux coordinate system on $\mathbb{M}_X$, known as the \textit{Fock-Goncharov coordinates} constructed in their work on higher Teichm\”uller theory (2006).  To prove that $\pi^\textup{ab}$ is an equivalence of categories, we introduce a new concept called the \textit{Voros class}.  It is a canonical cohomology class in $H^1$ of the base $X$ with values in the nonabelian sheaf $\mathcal{Aut} (\pi_\ast)$ of groups of natural automorphisms of the direct image functor $\pi_\ast$.  Any $1$-cocycle $v$ representing the Voros class defines a new functor $\mathsf{Conn}_\Sigma \to \mathsf{Conn}_X$ by locally deforming the pushforward functor $\pi_\ast$; the result is an explicit inverse equivalence to $\pi^\textup{ab}$, called a \textit{deabelianisation functor}.

We generalise the abelianisation equivalence to the case of \textit{quantum connections}: these are $\hbar$-families of meromorphic connections restricted to a sectorial neighbourhood in $\hbar$ with prescribed asymptotic regularity.   The Schr\”odinger equation is a quintessential example. The most important invariant of a quantum connection $\nabla$ is the Higgs field $\nabla^{\tiny(0)}$ obtained by restricting $\nabla$ to $\hbar = 0$ (the so-called \textit{semiclassical limit}).  Then abelianisation may be viewed as a natural extension to an $\hbar$-family of the spectral line bundle of $\nabla^{\tiny(0)}$.  That is, we show that for a given quantum connection $(\mathcal{E}, \nabla)$, the line bundle $\mathcal{E}^\textup{ab}$ obtained from $\mathcal{E}$ by abelianisation $\pi^\textup{ab}$ restricts at $\hbar = 0$ to precisely the spectral line bundle of the Higgs field $\nabla^{\tiny(0)}$.

Finally, in this thesis we explore the relationship between abelianisation and the WKB method, which is an asymptotic approximation technique for solving differential equations developed by physicists in the 1920s and reformulated by Voros in 1983 using the theory of Borel resummation.  We give an algebro-geometric formulation of the WKB method using vector bundle extensions and splittings. We then show that the output of the WKB analysis is precisely the data used to construct the abelianisation functor $\pi^\textup{ab}$.

A copy of the thesis can be found here:

Departmental PhD Thesis Exam – Rosemonde Lareau-Dussault

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

Wednesday, November 29, 2017
3:10 p.m.

PhD Candidate:  Rosemonde Lareau-Dussault
Supervisor:  Robert McCann
Thesis title:  Coupled Education and Labour Market Models



This study addresses the dynamics of the distribution of skills in a population over many generations. Two overlapping generation models are proposed: the first assumes complete information, which allows (and requires) all generations to be solved simultaneously, while the second assumes incomplete information, forcing the competitive equilibrium at each subsequent generation to be found iteratively. Both models combine a labour and an education matching problem. The skill distribution for each generation of adults is determined from that of the previous generation by the educational matching market.

We present conditions for the sequence of adult skills to converge. Then we study the asymptotic which is specific to each model. For the incomplete information model, we prove that, if the sequence of wage functions over the generation converges, the limiting steady state solves the steady state model of Erlinger, McCann, Shi, Siow, and Wolthoff (2015), which allows us to get an explicit formulation for solutions of the model of Erlinger et al. To study the limiting solution of the complete information model, we introduce a new steady state model, which includes a discounting factor to reduce the impact of future generations relative to how far in the future they are.

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