## Departmental PhD Thesis Exam – Fabian Parsch

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

Tuesday, June 18, 2019
11:10 a.m.
BA6183

PhD Candidate:  Fabian Parsch
Supervisor:   Alex Nabutovsky
Thesis title:  Geodesic Nets with Few Boundary Points

***

Geodesic nets on Riemannian manifolds form a natural class of stationary objects generalizing geodesics. Yet almost nothing is known about their classification or general properties even when the ambient Riemannian manifold is the Euclidean plane or the round 2-sphere.
We survey some results and open questions (old and new) about geodesic nets on Riemannian manifolds. A particular focus will be put on the question if the number of inner vertices (balanced vertices) in a geodesic net can be bounded by the number of boundary points (unbalanced vertices) or the total imbalance.
We prove that a geodesic net with three unbalanced vertices on a non-positively curved plane has at most one balanced vertex. We do not assume any a priori bound for the degree of unbalanced vertices. The result seems to be new even in the Euclidean case.
We demonstrate by examples that the result is not true for metrics of positive curvature on the plane, and that there are no immediate generalizations of this result for geodesic nets with four unbalanced vertices which can have a significantly more complicated structure. In particular, an example of a geodesic net with four unbalanced vertices and sixteen balanced vertices that is not a union of simpler geodesic nets is constructed. The previously known irreducible geodesic nets with four unbalanced vertices have at most two balanced vertices.
We provide a partial answer for a related question, namely a description of a new infinite family of geodesic nets on the Euclidean plane with 14 unbalanced vertices and arbitrarily many balanced vertices of degree three or more.

A copy of the thesis can be found here: parsch_thesis_2019-04-09

## Departmental PhD Thesis Exam – Zhifei Zhu

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

Tuesday, June 18, 2019
2:10 p.m.
BA6183

PhD Candidate:  Zhifei Zhu
Supervisor:   Regina Rotman
Thesis title: Geometric inequalities on Riemannian manifolds

***

In this thesis, we will show three results which partially answer several questions in the field of quantitative geometry. We first show that there exists Riemannian metric on a 3-disk so that the diameter, volume and surface area of the boundary is bounded, but during any contraction of the boundary of the disk, there exists a surface with arbitrarily large surface area. This result answers a question of P. Papasoglu.

The second result we will prove is that on any closed 4-dimensional simply-connected Riemannian manifolds with diameter <=D, volume > v > 0 and Ricci curvature |Ric| < 3, the length of a shortest closed geodesic can be bounded by some function f(v,D) which only depends on the volume and diameter of the manifold. This result partially answers a question of M. Gromov.

As an extension of our second result, we show that on 4-dimensional Riemannian manifolds satisfying the above conditions, the first homological filling function HF_1(l) <=f_1 (v,D)l +f_2 (v,D), for some functions f_1 and f_2 which only depends on v and D. And in particular, the area of a smallest minimal surface on the manifold can be bounded by some function which only depends on v and D.

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

## Departmental PhD Thesis Exam – Leonid Monin

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

Friday, April 26, 2019
11:10 a.m.
BA6183

PhD Candidate:  Leonid Monin
Thesis title: Newton Polyhedra, Overdetermined system of equations, and Resultants

***

In the first part of this thesis we develop Newton polyhedra theory for overdetermined systems of equations. Let A_1…A_k be finite sets in Z^n  and let be an algebraic subvariety of (C^*)^n  defined by a system of  Laurent polynomials with supports in A_1… A_k.  Assuming that  Laurent polynomials are sufficiently generic, the Newton polyhedron theory computes discrete invariants of Y in terms of their Newton polyhedra. It may appear that the generic system with fixed supports  is inconsistent. In this case one is interested in the generic consistent system. We extend Newton polyhedra theory to this case and compute discrete invariants generic non-empty zero sets. Unlike the classical situation, not only the Newton polyhedra of Laurent polynomials, but also their supports themselves appear in the answers.

We proceed then to the study of overdetermined collections of linear series on algebraic varieties other than algebraic torus. That is let E_1…E_k be a finite dimensional subspace of the space of  regular sections of line bundles on an irreducible algebraic variety X, so that the system

s_1 = … = s_k = 0,

where s_i is a generic  element of E_i does not have any roots on X. In this case we investigate the consistency variety  (the closure of the set of all systems which have at least one common root) and study general properties of zero sets Z of a generic consistent system. Then, in the case of equivariant linear series on spherical homogeneous spaces we provide a strategy for computing discrete invariants of such generic non-empty set Z.

The second part of this thesis is devoted to the study of Delta-resultants of (n+1)-tuple of Laurent polynomials with generic enough Newton polyhedra.  We provide an algorithm for computing Delta-resultant assuming that an n-tuple f_2, …, f_{n+1} is developed. We provide a relation between the product of f_1 over roots of  f_2 = … = f_{n+1} = 0 in (C^*)^n and the product of f_2 over roots of f_1=f_3 = … = f_{n+1} = 0 in (C^*)^n assuming that the n-tuple (f_1f_2, f_3…f_{n+1} is developed. If all n-tuples contained  in (f_1…f_{n+1}) are developed we provide a signed version of Poisson formula for Delta-resultant. Interestingly, the sign of the sparse resultant is nontrivial and is defined through Parshin symbols. Our proofs are based on a topological version of the Parshin reciprocity laws.
A copy of the thesis can be found here:  ut-thesis monin

## Departmental PhD Thesis Exam – Francisco Guevara Parra

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

Friday, March 15, 2019
1:30 p.m.
Fields Institute, room 210

PhD Candidate:  Francisco Guevara Parra
Supervisor:   Stevo Todorcevic
Thesis title:  Analytic spaces and their Tukey types

***

In this Thesis we study topologies on countable sets from the perspective of Tukey reductions of their neighbourhood filters. It turns out that is closely related to the already established theory of definable (and in particular analytic) topologies on countable sets. The connection is in fact natural as the neighbourhood filters of points in such spaces are typical examples of directed sets for which Tukey theory was introduced some eighty years ago. What is interesting here is that the abstract Tukey reduction of a neighbourhood filter $\mathcal{F}_{x}$ of a point to standard directed sets like $\mathbb{N}^\mathbb{N}$ or $\ell_1$ imposes that $\mathcal{F}_{x}$ must be analytic. We develop a theory that examines the Tukey types of analytic topologies and compare it by the theory of sequential convergence in arbitrary countable topological spaces either using forcing extensions or axioms such as, for example, the Open Graph Axiom. It turns out that in certain classes of countable analytic groups we can classify all possible Tukey types of the corresponding neighbourhood filters of identities. For example we show that if $G$ is a countable analytic $k$-group then $1=\{0\},$ $\mathbb{N}$ and $\mathbb{N}^\mathbb{N}$ are the only possible Tukey types of the neighbourhood filter $\mathcal{F}_{e}^{G}$. This will give us also new metrization criteria for such groups. We also show that the study of definable topologies on countable index sets has natural analogues in the study of arbitrary topologies on countable sets in certain forcing extensions.

A copy of the thesis can be found here: Francisco_PhD_Thesis

## Departmental PhD Thesis Exam – Duncan Dauvergne

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

Monday, March 25, 2019
10:10 a.m.
BA6183

PhD Candidate:  Duncan Dauvergne
Supervisor:   Balint Virag
Thesis title:  Random sorting networks, the directed landscape, and random polynomials

***

The first part of this thesis is on random sorting networks. A sorting network is a shortest path from $12 \cdots n$ to $n \cdots 21$ in the Cayley graph of the symmetric group $S_n$ generated by adjacent transpositions. We prove that in a uniform random $n$-element sorting network $\sigma^n$, all particle trajectories are close to sine curves with high probability. We also find the weak limit of the time-$t$ permutation matrix measures of $\sigma^n$. As a corollary, we show that if $S_n$ is embedded into $\mathbb{R}^n$ via the map $\tau \mapsto (\tau(1), \tau(2), \dots \tau(n))$, then with high probability, the path $\sigma^n$ is close to a great circle on a particular $(n-2)$-dimensional sphere. These results prove conjectures of Angel, Holroyd, Romik, and Vir\’ag. To prove these results, we find the local limit of random sorting networks and prove that the local speed distribution is the arcsine distribution on $[-\pi, \pi]$.

The second part of this thesis is on last passage percolation. The conjectured limit of last passage percolation is a scale-invariant, independent, stationary increment process with respect to metric composition. We prove this for Brownian last passage percolation. We construct the Airy sheet and characterize it in terms of the Airy line ensemble. We also show that the last passage geodesics converge to random functions with H\”older-$2/3^-$ continuous paths. To prove these results, we develop a new probabilistic framework for understanding the Airy line ensemble.

The third part of this thesis is on random sums of orthonormal polynomials. Let $G_n = \sum_{i=0}^n \xi_i p_i$, where the $\xi_i$ are i.i.d. non-degenerate complex random variables, and $\{p_i\}$ is a sequence of orthonormal polynomials with respect to a regular measure $\tau$ supported on a compact set $K$. We show that the zero measure of $G_n$ converges weakly almost surely to the equilibrium measure of $K$ if and only if $\mathbb{E}\log(1 + |\xi_0|) < \infty$. We also show that the zero measure of $G_n$ converges weakly in probability to the equilibrium measure of $K$ if and only if $\mathbb{P}(|\xi_0| > e^n) = o(n^{-1})$. Our methods also work for more general sequences of asymptotically minimal polynomials in $L^p(\tau)$, where $p \in (0, \infty]$.

A copy of the thesis can be found here: MainThesisPhD

## 2018 Malcolm Slingsby Robertson Prize winner and doctoral awards competition

The Department is happy to announce that the 2018 winner of the Malcolm Slingsby Robertson Prize in Mathematics for “a graduating PhD student who has demonstrated excellence in research” is:

• Alexander (Sacha) Mangerel

The Awards subcommittee of the Graduate Committee reviewed the theses and appraisal reports of several excellent graduating students for this prize.

Sacha wrote his thesis titled “Topics in Multiplicative and Probabilistic Number Theory under the supervision of John Friedlander.  One of his papers, joint with O. Klurman, of close to fifty pages, has been accepted for
publication in Mathematische Annalen.
He took on a Postdoctoral position at the
Centre de Recherches Mathématiques, Université de Montréal.

The prize carries a $500 monetary award. We congratulate Sacha for his excellent work and wish him great success! Malcolm Slingsby Robertson Prize winner 2018 Sacha is also the department’s sole nomination for the CMS Doctoral Prize. Our sole nomination for the CAIMS Cecil Graham Doctoral Dissertation Award (Applied Math) is Shuangjian Zhang, student of Robert McCann. Shuangjian is presently a postdoc at ENSAE ParisTech. We hope the nominations are successful. ## UTGSU accepting applications for the 2018 Conference Bursary (Fall Cycle) The UTGSU Conference Bursary was created in 2016 to financially assist UTGSU Members attending and/or presenting at academic conferences. The amount of a single bursary is$250, regardless of conference location or estimated expenses. A total of 120 bursaries are distributed each year, corresponding to 40 bursaries per each of the UTGSU’s three (3) Conference Bursary Cycles: Fall Cycle, Spring Cycle, and Summer Cycle.

Applications to the 2018 UTGSU Conference Bursary (Fall Cycle) will open on November 1, 2018 and will remain open until 11:59 PM on November 15, 2018.  This cycle is for conferences with start dates on or between December 1, 2018 and March 31, 2019.

Please note that you must be a UTGSU Member at the time of application for your application to be deemed eligible. Applications will only be accepted for conferences yet to be attended, not for conferences already attended. Additionally, applicants may only submit one application per Conference Cycle.

For more information and to access the Conference Bursary Application and Instructions please visit: https://www.utgsu.ca/funding/conference-bursary/ Contact Information and Accessibility If you require accessibility accommodations or have any questions related to the UTGSU Conference Bursary, please email the UTGSU Finance Commissioner at finance@utgsu.ca.

## Drop courses deadline – October 29, 2018

Drop courses absolute deadline: Monday, October 29, 2018

Students dropping courses on ACORN must also fill out a drop courses
form and submit to the Graduate Office.

Forms are also available on the counter in the math mailroom (BA 6290A).

## Departmental PhD Thesis Exam – Chia-Cheng Liu

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

Wednesday, December 5, 2018
11:10 a.m.
BA1170

PhD Candidate:  Chia-Cheng Liu
Co-Supervisors:   Joel Kamnitzer/Alexander Braverman
Thesis title:  Semi-innite Cohomology, Quantum Group Cohomology, and the Kazhdan-Lusztig
Equivalence

The Kazhdan-Lusztig tensor equivalence is a monoidal functor which sends modules over ane
Lie algebras at a negative level to modules over quantum groups at a root of unity. A positive
level Kazhdan-Lusztig functor is dened using Arkhipov-Gaitsgory’s duality between ane Lie
algebras of positive and negative levels. We prove that the semi-innite cohomology functor
for positive level modules factors through the positive level Kazhdan-Lusztig functor and the
quantum group cohomology functor with respect to the positive part of Lusztig’s quantum
group. This is the main result of the thesis.

Monoidal structure of a category can be interpreted as factorization data on the associated
global category. We describe a conjectural reformulation of the Kazhdan-Lusztig tensor equivalence
in factorization terms. In this reformulation, the semi-innite cohomology functor at
positive level is naturally factorizable, and it is conjectured that the factorizable semi-innite
cohomology functor is essentially the positive level Kazhdan-Lusztig tensor functor modulo the
Riemann-Hilbert correspondence. Our main result provides an important technical tool in a
proposed approach to a proof of this conjecture.

A copy of the thesis can be found here: thesis_chiachengliu-1