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

]]>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

]]>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.

]]>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.

Students dropping courses on ACORN must also fill out a drop courses

form and submit to the Graduate Office.

http://www.sgs.utoronto.ca/Documents/Add+Drop+Courses.pdf

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

]]>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

]]>Costumes are encouraged, we will have prizes available for the best costumes.

]]>Friday, October 19, 2018

2:10 p.m.

BA6183

PhD Candidate: Krishan Rajaratnam

Supervisor: Michael Sigal

Thesis title: Abrikosov lattice solutions of the ZHK Chern-Simons equations

****

In this thesis we study the ZHK Chern-Simons equations which occur in the study of the fractional quantum hall effect of condensed matter physics. After stating basic properties of these equations, we first prove the existence of Abrikosov lattice solutions of them. Among these solutions, we find the physically interesting one whose lattice shape minimizes the average energy per lattice cell. In addition to the Abrikosov lattice solutions, we find solutions of the ZHK Chern-Simons equations on Riemann surfaces of higher genus $g$, by utilizing similar results for the Ginzburg-Landau equations.

Finally, we study the orbital stability of the Abrikosov lattice solutions under perturbations which preserve the lattice.

A copy of the thesis can be found here:

]]>Tuesday, June 26 2018

1:10 p.m.

BA6183

PhD Candidate: Steven Amelotte

Supervisor: Paul Selick

Thesis title: Unstable Homotopy Theory Surrounding the Fibre of the $p^\text{th}$ Power Map on Loop Spaces of Spheres

***

In this thesis, we study the fibre of the $p^\text{th}$ power map on loop spaces of spheres with a view toward obtaining homotopy decompositions. For the prime $p=2$, we give an explicit decomposition of the fibre $\Omega^3S^{17}\{2\}$ of the squaring map on the triple loop space of the $17$-sphere, building on work of Campbell, Cohen, Peterson and Selick, who showed that such decompositions are only possible for $S^5$, $S^9$ and $S^{17}$. This induces a splitting of the mod $2$ homotopy groups $\pi_\ast(S^{17}; \mathbb{Z}/2\mathbb{Z})$ in terms of the integral homotopy groups of the fibre of the double suspension $E^2 \colon S^{2n-1} \to \Omega^2S^{2n+1}$ and refines a result of Cohen and Selick, who gave similar decompositions for $S^5$ and $S^9$. For odd primes, we find that the decomposition problem for $\Omega S^{2n+1}\{p\}$ is equivalent to the strong odd primary Kervaire invariant problem. Using this unstable homotopy theoretic interpretation of the Kervaire invariant problem together with what is presently known about the $3$-primary stable homotopy groups of spheres, we give a new decomposition of $\Omega S^{55}\{3\}$.

We relate the $2$-primary decompositions above to various Whitehead products in the homotopy groups of mod $2$ Moore spaces and Stiefel manifolds to show that the Whitehead square $[i_{2n},i_{2n}]$ of the inclusion of the bottom cell of the Moore space $P^{2n+1}(2)$ is divisible by $2$ if and only if $2n=2,4,8$ or $16$. As an application of our $3$-primary decomposition, we prove two new cases of a longstanding conjecture which states that the fibre of the double suspension is a double loop space.

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

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