## Departmental PhD Thesis Exam – Michal Kotowski

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

Tuesday, June 28, 2016
1:10 p.m.
BA6183

PhD Candidate:  Michal Kotowski
Co-Supervisors:  Balint Virag
Thesis title:  Return probabilities on groups and large deviations for permuton processes

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Abstract:

The topic of this thesis are random processes on finite and infinite groups. More specifically, we are concerned with random walks on finitely generated amenable groups and stochastic processes which arise as limits of trajectories of the interchange process on a line.

In the first part of the thesis we construct a new class of finitely generated groups, called bubble groups. Analysis of the random walk on such groups shows that they are non-Liouville, but have return probability exponents close to $1/2$. Such behavior was previously unknown for random walks on groups. Our construction is based on permutational wreath products over tree-like Schreier graphs and the analysis of large deviations of inverted orbits on such graphs.

In the second part of we analyze large deviations of the interchange process on a line, which can be thought of as a random walk in the group of all permutations, with adjacent transpositions as generators. This is done in the setting of random permuton processes, which provide a notion of a limit for a permutation-valued stochastic processes. More specifically, we provide bounds on the probability that the trajectory of the interchange process (as a permuton process) is close in distribution to a deterministic permuton process. As an application, we show that short paths joining the identity and the reverse permutation in the Cayley graph of $\mathcal{S}_{n}$ are typically close to the so-called sine curve process, which is the conjectured limit of random sorting networks. The analysis is done in the framework of interacting particle systems.

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

## Departmental PhD Thesis Exam – Marcin Kotowski

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

Tuesday, June 28, 2016
11:10 a.m.
BA6183

PhD Candidate:  Marcin Kotowski
Co-Supervisors:  Balint Virag
Thesis title:  Random Schroedinger operators with connections to spectral properties of groups and directed polymers

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Abstract:

This thesis studies random Schroedinger operators with connections to group theory and models from statistical physics. First, we study 1D operators obtained as perturbations of the standard adjacency operator on $\Bbb Z$ by putting random i.i.d. noise with finite logarithmic variance on the edges. We study their expected spectral measures $\mu_H$ near zero. We prove that the measure exhibits a spike of the form $\mu_H(-\varepsilon,\varepsilon) \sim \frac{C}{\sim{\log\varepsilon}^2}$, which was first observed by Dyson for a specific choice of the edge weight distribution. We prove the result in generality, without assuming any regularity of edge weights.

We also identify the limiting local eigenvalue distribution, obtained by counting crossings of the Brownian motion derived from the operator. The limiting distribution is different from Poisson and the usual random matrix statistics. The results also hold in the setting where the edge weights are not independent, but are sufficiently ergodic, e.g. exhibit mixing. In conjunction with group theoretic tools, we then use the result to compute Novikov-Shubin invariants, which are group invariants related to the spectral measure, for various groups, including lamplighter groups and lattices in the Lie group Sol.

Second, we study similar operators in the two dimensional setting. We construct a random Schroedinger operator on a subset of the hexagonal lattice and study its smallest eigenvalues. Using a combinatorial mapping, we relate these eigenvalues to the partition function of the directed polymer model on the square lattice. For a specific choice of the edge weight distribution, we obtain a model known as the log-Gamma polymer, which is integrable. Recent results about the fluctuations of free energy for the log-Gamma polymer allow us to prove Tracy-Widom type fluctuations for the smallest eigenvalue of the original random Schroedinger operator.

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

## 2016 Course Instructors Teaching Excellence Award Winner

Congratulations to our graduate student, Peter Crooks, winner of the 2016 CI Teaching Excellence Award.

In 2015, the Teaching Assistants’ Training Program’s (TATP) Teaching Excellence Award launched the first-ever TATP award specifically for graduate student Course Instructors.  This is a university-wide award that “recognizes one graduate student whose outstanding work as a sole-responsibility Course Instructor shows evidence of educational leadership, meaningful contributions to course and curriculum development, and impact on student learning.”

Peter is co-supervised by Lisa Jeffrey and John Scherk and is working in the area of Lie Theory and Equivariant Geometry.  Peter is defending his thesis this Wednesday, May 4, 2016.

## 2016 Daniel B. DeLury Teaching Assistant Awards winners

We are happy to announce that this year’s winners of the Daniel B. DeLury Teaching Assistant Awards for graduate students in mathematics are:

• Tracey Balehowsky
• Beatriz Navarro Lameda
• Nikita Nikolaev
• Asif Zaman

The selection committee consisted of Mary Pugh, Abe Igelfeld and Peter Crooks.  Nominations were made by faculty members, course instructors, and undergraduate students.

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 Tracey, Beatriz, Nikita and Asif!

## Ida Bulat Teaching Award for Graduate Students

The Department of Mathematics invites nominations for the inaugural:

Ida Bulat Teaching Award

This prize honors outstanding teaching by graduate students.  It is named in memory of Ida Bulat, former Graduate Administrator in the Department of Mathematics at the University of Toronto.

Nominations can be submitted online through this link, no later than Friday, May 13, 2016:

http://blog.math.toronto.edu/forms/nomination-form-2016/

## Departmental PhD Thesis Exam – Trefor Bazett

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

Wednesday, May 11, 2016
11:10 a.m.
BA6183

PhD Candidate:  Trefor Bazett
Co-Supervisors:  Lisa Jeffrey/Paul Selick
Thesis title: The equivariant K-theory of commuting 2-tuples in SU(2)

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Abstract:

In this thesis, we study the space of commuting n-tuples in SU(2), $Hom(\mathbb{Z}^n, SU(2))$. We describe this space geometrically via providing an explicit G-CW complex structure, an equivariant analog of familiar CW- complexes. For the n=2 case, this geometric description allows us to compute various cohomology theories of this space, in particular the G-equivariant K-Theory $K_G^*(Hom(\mathbb{Z}^2, SU(2)))$, both as an $R(SU(2))$-module and as an $R(SU(2))$-algebra. This space is of particular interest as $\phi^{-1}(e)$ in a quasi-Hamiltonian system $M\xrightarrow{\phi} G$ consisting of the G-space $SU(2)\times SU(2)$, together with a moment map $\phi$ given by the commutator map. Finite dimensional quasi-Hamiltonian spaces have a bijective correspondence with certain infinite dimensional Hamiltonian spaces, and we additionally compute relevant components of this larger picture in addition to $\phi^{-1}(e)=Hom(\mathbb{Z}^2, SU(2))$ for this example.

A copy of the thesis can be found here: TreforBazettThesis

## Departmental PhD Thesis Exam – Jennifer Vaughan

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

Monday, May 9, 2016
11:10 a.m.
BA6183

PhD Candidate:  Jennifer Vaughan
Co-Supervisors:  Yael Karshon
Thesis title:  Quantomorphisms and Quantized Energy Levels for Metaplectic-c Quantization

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Abstract:

Metaplectic-c quantization was developed by Robinson and Rawnsley as an alternative to the classical Kostant-Souriau quantization procedure with half-form correction.  This thesis extends certain properties of Kostant-Souriau quantization to the metaplectic-c context.  We show that the Kostant-Souriau results are replicated or improved upon with metaplectic-c quantization.

We consider two topics:  quantomorphisms and quantized energy levels.  If a symplectic manifold admits a Kostant-Souriau prequantization circle bundle, then its Poisson algebra is realized as the space of infinitesimal quantomorphisms of that circle bundle.  We present a definition for a metaplectic-c quantomorphism, and prove that the space of infinitesimal metaplectic-c quantomorphisms exhibits all of the same properties that are seen in the Kostant-Souriau case.

Next, given a metaplectic-c prequantized symplectic manifold $(M,\omega)$ and a function $H\in C^\infty(M)$, we propose a condition under which $E$, a regular value of $H$, is a quantized energy level for the system $(M,\omega,H)$.  We prove that our definition is dynamically invariant:  if two functions on $M$ share a regular level set, then the quantization condition over that level set is identical for both functions.  We calculate the quantized energy levels for the $n$-dimensional harmonic oscillator and the hydrogen atom, and obtain the quantum mechanical predictions in both cases.  Lastly, we generalize the quantization condition to a level set of a family of Poisson-commuting functions, and show that in the special case of a completely integrable system, it reduces to a Bohr-Sommerfeld condition.

The draft to the thesis can be found here: Vaughan-Draft

## Departmental PhD Thesis – Iva Halacheva

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

Wednesday, June 29, 2016
2:10 p.m.
BA6183

PhD Candidate:  Iva Halacheva
Co-Supervisors:  Dror Bar-Natan, Joel Kamnitzer
Thesis title:   Alexander-type invariants of tangles, Skew Howe duality for crystals and the cactus group

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Abstract:

In the first part of this work, we study generalizations of a classical link invariant–the multivariable Alexander polynomial–to tangles. The starting point is Archibald’s tMVA invariant for virtual tangles which lives in the setting of circuit algebras. Using the Hodge star map, we define a reduction of the tMVA to an invariant (rMVA) which is valued in matrices with entries equal to certain Laurent polynomials. When restricted to tangles without closed components, we show the rMVA has the structure of a metamonoid morphism and is further equivalent to another tangle invariant defined by Bar-Natan. This invariant also reduces to the Gassner representation on braids and has a partially defined trace operation for closing open strands of a tangle.

In the second part, we look at crystals and the cactus group. The  crystals for a finite-dimensional, complex, reductive Lie algebra $\mathfrak g$ encode the structure of its representations, yet can also reveal surprising new structure of their own. In this work, we construct a group $J_{\mathfrak g}$, the “cactus group”, using the Dynkin diagram of $\mathfrak g$ and show that it acts combinatorially on any $\mathfrak g$-crystal via the Sch\”{u}tzenberger involutions. For ${\mathfrak g} =\mathcal g l_n$, the cactus group was studied by Henriques and Kamnitzer, who construct an action of it on $n$-tensor products of $\mathfrak g$-crystals. We study the crystal corresponding to the $\mathfrak g l_n \times \mathfrak g l_m$-representation $\Lambda^N(\Bbb C^n \otimes \Bbb C^m)$, derive skew Howe duality on the crystal level and show that the two cactus group actions agree in this setting. An application of this result is discussed in studying a family maximal commutative subalgebras of the universal enveloping algebra, the shift of argument and Gaudin algebras, where an algebraically constructed monodromy action is expected to match that of the cactus group.

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

## Departmental PhD Thesis Exam – Tyler Holden

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

Wednesday, May 4, 2016
2:10 p.m.
BA6183

PhD Candidate:  Tyler Holden
Supervisor:  Lisa Jeffrey
Thesis title: Convexity and Cohomology of the Based Loop Group

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Abstract:

Let $K$ be a compact, connected, simply connected Lie group and define $\Omega K$ to be the loops on $K$. Let $\Omega_\text{alg}K$ be those loops which are the restriction of algebraic maps $\Bbb C^\times \to K_\Bbb C$.  Herein we establish two distinct but related results.  In the first, we demonstrate the module structure for various generalized abelian equivariant cohomology theories as applied to equivariantly stratified spaces. This result is applied to the algebraic based loop group for the cases of equivariant singular cohomology, $K$-theory, and complex cobordism cohomology.     Subsequently, we examine the image of the based loop group under the non-abelian moment map. We show that both the Kirwan and Duistermaat convexity theorems hold in this infinite dimensional setting.

A copy of the thesis can be found here: ThesisPreview

## Departmental PhD Thesis Exam – Peter Crooks

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

Wednesday, May 4, 2016
11:10 a.m.
BA1200

PhD Candidate:  Peter Crooks
Supervisor:  Lisa Jeffrey
Thesis title:  The Equivariant Geometry of Nilpotent Orbits and Associated Varieties

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Abstract:

Nilpotent orbits are highly structured algebraic varieties lying at the interface of algebraic geometry, Lie theory, symplectic geometry, and geometric representation theory. The interest in these objects has been long-standing, ranging from Kostant’s foundational work in the 1950s and 1960s to Kronheimer’s realization of nilpotent orbits as instanton moduli spaces. At the same time, nilpotent orbits are studied for the sake of understanding closely associated varieties, such as nilpotent Hessenberg varieties.

In this thesis, we study the equivariant algebraic geometry and topology of nilpotent orbits and related varieties. Our first group of results is principally concerned with presentations of $T$-equivariant cohomology rings. More specifically, we give concrete presentations of the $T$-equivariant cohomology rings of the regular and minimal nilpotent orbits, with the latter presentation providing an equivariant counterpart to existing work on the ordinary cohomology of the minimal nilpotent orbit. We also examine the family of Hessenberg varieties arising from the minimal nilpotent orbit, showing them to be GKM and obtaining presentations of their $T$-equivariant cohomology rings. In Lie type $A$, we explain how one would compute the Poincar\'{e} polynomials and irreducible components of these Hessenberg varieties.

Our second group of results includes a characterization of those semisimple real Lie algebras for which every complex nilpotent orbit contains a real one, building on Rothschild’s criterion for such a Lie algebra to be quasi-split. We also consider the role of nilpotent orbits in quaternionic K\”ahler geometry by giving a new, self-contained proof of the LeBrun-Salamon Conjecture for equivariant contact structures on partial flag varieties. This approach allows us to give an intrinsic description of the standard contact structure on the isotropic Grassmannian of $2$-planes in $\mathbb{C}^{2n}$.

We conclude on a somewhat different note by computing the generalized $T$-equivariant cohomology of a direct limit of smooth projective varieties. As a brief application, we obtain the generalized $T$-equivariant cohomology of the affine Grassmannian.

A copy of the thesis can be found here: Thesis3