## Departmental PhD Thesis Exam – Jeffrey Pike

Friday, August 14, 2020
11:00 a.m.

PhD Candidate:  Jeffrey Pike
Supervisor:   Eckhard Meinrenken
Thesis title:  Weil Algebras for Double Lie Algebroids

***

Given a double vector bundle D → M, we define a bigraded bundle of algebras W(D) → M called the ‘Weil algebra bundle’. The space W(D) of sections of this algebra bundle ‘realizes’ the algebra of functions on the supermanifold D[1, 1]. We describe in detail the relations between the Weil algebra bundles of D and those of the double vector bundles D′, D′′ obtained from D by duality operations. We show that VB-algebroid structures on D are equivalent to horizontal or vertical differentials on two of the Weil algebras and a Gerstenhaber bracket on the third. Furthermore, Mackenzie’s definition of a double Lie algebroid is equivalent to compatibilities between two such structures on any one of the three Weil algebras. In particular, we obtain a ‘classical’ version of Voronov’s result characterizing double Lie algebroid structures. In the case that D = T A is the tangent prolongation of a Lie algebroid, we find that W(D) is the Weil algebra of the Lie algebroid, as defined by Mehta and Abad-Crainic. We show that the deformation complex of Lie algebroids, the theory of IM forms and IM multivector fields, and 2-term representations up to homotopy, all have natural interpretations in terms of our Weil algebras.

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

## 2020 NSERC VANIER Winner

We are delighted to announce that our student Clovis Hamel Ascanio has been awarded a Vanier Canada Graduate Scholarship.

Clovis obtained his bachelor’s degree from Universidad Metropolitana, Caracas, Venezuela.  He is currently a second year doctoral student working in Set Theory and Model Theory under the supervision of Prof. Frank Tall.

Clovis writes: “The world-renowned Toronto Set Theory Seminar has been an ideal place for me, fostering my research and personal growth. Prof. Franklin D. Tall’s support and guidance have been essential”.

The prestigious Vanier CGS program aims to attract and retain world-class doctoral students by supporting students who demonstrate both leadership skills and a high standard of scholarly achievement in graduate studies in social sciences and humanities, natural sciences and engineering, and health.

The full list of Vanier CGS recipients is available at:

https://vanier.gc.ca/en/scholar_search-chercheur_recherche_2020.html

Congratulations Clovis!  NSERC VANIER winner 2020

## Departmental PhD Thesis Exam – Anne Dranowski

Monday, July 27, 2020
2:00 p.m.

PhD Candidate:  Anne Dranowski
Supervisor:   Joel Kamnitzer
Thesis title:  Comparing two perfect bases

****

We study a class of varieties which generalize the classical orbital varieties of Joseph. We show that our generalized orbital varieties are the irreducible components of a Mirkovic-Vybornov slice to a nilpotent orbit, and can be labeled by semistandard Young tableaux. Furthermore, we prove that Mirkovic-Vilonen cycles are obtained by applying the Mirkovic-Vybornov isomorphism to generalized orbital varieties and taking a projective closure, refining Mirkovic and Vybornov’s result. As a consequence, we are able to use the Lusztig datum of a Mirkovic-Vilonen cycle to determine the tableau labeling the generalized orbital variety which maps to it, and, hence, the ideal of the generalized orbital variety itself. By homogenizing we obtain equations for the cycle we started with, which is useful for computing various equivariant invariants such as equivariant multiplicity. As an application, we show that the Mirkovic-Vilonen basis differs from Lusztig’s dual semicanonical basis. This is significant because it is a first example of two perfect bases which are not the same. Our comparison relies heavily on the theory of measures developed by Baumann, Kamnitzer and Knutson (The Mirkovic-Vilonen basis and Duistermaat-Heckman measures) so we include what we need. We state a conjectural combinatorial ‘formula’ for the ideal of a generalized orbital variety in terms of its tableau.

A copy of the thesis can be found here:  dranowski_anne_yyyymm_phd_thesis

## Departmental PhD Thesis Exam – Khoa Pham

Monday, July 13, 2020
2:00 p.m.

PhD Candidate:  Khoa Pham
Supervisor:   Joel Kamnitzer
Thesis title:  Multiplication of generalized affine Grassmannian slices and comultiplication of shifted Yangians

***

Given a semisimple algebraic group $G$, shifted Yangians are quantizations of certain generalized slices in $G((t^{-1}))$. In this thesis, we work with these generalized slices and the shifted Yangians in the simply-laced case.

Using a presentation of antidominantly shifted Yangians inspired by the work of Levendorskii, we show the existence of a family of comultiplication maps between shifted Yangians. We include a proof that these maps quantize natural multiplications of generalized slices.

On the commutative level, we define a Hamiltonian action on generalized slices, and show a relationship between them via Hamiltonian reduction. This relationship is established by constructing an explicit inverse to a multiplication map between slices.

Finally, we conjecture that the above relationship lifts to the Yangian level. We prove this conjecture for sufficiently dominantly shifted Yangians, and for the $\mathfrak{sl}_2$-case.

A copy of the thesis can be found here: Thesis-Khoa-final

## Departmental PhD Thesis Exam – Xiao Ming

Thursday, June 25, 2020
2:00 p.m.

PhD Candidate:  Xiao Ming
Supervisor:   Stevo Todorcevic
Thesis title:  Borel Chain Conditions

***

The subject matter of this Thesis is an instance of the Chain Condition Method of coarse classification of Boolean algebras and partially ordered sets. This method has played an important role in the measure theory, the theory of forcing, and the theory of Martin type axioms.

We focus on the posets that are Borel definable in Polish spaces and investigate the connections between the chain condition method and the chromatic numbers, a classification scheme for graphs. We then introduce Borel version of some classical chain condition and show that the Borel poset $T(\pi\mathbb{Q})$, the Borel example Todorcevic used to distinguish $\sigma$-finite chain condition and $\sigma$-bounded chain condition, cannot be decomposed into countably many Borel pieces witnessing the $\sigma$-finite chain condition, despite the fact that the non-Borel such partition exists. Starting from there, we use the variations on the $G_0$-dichotomy analyzed to construct a number of examples of Borel posets of the form $\mathbb{D}(G)$ that the new hierarchy of Borel chain conditions is proper.

A copy of the thesis can be found here:  Thesis_revised_202006210421

## Departmental PhD Thesis Exam – Abhishek Oswal

Wednesday, June 3, 2020
2:00 p.m.

PhD Candidate:  Abhishek Oswal
Supervisor:   Jacob Tsimerman
Thesis title: A non-archimedean definable Chow theorem

***

O-minimality has had some striking applications to number theory. The utility of o-minimal structures originates from the remarkably tame topological properties satisfied by sets definable in such structures. Despite the rigidity that it imposes, the theory is sufficiently flexible to allow for a range of analytic constructions. An illustration of this `tame’ property is the following surprising generalization of Chow’s theorem proved by Peterzil and Starchenko – A closed analytic subset of a complex algebraic variety that is also definable in an o-minimal structure, is in fact algebraic. While the o-minimal machinery aims to capture the archimedean order topology of the real line, it is natural to wonder if such a machinery can be set up over non-archimedean fields. In this thesis, we explore a non-archimedean analogue of an o-minimal structure and prove a version of the definable Chow theorem in this context.

A copy of the thesis can be found here: thesis-draft-v4

## Departmental PhD Thesis Exam – Ren Zhu

Wednesday, May 27, 2020
2:00 p.m.

PhD Candidate:  Ren Zhu
Supervisor:   Kumar Murty
Thesis title:  The least prime whose Frobenius is an $n$-cycle

****
Let $L/K$ be a Galois extension of number fields.  We consider the problem of bounding the least prime ideal of $K$ whose Frobenius lies in a fixed conjugacy class $C$. Under the assumption of Artin’s conjecture we work with Artin $L$-functions directly to obtain an upper bound in terms of irreducible characters which are nonvanishing at $C$.   As a consequence we obtain stronger upper bounds for the least prime in $C$ when many irreducible characters vanish at $C$.  We also prove a Deuring-Heilbronn phenomenon for Artin $L$-functions with nonnegative Dirichlet series coefficients as a key step.
We apply our results to the case when $\Gal(L/K)$ is the symmetric group $S_n$.  Using classical results on the representation theory of $S_n$ we give an upper bound for the least prime whose Frobenius is an $n$-cycle which is stronger than known bounds when the characters which are nonvanishing at $n$-cycles are unramified, as well a similar result for $(n-1)$-cycles.
We also give stronger bounds in the case of $S_n$-extensions over $\mathbb{Q}$ which are unramified over a quadratic field.  We also consider other groups and conjugacy classes where unconditional improvements are obtained.

A copy of the thesis can be found here: Ren Zhu PhD Thesis

## Departmental PhD Thesis Exam – Mykola Matviichuk

Wednesday, May 20, 2020
1:00 p.m.

PhD Candidate:  Mykola Matviichuk
Supervisor:   Marco Gualtieri
Thesis title:  Quadratic Poisson brackets and co-Higgs fields

This thesis is devoted to studying the geometry of holomorphic Poisson brackets on complex manifolds. We concentrate on the case when the underlying manifold admits a structure of a vector bundle, and the Poisson bracket is invariant under the dilation action of the multiplicative group of the field of complex numbers. We call such a Poisson bracket quadratic, and associate to it a Higgs type tensor, which we call a co-Higgs field. We study the interplay between these two geometric structures. A parallel theory is developed for Poisson brackets on projective bundles. Using the classical tool of the spectral correspondence available for co-Higgs fields, we construct many new examples of Poisson brackets, and provide new classification results in low dimensional cases.

A copy of the thesis can be found here: Quadratic_Poisson_brackets_and_co_Higgs_fields

## Departmental PhD Thesis Exam – Beatriz Navarro Lameda

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

Wednesday, April 8, 2020
11:00 a.m.
BA6183

PhD Candidate:  Beatriz Navarro Lameda
Supervisor:   Kostya Khanin
Thesis title:  On Global Solutions of the Parabolic Anderson Model and Directed Polymers in a Random Environment

***

This thesis studies global solutions to the semidiscrete stochastic heat equation and the associated Cauchy problem known as Parabolic Anderson Model. Via a Feynman-Kac formula, it is linked with the analysis of directed polymers in random environment, and this thesis establishes a number of results for the corresponding partition function.
We consider a continuous-time simple symmetric random walk on the integer lattice $\Z^d$ in dimension $d \geq 3$, subject to a random potential given by two-sided Wiener processes. In the high-temperature regime, we prove the existence of the $L^2$- and almost sure limit of the partition function as time $t \to \pm \infty$. We show that the $L^2$-convergence rate is at least polynomial and that the limiting partition function is positive almost surely. Furthermore, we show that this limiting partition function defines a global stationary solution to the semidiscrete stochastic heat equation which is unique up to a rescaling, and which in some sense attracts solutions to the Parabolic Anderson Model for any subexponentially growing initial data. One of the primary tools in the proof of this uniqueness and attraction result is a factorization formula for the point-to-point partition function, which is related to the ones obtained by Sinai (1995) and Kifer (1997) for other polymer models, but valid not only on the diffusive scale but up to any sub-ballistic scale. This factorization formula allows us to obtain a uniqueness result for physical invariant probability measures of a certain skew product that can be naturally associated with the semidiscrete stochastic heat equation, which in turns gives uniqueness of global stationary solutions.

A copy of the thesis can be found here: Navarro-Lameda_PhDThesis

## Departmental PhD Thesis Exam – Debanjana Kundu

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

Wednesday, April 8, 2020
2:00 p.m. (sharp)
BA6183

PhD Candidate:  Debanjana Kundu
Supervisor:   Kumar Murty
Thesis title:  Iwasawa Theory of Fine Selmer Groups

****

Iwasawa theory began as a Galois module theory of ideal class groups, initiated by Kenkichi Iwasawa as part of the theory of cyclotomic fields.In the early 1970’s, Barry Mazur considered generalizations of Iwasawa theory to Selmer groups of elliptic curves (Abelian varieties in general).  At the turn of this century, Coates and Sujatha initiated the study of a subgroup of the Selmer group of an elliptic curve called the \textit{fine} Selmer group.

The focus of this thesis is to understand arithmetic properties of this subgroup. In particular, we understand the structure of fine Selmer groups and their growth patterns. We investigate a strong analogy between the growth of the $p$-rank of the fine Selmer group and the growth of the $p$-rank of the class groups. This is done in the classical Iwasawa theoretic setting of (multiple) $\ZZ_p$-extensions; but what is more striking is that this analogy can be extended to non-$p$-adic analytic extensions as well, where standard Iwasawa theoretic tools fail.

Coates and Sujatha proposed two conjectures on the structure of the fine Selmer groups. Conjecture A is viewed as a generalization of the classical Iwasawa $\mu=0$ conjecture to the context of the motive associated to an elliptic curve; whereas Conjecture B is in the spirit of generalising Greenberg’s pseudonullity conjecture to elliptic curves. We provide new evidence towards these two conjectures.

A copy of the thesis can be found here: Debanjana_thesis