Dissertations 2020

Friday, November 13, 2020

11:00 a.m.

PhD Candidate: Nathan Carruth

Supervisor: Spyros Alexakis

Thesis title: Focussed Solutions to the Einstein Vacuum Equations

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A copy of the thesis can be found here: thesis_comm

Wednesday, August 25, 2020

11:00 a.m.

PhD Candidate: Justin Ko

Supervisor: Dmitry Panchenko

Thesis title: The Free Energy of Spherical Vector Spin Glasses

We study a class of vector spin models with configurations restricted to subsets of the sphere. We will prove a constrained free energy formula for these models. This formula defines a large deviations principle for the limiting distribution of the overlaps under the asymptotic Gibbs measure. The thesis builds on the mathematical results used to prove free energy formulas for the classical Sherrington–Kirkpatrick spin glass, spherical spin models, and vector spin glass models. The free energy formula proved in this thesis are true generalizations of the classical results, in the sense that these vector spin formulas restricted to one dimension coincide with the known results for classical models.

The first contribution of this thesis is a variational formula for contrained copies of classical spherical spin glasses sampled at different temperatures. The free energy for multiple systems of spherical spin glasses with constrained overlaps was first studied by Panchenko and Talagrand. They proved an upper bound of the constrained free energy using Guerra’s interpolation. In this thesis, we prove this upper bound is sharp. Our approach combines the ideas of the Aizenman–Sims–Starr scheme and the synchronization mechanism used in the vector spin models. We derive a vector version of the Aizenman–Sims–Starr scheme for spherical spin glass and use the synchronization property of arrays obeying the overlap-matrix form of the Ghirlanda–Guerra identities to prove the matching lower bound.

The second contribution of this thesis is the simplification of this variational formula to the form originally discovered for the classical spherical spin glass model by Crisanti and Sommers. In particular, we prove the analogue of the Crisanti–Sommers variational formula for spherical spin glasses with vector spins. This formula is derived from the discrete Parisi variational formula for the limit of the free energy of constrained copies of spherical spin glasses. In vector spin models, the variations of the functional order parameters must preserve the monotonicity of matrix paths which introduces a new challenge in contrast to the derivation of the classical Crisanti–Sommers formula.

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

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.pdf 

Wednesday, August 5, 2020

4:00 p.m.

PhD Candidate: Afroditi Talidou

Co-Supervisors: Michael Sigal, Almut Burchard

Thesis title: Near-pulse solutions of the FitzHugh-Nagumo equations on cylindrical surfaces

****

In 1961, FitzHugh [19] suggested a model to explain the basic properties of excitability, namely the ability to respond to stimuli, as exhibited by the more complex HodgkinHuxley equations [24]. The following year Nagumo et al. [42] introduced another version based on FitzHugh’s model. This is the model we consider in the thesis. It is called the FitzHugh-Nagumo model and describes the propagation of electrical signals in nerve axons. Many features of the system have been studied in great detail in the case where an axon is modelled as a one-dimensional object. Here we consider a more realistic geometric structure: the axons are modelled as warped cylinders and pulses propagate on their surface, as it happens in nature.

The main results in this thesis are the stability of pulses for standard cylinders of small constant radius, and existence and stability of near-pulse solutions for warped cylinders whose radii are small and vary slowly along their lengths. On the standard cylinder, we write a solution near a pulse as the superposition of a modulated pulse with a fluctuation and prove that the fluctuation decreases exponentially over time as the solution converges to a nearby translation of the pulse. On warped cylinders, we write a solution near a pulse in the same way as in standard cylinders and prove bounds on the fluctuation of near-pulse solutions.

A copy of the thesis can be found here: Talidou-thesis-draft.pdf

Monday, July 27, 2020

2:00 p.m.

PhD Candidate: Anne Dranowski

Supervisor: Joel Kamnitzer

Thesis title: Comparing two perfect bases

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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_phd_thesis

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

Thursday, June 25, 2020

2:00 p.m.

PhD Candidate: Xiao Ming

Supervisor: Stevo Todorcevic

Thesis title: Borel Chain Conditions

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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.pdf

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.pdf

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

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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

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

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

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

Monday, March 30, 2020

10:00 a.m.

BA2139

PhD Candidate: Yvon Verberne

Supervisor: Kasra Rafi

Thesis title: Pseudo-Anosov homeomorphisms constructed using positive Dehn twists

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The mapping class group is the group orientation preserving homeomorphisms of a surface up to isotopy. The mapping class group encodes information about the symmetries of a surface. We focus on studying the pseudo-Anosov mapping classes, which are the elements of the group that mix the underlying surface in a complex way. These maps have applications in physics, notably in fluid dynamics, since we can stir a disk of fluid to create topological chaos, and in the study of magnetic fields since pseudo-Anosov maps create odd magnetic fields. Pseudo-Anosov maps also appear in industrial applications such as food engineering and polymer processing.

We introduce a construction of pseudo-Anosov homeomorphisms on $n$-times punctured spheres and surfaces with higher genus using only sufficiently many positive half-twists. These constructions can produce explicit examples of pseudo-Anosov maps with various number-theoretic properties associated to the stretch factors, including examples where the trace field is not totally real and the Galois conjugates of the stretch factor are on the unit circle.

We construct explicit examples of geodesics in the mapping class group and show that the shadow of a geodesic in mapping class group to the curve graph does not have to be a quasi-geodesic. We also show that the quasi-axis of a pseudo-Anosov element of the mapping class group may not have the strong contractibility property. Specifically, we show that, after choosing a generating set carefully, one can find a pseudo-Anosov homeomorphism $f$, a sequence of points $w_k$ and a sequence of radii $r_k$ so that the ball $B(w_k, r_k)$ is disjoint from a quasi-axis $a$ of $f$, but for any projection map from the mapping class group to $a$, the diameter of the image of $B(w_k, r_k)$ grows like $\log(r_k)$.

A copy of the thesis can be found here: Verberne_Yvon_ML_202006_PhD_thesis

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

Wednesday, February 19, 2020

1:00 p.m.

BA1210

PhD Candidate: Travis Ens

Supervisor: Dror Bar-Natan

Thesis title: On Braidors: An Analogue of the Theory of Drinfel’d Associators for Braids in

an Annulus

***

We develop the theory of braidors, an analogue of Drinfel’d’s theory of associators in which braids in an annulus are considered rather than braids in a disk. After defining braidors and showing they exist, we prove that a braidor is defined by a single equation, an analogue of a well-known theorem of Furusho [Furusho (2010)] in the case of associators. Next some progress towards an analogue of another key theorem, due to Drinfel’d [Drinfel’d (1991)] in the case of associators, is presented. The desired result in the annular case is that braidors can be constructed degree be degree. Integral to these results are annular versions \textbf{GT}$_a$ and \textbf{GRT}$_a$ of the Grothendieck-Teichm\”uller groups \textbf{GT} and \textbf{GRT} which act faithfully and transitively on the space of braidors.

We conclude by providing surprising computational evidence that there is a bijection between the space of braidors and associators and that the annular versions of the Grothendieck-Teichm\”uller groups are in fact isomorphic to the usual versions potentially providing a new and in some ways simpler description of these important groups, although these computations rely on the unproven result to be meaningful.

A copy of the thesis can be found here: ens_thesis