Graduate Blog

Thursday, August 19, 2021
10:00 a.m.

PhD Candidate: Jack Ding
Supervisor: Lisa Jeffrey
Thesis title:  The Atiyah-Bott Lefschetz formula applied to the based loops on
SU(2)

***

We prove two generalizations of localization formulae for finite-dimensional spaces to the infinite-dimensional based loop group $\Omega G$.

The Atiyah-Bott-Lefschetz Formula is a well-known formula for computing the equivariant index of an elliptic operator on a compact smooth manifold. We provide an analogue of this formula for the based loop group $\Omega SU(2)$ with respect to the natural $(T \times S^1)$-action. This is accomplished by computing certain equivariant multiplicities in the K-theory of affine Schubert varieties. From this result we also derive an effective formula for computing characters of certain Demazure modules.

The based loop group for a compact Lie group $G$ has been studied intensively since the work of Atiyah and Pressley and the book of Pressley and Segal. It is an infinite-dimensional symplectic manifold equipped with a Hamiltonian torus action, where the torus is the product of a circle and the maximal torus of $G$. When $G = SU(2)$,
the fixed points for this action are in bijective correspondence with the integers. Our final result is a Duistermaat-Heckman type oscillatory integral over the based loop group, expanded around the fixed points of the torus action. To accomplish this we use Frenkel’s results (1984) on pinned Wiener measure for orbital integrals on the affine Lie algebra, as well as the results of Urakawa (1975) on the heat kernel for a compact Lie group and Fegan’s inversion formula (1978) for orbital integrals.

A copy of the thesis can be found here: thesis

Monday, August 16, 2021
3:00 p.m. (sharp)

PhD Candidate: Roger Bai
Supervisor: Joel Kamnitzer
Thesis title:  Cluster Structure for Mirkovic-Vilonen Cycles and Polytopes

***

We look at the Mirkovic-Vilonen (MV) basis for semisimple Lie algebras and compare this to the associated cluster algebra to investigate the question of whether or not the cluster variables are in the MV basis.

We begin with finding analogues of the cluster structure among MV cycles and MV polytopes. In particular, we show the exchange relations correspond to an equation involving MV polytopes.

We extend a result of Baumann-Kamnitzer in relating valuations of an MV cycle and the dimension of homomorphism spaces of its associated preprojective algebra module. In doing so, we are able to give a partial result for an exchange relation involving MV cycles in low dimensions.

In joint work with Dranowski and Kamnitzer, we present a way to calculate the fusion product of MV cycles in type A through a generalization of the Mirkovic-Vybornov isomorphism.

Finally, we finish with examining the A_3 example and show directly that the cluster variables in this case are in the MV basis.

A copy of the thesis can be found here: Cluster_Structure_for_MV_Cycles_and_PolytopesFinal

Wednesday, August 18, 2021
10:00 a.m. (sharp)

PhD Candidate: Dylan Butson
Supervisor: Kevin Costello
Thesis title:  Equivariant Localization in Factorization Homology and Vertex
Algebras from Supersymmetric Gauge Theory

***

We develop a theory of equivariant factorization algebras on varieties with an action of a connected algebraic group $G$, extending the definitions of Francis-Gaitsgory [FG] and Beilinson-Drinfeld [BD1] to the equivariant setting. We define an equivariant analogue of factorization homology, valued in modules over $\textup{H}^\bullet_G(\text{pt})$, and in the case $G=(\mathbb{C}^\times)^n$ we prove an equivariant localization theorem for factorization homology, analogous to the classical localization theorem [AtB]. We establish a relationship between $\mathbb{C}^\times$ equivariant factorization algebras and filtered quantizations of their restrictions to the fixed point subvariety. These results provide a model for predictions from the physics literature about the $\Omega$-background construction introduced in [Nek1], interpreting factorization $\mathbb{E}_n$ algebras as observables in mixed holomorphic-topological quantum field theories.

We give an account of the theory of factorization spaces, categories, functors, and algebras, following the approach of [Ras1]. We apply these results to give geometric constructions of factorization $\mathbb{E}_n$ algebras describing mixed holomorphic-topological twists of supersymmetric gauge theories in low dimensions. We formulate and prove several recent predictions from the physics literature in this language:

We recall the Coulomb branch construction of [BFN1] from this perspective. We prove a conjecture from [CosG] that the Coulomb branch factorization $\mathbb{E}_1$ algebra $\mathcal{A}(G,N)$ acts on the factorization algebra of chiral differential operators $\mathcal{D}^{\text{ch}}(Y)$ on the quotient stack $Y=N/G$. We identify the latter with the semi-infinite cohomology of $\mathcal{D}^{\text{ch}}(N)$ with respect to $\hat{ \mathfrak{g}}$, following the results of [Ras3]. Both these results require the hypothesis that $Y$ admits a Tate structure, or equivalently that $\mathcal{D}^{\text{ch}}(N)$ admits an action of $\hat{\mathfrak{g}}$ at level $\kappa=-\text{Tate}$.

We construct an analogous factorization $\mathbb{E}_2$ algebra $\mathcal{F}(Y)$ describing the local observables of the mixed holomorphic-B twist of four dimensional $\mathcal{N} =2$ gauge theory. We identify $S^1$ equivariant structures on $\mathcal{F}(Y)$ with Tate structures on $Y=N/G$, and prove that the corresponding filtered quantization of $\iota^!\mathcal{F}(Y)$ is given by the two-periodic Rees algebra of chiral differential operators on $Y$. This gives a mathematical account of the results of [Beem4]. Finally, we apply the equivariant cigar reduction principle to explain the relationship between these results and our account of the results of [CosG] described above.

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

Wednesday, August 25 2021
2:00 p.m. (sharp)

PhD Candidate: Qin Deng
Supervisor: Vitali Kapovitch
Thesis title:  Hölder continuity of tangent cones and non-branching in RCD(K,N) spaces

***

This thesis is concerned with the study of the structure theory of metric measure spaces (X, d, m) satisfying the synthetic lower Ricci curvature bound condition RCD(K, N). We prove that such a space is non-branching and that tangent cones from the same sequence of rescalings are Holder continuous along the interior of every geodesic in X. More precisely, we show that the geometry of balls of small radius centred in the interior of any geodesic changes in at most a Holder continuous way along the geodesic in pointed Gromov-Hausdorff distance. This improves a result in the Ricci limit setting by Colding-Naber where the existence of at least one geodesic with such properties between any two points is shown. As in the Ricci limit case, this implies that the regular set of an RCD(K, N) space has m-a.e. constant dimension, a result recently established by Brue-Semola, and is m-a.e convex. It also implies that the top dimension regular set is weakly convex and, therefore, connected. In proving the main theorems, we develop in the RCD(K, N) setting the expected second
order interpolation formula for the distance function along the Regular Lagrangian flow of some vector field using its covariant derivative

A copy of the thesis can be found here:  Thesis Formatted

Wednesday, August 18, 2021
11:00 a.m. (sharp)

PhD Candidate: Andrew Colinet
Supervisor: Robert Jerrard
Thesis title: Geometric Behaviour of Solutions to Equations of Ginzburg-Landau
Type on Riemannian Manifolds

***

In this thesis, we demonstrate the existence of complex-valued solutions to the Ginzburg-Landau equation
\[
-\Delta{}u+\frac{1}{\varepsilon^{2}}u(|u|^{2}-1)=0\hspace{20pt}\text{on }M,
\]

for $\varepsilon\ll1$, where $M$ is a three dimensional compact manifold without boundary, that have interesting geometric properties. Specifically, we argue the existence of solutions whose vorticity concentrates about an arbitrary closed nondegenerate geodesic on $M$.
In doing this, we extend the work of \cite{JSt} and \cite{Mes} who showed that there are solutions whose energy converges, after rescaling, to the arclength of a geodesic as above.

An important ingredient in the proof is a heat flow argument, which requires detailed information about limiting behaviour of solutions of the parabolic Ginzburg-Landau equation. Providing the necessary limiting behaviour is the other contribution of this thesis. In fact, more is achieved. Provided that $N\ge3$, we give a structural description of the limiting behaviour of solutions to the parabolic Ginzburg-Landau equation on an $N$-dimensional compact manifold without boundary $(M,g)$. More specifically, we are able to show that the limit of the renormalized energy measure orthogonally decomposes into a diffuse part, absolutely continuous with respect to the volume measure on $M$ induced by $g$, and a concentrated vortex part, supported on a codimension $2$ surface contained in $M$. Moreover, the diffuse part of the limiting energy has its time evolution governed by the heat equation while the concentrated part evolves in time according to a measure theoretic version of mean curvature flow. This extends the work of \cite{BOS2} who proved this for $N$-dimensional Euclidean space provided that $N\ge2$.

A copy of the thesis can be found here: Andrew_Colinet_Thesis

Thursday, July 29, 2021
1:00 p.m. (sharp)

PhD Candidate:  Ivan Telpukhovskiy
Supervisor:   Kasra Rafi
Thesis title: On the geometry of the Thurston metric on Teichmüller spaces: geodesics
that disobey an analogue of Masur’s criterion

***

We construct a counterexample for an analogue of Masur’s criterion in the setting
of Teichmüller space with the Thurston metric. For that, we find a minimal, filling,
non-uniquely ergodic lamination λ on the seven-times punctured sphere with uniformly bounded annular projection distances. Then we show that a geodesic in the
corresponding Teichmüller space that converges to λ, stays in the thick part for the
whole time.

A copy of the thesis can be found here: ivan-telpukhovskiy-thesis-draft-1

Monday, July 12, 2021
1:00 p.m. (sharp)

PhD Candidate:  Artane Siad
Supervisor:   Arul Shankar
Thesis title: Monogenic Fields with Odd Class Number

***

We prove an upper bound on the average number of 2-torsion elements in the class group monogenised fields of any degree $n \ge 3$, and, conditional on a widely expected tail estimate, compute this average exactly. As an application, we show that there are infinitely many number fields with odd class number in any even degree and signature. This completes a line of results on class number parity going back to Gauss.

A copy of the thesis can be found here: thesis v3

Thursday, June 24, 2021
2:00 p.m. (sharp)

PhD Candidate:  Jamal Kawach
Supervisor:   Stevo Todorcevic
Thesis title: Approximate Ramsey Methods in Functional Analysis

***

We study various aspects of approximate Ramsey theory and its interactions with functional analysis. In particular, we consider approximate versions of the structural Ramsey property and the amalgamation property within the context of multi-seminormed spaces, Fréchet spaces and other related structures from functional analysis. Along the way, we develop the theory of Fraïssé limits of classes of finitedimensional Fréchet spaces, and we prove a version of the Kechris-Pestov-Todorčević correspondence relating the approximate Ramsey property to the topological dynamics of the isometry groups of certain
infinite-dimensional Fréchet spaces. Motivated by problems regarding the structural Ramsey theory of Banach spaces, we study various generalizations of the Dual Ramsey Theorem of Carlson and Simpson.
Specifically, using techniques from the theory of topological Ramsey spaces we obtain versions of the Dual Ramsey Theorem where ω is replaced by an arbitrary countable ordinal. Moving toward block Ramsey theory, we prove an infinite-dimensional version of Gowers’ approximate Ramsey theorem concerning the oscillation stability of S(c0), the unit sphere of the Banach space c0. We then show that results of this form can be parametrized by products of infinitely many perfect sets of reals, and we use this result to
obtain a parametrized version of Gowers’ c0 theorem.

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

Monday, April 12, 2021
1:00 p.m. (sharp)

PhD Candidate:  Seong Hyun Park
Supervisor:   Jérémie Lefebvre
Thesis title: Adaptive myelination and its synchronous dynamics in the Kuramoto network model with state-
dependent delays

***

White matter pathways form a complex network of myelinated axons that play a critical role in brain function by facilitating the timely transmission of neural signals. Recent evidence reveals that white matter networks are adaptive and that myelin undergoes continuous reformation through behaviour and learning during
both developmental stages and adulthood in the mammalian lifecycle. Consequently, this allows axonal conduction delays to adjust in order to regulate the timing of neuron signals propagating between different brain regions. Despite its newly founded relevance, the network distribution of conduction delays have yet
to be widely incorporated in computational models, as the delays are typically assumed to be either constant or ignored altogether. From its clear influence towards temporal dynamics, we are interested in how adaptive myelination affects oscillatory synchrony in the brain. We introduce a plasticity rule into the delays
of a weakly coupled oscillator network, whose positions relative to its natural limit cycle oscillations is described through a coupled phase model. From this, the addition of slowly adaptive network delays can potentially lead coupled oscillators to a more phase synchronous limit cycle. To see how adaptive white matter remodelling can shape synchronous dynamics, we modify the canonical Kuramoto model by enabling all connections with phase-dependent delays that change over time. We directly compare the synchronous behaviours of the Kuramoto
model equipped with static delays and adaptive delays by analyzing the synchronized equilibria and stability of the system’s phases. Our mathematical analysis of the model with dirac and exponentially distributed connection delays, supported by numerical simulations, demonstrates that larger, more widely varying distributions of delays generally impede synchronization in the Kuramoto network. Adaptive delays act as a stabilizing mechanism for the synchrony of the network by adjusting towards a more optimal distribution of delays. Adaptive delays also make global synchronization more resilient to perturbations and injury
towards network architecture. Our results provide insights about the potential significance of activity-dependent myelination. For future works, we hope that these results lay out the groundwork to computationally study the influence of adaptative myelination towards large-scale brain synchrony.

A copy of the thesis can be found here: SHP_Dissertation

Wednesday, March 24, 2021
11:00 a.m. (sharp)

PhD Candidate:  Larissa Richards
Supervisor:   Ilia Binder
Thesis title: Convergence rates of random discrete model curves approaching SLE curves in the scaling limit

***

Recently, A. Kempannien and S. Smirnov provided a framework for showing convergence of discrete
model interfaces to the corresponding SLE curves. They show that given a uniform bound on specific
crossing probabilities one can deduce that the interface has subsequential scaling limits that can be
described almost surely by Löwner evolutions. This leads to the natural question to investigate the
rate of convergence to the corresponding SLE curves. F. Johansson Viklund has developed a framework
for obtaining a power-law convergence rate to an SLE curve from a power-law convergence rate for the
driving function provided some additional geometric information along with an estimate on the growth
of the derivative of the SLE map. This framework is applied to the case of the loop-erased random
walk. In this thesis, we show that if your interface satisfies the uniform annulus condition proposed by
Kempannien and Smirnov then one can deduce the geometric information required to apply Viklund’s
framework. As an application, we apply the framework to the critical percolation interface. The first
step in this direction for critical percolation was done by I. Binder, L. Chayes and H.K. Lei where they
proved that the convergence rate of the Cardy-Smirnov observable is polynomial in the size of the lattice.
It relies on a careful analysis of the boundary behaviour of conformal maps and their discrete analytic
approximations as well as a Percolation construction of the Harris systems. Further, we exploit the
toolbox developed by D. Chelkak for discrete complex analysis on isoradial graphs to show polynomial
rate of convergence for the discrete martingale observables for harmonic explorer and the FK Ising
model to the corresponding continuum objects. Then, we apply the framework developed above to gain
a polynomial convergence rate for the corresponding curves.

A copy of the thesis can be found here:  LarissaRichardsThesis