Graduate Blog

Wednesday, June 28, 2023
10:00 a.m. (sharp)

PhD Defense – Raghavendran

Zoom Web Conference

PhD Candidate: Surya Raghavendran
Co-Supervisors: Kevin Costello/Marco Gualtieri
Thesis title:  Twisted eleven-dimensional supergravity and exceptional
simple infinite dimensional super-Lie algebras

****

We study a class of formal moduli problems associated to eleven-manifolds with a rank 6 transversely holomorphic foliation and a transverse Calabi-Yau structure. On R×C5, the (−1)-shifted tangent complex of this formal moduli problem is L∞-equivalent to a Lie-2 extension of an infinite dimensional exceptional simple super-Lie algebra called E(5|10). In the first part of the thesis, we equip this formal moduli problem with the structure of a perturbative classical field theory in the Batalin- Vilkovisky formalism. Conjecturally, this theory describes the minimal twist of eleven-dimensional supergravity. We present strands of evidence for this conjecture by computing dimensional reductions and comparing with expected descriptions of twists of supergravity in lower dimensions, and by identifying the residual symmetries of the putative twist of eleven-dimensional supergravity within the symmetries of our theory.

In the second half of the thesis, we construct particular backgrounds for our theory which we conjecture are twisted avatars of the AdS4×S7 and AdS7×S4 backgrounds of eleven-dimensional supergravity. To justify this conjecture, we study spaces of supergravity states on these backgrounds. We find that their characters match with prior expressions enumerating multi-gravitons in AdS4×S7 and AdS7×S4 respectively, and also admit specializations recovering generating functions of representation theoretic and enumerative significance, such as the MacMahon function. We study a decomposition of the state spaces we construct that exhibits them as direct sums of modules for two other exceptional linearly compact super-Lie algebras, E(1|6) and E(3|6) respectively. We conclude with some speculations about how our results can be used as input for holographic techniques.

****

The draft of the thesis can be found here: main

Thursday, May 18, 2023 at 10:00 a.m. (sharp)

Zoom Web Conference

PhD Candidate: Hyungseop Kim
Supervisor: Michael Groechenig
Thesis title: Descent techniques in algebraic K-theory

****

We investigate two different approaches to describing algebraic K-theory of schemes through descent techniques, one of global nature and the other of local nature.

The first half of the thesis is devoted to the study of an adelic descent statement for algebraic K-theory of Noetherian schemes, or more generally for any localizing invariants in place of algebraic K-theory. Given a Noetherian scheme $X$ of finite Krull dimension, Beilinson’s cosemisimplicial ring $A^{\bullet}_{\mathrm{red}}(X)$ of reduced adeles on $X$ provides a resolution of the structure sheaf of $X$. We prove that for any localizing invariant $E$ of small stable $\infty$-categories, e.g., nonconnective algebraic K-theory of Bass-Thomason, there is a natural equivalence $E(X)\simeq\lim_{\Delta_{s}}E(A^{\bullet}_{\mathrm{red}}(X))$. This can be viewed as a variant of the formal glueing problem for algebraic K-theory which concerns all irreducible closed subsets at once. We prove the descent statement by first converting the question to a cubical descent statement, and then constructing exact sequences of perfect module categories over adele rings.

In the second half of the thesis, we turn our attention to the study of $p$-adic K-theory of characteristic $p$ rings. Specifically, we provide an alternative proof of Kelly-Morrow’s generalization of Geisser-Levine theorem to the Cartier smooth case. Our approach puts emphasis on utilizing motivic filtration and descent spectral sequence. Using the homological smoothness of Cartier smooth rings, we first compute their prismatic cohomology and syntomic cohomology complexes. Through motivic filtration, this computation gives a description of topological cyclic homology for Cartier smooth rings. Then, we use the pro-\’etale descent spectral sequence for topological cyclic homology and rigidity properties of the cyclotomic trace and syntomic cohomology complexes to deduce the result, computing algebraic K-theory of local Cartier smooth rings in terms of their logarithmic de Rham-Witt groups. We also collect some direct consequences of our arguments to prismatic cohomology complexes of Cartier smooth rings and their $p$-torsion free liftings to mixed characteristic.

****

The draft of the thesis can be found here: main

Thursday, June 15, 2023 at 10:00 a.m. (sharp)

Zoom Web Conference/BA6183

PhD Candidate: Christopher Kennedy
Co-Supervisors: Catherine Sulem/Robert Haslhofer
Thesis title: Two Problems in Non-Linear Evolution Equations

PhD Defense – Kennedy

****

The theory of coupling between internal and surface waves for stratified fluid domains is a rich source of dispersive and non-linear model equations with broad applications to ocean engineering. We study the two-dimensional water wave problem consisting of two fluid domains, the lower of which is infinitely deep, separated by a sharp interface, which is due in practice to a temperature or salinity gradient, and analyze the coupling effect of free internal and surface waves. Starting from the incompressible, irrotational Euler equations of motion for a two-layered fluid consisting of two different densities, we use its Hamiltonian formulation and the corresponding canonical variables to derive a coupled system for the evolution of two waves, where the small amplitude, internal long wave is modelled by a Benjamin-Ono equation. The surface elevation, on the other hand, has a shorter wavelength and is modelled by a modulated monochromatic wave whose envelope satisfies a time-dependent, linear Schr\”odinger equation. The coefficients of the coupled system are evaluated in terms of the physical parameters. Our results extend previous work on the coupled Korteweg-de-Vries and modulational regime for coupling between internal and surface waves in shallow water by Craig, Guyenne and Sulem (\cite{CGS10}, \cite{CGS12}) to the case of deep water.

Part 2: Bochner formulas are often the starting point for the analysis of Riemannian manifolds with bounded Ricci curvature. We generalize the classical Bochner formula for the heat flow on evolving manifolds $(M,g_{t})_{t \in [0,T]}$ to an infinite-dimensional Bochner formula for martingales on parabolic path space $P\mathcal{M}$ of space-time $\mathcal{M} = M \times [0,T]$. Our new Bochner formula and the inequalities that follow from it are strong enough to characterize solutions of the Ricci flow. Specifically, we obtain characterizations of the Ricci flow in terms of Bochner inequalities on parabolic path space. We also obtain gradient and Hessian estimates for martingales on parabolic path space, as well as condensed proofs of the prior characterizations of the Ricci flow from Haslhofer-Naber \cite{HN18a}. Our results are parabolic counterparts of the recent results in the elliptic setting from \cite{HN18b}.

***

The draft of the thesis can be found here: Thesis___CPAK

Monday, May 15, 2023 at 10:00 a.m. (sharp)

Zoom Web Conference

PhD Candidate: Petr Kosenko
Supervisor: Giulio Tiozzo
Thesis title: Harmonic measures for random walks on cocompact Fuchsian groups

****

The singularity conjecture states that for an admissible finite-range random walk on a cofinite Fuchsian group $\Gamma$, the hitting measure is singular with respect to the Lebesgue measure on $S^1$. It has been known to hold for finite-range random walks on cofinite non-cocompact Fuchsian groups; however, none of the existing methods immediately generalize to the cocompact case. In this thesis we present a method which allows us to prove the singularity conjecture for a wide variety of finite-range random walks on cocompact Fuchsian groups, using the Blach\`ere-Ha\”{i}ssinsky-Mathieu criterion. In particular, we are able to prove the conjecture for a class of random walks on cocompact Fuchsian groups generated by the side-pairing translations identifying opposite sides of a centrally symmetric polygon. This includes a family of non-nearest-neighbour random walks supported on powers of side-pairing translations. Finally, we generalize our results to a similarly defined class of random walks on Coxeter groups, showing that the hitting measures for the Coxeter groups generated by reflections with respect to the sides of centrally symmetric polygons are singular in the nearest-neighbor case.

***

The draft of the thesis can be found here: thesis

Monday, May 15, 2023 at 11:00 a.m. (sharp)

Zoom Web Conference

PhD Candidate: Marios Apetroaie
Supervisor: Stefanos Aretakis
Thesis title: Instability of gravitational and electromagnetic perturbations of extremal Reissner–Nordström spacetime

PhD Defense – Apetroaie

****

We study the linear stability problem to gravitational and electromagnetic perturbations of the \textit{extremal},  $ |\mathcal{Q}|=M, $  Reissner–Nordstr\”om spacetime, as a solution to the Einstein-Maxwell equations. Our work uses and extends the framework \cite{giorgi2019boundedness,giorgie2020boundedness} of Giorgi, and contrary to the subextremal case we prove that instability results hold for a set of gauge invariant quantities along the event horizon $ \mathcal{H}^+ $. In particular, for associated quantities shown to satisfy generalized Regge–Wheeler equations we prove decay, non-decay, and polynomial blow-up estimates asymptotically along $ \mathcal{H}^+ $, the exact behavior depending on the number of translation invariant derivatives that we take. As a consequence, we show that for generic initial data, solutions to the generalized Teukolsky system of positive and negative spin satisfy both stability and instability results. It is worth mentioning that the negative spin solutions are significantly more unstable, with the extreme curvature component $ \underline{\alpha} $ not decaying asymptotically along the event horizon $ \mathcal{H}^+, $ a result previously unknown in the literature.

***

The draft of the thesis can be found here: thesis

Monday, April 17, 2023 at 1:00 p.m. (sharp)

Zoom Web Conference

PhD Candidate: Wenkui Du
Supervisor: Robert Haslhofer
Thesis title: Singularity Analysis in Mean Curvature Flow

*****

In this thesis, we investigate the formation of singularities in mean curvature flow. Specifically, we study ancient asymptotically cylindrical flows, i.e. ancient solutions whose tangent flow at $-\infty$ is a round shrinking cylinder $\mathbb{R}^{k}\times S^{n-k}(\sqrt{2(n-k)|t|})$, where $1\leq k\leq n-1$. While in the neck case, i.e. for $k=1$, a complete classification has been obtained in several breakthroughs, a classification for the case $2\leq k\leq n-1$ until recently seemed out of reach.

To analyze ancient asymptotically cylindrical flows for $2\leq k\leq n-1$, we consider the cylindrical profile function $u$ that measures the deviation of the renormalized flow from the round cylinder. We prove that for $\tau\to -\infty$ we have the asymptotics $u(y,\omega,\tau)= (y^\top Qy -2\textrm{tr}(Q))/|\tau| + o(|\tau|^{-1})$, where $Q$ is a constant symmetric $k\times k$-matrix whose eigenvalues are quantized to be either 0 or $-\sqrt{(n-k)/8}$.

We then focus on the extremal rank cases. Under the natural noncollapsing condition, we obtain a classification of all solutions with $\textrm{rk}(Q)=0$, and establish $\textrm{SO}(n-k+1)$-symmetry and unique asymptotics in the case $\textrm{rk}(Q)=k$, also known as the $k$-oval case.

Next, we confirm a conjecture by Angenent-Daskalopoulos-Sesum about uniqueness of $\textrm{O}(k) \times \textrm{O}(n-k+1)$-symmetric ancient ovals and more generally classify all $\textrm{O}(k) \times \textrm{O}(n-k+1)$-symmetric ancient noncollapsed solutions. On the other hand, for every $2\leq k\leq n-1$ we construct a $(k-1)$-parameter family of ancient ovals that are only $\mathbb{Z}^{k}_{2}\times \mathrm{O}(n-k+1)$-symmetric, giving counterexamples to another conjecture of Daskalopoulos. We then investigate ancient ovals without any symmetry assumption. Specifically, we prove that any $2$-oval in $\mathbb{R}^4$, up to scaling and rigid motion, either is the unique $\textrm{O}(2)\times \textrm{O}(2)$-symmetric ancient oval constructed by White and Haslhofer-Hershkovits, or belongs to our new one-parameter family of $\mathbb{Z}_2^2\times \textrm{O}(2)$-symmetric ancient ovals. In particular, this seems to be the first instance of a classification result for geometric flows that are neither cohomogeneity-one nor selfsimilar.

Finally, as an application of the theory of ancient solutions, we prove that for the mean curvature flow of closed embedded hypersurfaces the intrinsic diameter stays uniformly bounded as the flow approaches the first singular time, provided all singularities are of neck or conical type. In particular, our analysis yields sharp curvature bounds improving prior results by Head and Cheeger-Haslhofer-Naber.

The draft of the thesis can be found here: wenkui_du__thesis

Monday, April 10, 2023 at 9:00 a.m. (sharp)

BA2135
Zoom Web Conference

PhD Candidate: Matthew Sourisseau
Co-Supervisors: Mary Pugh/Hau-Tieng Wu
Thesis title: Statistics of the synchrosqueezing transform

****

We investigate the synchrosqueezing transform applied to complex gaussian white noise, as well as signals consisting of a single harmonic component contaminated with complex gaussian white noise. First, this involves analyzing the reassignment rule, which is built from a quotient of improper, correlated complex gaussians. We provide a new formula for the general density of said quotient and use this to carefully clarify the decay rate of the covariance of the reassignment rule.

Next, for a fixed time $t$, we analyze the synchrosqueezing integrand $Y_{\alpha,\eta}$ at different frequencies $\eta$ and resolutions $\alpha$. A detailed asymptotic analysis of the covariance between $Y_{\alpha,\eta}$ and $Y_{\alpha,\eta’}$ is provided. By appealing to an $M$-dependent approximation argument, we obtain a central-limit theorem for the synchrosqueezing transform at time $t=0$ and fixed frequency $\xi$ and give an interpretation within the context of kernel regression.

Finally, we provide a number of open problems whose resolution may lend themselves to further results in the vein of this work.

The draft o the thesis can be found here: Sourisseau – 997536199 – PhD Thesis

Monday, November 28, 2022 at 2:00 (sharp)

65 St George St, Room 101
Zoom Web Conference

PhD Candidate: Vincent Girard
Supervisor: Fiona Murnaghan
Thesis title: Relatively Supercuspidal Representations of the Symplectic p-adic Groups

***

In this thesis, we construct in a concrete manner a family of non-supercuspidal, relatively supercuspidal
representations of symplectic p-adic groups, based on the work of Murnaghan [Mur]. We cover
the symmetric pairs (Sp(4n, F), Sp(2n, E)), (Sp(2n, E), Sp(2n, F)) and (Sp(2n, F), Sp(2k, F) x Sp(2(n-k), F))
where F is a p-adic field of odd residual characteristic, and E is a quadratic field extension of F.

We also look at the symmetric pairs (Sp(2n, F), GL(n, F)) and (Sp(2n, F), U(n, E/F, ε)) for ε an invertible
Hermitian matrix over E/F. For these additional pairs, the above construction doesn’t result
in any non-supercuspidal, relatively supercuspidal representations (despite these pairs admitting
distinguished supercuspidals).

We end with an in-depth look at the case of Sp(2, F) = SL(2, F). We show that in this low-rank
example, for all of the above pairs, all irreducible relatively supercuspidal representations of SL2(F)
are either supercuspidal or obtained from our construction. In particular, all irreducible H-relatively
supercuspidal representations of SL(2, F), for H either GL(1, F) or U(1, E/F, ε), are supercuspidal.

A copy of the thesis can be found here: Girard_Vincent_202212_PhD_thesis

Monday, August 22, 2022 at 10:00 a.m. (sharp)

PhD Candidate: Lucas Ashbury-Bridgwood
Supervisor: Balint Virag
Thesis title: Random Canonical Products and the Secular Function of the Stochastic Airy Operator

***

Secular functions of random matrices and their limits are of recent interest in random matrix theory. Such functions are entire with zeros the spectra of the corresponding operators. For example, the general beta ensembles, extending the joint eigenvalue law of classical random matrix ensembles, have a universal soft edge limit upon rescaling called the Airy beta point process. This process also arises as eigenvalues of a random operator called the stochastic Airy operator. It is proven here that secular functions of the general beta ensembles converge in distribution to that of the stochastic Airy operator. Furthermore, this convergence is realized in the context of regularized determinants of operators. This is done by proving new asymptotics of the Airy process and rigidity estimates of the general beta ensembles and establishing this convergence for more general random sequences. These results extend the currently known case for the Gaussian ensembles in Lambert and Paquette (2020). Growth asymptotics are proven for the secular function of the stochastic Airy operator, and as an application some open questions in Lambert and Paquette (2020) are answered. By applying and extending the work in Valkó and Virág (2020) in the bulk case, the secular function is proven to be a unique limiting solution of an ordinary differential equation. Additionally, new convergence laws for discrete matrix models limiting to the stochastic Airy operator are proven, including convergence of the derivatives of eigenfunctions.

***

A copy of the thesis can be found here: thesis lucas ashbury-bridgwood

Wednesday, September 7 at 10:00 a.m. (sharp)

PhD Candidate: Tristan Milne
Supervisor: Adrian Nachman
Thesis title: Optimal Transport, Congested Transport, and Wasserstein Generative
Adversarial Networks

***

Generative Adversarial Networks (GANs) are a method for producing a distribution $\mu$ that one can sample which approximates a distribution $\nu$ of real data. Wasserstein GANs with Gradient Penalty (WGAN-GP) [GAA+17] were designed to update $\mu$ by computing and then minimizing the Wasserstein 1 distance between $\mu$ and $\nu$. In the first part of this thesis we show that in fact, WGAN-GP do not compute this distance. Instead, they compute the minimum of a different optimal transport problem, the so-called congested transport [CJS08]. We then use this result to offer explanations of the observed performance of WGAN-GP. Our discovery also elucidates the role of the gradient penalty sampling strategy in WGAN-GP, and we show that by modifying this distribution one can ameliorate a transient form of mode collapse in the optimal mass flows.

The second part of this thesis presents new algorithms for generative modelling based on insights from optimal transport theory. The basic idea is to transform one distribution into another via iterated descent with an adaptive step size on learned Kantorovich potentials computed with WGAN-GP. We provide an initial convergence theory for this technique, as well as guarantees of convergence for an extension of this procedure when the target distribution is supported on a submanifold of Euclidean space with codimension at least two. As a proof of concept, we demonstrate via experiments that this provides a flexible and effective approach for several generative modelling problems, including image generation, translation, and denoising.

Further analysis of this algorithm reveals that it is connected to image restoration techniques via learned regularizers, which generalize the classical total variation denoising technique of Rudin-Osher-Fatemi (ROF) [ROF92]. We provide analogues of the results of [Mey01] on ROF to the learned regularizer setting. Leveraging this connection, we provide optimal transport versions of the iterated denoising [AXR+15] and multiscale image decompositions [TNV04] associated with ROF.

***

A copy of the thesis can be found here: thesis_July26th