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Junior Analysis Seminar 2023-2024

Welcome to the Junior Analysis and Probability Seminar, a space for PG Students and Young Researchers to present and share their research in areas related to Mathematical Analysis and Probability.

We welcome talks in
  • Harmonic Analysis
  • Ordinary, Partial and Stochastic Differential Equations
  • Probability Theory
  • Measure Theory and Geometry

If you would like to give a talk, please contact one of the Organisers:

Jakub Takáč Ryan Acosta Babb

Term 3 (April – June 2024)

Talks will take place on Mondays 3pm-4pm in B3.03 (Note the room change from last term!)

Date Speaker Affiliation Title
Apr 22 No seminar
Apr 29 Pietro Wald Warwick A Kakeya maximal function in the (first) Heisenberg group
May 6 Bank Holiday
May 13 Hrit Roy Edinburgh Bochner–Riesz means associated with rough convex domains in R2
May 20 Benedetta Bertoli Imperial Stability and phase transitions of mean field particle systems
May 27 Bank Holiday
Jun 3 Zdeněk Mihula Czech Technical University TBC
Jun 10* Andrew Rout Rennes TBC
Jun 17 Artemis Aikaterini Vogiatzi Queen Mary Singularity Models for High Codimension Mean Curvature Flow
Jun 24* TBC
* On these dates only the seminar will be in MB0.07 in the Mathematical Sciencies Building (aka "Stats")

Abstracts

Bendetta Bertoli: Stability and phase transitions of mean field particle systems

Interacting particle systems are a key tool to understand the behaviour of multi-agent dynamics in many different areas, including, for example, mathematical biology, collective dynamics, machine learning and biophysics. In this talk we will analyse the system of stochastic differential equations describing such models, and its mean field limit. We derive rigorous results about the behaviour of the stationary solutions of the corresponding Fokker-Planck PDEs, and how study these change in relation to the interaction potential. We will also look at how to extend these and other results to the case where the interacting particles lie on a sequence of random graphs. This talk is based on joint work with Ben Goddard (University of Edinburgh) and Greg Pavliotis (Imperial College London).

Artemis Aikaterini Vogiatzi: Singularity Models for High Codimension Mean Curvature Flow

Mean curvature flow is a geometric evolution equation that describes how a submanifold embedded in a higher-dimensional space changes its shape over time. We establish a codimension estimate that enables us to prove at a singular time of the flow, there exists a rescaling that converges to a smooth codimension one limiting flow in Euclidean space, regardless of the original flow's codimension. Under a cylindrical type pinching, we show that this limiting flow is weakly convex and either moves by translation or is a self-shrinker. These estimates allow us to analyse the behaviour of the flow near singularities and establish the existence of the limiting flow. Considering spaces such as the Pn, we go beyond the finite timeframe of the mean curvature flow, by proving that the rescaling converges smoothly to a totally geodesic limit in infinite time. Our approach relies on the preservation of the quadratic pinching condition along the flow and a gradient estimate that controls the mean curvature in regions of high curvature.

Hrit Roy: Bochner–Riesz means associated with rough convex domains in R2

We consider generalized Bochner–Riesz operators associated with planar convex domains. We construct domains which yield improved Lp bounds for the Bochner–Riesz over previous results of Seeger–Ziesler and Cladek. The constructions are based on the existence of large Bm sets due to Bose–Chowla, and large Λ(p) sets due to Bourgain.

Pietro Wald: A Kakeya maximal function in the (first) Heisenberg group

The Kakeya conjecture is a prominent open problem in analysis, sitting at the intersection of geometric measure theory, harmonic analysis, and PDE. In this talk, I will introduce the (first) Heisenberg group and a notion of Kakeya set in this setting. I will then discuss the corresponding Kakeya conjecture (first solved by J. Liu) and present a new solution based on a suitable adaptation of the Kakeya maximal function to the Heisenberg group. This is based on joint work with K. Fässler and A. Pinamonti.

Term 2 (January – March 2024)

Talks will take place on Mondays 3pm-4pm in D1.07 (Complexity)

Date Speaker Affiliation Title
Jan 8 (W1) No speaker No speaker No speaker
Jan 17* (W2) Simon Gabriel Münster Rooted trees and singular SPDEs
Jan 22 (W3) Ricky Hutchins Birmingham Level sets of Lipschitz quotient mappings of the plane
Jan 29 (W4) Myles Workman UCL

Embedded min-max CMC Hypersurfaces

Feb 5 (W5) Harry Giles Warwick

Self-repelling Brownian polymer in the critical dimension

Feb 12 (W6) Andreas Koller Warwick Scaling limits of discrete gradient models
Feb 19 (W7) Muhammed Ali Mehmood Imperial

Duality solutions for the hard congestion model

Feb 26 (W8) Georgios Athanasopoulos Warwick

The Ising model and the Kac-Ward method

Mar 4 (W9) Andreas Mountakis Warwick

Finding product sets in some classes of amenable groups

Mar 11 (W10) Arjun Sobnack Warwick

Arjun's partially different geometry talk with bountiful cheerful facts about harmonic functions

*Note the change of date. The talk will take place in D1.07 on Wednesday 2-3pm

Abstracts

Arjun Sobnack: Arjun's partially different geometry talk with bountiful cheerful facts about harmonic functions

Partial differential equations which arise in geometry often exhibit surprisingly nice behaviour, which is best understood from the geometric view-point. This talk will attempt to informally highlight one particular instance of this phenomenon—the Bernstein Theorem in the theory of minimal surfaces. The talk will start by recalling some familiar properties of harmonic functions.

Andreas Mountakis: Finding product sets in some classes of amenable groups

In 2022, using methods from ergodic theory, Kra, Moreira, Richter, and Robertson resolved a longstanding conjecture of Erdős about sumsets in large subsets of the natural numbers. In this talk, we will discuss an extension of the result of the previous authors to several important classes of amenable groups, including finitely generated virtually nilpotent groups, and all abelian groups (G,+) with the property that the subgroup 2G has finite index. This is joint work with Dimitris Charamaras.

Georgios Athanasopoulos: The Ising model and the Kac-Ward method

Onsager proposed a closed-form expression for the free energy of the Ising model in 1944. In 1952, Kac and Ward introduced an alternative elegant method of combinatorial nature which became rigorous by Kager, Lis and Meester in 2013 followed by Aizenman and Warzel in 2018. We extend their result to the triangular lattice, with coupling constants of arbitrary sign.

This is joint work with Daniel Ueltschi.

Muhammed Ali Mehmood: Duality solutions for the hard congestion model

The hard-congestion model is an example of a two-phase free-congested system that finds many applications in the study of congestion phenomena, such as traffic flow, crowd dynamics and granular flows. Very little is known about the existence/uniqueness of weak or strong solutions, even in one dimension. In this talk I will discuss two recent works which prove the existence of weak and measure-valued solutions to the hard-congestion model. This is done by studying the 'hard-congestion limit' of the dissipative Aw-Rascle system, which is a traffic model that in fact resembles the compressible pressureless Navier-Stokes model. In the first half of the talk I will discuss the well-posedness of the system on the 1D torus and additionally the convergence towards a weak solution of the hard-congestion model. In the latter half of the talk, I will introduce the theory of the so-called 'duality solutions', which is a class of measure-valued solutions for conservation laws. In particular, I will provide a definition of duality solutions for our hard-congestion model and state an existence result for both weak and duality solutions in the case of the real line.

Andreas Koller: Scaling limits of discrete gradient models

Random fields of gradients are a class of model systems arising in the study of random interfaces, random geometry, field theory and elasticity theory. Scaling limits are an approach to understanding the large-scale behaviour of such models. Where the Hamiltonian (energy functional) of the system is determined by a strictly convex potential, good progress has been made on studying these models over the last two decades. For systems with non-convex energy fewer results are known. I will introduce the main objects of interest and give a (very much inexhaustive) survey of what is known about them. Time permitting, I will also discuss some recent advances using renormalisation group arguments and describe our result confirming the conjectured behaviour of the scaling limit for a class of non-convex potentials in the regime of low temperatures and small tilt. This is based on joint work with Stefan Adams.

Harry Giles: Self-repelling Brownian polymer in the critical dimension

We study a type of self-avoiding random motion in Rd. Its properties are dimension dependent. In dimension d = 3 and higher, scale the process diffusively, and in the limit you will see a Brownian motion. In d = 2, this is no longer the case; in fact, the process scales "super-diffusively", which means it is more difficult to study. Nonetheless, we show that you will still see a Brownian motion at large scales, provided that as we zoom out, we also tune down the strength of self-interaction (so called "weak-coupling"). I'll give an idea as to how we obtain this result. Strangely enough, while this is fundamentally a non-Markovian problem, our techniques are Markovian in nature, and are related to the theory of random walks in random environments. This is joint work with Giuseppe Cannizzaro.

Myles Workman: Embedded min-max CMC hypersurfaces

Given a compact manifold of dimension n + 1 (n greater than or equal to 2), and a positive constant c, C. Bellettini and N. Wickramasekera employed an Allen–Cahn min-max scheme to prove the existence of a two-sided immersion of constant mean curvature (given by the constant c). The immersion is quasi-embedded, in that any self-intersections are tangential, and the local picture is two smooth disks lying on opposite sides of each other. Moreover, the closure of the immersion is smooth away from a set of dimension at most n – 7. Now taking the ambient manifold to have positive Ricci curvature, in recent work with C. Bellettini, we rule out the presence of these non-embedded points for the immersion produced by C. Bellettini and N. Wickramasekera. The immediate geometric corollary is the existence of embedded constant mean curvature hypersurfaces, for any positive constant, in ambient manifolds as above.

Ricky Hutchins: Level sets of Lipschitz quotient mappings of the plane

Lipschitz quotient mappings, introduced at the close of the 20th century as non-linear analogues of linear quotient mappings between Banach spaces, have spawned a rich theory aimed at elucidating the extent to which these mappings emulate the characteristics of their linear counterparts. In this talk I will mainly discuss Lipschitz quotient mappings from the plane; in particular to Lipschitz quotient mappings f : ℝ2 → ℝ.

In doing so, a connection is established between the study of inscribed equilateral polygons and the structural nuances of the level sets associated with such Lipschitz quotient mappings. I shall discuss the relationship between the ratio of co-Lipschitz to Lipschitz constants for such mappings, as it relates to the presence and characteristics of inscribed equilateral polygons.

Simon Gabriel: Rooted trees and singular SPDEs

In this talk, we give an introduction on how rooted trees are used to encode iterated stochastic integrals, appearing for example in the study of singular stochastic PDEs. To this end, we first discuss what makes an SPDE singular and a possible approach on how to “solve” them in a certain (subcritical) regime. Here a notation using rooted trees comes in handy. Time permitting, we briefly discuss the (more ore less) open problem of treating so called critical singular SPDEs.

The talk is aimed at a non-specialist audience.


Term 1 (October – December 2023)

Date Speaker Affiliation Title
Oct 2 (W1) William O'Regan Warwick
On the discretised ring theorem
Oct 9 (W2) Lucas Lavoyer Warwick Ricci flow from spaces with edge type conical singularities
Oct 16 (W3) Julian Weigt Warwick Endpoint regularity bounds for maximal operators in higher dimensions
Oct 23 (W4) Federico Bertacco Imperial

Scaling limits of planar maps under the Smith embedding

Oct 30 (W5) Phoebe Valentine Warwick

Characterising 1-rectifiability via connected tangents

Nov 6 (W6) Arnaud Dumont Birmingham

Solvability of Schrödinger-type elliptic boundary value problems and operator-adapted Hardy spaces

Nov 13 (W7) Zofia Grochulska Warsaw

Homeomorphisms with (approximate) derivative

Nov 20 (W8) Mauricio Che-Moguel Durham

Isometric Rigidity and Flexibility of Wasserstein Spaces

Nov 27 (W9) Sotirios Kotitsas Warwick

The KPZ equation in dimensions d ≥ 2: a survey and recent results

Dec 4 (W10) Tom Sales Warwick

The Cahn–Hilliard equation on an evolving surface

Abstracts

Sotirios Kotitsas: The KPZ equation in dimensions d ≥ 2: a survey and recent results

The KPZ equation:

th(t,x) = ½∆h(t,x) + β|∇h(t,x)|2 + ξ(t,x)

where ξ is a random forcing term is one of the most important stochastic PDEs in mathematical physics. It is conjectured to encode the fluctuations of many natural models of randomly growing interfaces and it has been the study of intense research in the past decade. Due to its nonlinear nature it is hard to make sense of the equation directly and this has only been achieved in dimension d = 1. In this talk we will explain why the KPZ equation in d ≥ 2 is fundamentally different and we will survey some known results regarding its fluctuations and its connections to the theory of random polymers. Time permitting we will talk about some new work in progress in d = 2.

Tom Sales: The Cahn–Hilliard equation on an evolving surface

In recent years there has been interest on partial differential equations (PDEs) posed on domains which evolve in time, and in particular evolving surfaces. Applications for these systems can be found, for example, in the study of lipid biomembranes. In this talk we consider the Cahn–Hilliard equation on an evolving surface and discuss the corresponding analysis and numerical analysis. This includes a framework for PDEs on evolving domains, and techniques for the discretisation of PDEs on evolving surfaces via the evolving surface finite element method (ESFEM). Assuming a smooth potential function, we outline the main proofs for the well-posedness of the Cahn–Hilliard equation, as well as optimal order error bounds for a numerical scheme using backward-Euler time discretisation and isoparametric ESFEM.

Mauricio Che-Moguel: Isometric Rigidity and Flexibility of Wasserstein Spaces

The optimal transport theory has been used to model several phenomena both within mathematics and in other fields and is currently a very active area of research. In this talk, we will delve into a geometric aspect of this theory, specifically concerning the symmetries of Wasserstein spaces.

More precisely, given a real number $p\in [1,\infty)$ and a metric space $(X,d)$, the $p$-Wasserstein space over $X$ is the space $\mathbb{P}_p(X)$ consisting of Borel probability measures on $X$ with finite $p$-moment, endowed with the distance function induced by solving the optimal transport problem with the cost function $c(x,y)=d(x,y)^p$. We say that $X$ is isometrically rigid with respect to the $p$-Wasserstein distance if the group of isometries of $\mathbb{P}_p(X)$ is isomorphic to the group of isometries of $X$; otherwise, we say it is isometrically flexible.

In general, determining if $X$ is isometrically rigid with respect to the $p$-Wasserstein distance heavily relies on the geometry of $X$ and the value of $p$. In this talk, I will give an overview of this topic and present some recent results about the isometric rigidity and flexibility for some families of spaces, based on a collaboration with Fernando Galaz-García, Martin Kerin, and Jaime Santos-Rodríguez.

Zofia Grochulska: Homeomorphisms with (approximate) derivative

Homeomorphisms of subsets of R^n can be used to model deformations such as stretching a rubber band or bending a metal plate. This is one of the reasons why there has been an interest in studying homeomorphisms equipped with some notion of differentiability. I will talk mainly about almost everywhere (a.e.) approximately differentiable homeomorphisms and the interplay between their analytical and topological properties and discuss if these properties might be useful for potential applications. In particular, we will see that under mild assumptions on T, a measurable map defined on a unit cube in R^n with values in the space of invertible n x n matrices, there is an a.e. approximately differentiable homeomorphism of the unit cube whose approximate derivative equals T a.e. I will also show how closely this theorem is connected with the Homeomorphic measures theorem of Oxtoby and Ulam. This is joint work with Paweł Goldstein (University of Warsaw) and Piotr Hajłasz (University of Pittsburgh).

Arnaud Dumont: Solvability of Schrödinger-type elliptic boundary value problems and operator-adapted Hardy spaces

We will be considering Lp Dirichlet and Regularity boundary value problems for a class of non-smooth Schrödinger-type elliptic equations on the upper half-space.

Making the assumption of block coefficients, as well as independence with respect to the transversal direction, the equation can be written as a simple second-order evolution equation involving a Schrödinger-type operator on the boundary.

We will see how the spectral properties of this operator imply the existence of a functional calculus on L2. It turns out that this functional calculus is particularly well-behaved on L2, and allows for the solvability of the boundary value problems when the boundary data is in this space.

We will explain how to build on the well-behaved L2 theory for this operator to obtain existence results for Lp boundary data.

The central tool in this extrapolation from the L2 theory is a general construction of Hardy spaces that are adapted to a given operator with spectrum in a sector of the complex plane, and good functional calculus. The main reason for considering these spaces is that the functional calculus extends to them in the best possible way.

Finally, we will see how such abstract Hardy spaces can be identified with concrete Lp (or Sobolev) spaces in some range of p to construct solutions to the boundary value problems.

Phoebe Valentine: Characterising 1-rectifiability via connected tangents

A central concept in geometric measure theory is that of rectifiability. A set is called n-rectifiable if it can be covered almost everywhere by images of Lipschitz maps and is purely n-unrectifiable if its intersection with any rectifiable set has 0 measure. In this talk, we will start by motivating why tangents are a natural lens through which to view rectifiability. Indeed, the theory of Euclidean tangents has been well developed for some time, and in the case of 1-rectifiability we will discuss a geometric proof of a well known Euclidean linear approximability result. We will depend heavily on the inherent "gappiness" of purely 1-unrectifiable sets, as quantified by Besicovitch in 1938. We will then consider the problems in generalising this argument to hold in arbitrary metric spaces and have a gentle introduction to the theory of metric tangents. Finally, we will see how the construction of Besicovitch may be strengthened to show that the existence of connected metric tangents implies 1-rectifiability.

Federico Bertacco: Scaling limits of planar maps under the Smith embedding

Over the past few decades, there has been significant progress in the study of scaling limits of random planar maps. In this talk, I will provide motivation for this problem and then focus on the scaling limits of (random) planar maps under the Smith embedding. This embedding is described by a tiling of a finite cylinder by rectangles, where each edge of the map corresponds to a rectangle, and each vertex corresponds to a horizontal segment. I will argue that when considering a sequence of finite planar maps embedded in an infinite cylinder and satisfying a suitable invariance principle assumption, the a priori embedding is close to an affine transformation of the Smith embedding at larger scales. By applying this result, I will prove that the Smith embeddings of mated-CRT maps with the sphere topology converge to LQG. This is based on joint work with Ewain Gwynne and Scott Sheffield.

Julian Weigt: Endpoint regularity bounds for maximal operators in higher dimensions

The classical Hardy-Littlewood maximal function theorem states that maximal operators are bounded on L^p(ℝ^n) if and only if p>1. In 1997 Juha Kinnunen proved that for p>1 also the gradient of a maximal function is bounded on L^p(ℝ^n). It is an open question in the endpoint p=1. In one dimension this endpoint gradient bound is known to hold for most maximal operators due to Tanaka, Kurka and many others.We prove the endpoint gradient bound in all dimensions for the maximal operator that averages over uncentered cubes with any orientation. For the uncentered Hardy-Littlewood maximal operator we can prove the endpoint Sobolev bound only in the case of characteristic functions since some of our arguments only work for cubes and not for balls. Moreover, we prove the corresponding endpoint Sobolev bound for the fractional centered and uncentered Hardy-Littlewood maximal functions.The key arguments are of geometric nature and rely on the coarea formula, the relative isoperimetric inequality and covering lemmas.

Lucas Lavoyer: Ricci flow from spaces with edge type conical singularities

In this talk, we will construct a solution to Ricci flow coming out of spaces with edge type conical singularities along a closed, embedded curve, under the additional assumption that for each point of the curve, our space is locally modelled on the product of a fixed positively curved cone and a line. We also prove curvature estimates for the solution and, for edge points, we show that the tangent flow at these points is a positively curved expanding gradient Ricci soliton solution crossed with a line.

William O'Regan: On the discretised ring theorem
In 1966 Erdős and Volkmann showed that for any real number s between 0 and 1 there exists a Borel subgroup of the reals (under addition) with Hausdorff dimension s. They conjectured (without a lot of evidence) that the same is not true for Borel subrings of the reals, that is, if A is a Borel subring of the reals then it has dimension 0 or is the whole real line. This was eventually solved independently by Edgar - Miller and Bourgain independently. Bourgain's proof went via solving a discretised version of the problem - a conjecture of Katz & Tao. In this talk this problem will be slowly introduced and a short new proof will be given by making use information theory. This talk is based on joint work with András Máthé