Junior Analysis and Probability Seminar

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

Andrew Rout

Sotiris Kotitsas

Ryan Acosta Babb

Term 3 (April – June 2023)

Talks will take place on Thursdays 1pm-2pm, with rooms TBC.

Date	Speaker	Affiliation	Title
June 8 (W7) B3.02	Peter Hearnshaw	UCL	Diagonal Behaviour of the Density Matrix for Coulombic Wavefunctions
June 22 (W9) B3.02	Larry Read	Imperial	Weighted CLR type bounds for a magnetic Schrödinger operator in two dimensions

Term 2 (Jan – Mar 2023)

Date	Speaker	Affiliation	Title
Jan 19 (W2)	Iain Souttar	Heriot–Watt	Uniform in time approximations: Averaging
Jan 26 (W3) ONLINE	Eduardo Tablate Vila	ICMAT	Schur multipliers in Schatten von Neumann classes and noncommutative Fourier multipliers
Feb 9 (W5) B3.01	Will O'Regan	Warwick	A selected survey of projection theorems
Feb 16 (W6)	Guopeng Li	Edinburgh	Deep-water and shallow-water limits of the intermediate long wave equation: from deterministic and statistical viewpoints
Feb 23 (W7)	Dimitri Bytchenkoff	Vienna	Frames and kernel theorems for co-orbit spaces
Mar 2 (W8)	Martin Peev	Imperial	Localising Fermionic (S)PDEs
Mar 9 (W9)	Liam Hughes	Cambridge	Embeddability of Liouville Quantum Gravity metrics

Term 1 (Oct – Dec 2022)

Date	Speaker	Affiliation	Title
Nov 10 (W6)	Simon Gabriel	Warwick	On the Poisson–Dirichlet diffusion and Trotter–Kurtz approximations
Nov 17 (W7)	Jakub Takáč	Warwick	Norms in finite dimensions and rectifiability in metric spaces
Nov 24 (W8)	Giacomo Del Nin	Warwick	Isoperimetric shapes in Penrose tilings
Dec 1 (W9)	Milos Tasic	Warwick	Rigorous derivation and the propagation of chaos

Abstracts

Larry Read: Weighted CLR type bounds for a magnetic Schrödinger operator in two dimensions

In dimensions three and above, the CLR inequality provides an upper-bound on the number of negative eigenvalues of a Schrödinger operator, in terms of an integral of the potential. In dimensions one and two there is an absence of such bounds since even weakly attractive potentials generate at least one negative eigenvalue. In this talk I will consider the case of the two dimensional Schrödinger operator with an Aharonov-Bohm magnetic field. We will see that the addition of this magnetic field provides a significant amount of repulsion, which removes the problem with weak potentials, allowing us to derive a weighted form of the CLR bound.

Peter Hearnshaw: Diagonal Behaviour of the Density Matrix for Coulombic Wavefunctions

We consider the quantum system of N electrons in a fixed field of nuclei under a Coulombic interaction. A key role in approximation schemes for the energy eigenstates is played by the one-particle reduced density matrix. In this talk I will present new analytic results concerning the differentiability properties of this density matrix for eigenstates. It is known that the density matrix is real analytic away from the nuclei and the diagonal. We study the non-smoothness at the diagonal using derivative bounds and find that up to five bounded derivatives can be taken.

Liam Hughes: Embeddability of Liouville Quantum Gravity metrics

Introduced by Polyakov in the 1980s, Liouville quantum gravity (LQG) is in some sense the canonical model of a random fractal Riemannian surface. A host of discrete random objects in two dimensions are known to converge in the scaling limit to LQG, and many more such scaling limits are conjectured – in particular, certain discrete conformal embeddings of random planar maps converge to canonical (up to conformal reparametrization) embeddings of LQG surfaces into two-dimensional Euclidean space. Though one might expect these continuum embeddings to retain some vestige of conformality, in the special case of sqrt(8/3)-LQG (corresponding to uniform random planar maps) Troscheit showed that no embedding into Rⁿ can be quasisymmetric. I will explain how to generalize this result, and potentially discuss other questions of metric embeddability for LQG.

Martin Peev: Localising Fermionic (S)PDEs

When solving singular (bounded) operator-valued SPDEs describing the stochastic quantisation of Fermions, unbounded operators naturally appear. However, these cannot be directly equipped with a topology compatible with a fixed-point argument. I will discuss a novel construction, based on ideas from non-commutative geometry, allowing one to find non-commutative “points” in an extension of the algebra of operators. Using these, one can then construct local solutions analogous to pathwise solutions in stochastic PDEs.
This talk will be based on joint work with Ajay Chandra and Martin Hairer.

Dimitri Bytchenkoff: Frames and kernel theorems for co-orbit spaces

The seminar will begin with a reminder of the notion of the frame – a generalisation of the notion of the orthonormal base – for Hilbert spaces and of those of its properties that give it certain advantages over orthonormal bases. The notion of the frame will then be generalised for Banach spaces and it will be explained how a whole scale of Banach spaces, known as co-orbit spaces, can be generated from a single frame for a Hilbert space. Finally kernel theorems – establishing the possibility of expressing operators as integrals – for linear bounded operators on co-orbit spaces will be reported.

Guopeng Li: Deep-water and shallow-water limits of the intermediate long wave equation: from deterministic and statistical viewpoints

In this talk, we will discuss the convergence problem for the intermediate long wave equation (ILW) from deterministic and statistical viewpoints. ILW models the internal wave propagation of the interface in two-layer fluid of finite depth, providing a natural connection between the Korteweg-de Vries equation (KdV) in the shallow-water limit and the Benjamin-Ono equation (BO) in the deep-water limit.

In the first part of this talk, we discuss the convergence problem for ILW in the low regularity setting from a deterministic viewpoint. In particular, by establishing a uniform (in depth) a priori bound, we show that a solution to ILW converges to that to KdV (and to BO) in the shallow-water limit (and in the deep-water limit, respectively).

In the second part of this talk, we discuss an analogous convergence result from a statistical viewpoint. More precisely, we study convergence of invariant Gibbs dynamics for ILW in the shallow-water and deep-water limits. After a brief review on the construction of the Gibbs measure for ILW, we show that the Gibbs measures for ILW converge in total variation to that for BO in the deep-water limit, while in the shallow-water limit, we can only show weak convergence of corresponding Gibbs measures for ILW to that for KdV. In terms of dynamics, we use a compactness argument to construct invariant Gibbs dynamics for ILW (without uniqueness) and show that they converge to invariant Gibbs dynamics for KdV and BO in the shallow-water and deep-water limits, respectively. The second part of the talk is based on joint work with Tadahiro Oh (The University of Edinburgh) and Guangqu Zheng (University of Liverpool).

Will O'Regan: A selected survey of projection theorems

In 1954 Marstrand proved that for a subset of Euclidean space its Hausdorff dimension was preserved under projection to ‘almost every’ line. This was the first time the interplay between the geometric and dimensional properties of general fractals were explored, and 20 years before the word ‘fractal’ was coined. However, besides a new proof by Kaufman in 1968, and a generalisation by Mattila in 1975, the paper received very little attention until about 30 years ago. It now has a vast number citations, variants, generalisations, and specialisations. In this talk I will survey some projection theorems, new and old, along with some of the techniques used. The talk will be accessible to all those with some background in analysis.

Eudardo Tablate Vila: Schur multipliers in Schatten von Neumann classes and
noncommutative Fourier multipliers

We will introduce Schur multipliers and present some new results related to the boundedness of Schur multipliers in the Schatten classes. We also establish some relationships of them with the theory of Fourier multipliers in noncommutative groups and with the theory of operator valued functions. This is a joint work with José Conde Alonso, Adrián González-Pérez and Javier Parcet.

Iain Souttar: Uniform in time approximations: Averaging

Systems often exhibit homogeneity of time scales, but this provides a complicated platform for any analysis to take place. A popular way of reducing the dynamics to be more amenable in the setting of SDEs (as well as ODEs and PDEs) is through averaging, and the talk will begin with an overview of this. Historically, averaging is known to provide a good approximation to the fully coupled system over finite time horizons. I will consider a coupled system of SDEs where the 'fast' component is ergodic, and discuss our result which gives sufficient conditions such that the approximation obtained through the method of averaging provides a good approximation of this system across all times. That is, the distance between the averaged system and the fully coupled system is bounded uniformly in time. I will finally, time allowing, discuss the scenario in which the fast dynamics is non-ergodic (in the sense that it has multiple invariant measures).

Milos Tasic: Rigorous derivation and propagation of chaos

Deriving continuum models from atomistic descriptions is a long lasting and fundamental problem in mathematical physics. We will discuss what we mean by derivation of a continuum model and give a few examples. Finally, we will discuss a result by Matthies and Theil which gives a long time derivation for the gainless Boltzmann equation and explain their approach via marked trees. For simplicity we shall restrict ourselves to 3 dimensions and impose spatial homogeneity.

Giacomo Del Nin: Isoperimetric shapes in Penrose tilings

Given a tiling of the plane, for any fixed integer N we consider the configuration of N tiles whose union has minimal perimeter. When N goes to infinity, is the best configuration (once suitably translated and rescaled down) approaching some specific shape? In this talk I will explain how to answer this question for a large class of quasi-periodic tilings, such as the Penrose tiling. Before that, I will describe the connection with the closely related problem of interacting particle systems, that will help us obtain the solution. Based on a joint work with Mircea Petrache (Pontificia Universidad Católica de Chile).

Jakub Takáč: Norms in finite dimensions and rectifiability in metric spaces

How does fitting a linearly squished convex set into another convex set (in $Rn$ ) relate toproperties of n-rectifiable sets in (completely general) metric spaces? We call a subset of a metric space $X$ n-rectifiable if it can be covered, up to a set of $HnX$ -null measure, by a countable number of Lipschitz images of pieces of $Rn$ . We will first discuss a recent result of Bate on characterising n-rectifiable subsets of metric spaces, and then we describe the connection of these to finite-dimensional norms (due to Kirchheim). In the bulk of the talk, we shall study certain geometric (“squishing”) properties of unit balls of these norms which can be transferred into general n-rectifiable metric spaces, strengthening thereupon some of the results of Bate. Particular attention will be paid to the case of 2-dimensional norms in which case the arising problems are a bit easier to understand.

Simon Gabriel: On the Poisson–Dirichlet diffusion and Trotter–Kurtz approximations

The aim of the talk is to shed light on the Poisson–Dirichlet diffusion, which is a stochastic process arising in population genetics. After introducing briefly the connection between stochastic processes and generators of semigroups, I will discuss how approximating the process is helpful in understanding the underlying dynamics. We use stochastic particle systems and the Trotter–Kurtz theorem to prove convergence to the aforementioned model.
The talk is based on joint work with Paul Chleboun and Stefan Grosskinsky, but does not assume knowledge on stochastic processes.