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 RoutLink opens in a new window  Sotiris KotitsasLink opens in a new window  Ryan Acosta BabbLink opens in a new window 
Term 3 (April – June 2023)
Talks will take place on Thursdays 1pm2pm, 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  Deepwater and shallowwater limits of the intermediate long wave equation: from deterministic and statistical viewpoints 
Feb 23 (W7)  Dimitri Bytchenkoff  Vienna  Frames and kernel theorems for coorbit 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 upperbound 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 AharonovBohm 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 oneparticle 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 nonsmoothness 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 twodimensional 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 R^{n} 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) operatorvalued SPDEs describing the stochastic quantisation of Fermions, unbounded operators naturally appear. However, these cannot be directly equipped with a topology compatible with a fixedpoint argument. I will discuss a novel construction, based on ideas from noncommutative geometry, allowing one to find noncommutative “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 coorbit 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 coorbit 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 coorbit spaces will be reported.
Guopeng Li: Deepwater and shallowwater 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 twolayer fluid of finite depth, providing a natural connection between the Kortewegde Vries equation (KdV) in the shallowwater limit and the BenjaminOno equation (BO) in the deepwater 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 shallowwater limit (and in the deepwater 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 shallowwater and deepwater 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 deepwater limit, while in the shallowwater 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 shallowwater and deepwater 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 GonzálezPérez 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 nonergodic (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 quasiperiodic 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 ) relate toproperties of nrectifiable sets in (completely general) metric spaces? We call a subset of a metric space nrectifiable if it can be covered, up to a set of null measure, by a countable number of Lipschitz images of pieces of . We will first discuss a recent result of Bate on characterising nrectifiable subsets of metric spaces, and then we describe the connection of these to finitedimensional 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 nrectifiable metric spaces, strengthening thereupon some of the results of Bate. Particular attention will be paid to the case of 2dimensional 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.