2024-2025

Junior Analysis and Probability Seminar (2024–2025)

The Junior Analysis and Probability Seminar hosts talks from early-career researchers working broadly in analysis and probability.

Organisers: Hefin Lambley ( ), Grega Saksida ( ), Phoebe Valentine ( ), and Pietro Wald ( ).

Time: Mondays, 15:00–16:00, weeks 1–10.

Room: MB0.08, Mathematical Sciences Building, followed by discussions in Zeeman Common Room.

Autumn Term 2024–2025

• 30 September 2024: Anna Skorobogatova (SLMath). Regularity for critical points of semilinear elliptic variational problems with a topological constraint.
  I will discuss the regularity of critical points for a free boundary problem arising from the diffuse interface/Allen-Cahn approximation of the set-theoretic Plateau problem recently introduced by Maggi-Novack-Restrepo. Here, a homotopic spanning constraint, first considered by Harrison-Pugh, forces the surfaces (and also the corresponding interface for the diffuse approximation), to remain attached to the given wire frame. The presence of the spanning condition allows for minimizers of this problem to exhibit codimension 1 singularities such as triple junctions and tetrahedral singularities, in stark contrast to the work of Tonegawa-Wickramasekera which shows that any stable minimal hypersurface arising as a limit of interfaces for stable critical points of classical Allen-Cahn is smooth outside of a codimension 7 set. I will further discuss free boundary regularity for minimizers of this problem. This is a joint work with Mike Novack (Louisiana State University) and Daniel Restrepo (Johns Hopkins University).
• 7 October 2024: Ryan Acosta Babb (Warwick). The Riemann–Lebesgue Lemma: old and new.
  The Riemann–Lebesgue Lemma provides the basic information about the decay of the Fourier coefficients of an integrable function: they vanish as the frequencies tend to infinity. There is a lesser-known converse to this: for any rate of decay, one can construct an integrable function whose Fourier coefficients decay slower than the given rate. After presenting these classical results, we will examine the question for Bessel–Fourier series, for which there is a 'Riemann–Lebesgue'-type theorem. Our first result is a partial converse to this theorem, which assumes that a certain kernel is non-negative. Our second result does away with this restriction, at the cost of some regularity for the resulting function. Time permitting, we will discuss some implications of these results for the L1 convergence of 2-dimensional Bessel–Fourier series on the disc. This is joint work with James C. Robinson.
• 14 October 2024: Jakub Takáč (Warwick). Some remarks on dual Gelfand algebras and operators satisfying the Leibniz rule.
  A dual Gelfand algebra is a dual Banach space X* that is also an algebra, with weak*-continuous multiplication. One can consider operators Φ: X* → X that satisfy the Leibniz product rule: Φ(fg)=fΦ(g)+gΦ(f), where we consider X as a module over X* in the natural way. Such operators are a generalization of differential operators and can also be thought of as a generalization of distributional vector fields. Any such operator that is also compact corresponds to a metric 1-current (and this correspondence is a bijective isometry). In particular, if X* is the algebra of Lipschitz functions on the unit cube, say [0,1]ᵈ, every such operator is naturally identified with a genuine vector valued distribution on [0,1]ᵈ - the distribution must necessarily have some additional regularity, but, for example, it might not be of finite mass. Being able to identify the level of regularity of such a distribution is equivalent to solving a particular case of the so-called Flat Chain Conjecture. In more general settings, the framework of Leibniz operators can be used to study properties of general vector fields in metric spaces.
• 21 October 2024: Social event
• 28 October 2024: Luca Gennaioli (SISSA). On the Fourier transform of BV functions.
  In this talk we shall investigate the relation between the Fourier transform of BV (bounded variation) functions and their singularities. We will discuss some sort of weighted Plancherel identities for BV functions and a new characterisation of sets of finite perimeter in terms of their Fourier transform. Finally we will improve on a result of Herz concerning the set-theoretic derivative of the Fourier transform of characteristic functions of sets.
• 4 November 2024: Cancelled
• 11 November 2024: Henry Popkin (Bath). On rough Calderón solutions to the Navier–Stokes equations and applications to the singular set.
  One of the millennium problems concerns the global well-posedness of the 3D Navier–Stokes equations. It is known that, for smooth enough initial data, we have short time existence and uniqueness of solutions. Using a weak formulation, we can get global existence of a weak solution arising from L2 initial data (Leray 1934). This leads one to ask: what classes of initial data give rise to global weak solutions to the Navier–Stokes equations? I will discuss my recent paper about weak “Calderón” solutions, corresponding to very rough (Besov space) initial data, as well as some applications of the Calderón splitting method towards quantifying the structure of the putative singular set (where weak solutions may first lose their regularity).
• 18 November 2024: Toby Sheldon (Warwick). Infinite-dimensional constrained optimization with the primal-dual method.
  In machine learning, one often wants to find a function that minimises some cost functional. Further, one may want a minimiser to satisfy some (possibly infinitely many) constraints. Such an optimisation problem is usually non-convex and does not have an explicit solution. However, using a primal-dual method, one can solve the non-convex problem by solving its convex dual problem. I will give an outline of the primal-dual method for infinite dimensions, and show that it can be applied to obtain some terms in an eigenvalue problem from data about the eigenvalues.
• 25 November 2024: Oskar Olander (Gothenburg). Phase transitions in a N-partite Heisenberg model.
  This talk in statistical physics explores a quantum spin model based on a complete N-partite graph. Each node corresponds to a spin-1/2​ particle, and each edge represents an antiferromagnetic interaction. At high temperatures, the magnetisation is random, but at a certain critical temperature magnetisation occurs. We examine the optimisation problem of minimising the free energy to determine the critical temperature, which is found to be the solution of an N-degree polynomial.
• 2 December 2024: Carla Rubiliani (Tübingen). Particle Propagation Bounds for Lattice Bosons Under Long-Range Interactions.
  In this talk we explore locality properties of a Schrödinger-type PDE describing the dynamics of a many-body bosonic system on a finite volume lattice. Such locality properties translate into two types of propagation bounds:
  1. Information propagation bounds (IPB) or Lieb-Robinson bounds, which control the dynamics of local operators.
  2. Particle propagation bounds (PPB), which control the dynamics of particles through the lattice.
  Although the PDE under consideration is linear, we have to face two mathematical challenges. First, the dimension of the underlying Hilbert space is very high, and we want to derive bounds that are thermodynamically stable, i.e. are uniform in such dimension. Second, bosonic interactions are unbounded in the operator norm. Thanks to a multi-scale adaptation of the ASTLO (adiabatic space-time localisation observables) method, which allows to remove the dependence of the error term on far-away particles, we establish the first thermodynamically stable bosonic PPB for a class of long-range interactions. We are also able to control higher moments of the number operator. This opens the door to proving the first thermodynamically stable IPB for such bosonic systems. Establishing these bounds for bosonic systems is a long-standing open problem due to the lack of existing methods to control the unbounded nature of the bosonic operators when deriving IPB.

Spring Term 2024–2025

• 13 January 2025: Spyros Garouniatis (Warwick). Large systems of symmetrized Brownian motions and their connection with a probabilistic interpretation of the Bose-Einstein condensation.
  We study a model of N Brownian motions on a fixed time interval [0,β], confined to a specified region of space and subject to symmetrized initial-terminal conditions. Specifically, the terminal position of the i-th Brownian motion is matched to the initial position of the σ(i)-th Brownian motion, where σ is a uniformly distributed permutation of {1,…,N}. This model is significant in quantum physics for describing Boson systems at a positive temperature 1/β. The underlying probability distribution is referred to as the "symmetrized measure." We analyze the asymptotic behavior of certain empirical functionals under the symmetrized measure using the framework of Large Deviations Theory. Specifically, we examine the empirical path measures and the mean of occupation measures for large N. Additionally, we explore the connection between this model and an entropic optimal transport problem, known as the Schrödinger bridge problem. Our results become explicit in the special case where the initial distribution of the Brownian motions is the Lebesgue measure. In particular, we show that the rate function governing the mean of occupation measures corresponds to the well-known Donsker-Varadhan rate function, while the empirical path measures concentrate around a Nelson diffusion as N grows large.
• 20 January 2025: Alessandro Cucinotta (Oxford). A splitting theorem for manifolds with a convex boundary component.
  The celebrated Splitting Theorem by Cheeger-Gromoll states that a manifold with non-negative Ricci curvature which contains a line is isometric to a product, where one of the factors is the real line. A related result was later proved by Kasue. He showed that a manifold with non-negative Ricci curvature and two mean convex boundary components, one of which is compact, is also isometric to a product. In this talk, I will present a variant of Kasue’s result based on joint work with Andrea Mondino. We consider manifolds with non-negative Ricci curvature and disconnected mean convex boundary. We show that if one boundary component is parabolic and convex, then the manifold is a product, where one of the factors is an interval of the real line. The result is an application of recently developed tools in synthetic geometry and exploits the interplay between Ricci curvature and optimal transport.
• 27 January 2025: no seminar

• 3 February 2025: Stefano Fronzoni (Oxford). Finite Element Approximation of fractional nonlinear PDEs .
  We present a finite element method for the numerical solution of the fractional porous medium equation on a domain $\Omega \subset \mathbb{R}^d$. After introducing the fractional Laplacian operator, in its several definitions, we present a rigorous passage to the limit in the fully discrete approximation of the fractional porous medium equation. As the spatial and temporal discretization parameters tend to zero, we show convergence for a subsequence of finite element approximations to a weak solution of the initial boundary-value problem, with an argument based on compactness techniques for nonlinear partial differential equations. We then show the algorithm used for the implementation of the method and its application to other problem of interest, involving the fractional Laplacian in interaction potentials, including the fractional Keller-Segel model and blow-up of its solution

• 10 February 2025: Jethro Warnett (Oxford). The Stein-log-Sobolev inequality and the exponential rate of convergence for the continuous Stein variational gradient descent method .
  The Stein Variational Gradient Descent method is a variational inference method in statistics that has recently received a lot of attention. The method provides a deterministic approximation of the target distribution, by introducing a nonlocal interaction with a kernel. Despite the significant interest, the exponential rate of convergence for the continuous method has remained an open problem, due to the difficulty of establishing the related so-called Stein-log-Sobolev inequality. Here, we prove that the inequality is satisfied for each space dimension and every kernel whose Fourier transform has a quadratic decay at infinity and is locally bounded away from zero and infinity. Moreover, we construct weak solutions to the related PDE satisfying exponential rate of decay towards the equilibrium. The main novelty in our approach is to interpret the Stein-Fisher information, also called the squared Stein discrepancy, as a duality pairing between H−1(Rd) and H1(Rd), which allows us to employ the Fourier transform. We also provide several examples of kernels for which the Stein-log-Sobolev inequality fails, partially showing the necessity of our assumptions.
• 17 February 2025: Gabriel Flath (Oxford). A picture of particles at sublinear distance from the rightmost one in the branching Brownian motion.
  The object of study is the binary branching Brownian motion. It is well known that the rightmost particles shifted by m(t) = √2 t − 3/2 √2 log(t) converges in distribution. We will study the density of particles at sublinear distance from m(t) and presents results on the almost sure convergence and the convergence in probability of the number of particles at distance m(t) − x, for x = o(t). Moreover, the techniques developed to prove the results allow us to guess the trajectory of a particle chosen uniformly at random to the right of m(t) − x, as well as to better understand how the derivative Martingale, and its fluctuations, impacts the particles at the front up to m(t) − o(t), allowing us to draw a picture of the front of the branching Brownian motion.
• 24 February 2025: Andreas Klippel (TU Darmstadt). Strict Inequalities: Loops vs. Percolation.
  In recent years, many models in mathematical physics have been encoded into graphical models, which are more accessible through the lens of probability theory. These graphical models often exhibit a natural percolation structure. One such model is the Random Loop Model introduced by Daniel Ueltschi. Peter Mühlbacher showed that the loop threshold for the Random Loop Model with θ=1 is larger than the percolation threshold. This is due to so-called blocking events in graphs with uniformly bounded degree. The proof primarily relies on a coupling method.
  In my talk, I will introduce the model and the basic proof techniques. Furthermore, I will discuss a recent result where we generalize the method to obtain new results for general trees.
  I will explain why the tree case differs from the case of a general graph. If time permits, I will use the Galton-Watson case to illustrate how the coupling in the proof works.
  This talk is based on joint work with V. Betz, M. Kraft, B. Lees and C. Mönch.
• 3 March 2025: Sotirios Kotitsas (Pisa). Large-scale fluctuations of random walks in space-time random environments.
  In this talk, we will consider the large-scale behavior of the quenched density of random walk models in a space-time random environment. Our main example is the following diffusion:
  dXt = V(t,Xt)dt+√κdBt,
  where V is a random Gaussian field, white in time and smooth in space, and (Bt)t≥0 is a d-dimensional Brownian motion. Focusing on d = 1, we will review known results about three different scaling regimes: the diffusive regime, the moderate deviation regime, and the large deviation regime, highlighting a connection to the KPZ equation. In d ≥ 2, we will discuss the model’s relevance to the statistical theory of turbulence and explore several physics conjectures regarding its behavior. Finally, we will present ongoing work with Mario Maurelli and Dejun Luo about the fluctuations around the quenched local central limit theorem.
• 10 March 2025: Zhengang Zhong (Warwick Statistics). Analysis of Laplacian Regularization in Semi-Supervised Learning.
  We investigate a family of regression problems in a semi-supervised setting, where the goal is to assign real-valued labels to n sample points, given a small subset of N labeled points. A goal of semi-supervised learning is to take advantage of the (geometric) structure provided by the large number of unlabeled data when assigning labels. To capture this structure, we model the data set using random geometric graphs with a connection radius.
  In this talk, we consider two problem settings: (1) In the context of variational problems, the objective functionals, which model the task, reward the regularity of the labeling function. We analyze the point-wise asymptotic behavior of such objective functionals in the setting of functional sample data, when n goes to infinity in the meanwhile the connection radius goes to zero. (2) In the setting of graph-based Bayesian inference problems, we study the convergence of the graph posterior measures over functions in the discrete domain to a posterior over functions in the continuous domain that solves a Bayesian inference problem.

Summer Term 2024–2025

• 28 April 2025: Spyros Garouniatis (University of Warwick). Large Deviations for Integer Partitions: Multiplicative Models and Gibbs Ensembles.
  Seminars in MB0.08 this term.
  We investigate large deviations for models of integer partitions under multiplicative statistics and Gibbs ensembles. In the multiplicative framework, partition structures are governed by probability measures derived from a specific class of generating functions, which induce statistical independence among part sizes. Under mild conditions on the generating function, we establish a full Large Deviation Principle. Additionally, we consider a class of canonical and grand canonical Gibbs measures arising in models of aggregation and interacting particle systems, where the asymptotic behavior of internal energies plays a central role. While both canonical and grand canonical measures exhibit similar asymptotic behavior, we demonstrate that the decay of their respective rare events differs significantly.
• Tuesday 6 May 2025: Charlie Wilson (University of Exeter). Some new results on hitting points infinitely often and limsup sets.
  Seminar on Tuesday 6 May due to bank holiday. Note change of time to 14:00.
  We examine some fairly easily stated measure theory problems about covering a space with sets and seeing which points are hit infinitely often. That is if we fix the measures of the finite intersections, what can we say about the measure of the limsup. This will tie in to the Divergence Borel Cantelli Lemma and Erdos Chung inequality and show that there are limits to the use of such inequalities. We will go on to see that what we may feel is intuitively true and also what the current state of the art may indicate actually isn't- via some counterexamples that will be shown. The result, though a nice problem in isolation, does indeed have ramifications in many other areas of maths such as Probability theory, Diophantine approximation and Dynamical systems.
• 12 May 2025: Cillian Doherty (University of Cambridge). On the Sobolev removability of the graph of one-dimensional Brownian motion.
  A compact subset K of the complex plane is said to be W(1, p) removable if any continuous real-valued function f on C which is in the Sobolev space W(1, p)(C \ K) is automatically also in the Sobolev space W(1, p)(C). This property is true for all values of p for points and line segments, but false for sets with non-empty interior, and in general there is no simple condition to determine whether a given set is removable or not. In certain cases, there is a link between removability and how “rough” the set K is. In particular, if K is the graph of a function, it is known that its Hölder continuity is related to its removability. We will present new results on the removability of the graph of a one-dimensional Brownian motion on an interval and show that it is almost surely not W(1, p) removable for finite p, but is removable for p = ∞. This talk is based on joint work with Jason Miller.
• 19 May 2025: Oskar Vavtar (King's College London). Stochastic Ising model on dynamical percolation.
  Ising Glauber dynamics are a class of Markov chains originally constructed to simulate the Ising model. The model exhibits a so-called dynamical phase transition in the speed of convergence to its invariant measure as one varies the inverse temperature. Specifically, as the parameter crosses the critical point from high- to low-temperature regime, the mixing time shifts from subpolynomial to exponentially slow. We consider a version of Glauber dynamics defined on subcritical dynamical percolation and show that, for a sufficiently 'slow' random environment, such phase transition does not occur. My goal is to give some insights into proving results for Markov chains of such nature, which, due to the presence of a random dynamical environment, are non-reversible with an unknown invariant measure. Based on joint work with Alexandre Stauffer.
• 20 May 2025: William O'Regan (University of British Columbia). Current affairs in fractal geometry.
  Additional seminar this week.
  I will give an overview of some of the very much in-vogue topics in fractal geometry/ geometric measure theory. Time allowing, this may include (not in order): - An introduction to discrete models of fractals. - Bourgain's projection theorem. - Sum-product phenomena. - Kakeya sets: the short proof of the Kakeya conjecture in two dimensions, and a few words why this proof fails in higher dimensions. - Furstenberg sets. - The deeply interwoven connection between all of the above. Very limited prior knowledge will be expected or required, but knowing what a measure is could be helpful. Some of the topics of this talk could be of interest to those studying dynamics (just don't ask me why). This talk could be viewed as a pre-seminar for the talk at 4pm on Thursday.
• Tuesday 27 May 2025: Emanuele Caputo (University of Warwick). Sobolev spaces and differentiation in metric spaces: old and new..
  Seminar on Tuesday 27 May due to bank holiday. Note change of time to 14:00 and change of room to MB0.07.
  We provide an overview of the metric theory of Sobolev spaces. This is an extension of the classical Euclidean Sobolev theory that incorporates calculus in several mathematical structures at the same time: Riemannian manifolds, singular spaces, metric trees, and fractals.
  We review three equivalent approaches. The first two of them concern the behavior of functions along curves, due respectively to Shanmugalingam, and Ambrosio, Gigli and Savarè. The third one, due to Cheeger, amounts at defining a notion of Dirichlet energy, starting from Lipschitz functions.
  If time allows, we will present an alternative approach by using chains of points instead of curves and its equivalence to the previous ones (jw with N. Cavallucci (EPFL)).
• 2 June 2025: Andrew Rout (University of Rennes). The nonlinear Schrödinger equation: What's the Gibb idea.
  In this talk I will give an introduction to Gibbs measures for the nonlinear Schrödinger equation. The construction of global solutions to dispersive PDEs usually relies on the conservation of quantities like the energy and the mass. For lower regularity functions, these quantities are infinite, and so cannot be used. Instead, one introduces an invariant measure, which can act as a substitute for the conserved quantities.
  I will give the heuristic ideas for the construction of the Gibbs measure, and also sketch the details of the rigorous construction. I will also discuss how to use the Gibbs measure to construct global solutions to the nonlinear Schrödinger equation. The talk is aimed at a non-specialist audience.
• 9 June 2025: Natasha Diederen (University College London). Increase in topological complexity along the mean curvature flow.
  A hypersurface evolves by mean curvature flow if it moves with velocity equal to its mean curvature vector at each point. This process is described by a non-linear parabolic PDE, and solutions form singularities in finite time. A central question in the study of mean curvature flow is the classification of these singularities, and the behaviour of the flow as it moves through them. For surfaces in three-dimensional space, the picture is now well understood: all singularities are modelled by self-shrinkers of multiplicity one, and the genus is non-increasing with time. However, as we will demonstrate in this talk, in higher dimensions the topology can become more complicated. In particular, we present an example of a mean convex rotationally symmetric hypersurface in four-dimensional space whose first and second Betti numbers increase at the first singular time.
• 16 June 2025: Luca Seemungal (University of Leeds). The Index of Constant Mean Curvature Surfaces in Three-Manifolds.
  Constant mean curvature (CMC) surfaces are special geometric variational objects, closely related to minimal surfaces. The key properties of a CMC surface are its area, mean curvature, genus, and index. The index of a CMC surface measures its stability: the index counts how many ways one can perturb the surface to decrease the area while keeping the enclosed volume constant. In this talk we discuss relationships between these key properties. In particular we present recent joint work with Ben Sharp, where we bound the index of CMC surfaces linearly from above by genus and the correct scale-invariant quantity involving mean curvature and area.
• 23 June 2025: CANCELLED Rohan Shiatis (King's College London). The Integrable Snake Model. This talk will likely be rescheduled to next term.
  On the integer lattice ℤ², a "pure snake configuration" is a permutation on the vertices containing no two-cycles and such that every vertex is mapped to either itself, the vertex to the right, the vertex above or the vertex below. As such, a pure snake configuration admits an interpretation as a collection of snaking, non-intersecting paths on ℤ². Pure snake configurations are a generalisation of lozenge tilings, which are in natural correspondence with paths that only travel right or up. We introduce a partition function on a finite version of this model and study the probabilistic properties of random pure snake configurations chosen according to their contribution to this partition function. Under a suitable weighting, the model is integrable in the sense that we have access to explicit formulas for its partition function and correlation function. We utilise the integrable and determinantal structure of this model in several applications through its various scaling limits, such as to prove a traffic representation of ASEP on the ring.