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: D1.07, Zeeman, 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
• 20 January 2025
• 27 January 2025
• 3 February 2025
• 10 February 2025
• 17 February 2025: Gabriel Flath (Oxford)
• 24 February 2025: Andreas Klippel (Technical University of Darmstadt)
• 3 March 2025
• 10 March 2025

Summer Term 2024–2025

• 28 April 2025
• 5 May 2025: No seminar
• 12 May 2025
• 19 May 2025
• 26 May 2025: No seminar
• 2 June 2025
• 9 June 2025
• 16 June 2025
• 23 June 2025