Analysis and Optimal Transport

15.-16. September 2025

This two-day workshop focuses on the rich interactions between analysis, partial differential equations, spectral theory and the theoretical and computational aspects of optimal transport. Graduate students and researchers of all career stages are welcome to attend. The event is supported by an LMS Celebrating New Appointments (Scheme 9) grant and the University of Sussex.

Location: The workshop will take place in Pevensey 1A6, Falmer Campus, University of Sussex.

Speakers:

• Bence Borda (University of Sussex)
• David Bourne (Heriot-Watt University)
• Jean-Claude Cuenin (Loughborough University)
• Federica Dragoni (Cardiff University)
• Ujué Etayo (CUNEF Universidad)
• Lukas Koch (University of Sussex)
• Matthew Thorpe (University of Warwick)

Schedule:

Monday 15 September	1:30–2:30	Borda
	2:30–3:30	Thorpe
	3:30–4:00	Coffee break
	4:00–5:00	Cuenin
	5:00–6:00	Koch
	7:30	Social dinner
Tuesday 16 September	9:00–10:00	Etayo
	10:00–10:30	Coffee break
	10:30–11:30	Bourne
	11:30–12:30	Dragoni

A group of us will go for lunch together at IDS on Monday, leaving the maths department at 12, and Tuesday after the last talk of the workshop. Please feel free to join us. There will be a social dinner on Monday 15 September.

Registration: Please register here if you are planning to attend the conference.

Titles and Abstracts:

• Bence Borda- Equidistribution in Wasserstein metric: The theory of optimal transport and in particular the Wasserstein metric provide a natural way to measure how evenly distributed a finite point configuration is. In this talk, we present a flexible framework to estimate the Wasserstein metric using harmonic analysis in compact spaces including the torus, the Euclidean unit sphere and general compact Riemannian manifolds. We survey applications to i.i.d. random points, random walks, eigenvalues of random matrices, and spherical designs.
• David Bourne- Semi-discrete optimal transport: In this talk I will describe an application of semi-discrete optimal transport theory in the steel industry, and how in turn this very applied problem has led to new pure mathematics, to results in convex geometry. In particular, I will discuss necessary and sufficient conditions for the existence of convex partitions with cells of prescribed volumes and barycentres.
• Ujué Etayo- Bombieri-Type Inequalities & Zero-Packing on Spheres and Tori: We revisit the classical Bombieri inequality for products of homogeneous polynomials and present the sharper, geometric version recently obtained by Etayo. By reformulating the inequality through the unipolar Green function of the Riemann sphere, the constant on the new inequality is replaced by a packing number that captures how well zeros can be “spread out’’ on a compact surface. I will outline the optimal lower and upper bounds of this number on the sphere, explain why the problem naturally generalises to compact Riemann surfaces, and then focus on the torus, where a clever lattice refinement yields uniform bounds independent of the number of zeros. Finally, I will introduce toroidal pseudopolynomials—Weierstrass-σ analogues of polynomials—and translate the inequality in terms of Lp norms of such pseudopolynomials. This is joint work with H. Hedenmalm and J. Ortega-Cerdà.
• Jean-Claude Cuenin- Spectral Theory meets Harmonic Analysis: I will give an overview of recent research at the interface of spectral theory and harmonic analysis. In particular, I will explain how ideas from Fourier restriction theory can be used to derive sharp eigenvalue bounds for Schrödinger operators with complex potentials.
• Federica Dragoni- Semiconcavity of the square distance in Carnot groups: Semiconcavity and semiconvexity are key regularity properties for functions with many applications in a broad range of mathematical subjects. The notions of semiconcavity and semiconvexity have been adapted to different geometrical contexts, in particular in sub- Riemannian structures such as Carnot groups, where they turn out to be extremely useful for the study of solutions of degenerate PDEs. In this talk I will show that, for a suitable class of Carnot groups, the Carnot-Carathéodory distance is semiconcave, in the sense of the group, in the whole space. I will also give some applications to solutions of non-coercive Hamilton-Jacobi equations. Joint work with Qing Liu and Ye Zhang.
• Lukas Koch- Uniform Lipschitz regularity for entropic optimal transport: Entropic optimal transport has received a lot of interest recently as it can be efficiently computed via Sinkhorn's algorithm. I will present recent regularity results that establish Lipschitz regularity of a map that is naturally associated to an entropic optimal transport plan, independent of the regularisation parameter. This is joint work with R. Gvalani.
• Matthew Thorpe- Linearising Optimal Transport: Optimal transport distances are great, but expensive and lack off-the-shelf data analysis tools such as PCA. The idea behind linearisation is to find an approximate isometry/equivalence from the space of probability measures (or positive measures) to a Euclidean space where the Euclidean distance in the latter space is approximately the optimal transport distance in the former space. Formally linear optimal transport projects onto the tangent space. In general this is not even equivalent (in the metric sense), however in the first part of the talk we give conditions sufficient for the linear approximation of the Wasserstein distance to be equivalent to the restricted shortest-path Wasserstein distance on the submanifold. We then show how the latent manifold structure of the submanifold can be learned from samples and pairwise extrinsic Wasserstein distances. In particular, we show that the submanifold metric space can be asymptotically recovered in the sense of Gromov-Hausdorff from an appropriate graph. In the second part of the talk we linearise the Hellinger--Kantorovich distance, an extension of the Wasserstein distance to positive measures.

This workshop is organised by Bence Borda and Lukas Koch (University of Sussex) with support from a LMS Celebrating New Appointments (Scheme 9) grant and the University of Sussex.