Abstracts Spring Semester 2025

Title: Area-minimizing currents mod an integer

Abstract: Currents mod p are a suitable generalization of classical chains mod p, i.e. of finite combinations of smooth submanifolds with coefficients in the cyclic group $\mathbb Z_p$. By the pioneering work of Federer and Fleming it is possible to minimize the area in this context and, for instance, represent mod p homology classes with area minimizers. For $p>2$ typically (i.e. away from a small set of exceptional points) one would expect such minimizers to be a union of smooth minimal surfaces joining together (``in multiples of $p$'') at some common boundary. This is however surprisingly challenging to prove, especially for even $p$'s, and up until recently only known for $p=3$ and $4$ in codimension $1$. In this talk I will explain the outcome of a series of more recent works (some joint of the speaker with Hirsch, Marchese, Stuvard and Spolaor, some by Wickramasekera and Minter-Wickramasekera, and some joint of the speaker with Minter and Skorobogatova) which confirms this picture, with varying degrees of precision in a variety of situations.

Title: Coercivity and Gamma-convergence of the p-energy of sphere-valued Sobolev maps

Abstract: In this talk, we explore the asymptotic behavior of sphere-valued Sobolev maps as their p-energy approaches the critical Sobolev exponent. Based on a recent work (to appear) joint with Mattia Freguglia and Nicola Picenni, we show compactness and Gamma-convergence of the p-energy to the area functional of the suitable dimension. Our result establishes the analog for the p-energy of a celebrated work by Alberti, Baldo, and Orlandi for the Ginzburg-Landau energy in general dimension and codimension.

Title: Asymptotic behavior of a diffused interface volume-preserving mean curvature flow

Abstract: We consider a diffused interface version of the volume-preserving mean curvature flow in the Euclidean space, and prove, in every dimension and under natural assumptions on the initial datum, exponential convergence towards single "diffused balls". The relation with the problem of determining the long time behavior of the standard volume-preserving mean curvature flow is also discussed. This is based on a joint works with Matteo Bonforte (U. Autonoma Madrid) and Daniel Restrepo (Johns Hopkins U.); see https://arxiv.org/pdf/2202.11583 andhttps://arxiv.org/pdf/2407.18868.

Title: Sharp Lp bounds for the centered Hardy-Littlewood maximal function on hyperbolic spaces 

Abstract: We will provide sharp weak type estimates for the Hardy-Littlewood maximal operator in the context of Gromov hyperbolic metric measure spaces, which satisfy a locally doubling condition and the measures of balls grows exponentially for large radii. This result generalizes previous results on symmetric spaces of non compact type and rank 1, Damek-Ricci spaces, and Riemannian manifolds of pinched negative curvature. This is a joint work with S. Meda and F. Santagati.

Title: Concentrations in Bernoulli's free boundary problem

Abstract: Bernoulli's free boundary problem is an overdetermined boundary value problem in which one seeks an annular domain such that the capacitary potential satisfies an extra boundary condition.

I will talk about recent progress on a conjecture of Flucher and Rumpf that asserts the existence of a family of free boundaries concentrating at non-degenerate local minima of the Robin function.

Title: A solution of Enflo’s problem 

Abstract: To what extent we need linearity in Banach spaces, in particular in embedding questions, is a hot topic in mathematics/theoretical computer science popularized By Bourgain, Pisier, Naor.

One of unsolved (since 1978) problems in this program (called Ribe’s program) was a problem of Per Enflo: is it true that Rademacher type p of a space implies that it has metric type p?

There were many particular results, e.g. Bourgain—Milman—Wolfson proved that metric type p-\epsilon is ensured. Paata Ivansivili, Ramon Van Handel and myself proved that indeed metric type p holds. I will present the story and a sketch of the proof.

Title: Mean curvature flow of entire graphs 

Abstract: We consider the graphical mean curvature flow of maps f: R^m\to R^n, m>1, and derive estimates on the growth rates of the evolved graphs.

In the case of uniformly area decreasing maps f: R^m\to R^2, m>1, we show that the graphicality and the area decreasing property are preserved.

Moreover, if the initial graph is asymptotically conical at infinity, we prove that the normalized mean curvature flow smoothly converges to a self-expander.

The talk is based in a joint work with Knut Smoczyk.

Title: Rigidity for stable minimal hypersurfaces 

Abstract: In this talk I will present some recent results concerning the rigidity of complete, immersed, stable (or delta-stable) minimal hypersurfaces in the Euclidean space. The approach is based on a conformal method, inspired by classical papers of Schoen-Yau and of Fischer-Colbrie.

This is based on joint works with G. Catino, P. Mastrolia and L. Mari.

Title: Stability estimates for inverse boundary value problems

Abstract: Inverse boundary value problems involve the retrieval of unknown parameters of a partial differential equation (PDE) from boundary data. In practical scenarios, this entails reconstructing internal properties of a medium (e.g., conduction, stiffness, density) based on observations made at its boundary. Typically, parameter estimation problems  are ill-posed according to Hadamard's definition: small errors in the data may result in uncontrollable errors in the unknowns. In view of the many applications, this leads to the search for appropriate methods to contain such instability. In this talk I will introduce the conductivity problem, a prototypical example of ill-posed nonlinear inverse problem at the basis of Electrical Impedence Tomography (EIT). I will show that by introducing, mathematically suitable but physically relevant, a-priori assumptions on the unknown parameters one can mitigate the ill-posedeness. Quantitative Unique Continuation Estimates will emerge as a crucial tool in this endeavor.

Title: Stability for overdetermined problems involving the fractional Laplacian 

Abstract: Overdetermined problems are a type of boundary value problem where `too many' conditions are imposed on the solution. In general, such a problem is ill-posed, so the main objective is to classify sets in which the problem admits a solution. A classical result due to J. Serrin says that a bounded domain in R^n that admits a function with constant Laplacian, zero Dirichlet data, and constant Neumann data must be a ball. We will discuss rigidity and stability results for two related generalisations of Serrin's problem where we replace the usual Laplacian with the fractional Laplacian.

Title: The codimension 4 conjecture under Kato bounds on the Ricci curvature 

Abstract: I will present a joint work with Gilles Carron (Nantes Université) and Ilaria Mondello (Université Paris-Est Créteil) where we prove that the singular set of any Gromov-Hausdorff limit point of a sequence of smooth Riemannian manifolds satisfying a uniform Kato bound on the norm of the Ricci tensor has codimension 4. This extends a celebrated result of Cheeger and Naber obtained under a uniform bound on the norm of the Ricci curvature. 

Title: Scaling Limits of Bayesian Neural Networks: Gaussian Processes and Mixtures

Abstract: This talk offers a gentle introduction to Bayesian neural networks and the concept ofGaussian equivalence. In large neural networks, key theoretical insights emerge in the infinite-width limit, where the number of neurons per layer grows while depth stays fixed. In this regime, networks with Gaussian-initialized weights define a mixture of Gaussian processes with random covariance, which converges in the infinite-width limit to a pure Gaussian process (NNGP) with deterministic covariance.

If time permits, we’ll also explore recent results on deep linear networks in the proportional limit, where both depth and width diverge at a fixed ratio. In this setting, the network converges to a nontrivial Gaussian mixture, both at the prior and posterior level. This structure allows the network to capture dependencies in the outputs—an ability lost in the infinite-width limit but retained in finite networks.

Title: On the smooth approximation of integral cycles

Abstract: The natural question of how much smoother integral currents are with respect to their initial definition goes back to the late 1950s and to the origin of the theory with the seminal article of Federer and Fleming. In this seminar I will explain how closely one can approximate an integral current representing a given homology class by means of a smooth submanifold. This is a joint study with William Browder and Camillo De Lellis, based on some previous preliminary work of the former author together with Frederick Almgren.

Title: BV curves of measures

Abstract: Representation results for  absolutely continuous curves with values inthe Wasserstein space of Borel probability measures in $\R^d$ with finite $p$-moment, $p>1$, provide a crucial tool to study evolutionary PDEs in a measure-theoretic setting. They are strictly related to the superposition principle for measure-valued solutions to the continuity equation.

This talk revolves around the extension of these results to the case $p=1$, and to curves that are only of bounded variation in time. Based on a  joint collaboration with Stefano Almi (Napoli) and Giuseppe Savaré (Milano). 

Title: Global existence results for the 2D Kuramoto-Sivashinsky equation 

Abstract: I will present recent results concerning global existence for the Kuramoto-Sivashinsky equation in 2 space dimensions with and without advection in the presence of growing modes. The KSE is a model of long-wave instability in dissipative systems, derived to describe flame-front propagation in combustion. This is joint work with David Ambrose (Drexel),  and also with Michele Coti-Zelati  (Imperial College), Michele Dolce (EPFL), and Yuanyuan Feng (East China Normal University). 

Title: Lattice tilings with minimal perimeter and unequal volumes

Abstract: I will consider tessellations of the Euclidean space with unequal cells arising from the minimization of local or non-local perimeters, discussing existence, regularity and qualitative properties of minimizers. I will show the optimality of the hexagonal tiling among periodic partitions with almost equal areas.