Seminari in programmazione durante l'anno accademico 2025-2026

Alessandro Cosenza (Paris - Saclay)

"Fractal" conjectures in branched transport type problems

14 maggio 2026 alle ore 14:30: in aula 2BC30

Branched optimal transport is a variant of optimal transport in which an economy of scale principle is present. Grouped transportation is favoured, leading to mass moving on networks, instead of following geodesics.

This talk is concerned with several conjectures concerning fractal measures, which are related to two different branched transport type models. The first model is the standard branched optimal transport problem developed by Xia and by Bernot, Caselles and Morel as a generalization of the classical model by Gilbert for communication networks. The second one is a model for pattern formation of the magnetic field in type-I superconductors, which was introduced as a reduced model for the Ginzburg-Landau functional by Conti, Goldman, Otto and Serfaty. To both models we can associate an irrigation energy, which is the cost of transporting a Dirac delta to a target measure.

In recent years, several conjectured were formulated concerning problems in which the irrigation energy, which favours mass concentration, is paired with a non local energy or constraint, which favours a diffused behaviour. In all cases, the optimal measure seems to exhibit fractal behaviour. I this talk I present some of these conjectures and recent advances in solving them. In particular, concerning the first model, I introduce the conjecture proposed by Xia, Santambrogio and Pegon for the "unit" ball for branched optimal transport, and the conjecture on the optimal shape of tree roots proposed by Bressan. For the second model, I present a conjecture by Conti, Otto and Serfaty on the behaviour of the magnetic field on the boundary of a superconductor, when the adherence to the external magnetic field is imposed weakly. While the models and the techniques used are different, the spirit of the results is similar. Hence, analogies between the two models are an interesting source of ideas for future research and will be underlined. This talk is based on some works with Michael Goldman, Melanie Koser, Felix Otto and Paul Pegon.

-----------------------------------------------

Federico Luigi Dipasquale (Scuola Superiore Meridionale)

Lifting of Sobolev maps into closed Riemannian manifolds via double coverings and minimal connections relative to planar sets

7 maggio 2026 alle ore 12:30: in aula 2BC30

In this talk, we consider Sobolev maps from a planar domain into a closed Riemannian manifold and their BV liftings via a double covering of the target. We present a sharp lower bound on the jump length of the lifting, established in a recent joint work with G. Canevari (Università di Verona) and B. Stroffolini (Università degli studi di Napoli Federico II), and expressed in terms of a geometric quantity: the minimal connection, relative to the domain, of the non-orientable singularities.

-----------------------------------------------

Alessio Figalli (ETH Zürich)

The De Giorgi Conjecture for the Free Boundary Allen–Cahn Equation

24 aprile 2026 alle ore 15:30: in aula 1AD100

The Allen–Cahn equation is well known as a diffuse-interface model that approximates minimal surfaces. This connection led to the conjecture that, in dimensions up to seven, all global stable solutions of the Allen–Cahn equation are one-dimensional. If true, this would in particular imply the celebrated De Giorgi conjecture for monotone solutions.

Motivated by the geometric character of the equation, David Jerison has long advocated that a free-boundary formulation offers a more natural framework for capturing its relationship with minimal surfaces. From this viewpoint, one is led to revisit the conjecture in a free-boundary setting.

In recent joint work with Chan, Fernández-Real, and Serra, we classify all global stable solutions to the one-phase Bernoulli free-boundary problem in three dimensions. As a consequence, we prove that global stable solutions to the free-boundary Allen–Cahn equation in three dimensions are necessarily one-dimensional.

-----------------------------------------------

Mikaela Iacobelli (ETH Zürich)

Challenges and Breakthroughs in the Mathematics of Plasmas

24 aprile 2026 alle ore 13:30: in aula 2BC60

In this talk I will present some recent advances in the mathematics of collisionless plasmas. After a general introduction to the theory of Vlasov-type equations, I will discuss questions of well-posedness, stability, and singular limits. 

I will present almost-optimal stability for the quasineutral limit for the Vlasov-Poisson and Vlasov-Poisson with massless electrons. Finally, for magnetized plasmas, I will describe a new result on the quasineutral limit from the relativistic Vlasov-Maxwell system to electron-MHD in an analytic-regularity setting. Here the analysis provides estimates uniform in the quasineutral parameter and strong (filtered) convergence to the limiting dynamics. 

-----------------------------------------------

Antonio Marigonda (Università di Verona)

A multiagent problem with aggregation terms

10 aprile 2026 alle ore 12:30: in aula 2BC30

In this talk we introduce and study an optimal control problem for multiagent systems, modeled in the Wasserstein space of probability measure, where the cost function fosters the aggregation of the agents, by mean of a suitable reformulation of the classical notion of "multiplicity" used in branched optimal transport, adapted to a dynamical setting with nonholonomic constraints. A Lagrangian approach is used to carry the analysis. The main motivation for this study is related to supply chains design. The results proved encompass existence of minimizing cost network in a wide class of cases, and an analysis of the value function associated to the problem.

-----------------------------------------------

Xavier Lamy (Université de Toulouse - Institut Universitaire de France)

Hyperbolic regularization effects for degenerate elliptic equations

8 aprile 2026 alle ore 14:30: in aula 2BC30

I will report on joint work with Riccardo Tione, where we establish partial regularity results for a large class of planar equations div G(Du)=0 which are only qualitatively elliptic, using tools from hyperbolic conservation laws and Hamilton-Jacobi equations.

-----------------------------------------------

Yizhou Zhou (IGPM RWTH Aachen University) 

Boundary conditions for first-order hyperbolic relaxation systems

25 marzo 2026 alle ore 13:30: in aula 2AB40

The first-order hyperbolic relaxation system is a class of time-dependent partial differential equations which model various non-equilibrium phenomena. For such systems, the main interest is to understand the zero relaxation limit. The initial-value problem for the relaxation system has been well-developed and a systematical framework has been built. However, the initial-boundary value problem of the relaxation system is still in the developing stage. In this talk, I will first introduce the theory of boundary conditions for general relaxation systems. Then I will present the recent results for the initial-boundary-value problem with characteristic boundaries. Specifically, we redefine a characteristic Generalized Kreiss condition (GKC) which is essentially necessary to have a well-behaved relaxation limit. Under this characteristic GKC and a Shizuta-Kawashima-like condition, we derive reduced boundary conditions for the relaxation limit solving the corresponding equilibrium systems and justify the validity.

-----------------------------------------------

Benedetto Piccoli (Rutgers University-Camden)

Measure differential equations and measure controls

18 marzo 2026 alle ore 11:30: in aula 2AB40

We discuss the concept of MDE and various recent results. Then we turn to introduce the concept of measure control, its relation to MDEs, basic properties, and some interesting examples. Finally, we show some potential applications to multi-agent systems.

-----------------------------------------------

Gianmarco Caldini (Università di Trento)

On regularity theory for generalized surfaces

26 febbraio 2026 alle ore 12:30: in aula 2AB40

The natural question of how much smoother integral currents are with respect to their initial definition goes back 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 a smooth submanifold. Part of what will be discussed is derived from a joint study with William Browder and Camillo De Lellis, and builds on earlier preliminary work by the former author together with Frederick Almgren.

-----------------------------------------------

Matilde Gianocca (ETH Zürich)

Rigidity in the Ginzburg–Landau equation from S2 to S2

17 febbraio 2026 alle ore 12:30: in aula 2AB45

 The Ginzburg–Landau energy is often used to approximate the Dirichlet energy. As the perturbation parameter tends to zero, critical points of the Ginzburg–Landau energy converge, in an appropriate (bubbling) sense, to harmonic maps. In this talk I will first explain key analytical properties of this approximation procedure, then show that not every harmonic map can be approximated in this way. This is based on a rigidity theorem: under the energy threshold of 8pi, we classify all solutions of the associated nonlinear elliptic system from S2 to S2, thereby identifying exactly which harmonic maps can arise as Ginzburg–Landau limits in this regime.

-----------------------------------------------

Albachiara Cogo (SNS Pisa)

Sobolev conformal structures on closed manifolds

6 febbraio 2026 alle ore 12:30: in aula 2AB45

Low-regularity metrics arise naturally in the realm of geometric PDEs, especially in relation to physical models. It is well known in Riemannian geometry that the components of such metrics have the best regularity in harmonic coordinates. In this talk, we introduce a novel approach to globalize this idea and study conformal classes of rough Riemannian metrics on closed 3-manifolds. Given a Riemannian metric in the Sobolev class W^{2, q} with q > 3, we characterize when a more regular representative exists in its conformal class. We highlight a deep link to the Yamabe problem for rough metrics, which we fully resolve. In particular, for Yamabe positive metrics, the Yamabe problem in this low-regularity setting requires developing new elliptic theory for the conformal Laplacian, including existence, regularity and a fine blow-up analysis of its Green function, which we provide in any dimension n  \geq 3 and for q>n/2.

This is based on joint work with R. Avalos and A. Royo Abrego.

-----------------------------------------------

Andrea Merlo (Universidad del País Vasco)

Marstrand's density theorem for all norms in the plane

29 gennaio 2026 alle ore 12:00: in aula 1BC50

This talk is devoted to generalizing, in the plane, the following classical result of J. Marstrand: if, for a measure, the m-upper and m-lower densities with respect to Euclidean balls coincide almost everywhere, then m is an integer. We will show that, in the plane, this result extends to densities defined with respect to an arbitrary norm. As a consequence, we will prove that Preiss’s theorem holds for every norm in the plane. These results were obtained in collaboration with G. Del Nin.

-----------------------------------------------

Salvatore Stuvard (Università di Milano Statale)

New advances on the existence and regularity of Brakke’s mean curvature flows

23 gennaio 2026 alle ore 12:00: in aula 2AB45

In the first part of this talk, I will give an overview of what is known about the existence and the regularity properties of Brakke flows. These are measure-theoretic, weak solutions to the mean curvature flow capable of describing the evolution of surfaces through singularities and topology changes. In the second part, I will focus on a recent result, obtained jointly with Yoshihiro Tonegawa (Institute of Science Tokyo), on the regularity of Brakke flows that are weakly close to a multiplicity one static triple junction.

-----------------------------------------------

Alessandro Pigati (Università Bocconi)

Anisotropic Allen-Cahn and convergence to anisotropic area

5 dicembre 2025 alle ore 12:30: in aula 1BC45

In this talk we will introduce a PDE way to construct hypersurfaces which are critical for a generalization of area, based on an anisotropic integrand. Namely, we study energy concentration for rescalings of an anisotropic version of Allen-Cahn.

Besides a Gamma-convergence result, we will sketch a proof of the fact that energy of stable critical points (of the rescaled Allen-Cahn) concentrates along an integer rectifiable varifold, a weak notion of hypersurface, using stability (or finite Morse index) to compensate for the lack of monotonicity formulas.

Among the technical ingredients, we will see a generalization of Modica's bound and a diffuse version of the stability inequality for hypersurfaces.

This is joint work with Antonio De Rosa (Bocconi University).

-----------------------------------------------

Cyrill Muratov (Università di Pisa)

Skyrmions in ultrathin magnetic films: an overview

19 novembre 2025 alle ore 11:30: in aula 2AB45

I will present an overview of the current results on existence and asymptotic properties of magnetic skyrmions defined as topologically nontrivial maps of degree +1 from the plane to a sphere which minimize a micromagnetic energy containing the exchange, perpendicular magnetic anisotropy  and interfacial Dzyaloshinskii-Moriya interaction (DMI) terms. In ultrathin films, the stray field energy simply renormalizes the anisotropy constant at leading order, but in finite samples it also produces additional non-trivial contributions at the sample edges, promoting nontrivial spin textures. Starting with the whole space problem, I will first discuss the existence of single skyrmions as global energy minimizers at sufficiently small DMI strength. Then, using the quantitative rigidity of the harmonic maps I will present the asymptotic characterization of single skyrmion profiles both in infinite and finite samples. Lastly, I will touch upon the question of existence of multi-skyrmion solutions as minimizers with higher topological degree and present recent existence results obtained jointly with T. Simon and V. Slastikov.

-----------------------------------------------

Michele Caselli  (SNS Pisa)

Saddle-type solutions to phase-transition models

28 ottobre 2025 alle ore 11:00: in aula 1BC45

In this talk, we will provide an overview of results on entire saddle-type solutions to phase transition models, as the Allen–Cahn equation in codimension one or the Ginzburg–Landau system in codimension two, highlighting similarities and differences. Then, we will present a new result, obtained in a joint work with Nicola Picenni, regarding the existence of an entire saddle solution of the complex Ginzburg–Landau system in 3D whose zero set is a union of two orthogonal, intersecting lines.

-----------------------------------------------

Eugenio Bellini  (Università di Padova)

Curvature measures and the sub-Riemannian Gauss-Bonnet Theorem

8 ottobre 2025 alle ore 12:30: in aula 1BC45

 It is not uncommon for curvature to concentrate at the singularities of geometric spaces. In this talk, we show how this phenomenon occurs for surfaces immersed in 3D contact sub-Riemannian manifolds. Adopting a measure-theoretic viewpoint on the Riemannian approximation scheme, we prove that the Gaussian curvature measure of such a surface is singular and supported on its isolated characteristic points. We identify natural geometric conditions under which this behavior occurs, namely when the surface admits characteristic points of finite order of degeneracy.

-----------------------------------------------

Isabeau Birindelli  (Università di Roma Sapienza)

Fully nonlinear equations in thin domains, à la Evans

25 settembre 2025 alle ore 11:30: in aula 2AB45

We will present some recent works and works in progress with Ariela Briani and Hitoshi Ishii, where we consider the asymptotic behaviors of solutions of equations in family of domains which "lose" one dimension at the limit. Starting from new proofs for old results and obtaining new insights in particular for general oblique boundary conditions. The analysis is done through the perturbed test function approach à la Evans.

-----------------------------------------------

Luca Capogna  (Smith College)

The Neumann problem in metric measure spaces and  applications to non-local p-Laplacians

18 settembre 2025 alle ore 12:30: in aula 1BC50

We review the ongoing program joint with Gibara, Korte, and Shanmugalingam, aimed at extending the Caffarelli-Silvestre approach to non-local operators from the Euclidean space to general doubling metric measure spaces. An integral component of this program is the study of the well-posedness of the Neumann problem in uniform domains in metric measure spaces that satisfy a Poincare' inequality and a doubling condition.

Seminari anno accademico 2024-2025

Seminari anno accademico 2023-2024 

Seminari anno accademico 2022-2023

Seminari anno accademico 2021-2022

Archivio seminari Anni Accademici Passati