A.A. 2025/26

Abstract

In this talk we are interested in nontrivial solutions, so called zero modes, to the Dirac equation on closed Riemannian spin manifolds. We begin by recalling the necessary background material from spin geometry. Then we present a necessary condition for the existence of zero modes, relating the norm of a certain vector field to the Yamabe constant of the manifold.

Abstract

A model for the modes of vibration of an idealized drumhead is given by Dirichlet eigenfunctions. To each of them it is associated a positive number called eigenvalue. This quantity is related to the frequency of the sound produced by the drum when the drumhead vibrates in the corresponding mode.

In this seminar we introduce a new type of eigenfunctions that satisfy an orthogonality constraint with respect to a given function. Finally we treat some shape optimization problems relative to the corresponding eigenvalues.

Based on a joint work with D. Zucco.

Abstract

Motivated by the Onsager statistical mechanics description of turbulent Euler flows with point singularities, we generalize the mean field theory developed by Caglioti, Lions, Marchioro and Pulvirenti to the case where the total vorticity is influenced by a fixed co-rotating/counter-rotating vortex. This study leads to the analysis of a class of mean field equations which exhibit, in this context, a new blow up phenomenon, which we call "blow up and concentration without quantization", where the mass associated with blow up is not anymore quantized but rather allowed to take values in a full interval of real numbers. We provide a refined description of this phenomenon and obtain the asymptotic profiles of the solutions in the different singular limits allowed in this non standard setting. 

Joint work with D. Bartolucci and L. Wu.

Abstract

In 1950, Milnor introduced a definition of total curvature for rectifiable (hence Lipschitz) curves in R^n and proved that if the curve is C^2-regular and parametrized by arc length, this definition coincides with the integral of the norm of the geodesic curvature of the curve. In 2023, Mucci and Saracco proposed a definition of p-curvature for any exponent p greater than or equal to 1, showing that a rectifiable curve parametrized by arc length belongs to the Sobolev space W^{2,p} if and only if its p-curvature is finite. Moreover, in this case, the p-curvature equals the integral on the curve of the p-th power of the norm of its geodesic curvature. In this seminar,  I will explain how the concept of p-curvature can be extended to rectifiable curves in the sphere and how analogous results can be obtained in this setting. 

This is joint work with D. Mucci and A. Saracco.

Abstract

In this talk I will introduce the notion of fractional perimeter of a set, show its main properties and motivate the interest in this object. Then, I will discuss an ongoing project in which we prove that the ball is a local minimizer of the scale invariant isoperimetric ratio between two fractional perimeters under C^1-small graph perturbations: this parallels the results known in the literature where one of the two fractional perimeters is replaced by the volume. 

This is a joint work with Annalisa Massaccesi (University of Padova),  Giovanni Alberti and Jeremy Mirmina (University of Pisa).

Abstract

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 branching networks. 

This talk is concerned with two different branched transport type models. The first one is the standard branched optimal transport problem, introduced by Gilbert for communication networks and extended by Xia and by Bernot, Caselles and Morel. The second one is a model for pattern formation in type-I superconductors, which was introduced as a reduced model for the Ginzburg-Landau functional by Conti, Goldman, Otto and Serfaty. Both models naturally define an irrigation energy, which is the cost of transporting a Dirac delta to a target measure.

In recent years, several conjectures 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. In this talk I present some of these conjectures and recent advances in solving them. For the first model, I introduce a conjecture by Xia, Santambrogio and Pegon on the "unit ball" for branched transport, and another by Bressan on the optimal shape of tree roots. For the second model, I present a conjecture by Conti, Otto and Serfaty on the behaviour of the magnetic field at the boundary of a superconductor, when the adherence to the external magnetic field is imposed weakly. This talk is based on some works with Michael Goldman, Melanie Koser, Felix Otto and Paul Pegon.

Abstract

Following a model proposed by Alberto Bressan and Maria Teresa Chiri, we study a reaction–diffusion equation describing the evolution of an invasive population. Motivated by a pest–eradication problem, we introduce a control term to be chosen optimally so as to simultaneously minimize the control cost and the population size over time. After recalling classical results on parabolic reaction-diffusion equations, we focus our analysis on the optimal control of one-dimensional traveling waves and provide an explicit characterization of the unique optimal profile, removing all assumptions previously made in the literature.

Moreover, we investigate the regularity of the effort function with respect to the traveling-wave speed. Finally, we explain a possible restatement of the parabolic problem in terms of controlled set evolution and present a Γ-convergence conjecture formulated by Bressan. 

All results are obtained in collaboration with Stefano Bianchini (SISSA).