Per partecipare agli eventi (telematici), contattare lo Staff di Dipartimento.

## Prossimo appuntamento

### Martedì 2 febbraio ore 14:30

**Mattia Fogagnolo**(Centro De Giorgi)

**The isoperimetric problem on manifolds with nonnegative Ricci curvature: geometric inequalities and asymptotic Gromov-Hausdorff analysis**

In this talk we discuss two basic tools in order to deal with the isoperimetric problem on manifolds with nonnegative Ricci curvature, namely the sharp isoperimetric inequality and the asymptotic behaviour of the minimizing sequences on these manifolds. The first implies that the sharp isoperimetric constant is given by the Asymptotic Volume Ratio of the manifold and that it is saturated only on balls of the Euclidean space. For what it concerns the latter, we show that the runaway volume is recovered in nonsmooth spaces arising as pointed Gromov-Hausdorff limits.

Examples of Riemannian manifolds enjoying existence and nonexistence of isoperimetric sets will be provided too.

The results are based on joint works with G.Antonelli, L. Mazzieri and M. Pozzetta.

## Il calendario

### Martedì 9 febbraio ore 14:30

**David Bate**(Warwick)

### Martedì 16 febbraio ore 14:30

**Ihsan Topaloglu**(Virginia Commonwealth University)

**A nonlocal isoperimetric problem with density perimeter**

In this talk I will present recent results on a variant of Gamow's liquid drop model where we consider the mass-constrained minimization of an energy functional given as the sum of a density perimeter term and a nonlocal interaction term of Riesz type. In particular, I will show that for a wide class of density functions this energy admits a minimizer for any choice of parameters, and that for monomial densities the unique minimizer is given by the ball of fixed volume when the nonlocal effects are sufficiently small. This is a joint work with S. Alama, L. Bronsard, and A. Zuniga.

### Martedì 23 febbraio ore 14:30

**Robin Neumayer**(Northwestern University)

### Martedì 2 marzo ore 14:30

**Gareth Speight**(University of Cincinnati)

### Martedì 9 marzo ore 14:30

**Davide Vittone**(Università di Padova)

### Martedì 16 marzo ore 14:30

**Robert Young**(NYU Courant)

### Martedì 23 marzo ore 14:30

**Andrea Malchiodi**(SNS Pisa)

### Martedì 30 marzo ore 14:30

**Andrea Mondino**(Oxford)

### Martedì 13 aprile ore 14:30

**Giorgio Saracco**(SISSA)

### Martedì 20 aprile ore 14:30

**Sara Daneri**(GSSI)

### Martedì 27 aprile ore 14:30

**Antonio De Simone**(SISSA)

### Martedì 4 maggio ore 14:30

**Maria Colombo**(EPFL Losanna)

### Martedì 11 maggio ore 14:30

**Anna Sakovich**(Uppsala University)

### Martedì 18 maggio ore 14:30

**Paolo Bonicatto**(Warwick)

### Martedì 25 maggio ore 14:30

**Chao Xia**(Xiamen University)

### Martedì 1 giugno ore 14:30

**Daniele Semola**(Oxford)

### Martedì 8 giugno ore 14:30

**Ivan Violo**(SISSA)

## Eventi passati

**Martedì 1 dicembre ore 14:30**

**Simone Di Marino**(Università di Genova)

**Minimal charge configurations, total charge discrepancy and grand-canonical optimal transport**

Based on a work in progress with L. Nenna and M. Lewin. Motivated by mathematical physics, in particular by the semi-classical limit of the grand-canonical density functional theory, we introduce the grand-canonical optimal transport. We minimize the expectation of the interaction energy between electrons, among probabilities in the configuration space which fix the average density of electrons. The key new feature of this model is that the number of electrons is not fixed. In discussing the features of this problem, we will describe the new phenomenon that is appearing, regarding the estimation of the number of electrons for the optimal configuration; we will see that that the estimate of the discrepancy of this number with respect to the average number of electrons is related to the problem of total charge discrepancy in minimal charge configurations.

**Martedì 15 dicembre ore 15:00**

**Francesco Rossi**(Università di Padova)

**From control of deterministic systems to control of transport partial differential equations**

Take a system of N interacting particles and control them with an external control. What is the corresponding mean-field problem, i.e. the problem of controlling the dynamics when N tends to infinity? It turns out that the answer only makes sense when the control is feedback and sufficiently regular. I will then answer two different questions: - what are the hypothesis on the original problem ensuring the required control regularity? - how can we translate our knowledge of control of finite dimensional deterministic systems into control of the mean-field PDE? These results are obtained with B. Bonnet, M. Duprez, M. Morancey.

**Martedì 12 gennaio ore 14:30**

**Paolo Baroni**(Università di Parma)

**On the relation between Morrey-type spaces of measures and regularity of solutions to related measure data problems**

In the talk I will show implementations, in different settings and various directions, of a general principle connecting regularity properties of measures (in terms of their decay) and good properties for solutions to related measure data problems; in particular, I will focus on quantitative integrability gain for the gradient.

Elliptic and parabolic equations, all modeled after the p-Laplacian, will be considered, as well as generalized Morrey conditions, where the total variation of balls decays in terms of a generic function of their radius.

**Martedì 19 gennaio ore 14:30**

**Nicholas Edelen**(Notre Dame)

**Regularity of minimal surfaces near quadratic cones**

Hardt-Simon proved that every area-minimizing hypercone C having only an isolated singularity fits into a foliation of the Euclidean space by smooth, area-minimizing hypersurfaces asymptotic to C. We prove that if a minimal hypersurface M in the unit ball lies sufficiently close to a minimizing quadratic cone (for example, the Simons' cone), then, in the ball of radius 1/2, M is a C^{1,\alpha} perturbation of either the cone itself, or some leaf of its associated foliation. In particular, we show that singularities modeled on these cones determine the local structure not only of M, but of any nearby minimal surface. Our result also implies the Bernstein-type result of Simon-Solomon, which characterizes area-minimizing hypersurfaces in the Euclidean space asymptotic to a quadratic cone as either the cone itself, or some leaf of the foliation. This is joint work with Luca Spolaor.

**Martedì 26 gennaio ore 16:00**

**Francesco Maggi**(University of Texas at Austin)

### Soap films and capillarity theory

Soap films are modeled, rather than as surfaces with zero mean curvature, as regions with small volume satisfying a spanning condition of homotopic type. We discuss qualitative properties of such soap films, and their convergence towards minimal surfaces, when the volume goes to zero. This talk is based on a series of joint works with Antonello Scardicchio (ICTP), Darren King (UT Austin) and Salvatore Stuvard (UT Austin).