Seminars

Abstract: TBA.

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

Abstract: TBA.

Abstract: The goal of this talk is to showcase how we can use stochastic processes to study the geometry of surfaces. More precisely, I will recall the basic facts about surfaces with constant curvature and Brownian motion on them. Then, we use the Brownian loop measure to express the lengths of closed geodesics on a hyperbolic surface and zeta-regularized determinant of the Laplace-Beltrami operator. This gives a tool to study the length spectra of a hyperbolic surface and we obtain a new identity between the length spectrum of a compact surface and that of the same surface with an arbitrary number of additional cusps. This is a joint work with Yuhao Xue (IHES).  

Abstract: I will discuss the regularity of solutions to a class of semilinear free boundary problems in which admissible functions have a topological constraint, or spanning condition, on their 1-level set. This constraint forces the 1-level set, which is a free boundary, to behave like a surface with singularities, attached to a fixed boundary frame, in the spirit of the set-theoretic Plateau problem. Two such free boundary problems that have been well-studied are the minimization of capacity among surfaces sharing a common boundary and an Allen-Cahn approximation of the set-theoretic Plateau problem. We establish optimal Lipschitz regularity for solutions, and analytic regularity for the free boundaries away from a codimension two singular set. We further characterize the singularity models for these problems as conical critical points of the minimal capacity problem, which are closely related to spectral optimal partition and segregation problems. This is joint work with Mike Novack and Daniel Restrepo. 

Abstract: Optimal transport tools have been extremely powerful to study Ricci curvature, in particular Ricci lower bounds in the non-smooth setting of metric measure spaces (which can be been as a non-smooth extension of Riemannian manifolds). Since the geometric framework of general relativity is the one of Lorentzian manifolds (or space-times), and the Ricci curvature plays a prominent role in Einstein’s theory of gravity, a natural question is whether optimal transport tools can be useful also in this setting. The goal of the talk is to introduce the topic and to report on recent progress. More precisely: After recalling some basics of optimal transport, we will define "timelike Ricci curvature and dimension bounds" for a possibly non-smooth Lorentzian space in terms of displacement convexity of suitable entropy functions and discuss applications. Some cases of such bounds have remarkable physical interpretations (like the attractive nature of gravity) and can be used to give a characterisation of the Einstein's equations for a non-smooth space, extend classical singularity theorems to settings of low regularity, and prove new results even for smooth Lorentzian manifolds (such as new isoperimetric-type inequalities). Based partly on joint work with S. Suhr and partly on joint work with F. Cavalletti. 

Abstract: In recent decades, the theory of optimal transport has advanced rapidly, finding an ever-growing range of applications. The original problem of Monge is to find the cheapest way to transform one probability distribution into another when the cost is proportional to the distance. The most important metric structure that is related to optimal transport is the so-called p-Wasserstein space [denoted by Wp(X)] over the metric space X. The pioneering work of Bertrand and Kloeckner started to explore fundamental geometric features of 2-Wasserstein spaces, including the description of complete geodesics and geodesic rays, determining their different types of ranks, and understanding the structure of their isometry group. In this talk I will focus on isometry groups. A notable and useful property of p-Wasserstein spaces is that X embeds isometrically into Wp(X), moreover an isometry of X induces an isometry of Wp(X) by the push-forward operation. These induced isometries are called trivial isometries, and we say that Wp(X) is isometrically rigid if all its isometries are trivial. The question is: are there non-rigid Wasserstein spaces? What does a non-trivial isometry look like? Until very recently, only a few non-rigid examples were known such as the 2-Wasserstein space over R^n, and the 1-Wasserstein space over [0,1]. In the first part of the talk, I will introduce some key concepts and notation. The main focus will then shift to exploring results concerning both rigidity and non-rigidity in Wasserstein spaces. 

Abstract: In 1928 Besicovitch formulated the following conjecture. Let E be a Borel subset of the plane with finite length and assume its length is more than half of the diameter in all sufficiently small disks centered at a.a. its points. Then E is rectifiable, i.e. it lies in a countable union of C^1 arcs with the exception of a null set. 1/2 cannot be lowered, while Besicovitch himself showed that the statment holds if it is replaced by 3/4. His bound was improved only once by Preiss and Tiser in the nineties to a number which is approximately 0.735. In this talk I will report on further progress stemming from a joint work with Federico Glaudo, Annalisa Massaccesi, and Davide Vittone. Besides improving the bound of Preiss and Tiser to a substantially lower number, our work uncovers an interesting connection with a class of linear programming problems.  

Abstract: Gradient-flow dynamics in abstract spaces have attracted increasing attention in the last decades, especially in connection with Wasserstein metrics. In absence of a linear structure, dissipative evolutions of gradient-descent type are usually reformulated in variational terms. Generalized minimizing movements and evolution variational inequalities are two important concepts arising in this context. The starting point for both concepts is to specify the cost of evolving between two distinct states of the system. A quite complete theory is currently available for costs being the power of the distance in a complete metric space. In this lecture, I aim at presenting some first results for the case of more general costs, eventually leaving the metric-space setting. In particular, I will present an existence theory of evolution variational inequalities. This is work in collaboration with Pierre-Cyril Aubin-Frankowski (TU Wien) and Giacomo Enrico Sodini (University of Vienna). 

Abstract: A celebrated result by S.N. Bernstein asserts that the planes are the only entire minimal (i.e. critical points of the area functional) graphs in R^3 . This result, known as the Bernstein problem, has been generalized for entire minimal graphs in R^(n+1), with n ≤ 7 and it turns out to be false for n ≥ 8. Motivated by the fact that minimal graphs are actually stable (i.e. the second variation of the area functional is non-negative), the following natural generalization of the Bernstein problem is still an open and fascinating question: if M is a complete, orientable, immersed, stable, minimal hypersurface in R^(n+1), does it have to be necessarily a hyperplane? In a recent paper Chodosh and Li proved that this is true in R^4 . In this talk I will discuss an alternative proof of the result by Chodosh and Li obtained in collaboration with G. Catino and P. Mastrolia.

Abstract: From recent progresses in the study of smooth and nonsmooth Lorentzian structures it emerges the need of a functional-analytic theory where, among other things, the relevant norms satisfy a reverse triangle inequality. Aim of the talk is to show that perhaps such a theory is possible. 

Abstract: In this talk, I shall discuss two results that show how the isoperimetric structure of a space is connected to its geometry. First, I will show a sharp concavity property of the isoperimetric profile of manifolds with Ricci lower bounds. Although the statement is set in the smooth context, its proof relies on tools from non-smooth geometry that have been developed in recent years. I will explain how this concavity result interplays with the existence, non-existence, and uniqueness of isoperimetric regions in spaces with lower curvature bounds.Next, I will present a sharp and rigid generalization of the Bishop-Gromov volume comparison theorem. The proof of this result builds on a concavity property of an unequally weighted isoperimetric profile on the manifold, similar to the one mentioned above. Time permitting, I will discuss how this volume estimate has been recently used by L. Mazet, following contributions by O. Chodosh, C. Li, P. Minter, and D. Stryker, to settle a well-known open problem: the stable Bernstein problem in R^n, with n<=6.

Abstract: In this talk we consider a class of scalar nonlinear models describing crowd dynamics. The congestion term appears in the transport equation in the form of a compactly supported nonlinear mobility function, thus making standard weak-type compactness arguments and uniqueness of weak solutions fail. We introduce two different approaches to the problem and discuss their connections with the wellposedness of entropy solutions of the target pde in the sense of Kruˇzkov. A deterministic particle approach relying on suitable generalisations of the Follow-the-leader scheme, which can be interpreted as the Lagrangian discretisations of the problem; and a variational approach in the spirit of a minimising movement scheme exploiting the gradient flow structure of the evolution in a suitable metric framework. 

Abstract: In this talk I will outline the iterative construction of a C^1, bounded function u, whose gradient coincides with a prescribed vectorfield F on a Cantor-type set C. It is transparent from the construction the presence of a trade off between the size of C and the Hölder regularity of the gradient of u. This type of construction is the building block for counterexamples to Frobenius theorem when the tangency set is not regular enough. 

Abstract: In gradient flows, the steady states are given by the critical points of the driving functional. Hence, the associated fluxes vanish.  If a gradient system is coupled to the environment via so-called ports, then steady states may have non-zero fluxes and are called Non-Equilibrium Steady States (NESS).  We consider so-called port gradient systems, where boundary conditions act as constraint which generate suitable fluxes as Lagrange multipliers. In general, steady states of the associated port gradient-flow equation are NESS. We provide a saddle-point characterization of NESS via a so-called BER function. We discuss a few examples of this characterization and show how it appears naturally when performing the EDP-convergence of slow-fast gradient systems, where the fast part stays in NESS connected via ports to the slow system. Prigogine's dissipation principle of 1947 states that under suitable conditions, NESS are minimizers of the dissipation. We will discuss how the saddle-point formulation provides a mathematically rigorous generalization of this principle. 

Abstract: At least two connections can be made between optimal transport theory  and gravitation: 1) Monge-Ampere gravitation can be seen as a nonlinear  correction to classical Newtonian gravitation and present several interesting features:  a) it makes exact the approximate model made by Zeldovich for the modeling of the  early universe in computational cosmology, b) it can be rather directly derived (through  large deviations and Gamma-convergence) from the purely stochastic model of a brownian cloud.  2) Einstein equations in vacuum describe special solutions of a  generalized optimal transport  problem with quadratic cost in a matrix-valued setting.  I will try to review some of these topics during the seminar. 

Abstract: In the early 90's Yong-Geun Oh introduced the problem of studying critical points of the area among Lagrangian surfaces in a symplectic riemannian manifold. Such surfaces are called Hamiltonian stationary or sometimes H-minimal surfaces. This variational problem  is motivated by natural questions such as the study of the Plateau problem in Lagrangian homology classes, the construction of calibrated minimal surfaces in Calabi Yau geometry, the minmax construction of minimal surfaces in spheres....etc. We will first present the difficulties with dealing with the Hamiltonian Stationary equation in general and present the construction of ``pathological solutions'' to this equation in 2 dimension which are nowhere continuous. Then we will turn to the special case of area minimizing H-minimal surfaces and the discovery in the early 2000 of a family of singularities of conical type by Schoen and Wolfson. We will present a direct method for constructing Hamiltonian Stationary discs with prescribed Schoen Wolfson cones. If time permits we will introduce a weak formulation of the H-minimal equations excluding a-priori pathological solutions and compatible with min-max variational constructions. We will then formulate a conjecture relative to the singular set of general solutions to this new variational formulation of H-minimal surfaces.

Abstract: Characterizing the function spaces corresponding to neural networks can provide a way to understand their properties.  The talk is devoted to show how the theory of reproducing kernel Banach spaces can be used to characterize the function spaces corresponding to neural networks. In particular, I will show a representer theorem for a class of reproducing kernel Banach spaces, which includes one hidden layer neural networks of possibly infinite width. Furthermore, I will prove that, for a suitable class of ReLU activation functions, the norm in the corresponding reproducing kernel Banach space can be characterized in terms of the inverse Radon transform of a bounded real measure. The talk is based on on joint work with F. Bartolucci, L. Rosasco and S. Vigogna.

Abstract: In metric geometry and analysis on nonsmooth spaces, it is often convenient (or necessary) to take limits of sequences of metric measure spaces, for example to define tangents or asymptotic cones. This amounts to introducing a topology, or at least a notion of convergence, on the so-called "space of spaces". In the first part of the talk, I will give an overview on several such notions of convergence, focussing mostly on the pointed-measured-Gromov-Hausdorff topology (pmGH for short) and its variants, but briefly mentioning also the concentration topology and the convergence in the sense of pyramids. The emphasis will be on the role of these notions in the study of nonsmooth spaces verifying synthetic curvature-dimension bounds. In the second part of the talk, I will discuss a joint work with Dr. Timo Schultz, where we constructed ultralimits of pointed metric measure spaces and we investigated their relation with pmGH limits. We also used the ultralimit machinery to obtain new results on the category of pointed metric measure spaces.

Abstract: Kolmogorov's K41 theory of turbulence advances quantitative predictions on anomalous dissipation in incompressible fluids. This phenomenon can be described as follows: although smooth solutions of the Euler equations conserve the kinetic energy, in turbulent fluids the energy can be transferred  to high frequencies and anomalously dissipated. Hence turbulent solutions of the Navier-Stokes equations are expected to converge, in the vanishing viscosity limit, to irregular solutions of the Euler equations, with decreasing kinetic energy. In rigorous analytical terms, however, this phenomenon is little understood. In this talk, I will present the recent developments on this topic and focus on a joint work with G. Crippa and M. Sorella which considers the case of passive-scalar advection, where anomalous dissipation is predicted by the Obukhov-Corrsin theory of scalar turbulence. I will discuss the construction of a velocity field and a passive scalar exhibiting anomalous dissipation in the supercritical Obukhov-Corrsin regularity regime. The techniques developed in this context allow also to answer the question of (lack of) selection for passive-scalar advection under vanishing diffusivity. Finally, I will present a joint work with E. Brue’, G. Crippa, C. De Lellis, and M. Sorella, where we use the previous construction to give example of anomalous dissipation for the forced Navier-Stokes equations in the supercritical Onsager regularity regime.

Abstract: Deep neural networks are commonly initialized with random parameters from Gaussian distributions. This talk analyzes the relationship between the output distribution of such a randomly initialized neural network and an equivalent Gaussian process distribution. Explicit inequalities are derived bounding the quadratic Wasserstein distance between the network outputs and a Gaussian distribution as a function of the network architecture. As the hidden layer sizes increase, the bounds quantify how the network output distribution converges to a Gaussian in the so-called "wide limit". Furthermore, the bounds can be extended to characterize the Gaussian approximation of the exact Bayesian posterior distribution over the weights. The results provide a quantitative mathematical understanding of when and why random neural networks exhibit Gaussian-like behavior, with implications for modeling and analysis in machine learning applications. Joint work with A. Basteri (arXiv:2203.07379). 

Abstract: I will survey a serie of recent results on the two phase Bernoulli problem and on the structure of the free boundaries. In particular I will show some recent regularity results for the free boundaries and for the structure of the branch set in 2 dimension. 

Abstract: We study a popular kinetic model introduced by Saintillan and Shelley for the dynamics of suspensions of active elongated particles. We focus on the linear analysis of incoherence, that is on the linearized equation around the uniform distribution, in the regime of parameters corresponding to spectral (neutral) stability. We show that in the absence of rotational diffusion, the suspension experiences a mixing phenomenon similar to Landau damping. We show that this phenomenon persists for small rotational diffusion, and is combined with an enhanced dissipation at time scale at a faster time scale than the diffusive one. 

Abstract: In this talk, we present a comparison principle for the Hamilton Jacobi (HJ) equation corresponding to a Mean Field control problem in which one linearly controls the gradient flow of an energy functional defined on a metric space. The main difficulties are given by the fact that the geometrical assumptions we require on the energy functional do not give any control on the growth of its gradient flow nor on its regularity. Therefore this framework is not covered by previous results on HJ equations on infinite dimensional spaces (whose study has been initiated in a series of papers by Crandall and Lions). Our proof of the comparison principle combines some rather classical ingredients, such as Ekeland’s perturbed optimization principle, with the use of the Tataru distance and of the regularizing properties of gradient flows in evolutional variational inequality formulation, that we exploit for constructing rigorous upper and lower bounds for the formal Hamiltonian.  Some hints on the existence of solutions will also be given. 

Abstract: The transfer operator is an elegant way to capture the behaviour of a (stochastic) dynamical system as a linear operator. Spectral analysis can then in principle reveal (almost) invariant measures, cyclical behaviour, as well as separation of the dynamics into different time scales. In practice this analysis can rarely be done analytically, due to the complexity of the operator or since it may not be known in closed form. A central objective is therefore to numerically approximate this operator (or its adjoint: the Koopman operator) or to estimate it from data. In this talk we introduce a new estimation method based on entropic optimal transport and show convergence to a smoothed version of the original operator as more data becomes available. This involves an interplay between three different length scales: the discretization scale given by the data, the blur scale introduced by entropic transport, and the spatial scale of eigenfunctions of the operator. 

Abstract: We discuss recent developments in the use of interacting particles for solving high-dimensional optimization problems, and provide insights into analytical guarantees for convergence in the particle mean-field limit. 

Abstract: In this talk, I will describe a variant of the celebrated de Finetti, Hewitt and Savage theorem for finite exchangeable laws and discuss possible applications to some symmetric optimal transport problems. I will emphasize the role of some universal polynomials of quantized measures which capture correlated corrections and the convex geometry of symmetric laws. This is a joint work with Gero Friescke and Daniela Vögler (Munich).