OPTiMiSE — Talks & Presentations
2026
- Graduate School in Analysis and Applications, ESI, Vienna
June 22–26, 2026
4 Lectures on Variational methods for evolution - 5th Italian Meeting on Probability and Mathematical Statistics (IMPMS 2026), University of Palermo
June 8–12, 2026
From the Monge–Kantorovich problem to optimal transport for random measures — Plenary talkThe Monge–Kantorovich problem, originally posed as the optimal rearrangement of a mass distribution and later relaxed by Kantorovich into a problem of optimal coupling between two probability measures, has grown over the last decades into a rich theory that provides a natural geometric language for probability measures. I will first recall the main ideas of optimal transport and the geometry of the Wasserstein space, working throughout in the \(L^2\) setting, with emphasis on couplings and the role of convex functions.
I will then turn to optimal transport for random measures, namely the laws of random variables taking values in a space of probability measures. The theory exhibits striking analogies with the Euclidean case and highlights, on the one hand, the role of random measures admitting a Gaussian lift in the space of maps, and, on the other, the role of the distinguished class of totally convex functions. This offers a new perspective on differential tools already developed in classical optimal transport. I will illustrate this picture through a few examples and comment on some of the questions and perspectives it raises. - UMI — Incontro Scientifico di Maggio, University of Bologna
May 22, 2026
Strutture variazionali nei problemi di evoluzioneMolti problemi di evoluzione sono formulati come equazioni differenziali che descrivono localmente la dinamica nel tempo. In casi significativi, la stessa dinamica ammette anche una formulazione variazionale, fondata su principi globali che mettono in relazione energia, dissipazione e ottimalità delle traiettorie. Questo cambio di prospettiva permette di cogliere aspetti strutturali comuni a problemi molto diversi, e si rivela particolarmente efficace nello studio della stabilità, dell’approssimazione e dei passaggi al limite. La conferenza si propone di presentare alcune di queste idee a un livello introduttivo, con un cenno finale a sviluppi più recenti in contesti metrici e probabilistici, in cui l’evoluzione di misure svolge un ruolo centrale.
- Distinguished Lecture 2026, Durham University, UK
May 7, 2026
Diffusion, Optimal Transport and Ricci CurvatureThe interplay between diffusion, optimal transport and Ricci curvature has been an active theme at the interface of analysis, geometry and probability over the last two decades. On the analytic side, Bakry–Émery’s \(\Gamma\)-calculus characterises lower Ricci curvature bounds through gradient estimates for the heat semigroup, within the framework of Dirichlet forms. On the geometric side, the work of Lott, Sturm and Villani shows that the same curvature information is encoded in the displacement convexity of the relative entropy along Wasserstein geodesics, leading to a synthetic notion of Ricci curvature that makes sense on general metric measure spaces.
The two viewpoints meet on the class of RCD spaces, developed in a series of papers with L. Ambrosio and N. Gigli. On such spaces the heat flow is linear — equivalently, the Cheeger energy is quadratic — and admits a double description as the \(L^2\)-gradient flow of the Dirichlet energy and as the Wasserstein EVI flow of the entropy. Together with stability under measured Gromov–Hausdorff convergence, this allows RCD spaces to inherit many of the structural properties and calculus tools available on smooth Riemannian manifolds.
The lecture will present the main ideas behind this circle of results and will close with some recent developments along related directions: gradient flows for convex functionals on RCD spaces, Hellinger–Kantorovich contractions, and metric Sobolev structures on spaces of random measures. - Multiscale Stochastics, Patterns, and Analysis of Combinatorial Environments (mSPACE Kickoff Meeting), Bocconi University, Milano
March 16–19, 2026
The De Giorgi Variational Principle for Gradient Flows: A Direct ApproachDe Giorgi's variational principle provides a global-in-time characterization of gradient flows of a functional \(\phi\) as curves minimizing an energy-dissipation functional. While the existence of minimizers is relatively easy to establish, the original formulation does not immediately imply that the minimum value coincides with the initial energy \(\phi(x_0)\). This identification is crucial, since only minimizers attaining this value correspond to genuine solutions of the gradient flow. We present a direct variational approach showing how to prove that the minimum equals \(\phi(x_0)\), without relying on time-discretization or iterative constructions such as the minimizing movements scheme. The key idea is to replace iterative arguments with a global method based on relaxation, convexification, and duality, allowing one to identify the value of the minimum directly at the continuous level. Although the presentation focuses on the simplest case of gradient flows in Hilbert spaces, the same strategy extends to more general frameworks, including doubly nonlinear equations in reflexive Banach spaces, metric gradient flows, and time-dependent energies.
Joint work with Alessandro Pinzi and Filippo Riva. - Horizons in Nonlinear PDEs — Spring School 2026, Universität Ulm, Germany
March 9–13, 2026
Variational Principles for Evolution ProblemsThe course provides an introduction to variational methods for evolution equations in Hilbert and metric spaces. After reviewing the main generation results for gradient flows and contraction semigroups, we will discuss and apply the abstract theory to the paradigmatic example of the Derrida-Lebowitz-Speer-Spohn (DLSS) equation.
- Recent Advances in PDEs and Geometric Analysis, Politecnico di Milano
February 17–18, 2026
Variational Convergence of Metric Gradient FlowsIn this talk we address variational convergence and stability properties of gradient flows in metric spaces. We begin by introducing the notion of metric gradient flow in the sense of Evolution Variational Inequalities (EVI), which provides a robust and intrinsic characterization of dissipative evolutions beyond the Hilbertian setting. Given a sequence of functionals \((\phi_h)\) defined on a metric space and generating corresponding gradient flows, we study the problem of understanding when and how the associated evolutions converge as the functionals vary. In the classical Hilbert space framework, convergence of gradient flows is closely related to Mosco (i.e. \(\Gamma\)-convergence with respect to the strong and weak topology) of convex energies and to the convergence of resolvent operators. In a general metric setting, however, several key tools are missing: resolvents may fail to be well defined or contractive, compactness is often unavailable, and no natural weak topology is present. We present a variational approach to stability that avoids strong coercivity assumptions and relies instead on quantitative error estimates for minimizing movement schemes and their Ekeland relaxations. These estimates provide a robust bridge between discrete variational approximations and continuous EVI flows, and allow one to transfer convergence properties from the energies to the evolutions under suitable variational assumptions. The results apply to a broad class of metric spaces admitting EVI gradient flows and yield general convergence criteria that extend classical Hilbert space results to genuinely metric frameworks, with applications for instance to Wasserstein and RCD spaces.
Joint work with Matteo Muratori.