Dipartimento di Matematica

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The notion of a holomorphic motion was introduced by Mané, Sad and Sullivan in the 1980's, motivated by the observation that Julia sets of rational maps often move holomorphically with holomorphic variations of the parameters. In the years that followed, the study of the behavior of various set-functions under holomorphic motions became an area of significant interest. For instance, holomorphic motions played a central role in the work of Astala on distortion of Hausdorff dimension and area under quasiconformal mappings.

In this talk, I will first review the basic notions and results related to analytic capacity and holomorphic motions, including the extended lambda lemma. I will then present some recent results on the behavior of analytic capacity under holomorphic motions. The proofs involve different notions such as conformal welding, quadratic Julia sets and harmonic measure. This is joint work with Tom Ransford and Wen-Hui Ai.

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ABSTRACT

The minimum spanning tree (MST) problem is a combinatorial optimization problem with many applications, well beyond its historical introduction for network design. The study of its random instances on Euclidean models, e.g., on complete graphs obtained by sampling i.i.d. uniform points on a d-dimensional cube, is classical, with many limit results as the number of the points grows. In this talk, I will present two new results for its bipartite counterpart, i.e., with an additional colouring (red/blue) of the points and allowing connections only between different colours. First, we prove that the maximum vertex degree of the MST grows logarithmically, in contrast with the non-bipartite case, where a uniform bound holds, depending on d only -- a fact crucially used in many classical results. Despite this difference, we then argue that the cost of the MST, suitably normalized, converge a.s. to a limiting constant that can be represented as a series of integrals, thus extending a result of Avram and Bertsimas to the bipartite case and confirming a conjecture by Riva, Malatesta and Caracciolo. Joint work with M. Correddu, Università di Pisa.

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Interpolation inequalities play an essential role in Analysis with fundamental consequences in Mathematical Physics, Nonlinear Partial Differential Equations (PDEs), Markov Processes, etc., and have a wide range of applications in various areas of Science. Research interests have evolved over the 80 years: while mathematicians were originally focussed on abstract properties (like notions of weak solutions and Cauchy problem in PDEs), more qualitative questions (for instance, bifurcation diagrams, multiplicity of the solutions in PDEs and their qualitative behaviour) progressively emerged. Entropy methods for nonlinear PDEs is a typical example: in some cases, the optimal constant in the inequality can be interpreted as the optimal rate of decay of an entropy for an associated evolution equation. Much more can be learned on the way.
This lecture is intended to give an overview of various results on some Gagliardo-Nirenberg-Sobolev and Caffarelli-Kohn-Nirenberg inequalities obtained during the last decade. It will not be a global picture of an active area of research but more a series of snapshots aiming at the illustration of some emerging tools and new directions of research.

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A pair of particles, one from inside, one from outside the Fermi sea, can be considered as a boson. Such a simplifying picture allows to obtain the correlation energy for both high density and low density Fermi systems, despite their completely different behavior.

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We provide a simple abstract formalism of integration by parts under which we obtain some regularization lemmas. These lemmas apply to any sequence of random variables which are smooth and non-degenerated in some sense and enable one to upgrade the distance of convergence from a smooth (Wasserstein e.g.) distance to the total variation in a quantitative way. We provide a result removing the costly assumption that some non-degeneracy is required along the whole sequence, as we require only non-degeneracy at the limit. The price to pay is to control the smooth distance between the Malliavin matrix of the sequence and the Malliavin matrix of the limit which is particularly easy in the context of Gaussian limits as their Malliavin matrix is deterministic. We show some applications of this result.
From joint papers with Vlad Bally and Guillaume Poly.

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Convex billiards are a classical topic in conservative dynamics. Typically, their dynamics is qualitatively very intricate, since it showcases a coexistence of hyperbolic dynamics and KAM phenomena. Understanding long-term statistical properties of the dynamics with the current technology is essentially an intractable problem.

Here I venture in the opposite direction and I will discuss dynamical inverse problems: how much geometrical information can be extracted from the dynamics?

More precisely: what can be deduced about the billiard table if one knows the lengths of all periodic orbits? The quantum version of this question has been famously stated as "Can one hear the shape of a drum?"

In this talk I will review the latest results and describe the next steps in this direction. This is a joint project with Vadim Kaloshin.

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We first review both the theory of discrete analytic functions and the main features of Schur analysis (a collection of problems pertaining to functions analytic and contractive in the open unit disk, and with a wide range of applications). Then, we present new connections between the theory of discrete analytic functions and Schur analysis. This allows us to define a new class of problems pertaining to discrete analytic functions.

References:

D. Alpay, P. Jorgensen, R. Seager, and D. Volok. On discrete analytic functions: Products, Rational Functions, and Reproducing Kernels. Journal of Applied Mathematics and Computing. Volume 41, Issue 1 (2013), Page 393-426.

D. Alpay and D. Volok, Discrete analytic functions and Schur analysis. Preprint, 2021.

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We discuss the existence, uniqueness, and regularity of invariant measures for the damped-driven stochastic Korteweg-de Vries equation, where the noise is additive and sufficiently non-degenerate. It is shown that a simple, but versatile control strategy, typically employed to establish exponential mixing for strongly dissipative systems such as the 2D Navier-Stokes equations, can nevertheless be applied in this weakly dissipative setting to establish elementary proofs of both unique ergodicity, albeit without mixing rates, as well as regularity of the support of the invariant measure. Under the assumption of large damping, however, a one-force, one-solution principle is established, from which we are able to deduce the existence of a spectral gap with respect to a Wasserstein distance-like function.