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08:00
09:00
10:00
11:00
12:00
13:00
14:00
15:00
16:00
17:00
Opening, registration
Ecole Française de Rome
08:30 - 09:00
Official words:: Christophe LEMOINE, Ministre Conseiller de l'Ambassade de France en Italie, Jean-Stéphane DHERSIN, Directeur Adjoint Scientifique de l'INSMI, Giorgio PATRIZIO, Presidente dell'INdAM, Alessandro GUILIANI. Directeur du LYSM: Official words: Christian MASSET, Ambassadeur de France en Italie, Jean-Stéphane DHERSIN, Directeur Adjoint Scientifique de l'INSMI, Giorgio PATRIZIO, Presidente dell'INdAM, Alessandro GUILIANI. Directeur du LYSM
Salle de Conférences de l'Ecole Française de Rome
09:00 - 09:30
Serge CANTAT "The Tits Alternative"
09:30 - 10:30
Abstract: The Tits alternative, initially proven by Jacques Tits around 1972, concerns the structure of groups of matrices, more precisely of subgroups of GL(V ) for any finite dimensional vector space V . As we shall see, there are three interacting perspectives in the Tits alternative, coming from algebra, geometry, and dynamics. What is the precise statement and the meaning of this alternative? How is it proven? Does it hold in other groups, for instance in groups of diffeomorphisms of compact manifolds, or in groups of algebraic transformations? I will discuss these questions at an elementary level, with a focus on explicit examples and an emphasis on the dynamical systems viewpoint.
09:30 - 10:30
Coffee break and end of registration
Galerie de l'Ecole Française de Rome
10:45 - 11:15
AMP: P. Bonicato "Moving currents: on the Lie transport equation"
11:15 - 11:35
Abstract: In the classical theory, one usually studies the transport equation looking for solutions in the class of functions. In the seminar I will talk about recent efforts, motivated by the modeling of defects in plastic materials, aimed at extending this theory to the case where the unknown is represented by $k$-currents in $R^d$ , i.e. generalised $k$-dimensional surfaces. I will explain the main challenges this problem presents and some recent results based on an ongoing project with G. Del Nin and F. Rindler (University of Warwick).
11:15 - 11:35
AMP,: Chiara Saffirio "Mean-field evolution of many interacting fermions"
11:35 - 11:55
Abstract: We will review recent progresses in the derivation of effective evolution equations for the dynamics of many weakly interacting fermions. We will focus on the mean-field regime and couple it with a semiclassical limit to obtain the Hartree-Fock and the Vlasov equations. We will compare the different methods used in this context and draw a comparison with results and methods employed in the case of classical particles. A particular emphasis will be given on the class of interactions and quantum states that we are able to treat.
11:35 - 11:55
AMP,: L. Lafleche "From many-body quantum dynamics with singular potentials to Vlasov equation: regularity and weak-strong uniqueness"
11:55 - 12:15
Abstract: In this talk I will present several techniques and concepts used in the context of the mean-field and the classical limit allowing to go from the N -body Schrödinger equation to the Hartree-Fock and Vlasov equation, linked to works in collaboration with Chiara Saffirio and Jacky Chong. To understand how close these equations are in the case of singular potentials such as the Coulomb potential, one possibility is to use weak-strong uniqueness principles and understand the similarity of the models to prove the propagation of a semiclassical notion of regularity uniformly in N and h. A typical obstacle is then the lack of positivity of the Wigner transform and its few conserved quantities. A solution to this problem is to consider operators and a quantum analogue of Sobolev spaces defined using Schatten norms. The absence of commutation requires however sharp bounds on commutators of operators without assuming high regularities.
11:55 - 12:15
Lucia CAPORASO "Moduli spaces and their completions"
12:30 - 13:30
Abstract: In algebraic geometry, moduli spaces parametrize equivalence classes of certain geometric objects (curves of given genus, abelian varieties of given dimension, ...). They carry a remarkable geometric structure governed by the deformation theory of the parametrized objects, and as such they are rarely complete. Constructing a geometrically meaningful completion for them is crucial for applications, and has been a widely investigated problem since the half of the 20th century. The talk will describe some general aspects and recent developments, focusing on the cases of curves and abelian varieties.
12:30 - 13:30
Lunch offered to all participants, avec une presentation of the École françaisw de Rome par Brigitte MARIN, Directrice.
Galerie de l'Ecole Française de Rome
13:45 - 15:15
AGT: S. Torelli "Holomorphic one forms on moduli of curves",
15:15 - 15:35
Abstract: The moduli space $\mathcal{M}_g$ of smooth projective curves of genus $g$ is a quasi-projective variety, singular on loci of dimension at most $2g-1$. Let $\mathcal{M}^0_g$ denote its smooth locus. Not much is known about the cohomology $H^i(\mathcal{M}^0_g, \mathbb{C})$ and even less about the spaces of holomorphic forms $H^i(\Omega^j_{\mathcal{M}^0_g})$. Notice that $\mathcal{M}_g$ is not compact, so in particular it doesn't carry a Hodge decomposition and thus $H^i(\Omega^j_{\mathcal{M}^0_g})$ can't be recovered from $H^i(\mathcal{M}^0_g, \mathbb{C})$ just using Hodge theory. In the talk I will present the result for $i=1,j=0$, namely that $\mathcal{M}_g$ do not admit holomorphic 1-forms, and I will briefly discuss its generalization to other moduli spaces realized as finite coverings of $\mathcal{M}_g$ (e.q. spin curves). The techniques comes from Hodge theory on the Deligne-Mumford compactification and intersection theory on the Satake compactification of $\mathcal{M}_g$. The work is joint with F.F. Favale and G.P.Pirola.)
15:15 - 15:35
AGT, O. Mohsen; "An introduction on maximally hypoelliptic differential operators".
15:35 - 15:55
Abstract:Maximally hypoelliptic operators are differential operators which enjoy regularity properties very similar to that of elliptic operators. I will give a brief introduction on maximally hypoelliptic differential operators, focusing on recent developments in the subject like index theory and principal symbol. No prior knowledge of elliptic operators is assumed for the talk.
15:35 - 15:55
AGT,: Y. Tanimoto "Unitary vertex algebras and Wightman conformal field theories",
15:55 - 16:15
Abstract: We report recent progress on the axiomatic approaches to two-dimensional conformal field theory. We prove the equivalence between unitary vertex operator algebras and Moebius-covariant Wightman fields with some analytic conditions. Sufficient conditions to construct conformal nets will also be discussed.
15:55 - 16:15
François GOLSE "Optimal transport distances in quantum mechanics"
16:30 - 17:30
Abstract: The optimal transport problem studied by Monge (1781) and Kantorovich (1942) provides a general method for metrizing the set of Borel probability measures on $\mathbf R^d$. The purpose of this talk is to present an analogous method for comparing density operators on $L^2(\mathbf R^d)$, which are the quantum mechanical analogues of probability measures on the phase space $\mathbf R^d\times\mathbf R^d$. We shall discuss some properties of the ``pseudo-distance’’ on quantum states obtained in this way, and show applications to some problems in quantum dynamics. (Based on joint works with E. Caglioti, C. Mouhot, T. Paul).
16:30 - 17:30
End of the first day
17:45 - 17:46
Mise à jour de l'ordre du jour...
