In Lie theory we define root systems in several contexts: Lie algebras, superalgebras, affine algebras, etc. There is even more: Kostant defines a more general notion of root systems, by taking roots with respect to a generic toral subalgebra (i.e. not necessarily maximal). All these notions of root systems do not behave well with respect to quotients: the quotient (or projection) of a root systems is not in general a root system. We present here a more general notion of root system, inspired by Kostant, which accomodates all of the above examples and behaves well with respect to quotients and projections. <br>   We give a classification theorem for rank 2 generalized root system: there are only 14 of them up to combinatorial equivalence, moreover they are all quotients of Lie algebra root systems. We also prove that root systems of contragredient Lie superalgebras are quotients of root systems of Lie algebras, up to combinatorial equivalence. <br>   In the end, we relate our construction with the problem of determining the conjugacy class of two Levi subgroups in a Lie (super)algebra. <br>   <strong>N.B.:</strong> please <a href="https://teams.microsoft.com/l/meetup-join/19%3a342c23eced2540e4bcbb6f938e999db6%40thread.tacv2/1614163430980?context=%7b%22Tid%22%3a%2224c5be2a-d764-40c5-9975-82d08ae47d0e%22%2c%22Oid%22%3a%22cf31dbee-7758-4272-af72-503d7694f2ea%22%7d" target="Teams">click <strong>HERE</strong> to attend the talk</a> in streaming.

A morphism of smooth varieties of the same dimension is called real fibered if the inverse image of the real part of the target is the real part of the source. It goes back to Ahlfors that a real algebraic curve admits a real fibered morphism to the projective line if and only if the real part of the curve disconnects its complex part. Inspired by this result, in a joint work with Mario Kummer and Cédric Le Texier, we are interested in characterising real algebraic varieties of dimension n admitting real fibered morphisms to the n-dimensional projective space. We present a criterion to construct real fibered morphisms that arise as finite surjective linear projections from an embedded variety; this criterion relies on topological linking numbers. We address special attention to real algebraic surfaces. We classify all real fibered morphisms from real del Pezzo surfaces to the projective plane and determine when such morphisms arise as the composition of a projective embedding with a linear projection.

In this talk I will report on a joint work with S. Daudin (Paris Dauphine), Joe Jackson (U. Texas) and P. Souganidis (U. Chicago). We are interested in the convergence problem for the optimal control of McKean-Vlasov dynamics, also known as mean field control. We establish an algebraic rate of convergence of the value functions of N-particle stochastic control problems towards the value function of the corresponding McKean-Vlasov problem. This convergence rate is established in the presence of both idiosyncratic and common noise, and in a setting where the value function for the McKean-Vlasov problem need not be smooth. Our approach relies crucially on Lipschitz and semi-concavity estimates, uniform in N, for the N-particle value functions, as well as a certain concentration inequality. <br> <br> <a href="https://teams.microsoft.com/l/meetup-join/19%3a7f273a2aff1145dfae91372d186b1cd9%40thread.tacv2/1645553134652?context=%7b%22Tid%22%3a%2224c5be2a-d764-40c5-9975-82d08ae47d0e%22%2c%22Oid%22%3a%22d37d6fea-2e4d-4c35-88e4-99bf4cf68fe9%22%7d"> <strong> MS Teams Link for the streaming </strong> <em> </strong> </a> <br> <strong> Note: </strong> This talk is part of the activity of the MIUR Department of Excellence Project MATH@TOV CUP E83C18000100006

We will present joint work with Olivier Debarre, Daniel Huybrechts and Claire Voisin on the SYZ hyper-Kähler conjecture for fourfolds under certain topological assumptions. As application, this proves a conjecture by O'Grady that a hyper-Kähler fourfold whose cohomology ring is isomorphic to the one of the Hilbert square of a K3 surface is a deformation of a Hilbert square.

  Let <strong>g</strong> be a simple Lie algebra, with a fixed Borel subalgebra <strong>b</strong> = <strong>t</strong>+<strong>n</strong> , and let <em>W</em> be the associated Weyl group. The Steinberg map associates to any element of <em>W</em> a nilpotent orbit in <strong>g</strong>, which is defined by the corresponding set of inversions. Extending on previous work of Fan and Stembridge, in this talk I will compare two different notions of "smallness", one available in the Weyl group and the other one for nilpotent orbits. <br>   <strong>N.B.:</strong> please <a href="https://teams.microsoft.com/l/meetup-join/19%3a342c23eced2540e4bcbb6f938e999db6%40thread.tacv2/1614163430980?context=%7b%22Tid%22%3a%2224c5be2a-d764-40c5-9975-82d08ae47d0e%22%2c%22Oid%22%3a%22cf31dbee-7758-4272-af72-503d7694f2ea%22%7d" target="Teams">click <strong>HERE</strong> to attend the talk</a> in streaming.

&nbsp; In this talk we will discuss the nature of the relationship between the representations of a finite group <em>G</em> and those of a Sylow subgroup <em>P</em> of <em>G</em>. <br> &nbsp; We will introduce Sylow Branching Coefficients (SBCs) and we will show how the study of these numbers led us to prove a conjecture proposed by Malle and Navarro in 2012. We will conclude by presenting new results on SBCs in the case where <em>G</em> is the symmetric group. <br> &nbsp; The talk is based on joint works with Law, Long, Navarro, Vallejo and Volpato. <br>   <strong>N.B.:</strong> please <a href="https://teams.microsoft.com/l/meetup-join/19%3a342c23eced2540e4bcbb6f938e999db6%40thread.tacv2/1614163430980?context=%7b%22Tid%22%3a%2224c5be2a-d764-40c5-9975-82d08ae47d0e%22%2c%22Oid%22%3a%22cf31dbee-7758-4272-af72-503d7694f2ea%22%7d" target="Teams">click <strong>HERE</strong> to attend the talk</a> in streaming.

The analysis of the ergodic behavior of solutions to Hamilton-Jacobi-Bellmann equations has a long history going back to the seminal paper by [Lions, P.-L., Papanicolaou, G. and Varadhan, S.R.S]. Since this work, the subject has grown very fast and when the Hamiltonian is of Tonelli type a large number of results have been proved. A full characterization of the ergodic behavior of solutions to Tonelli Hamilton-Jacobi equations can be found in the celebrated weak KAM theory and Aubry-Mather theory. However, few results are available if the Hamiltonian fails to be Tonelli, i.e., the Hamiltonian is neither strictly convex nor coercive with respect to the momentum variable. In particular, such results cover only some specific structure and so, the general problem is still open. In this talk, I will present some recent results obtained in collaboration with Piermarco Cannarsa and Pierre Cardaliaguet concerning the long time-average behavior of solutions to Hamilton-Jacobi-Bellman equations. We will look, first, to the case of control of acceleration and, then, to sub-Riemannian control systems. Finally, we conclude this talk showing how the previous analysis applies to mean field game systems. <br> <br> <a href="https://teams.microsoft.com/l/meetup-join/19%3a7f273a2aff1145dfae91372d186b1cd9%40thread.tacv2/1644948272731?context=%7b%22Tid%22%3a%2224c5be2a-d764-40c5-9975-82d08ae47d0e%22%2c%22Oid%22%3a%22d37d6fea-2e4d-4c35-88e4-99bf4cf68fe9%22%7d"> <strong> MS Teams Link for the streaming </strong> <em> </strong> </a> <br> <strong> Note: </strong> This talk is part of the activity of the MIUR Department of Excellence Project MATH@TOV CUP E83C18000100006

Let G be a complex (connected) reductive group and C be a complex smooth projective curve of genus at least four. It is known that the moduli space of semistable G-bundles over C is a projective variety. The automorphism group of this variety contains the so-called tautological automorphisms: they are induced by the automorphisms of the curve C, outer automorphisms of G and tensorization by Z-torsors, where Z is the center of G. It is a natural question to ask if they generate the entire automorphism group. Kouvidakis and Pantev gave a positive answer when G=SL(n). An alternative proof has been given by Hwang and Ramanan. Later, Biswas, Gomez and Muñoz, after simplifying the proof for G=SL(n), extended the result to the symplectic group Sp(2n). All the proofs rely on the study of the singular fibers of the Hitchin fibration. In this talk, we present a recent work where, by adapting the Biswas-Gomez-Muñoz strategy, we describe the automorphism group of the connected components of the moduli space of semistable G-bundles over C, for any almost-simple group G.

In a first part of the talk, I will present some geometric techniques allowing to control quantum systems using slowly-varying controls, in the so-called adiabatic regime. The latter provides strong control results only when the system is driven by at least two controls, which is a strong requirement in practice. The second part of the talk will be dedicated to an averaging approximation (Rotating Wave Approximation) which allows to duplicate controls in this setting.<br> <b>NB</b>:<i>This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C18000100006</i>