&nbsp; After explaining some known basic facts about elliptic root systems (ERS) of reduced type, I will show the classification and automorphism groups of ERS of non-reduced type. Some future problems will be discussed. <br> &nbsp; These results are obtained in collaboration with A. Fialowski and Y. Saito. <br> &nbsp;  <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; An amazing nabla operator discovered by Bergeron and Garsia is a cornerstone of the theory of Macdonald polynomials. Applying it to various symmetric functions produces interesting generating functions of Dyck paths and parking functions. These kind of results are sometimes known as "shuffle theorems". I will try to give an overview of these results and explain how working with affine permutations and certain generalized P-tableaux allows to view them from a uniform point of view. The "new" in the title refers to the formula conjectured by Loehr and Warrington giving an explicit expansion of nabla of a Schur function in terms of nested Dyck paths. <br> &nbsp; This is a joint work with Erik Carlsson. <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

In control theory, chattering refers to fast oscillations of controls, such as accumulation of switchings in finite time. This behavior is rather typical, as it is the case for the class of single-input control-affine problems, and may be a serious obstacle to convergence of standard numerical methods to detect optimal solutions. We propose a general regularization procedure, consisting of penalizing the cost functional with a total variation term. Under appropriate assumptions of small-time local controllability, we prove that the optimal cost and any optimal solution of the regularized problem converge respectively to the optimal cost and an optimal solution of the initial problem. Our approach is valid for general classes of nonlinear optimal control problems and applies to chattering phenomena appearing in constrained problems as well as to switching systems. We also quantify the error in terms of the rate of convergence of the sequence of switching times, for systems with regular time-optimal map. <br> <b>NB</b>:<i>This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C18000100006</i>

&nbsp;  The talk is based on work in progress with Francesco Esposito and Lleonard Rubio y Degrassi. <br> &nbsp; Recently Kapranov and Schechtman have settled an equivalence between the category of graded connected co-connected bialgebras in a braided monoidal category <em>W</em> and the category of factorizable systems of perverse sheaves on all symmetric products Sym^<em>n</em>(<strong>C</strong>) with values in <em>W</em>. The Nichols (shuffle) algebra associated with an object <em>V</em> corresponds to the system of intersection cohomology extensions of a precise local system on the open strata. <br> &nbsp; Motivated by the study of Fomin-Kirillov algebras and their relation with Nichols algebras, we describe the factorizable perverse sheaves counterpart of some algebraic constructions, including the <em>n</em>-th approximation of a graded bialgebra, and we translate into geometric statements when a Nichols algebra is quadratic. <br> &nbsp; <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 dynamics of an inviscid and incompressible fluid flow on a Riemannian manifold is governed by the Euler equations. Recently, Tao launched a programme to address the global existence problem for the Euler and Navier Stokes equations based on the concept of universality. Inspired by this proposal, we show that the stationary Euler equations exhibit several universality features, In the sense that, any non-autonomous flow on a compact manifold can be extended to a smooth stationary solution of the Euler equations on some Riemannian manifold of possibly higher dimension. These results can be viewed as lending support to the intuition that solutions to the Euler equations can be extremely complicated in nature. A key point in the proof is looking at the h-principle in contact geometry through a contact mirror, unveiled by Sullivan, Etnyre and Ghrist more than two decades ago. We end up this talk addressing an apparently different question: What kind of physics might be non-computational? Using the former universality result, we can establish the Turing completeness of the steady Euler flows, i.e., there exist solutions that encode a universal Turing machine and, in particular, these solutions have undecidable trajectories.. This talk is based on joint work with Robert Cardona and Fran Presas. <br> <br> <strong> Note: </strong> <em> </strong> <em> The zoom link to the seminar will be posted on the <a href="https://www.dinamici.org/dai-seminar/"> DinAmicI website </a> and on <a href="https://mathseminars.org/seminar/DinAmicI"> Mathseminars.org</a>. Moreover, it will be also streamed live via the youtube <a href="https://www.youtube.com/channel/UCyNNg155G3iLS7l-qZjboyg"> DinAmicI channel</a>. </em> </em>

In this talk, I will present some recent results concerning the existence of prescribed norm solutions in problems where the associated nonlinearity is the sum of two powers, one which is mass-subcritical and one mass-supercritical. This leads to consider a constrained variational problem presenting a so-called convex-concave geometry. The issues of existence, multiplicity and orbital stability of solutions will be addressed with a special emphasize on the cases where the mass-supercritical power is Sobolev critical. The content of this talk is based on some jointed work with J. Jendrej, T. T. Le and N. Visciglia.

&nbsp; The sheets of a variety <em>X</em> under the action of an algebraic group <em>G</em> are the irreducible components of subsets of elements of <em>X</em> with equidimensional <em>G</em>-orbits. For <em>G</em> complex connected reductive, the sheets for the adjoint action of <em>G</em> on its Lie algebra <strong>g</strong> were studied by Borho and Kraft in 1979. In 2016, Losev introduced finitely many subvarieties of <strong>g</strong> consisting of equidimensional orbits, called birational sheets: their definition is less immediate than the one of a sheet, but they enjoy better geometric and representation-theoretic properties and are central in Losev's suggestion of an Orbit method for semisimple Lie algebras. <br> &nbsp; In the opening part of the seminar we give a brief overview of sheets and recall some basics about Lusztig-Spaltenstein induction of conjugacy classes in terms of the so-called Springer generalized map and analyse its interplay with birationality. This will give the instruments to introduce Losev's birational sheets in <strong>g</strong>. <br> &nbsp; The main part is aimed at investigating analogues of birational sheets of conjugacy classes in <em>G</em>. To conclude, assuming that the derived subgroup of <em>G</em> is simply connected, we illustrate the main features of these varieties, comparing them with the objects defined by Losev. <br> &nbsp; Part of the talk is based on joint works with G. Carnovale and F. Esposito, and M. Costantini. <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.

  Let <em>G</em> be a complex reductive group, <em>G</em>* its dual Poisson-Lie group, and <strong>g</strong> the Lie algebra of <em>G</em>. <em>G</em>-valued Stokes phenomena were exploited by P. Boalch to linearise the Poisson structure on <em>G</em>*. I will explain how <em>U</em>(<strong>g</strong>)-valued Stokes phenomena can be used to give a purely transcendental construction of the quantum group <em>U_h</em>(<strong>g</strong>), and show that the semiclassical limit of this construction recovers Boalch's. The latter result is joint work with Xiaomeng Xu. <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

When one considers manifolds with boundary, billiard dynamics are the natural analogue of standard geodesic dynamics. Namely, instead of having geodesics escape at the boundary, we force them back into the manifold using the reflection law. In other dynamical settings, similar constructions are possible: In 2006, B. Khesin and S. Tabachnikov initiated the study of billiards in the semiriemannian setting, studying the integrability of various tables. In recent years we have also seen the appearance of several billiard setups of symplectic nature. In this talk I will discuss recent work with L. Dahinden in which we look at billiards in subriemannian geometry. I will sketch how the reflection law arises naturally both from the control-theoretical and symplectic perspectives, how the reflection is problematic at tangency points between the distribution and the boundary of the table, and I will introduce some concrete examples. My ultimate goal will be to pose several intriguing open questions. <br> <br> <strong> Note: </strong> <em> </strong> <em> The zoom link to the seminar will be posted on the <a href="https://www.dinamici.org/dai-seminar/"> DinAmicI website </a> and on <a href="https://mathseminars.org/seminar/DinAmicI"> Mathseminars.org</a>. Moreover, it will be also streamed live via the youtube <a href="https://www.youtube.com/channel/UCyNNg155G3iLS7l-qZjboyg"> DinAmicI channel</a>. </em> </em>

We consider the bilinear Schrödinger equation with a discrete-spectrum free Hamiltonian and we present a general method for the approximate controllability based on Lie-algebraic control techniques applied to suitable finite dimensional approximation of the Galerkin-type. Under some regularity assumptions on the Hamiltonians and generic conditions on the controllability of the finite dimensional Galerkin approximations we show exact controllability in projection on an arbitrary given number of eigenstates. <br> <b>NB</b>:<i>This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C18000100006</i></div>