The celebrated Pontryagin Maximum Principle (PMP) provides a (first order) necessary condition for the optimality of trajectories of optimal control problems. In most cases, however, a trajectory satisfying PMP is not optimal. For these reasons, additional optimality conditions are required. In this context, Hamiltonian methods are quite effective in establishing sufficient optimality conditions. In this talk, after a brief review of the main ideas of the general method, we will focus on optimal control problems associated with control-affine dynamics and costs of the form <br> $$ int_0^T |u(t)| |varphi( X(t))| dt $$ <br> Costs of these form are very common in problems modeling neurobiology, mechanics and fuel-consumption. <br> <br> <a href="XXXsitoX"> <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

The treatment of non-relativistic interacting bosonic systems, exhibiting condensation in the limit of large particle numbers, is commonly based on studies of single particle density matrices, determined from the microscopic equilibrium states. In order to exhibit more detailed properties of these states, such as correlations between observables, one needs an algebra that is stable under the underlying dynamics and remains meaningful in the limit. In the present talk it is shown that the resolvent algebra of canonical quantum systems provides such a framework. The popular mean field, dilute gas and Gross-Pitaevskii approximations of the interactions lead to C*-dynamical systems based on the resolvent algebra. This fact implies that the limits of equilibrium states are still in equilibrium, satisfying the KMS condition. Moreover, the resolvent algebra contains all observables needed to study the condensates and their thermal background. If time permits, these results are illustrated by examples. <br> <br> This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C18000100006. <br> <br> Send an email to v.morinelli@gmail.com to get the link to the seminar.

&nbsp; In this talk I will introduce the category of perverse sheaves on a topologically stratified space <em>X</em> and give some examples. Then, I will show that when <em>X</em> has finitely many strata, each with finite fundamental group, such category is equivalent to a category of modules over a finite dimensional algebra <em>A</em>. Finally, I will discuss some algebraic approaches one can use in order to describe the algebra <em>A</em>. <br> &nbsp; This talk is based on joint work with Jon Woolf. <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

Let N be a compact Riemannian manifold of dimension 3 or higher, and g a Lipschitz non-negative (or non-positive) function on N. We prove that there exists a closed hypersurface M whose mean curvature attains the values prescribed by g (joint work with Neshan Wickramasekera, Cambridge). Except possibly for a small singular set (of codimension 7 or higher), the hypersurface M is C^2 immersed and two-sided (it admits a global unit normal); the scalar mean curvature at x is g(x) with respect to a global choice of unit normal. More precisely, the immersion is a quasi-embedding, namely the only non-embedded points are caused by tangential self-intersections: around such a non-embedded point, the local structure is given by two disks, lying on one side of each other, and intersecting tangentially (as in the case of two spherical caps touching at a point). A special case of PMC (prescribed-mean-curvature) hypersurfaces is obtained when g is a constant, in which the above result gives a CMC (constant-mean-curvature) hypersurface for any prescribed value of the mean curvature. The construction of M is carried out largely by means of PDE principles: (i) a minmax for an Allen--Cahn (or Modica-Mortola) energy, involving a parameter that, when sent to 0, leads to an interface from which the desired PMC hypersurface is extracted; (ii) quasi-linear elliptic PDE and geometric-measure-theory arguments, to obtain regularity conclusions for said interface; (iii) parabolic semi-linear PDE (together with specific features of the Allen-Cahn framework), to tackle cancellation phenomena that can happen when sending to 0 the Allen-Cahn parameter. <br> <br> <a href="https://teams.microsoft.com/l/meetup-join/19%3a7f273a2aff1145dfae91372d186b1cd9%40thread.tacv2/1621517541296?context=%7b%22Tid%22%3a%2224c5be2a-d764-40c5-9975-82d08ae47d0e%22%2c%22Oid%22%3a%22d37d6fea-2e4d-4c35-88e4-99bf4cf68fe9%22%7d"> <strong> Link </strong> <em> </strong> </a> <br> Note: Sponsored by the Grant MATH@TOV

Abstract: The aim of the talk is to provide conditions ensuring Connes' measurability of the canonical, $(1,infty)$-summable, spectral weight $ho(D)$, we associate to any self-adjoint operator $D$ with discrete spectrum and of its associated state $phi_D(T):={ m Tr,}_omega (T ho(D))$. The framework allows to discuss measurability for discrete groups with sub-exponential growth and $ heta$-summable spectral triples. Examples also include 1. the volume density of Euclidean domains of infinite volume 2. densities on the C$^*$-algebra of pseudo-differential operators of 0-order on a compact Riemannian manifold 3. densities on Toeplitz extensions. <br> <br> This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C18000100006. <br> <br> <a hrref=https://teams.microsoft.com/l/meetup-join/19%3a428ea736adc6424c8ae37f187c91b51b%40thread.tacv2/1621498100746?context=%7b%22Tid%22%3a%2224c5be2a-d764-40c5-9975-82d08ae47d0e%22%2c%22Oid%22%3a%22e6325df7-3e74-4c88-ab0b-65ff8a758e69%22%7d"> Link MS Teams</a>

&nbsp; An <em>R</em>-matrix is a solution to the Yang-Baxter equation (YBE), a central object in Statistical Mechanics, discovered in 1970's. The <em>R</em>-matrix also features prominently in the theory of quantum groups formulated in the eighties. In recent years, many areas of mathematics and physics have found methods to construct <em>R</em>-matrices and solve the associated integrable system. <br> &nbsp; In this talk I will present one such method, which produces meromorphic solutions to (YBE) starting from the representation theory of a family of quantum groups called Yangians. Our techniques give <em>(i)</em> a constructive proof of the existence of the universal <em>R</em>-matrix of Yangians, which was obtained via cohomological methods by Drinfeld in 1983, and <em>(ii)</em> prove that Drinfeld's universal <em>R</em>-matrix is analytically well behaved. <br> &nbsp; This talk is based on joint works with Valerio Toledano Laredo and Curtis Wendlandt. <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

After a very brief review of basics on elliptic curves and their families, we shall consider "sections" of such families, and especially their "torsion values". For instance, what can be said of the complex numbers b for which (2, sqrt{2(2-b)}) is torsion on the Legendre curve y^2=x(x-1)(x-b)? In particular, we shall recall results of "Manin-Mumford type" and focus to illustrate some applications to elliptical billiards. Finally, if time allows we shall frame these issues as special cases of a general question in arithmetic dynamics, which can be treated with different methods, depending on the context. (Most results refer to work with Pietro Corvaja and David Masser.) <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>

<a href="~ricerca/analis/seminario_Molica-Bisci.pdf">Link to the abstract</a> <br> <b>NB</b>:<i>This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C18000100006</i>