In this talk, we will provide a basic introduction to some of the fundamental ideas and results in machine learning, with emphasis on mathematical aspects. We will begin contrasting the modern data driven approach to modeling to classic mechanistic approaches. Then, we will discuss basic elements of machine learning theory connected to approximation theory, probability and optimization. Finally, we will discuss the need of new theoretical advances at the light of recent empirical observations while using deep neural networks.

  Given a root system and a prime number p we introduce a category X of “graded spaces with Lefschetz operators” over a ring A. Then we show that under a base change morphism from A to a field K this category specialises to representations of the hyperalgebra of a reductive group, if K is a field of positive characteristic, and of a quantum group at p^l-th root of unity, if K is the p^l-th cyclotomic field. In this category we then study torsion phenomena (over the ring A) and construct for any highest weight a family of universal objects with certain torsion vanishing condi-tions. By varying these conditions, we can interpolate between the Weyl modules (maximal torsion) and the tilting objects (no torsion). This construction might shed some light on the character generations philosophy of Lusztig and Lusztig-Williamson.
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In the talk a C¹ spline space over mixed meshes composed of triangles and quadrilaterals, suitable for FEM-based or isogeometric analysis, will be introduced. The mesh is considered to be a partition of a planar polygonal domain into triangles and/or quadrilaterals. The proposed space combines the Argyris triangle element with the C¹ quadrilateral element for polynomial degrees d ≥ 5. The spline space is assumed to be C² at all vertices and C¹ across edges, and the splines are uniquely determined by C²-data at the vertices, values and normal derivatives at chosen points on the edges, and values at some additional points in the interior of the elements. The motivation for combining the Argyris triangle element with a recent C¹ quadrilateral construction, inspired by isogeometric analysis, is two-fold: on one hand, the ability to connect triangle and quadrilateral finite elements in a C¹ fashion is non-trivial and of theoretical interest. On the other hand, the construction facilitates the meshing process by allowing more flexibility while remaining C¹ everywhere. This is for instance relevant when trimming of tensor-product B-splines is performed. As it will be demonstrated by numerical examples the presented spline space can be employed to various applications like interpolation and least square approximation of a given function or to solve the biharmonic equation via its weak form and Galerkin discretization. This talk is part of the activity of the MIUR Excellence Department Project CUP E83C18000100006.

For spectral actions made of the average number of particles and arising fromopen systems made of general free q-particles (including Bose, Fermi and classical ones corresponding to q= pm 1 and 0, respectively) in thermal equilibrium, we compute the asymptotic expansion with respect to the natural cut-off. We treat both relevant situations relative to massless and massive particles, where the natural cut-off is 1/eta=k_eta T and  1//sqrt{eta}, respectively. We show that the massless situation enjoys less regularity properties than the massive one. We also consider the passage to the continuum describing infinitely extended open systems in thermal equilibrium. We briefly discuss the appearance of condensation phenomena occurring for Bose-like q-particles, for which qin (0,1], after passing to the continuum. We also compare the arising results for the finite volume situation (discrete spectrum) with the corresponding infinite volume one (continuous spectrum).
NB:This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C18000100006

  In the work of Kang, Kashiwara, Kim, and Oh, the Schur-Weyl duality between quantum affine algebras and affine Hecke algebras is extended to certain Khovanov-Lauda-Rouquier (KLR) algebras, whose defining combinatorial datum is given by the poles of the normalised R-matrix on a set of representations.
  In this talk, I will review their construction and introduce a "boundary" analogue, consisting of a Schur-Weyl duality between a quantum symmetric pair of affine type and a modified KLR algebra arising from a (framed) quiver with a contravariant involution. With respect to the Kang-Kashiwara-Kim-Oh construction, the extra combinatorial datum we take into account is given by the poles of the normalised K-matrix of the quantum symmetric pair.
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  Hochschild cohomology is a fascinating invariant of an associative algebra which possesses a rich structure. In particular, the first Hochschild cohomology group HH¹(A) of an algebra A is a Lie algebra, which is a derived invariant and, among selfinjective algebras, an invariant under stable equivalences of Morita type. This establishes a bridge between finite dimensional algebras and Lie algebras, however, aside from few exceptions, fine Lie theoretic properties of HH¹(A) are not often used.
  In this talk, I will show some results in this direction. More precisely, I will explain how maximal tori of HH¹(A), together with fundamental groups associated with presentations of A, can be used to deduce information about the shape of the Gabriel quiver of A. In particular, I will show that every maximal torus in HH¹(A) arises as the dual of some fundamental group of A. By combining this, with known invariance results for Hochschild cohomology, I will deduce that (in rough terms) the largest rank of a fundamental group of A is a derived invariant quantity, and among self-injective algebras, an invariant under stable equivalences of Morita type. Time permitting, I will also provide various applications to semimonomial and simply connected algebras.
  This is joint work with Benjamin Briggs.
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Motivated by studies of axially symmetric stationary solutions of the Einstein vacuum equations in general relativity, we study singular harmonic maps from domains of the 3-dimensional Euclidean space to the hyperbolic plane, with bounded hyperbolic distance to extreme Kerr harmonic maps. We prove that every such harmonic map has a unique tangent map at the black hole horizon. As an application, we establish an explicit and optimal lower bound for the ADM mass in terms of the total angular momentum, in asymptotically flat, axially symmetric, and maximal initial data sets for the Einstein equations with multiple black holes. The talk is based on joint work with Marcus Khuri, Gilbert Weinstein, and Jingang Xiong.
NB:This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C18000100006

A Magnetic System describes the motion of a charged particle moving on a Riemannian Manifold under the influence of a magnetic field. Trajectories for this dynamics are called Magnetic Geodesics and one of the main tasks in the theory is to investigate the existence of Magnetic Geodesic which are closed. In general this depends on the magnetic system taken into account and on the topology of the base space. Inspired by the work of Bahri and Taimanov, I will introduce the notion of Magnetic Curvature which is a perturbation of the standard Riemannian curvature due to the magnetic interaction. We will see that Closed Magnetic Geodesic exist when the Magnetic Curvature is positive, which happens , for instance, when the magnetic field is sufficiently strong.

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Note: This talk is part of the activity of the MIUR Department of Excellence Project MATH@TOV CUP E83C18000100006

Isogeometric Analysis generalizes classical finite element analysis and intends to integrate it with the field of Computer-Aided Design. A central problem in achieving this objective is the reconstruction of analysis-suitable models from Computer-Aided Design models, which is in general a non-trivial and time-consuming task. This talk will present an overview of new piecewise-quadratic spline constructions that enable model reconstruction, as well as simulation of high-order PDEs on the reconstructed models. In particular, we will discuss splines on unstructured meshes in both two and three dimensions. This talk is part of the activity of the MIUR Excellence Department Project CUP E83C18000100006.

  I will discuss some projects in collaboration with D. Adamovic, V. Kac and P. Moseneder-Frajria regarding affine W-algebras. I will concentrate on the notion of collapsing level for not necessarily minimal W-algebras and I will illustrate some applications to the representation theory of affine algebras and, if time allows, to our conjectural classification of unitary representations for minimal W-algebras.
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