The digital signature schemes that have been proposed so far in the setting of the Supersingular Isogeny Diffie-Hellman scheme (SIDH) were obtained by turning an interactive identification protocol by De Feo, Jao and Plût into non-interactive schemes. The security of the resulting schemes is therefore deduced from that of the base identification protocol. In this talk, we revisit the proofs that have appeared in the literature for the special soundness property of the above-mentioned SIDH-based identification protocol. The existence of some special cycles in supersingular isogeny graphs make such previous proofs fail.

  In this talk, we will discuss the biclosedness of the monoidal categories of modules and comodules over a (left or right) Hopf algebroid, along with the bimodule category centres of the respective opposite categories and a corresponding categorical equivalence to anti Yetter-Drinfel'd contramodules and anti Yetter-Drinfel'd modules, respectively. This is directly connected to the existence of a trace functor on the monoidal categories of modules and comodules in question, which in turn allows to recover (or define) cyclic operators enabling cyclic cohomology.
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  Kraft and Procesi showed that the Zariski closure of the conjugacy classes of type A are all normal and, in type B, C and D, they have described which ones are normal. In their work the Lie group acts on its Lie algebra by the adjoint action. In types B, C, D, a similar question can be asked for the action of the Lie group on the odd part of the general linear Lie algebra; that is the orthogonal group acting on the symmetric matrices and the symplectic group acting on the symmetric-symplectic matrices. Ohta showed that in the latter case every orbit has normal closures while this conclusion is not valid in the former case. In this talk I will present the main result of my Ph.D. thesis which gives a combinatorial description of the orbit whose closures are normal in the orthogonal case.
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We will consider a nonlinear elliptic system in R^N. The interest of this problem is based on the presence of critical power nonlinearities and a nonlinear coupling, possibly critical, as well as Hardy-type singular potentials. By means of variational methods, we will focus on the existence of solutions. More precisely, we shall derive new results on bound and ground states of the underlying energy functional. Finally, we extend our results to the special case of ''Schrödinger-Korteweg-de Vries'' coupling type term. This seminar is based on a couple of works in collaboration with Eduardo Colorado and Alejandro Ortega (UC3M).

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

Let (S,L) be a general K3 surface of genus g. I will prove that the closure in |L| of the Severi variety parametrizing curves in |L| of geometric genus h is connected for h>=1 and irreducible for h>=4, as predicted by a well known conjecture. This is joint work with Andrea Bruno.

In the last few years, tropical methods have been applied quite successfully in understanding several aspects of the geometry of classical algebro-geometric moduli spaces. In particular, in several situations the combinatorics behind compactifications of moduli spaces have been given a tropical modular interpretation. Consequently, one can study different properties of these (compactified) spaces by studying their tropical counterparts. In this talk, which is based in joint work with Madeleine Brandt, Juliette Bruce, Melody Chan, Gwyneth Moreland and Corey Wolfe, I will illustrate this phenomena for the moduli space Ag of abelian varities of dimension g. In particular, I will show how to apply the tropical understanding of the classical toroidal compactifications of Ag to compute, for small values of g, the top weight cohomology of Ag. The techniques we use follow the breakthrough results and techniques recently developed by Chan-Galatius-Payne in understanding the topology of the moduli space of curves via tropical geometry.

Artificial neural networks are nowadays a widespread tool in applied mathematics for approximation and classification purposes. In this talk, we will survey recent results on the use of neural networks in computing the solution of PDEs. First of all, we introduce the use of network functions for the computation of forward and inverse problems for PDEs. Then, we will focus on collocation methods with feedforward neural network with a single hidden layer and sigmoidal transfer functions randomly generated, the so-called extreme learning machines. We will present results on elliptic problems both in the linear (with sharp gradient) case and for the construction of bifurcation diagrams of nonlinear problems. The results are obtained in collaboration with Gianluca Fabiani and Costantinos Siettos. This talk is part of the activity of the MIUR Excellence Department Project CUP E83C18000100006.
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  The notion of Zeta-function for groups was introduced in a seminal paper from Grunewald, Segal and Smith and proved to be a powerful tool to study the subgroup growth in some classes of groups.
  In this seminar I will introduce this Zeta-function presenting some general properties for this object. I will then focus on some results obtained for class two nilpotent groups.
  I will in particular describe some combinatorial tecniques used to tackle this problem, namely the study of series associated to polyhedral integer cones.
  This is a joint work with Christopher Voll and Marlies Vantomme.
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  Let G be a simple complex algebraic group and let K be a reductive subgroup of G such that the coordinate ring of G/K is a multiplicity free G-module. We consider the G-algebra structure of C[G/K], and study the decomposition into irreducible summands of the product of irreducible G-submodules in C[G/K]. We will present a conjectural decomposition rule for some special reductive pairs together with some partial results supporting the conjecture. We will explain how our conjecture would actually follow from an old conjecture of Stanley on the multiplication of Jack symmetric functions. We will also present a few new basic results related to Stanley's conjecture itself. The talk is based on a collaboration with Jacopo Gandini.
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We investigate discrete thin objects which are described by a subset $X$ of $mathbb{Z}^d imes {0,dots, M-1 }^k$, for some $Minmathbb{N}$ and $d,kgeq 1$. We only require that $X$ is a connected graph and periodic in the first $d$-directions. We consider quadratic energies on $X$ and we perform a discrete-to-continuum and dimension-reduction process for such energies. We show that, upon scaling of the domain and of the energies by a small parameter $varepsilon$, the scaled energies $Gamma$-converges to a $d$-dimensional functional. The main technical points are a dimension-lowering coarse-graining process and a discrete version of the p-connectedness approach by Zhikov. This is a joint work with A. Braides.

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