The only known examples of hyper-Kähler manifolds are constructed from moduli spaces of sheaves on symplectic surfaces. One would expect that moduli spaces of sheaves on hyper-Kähler manifolds should be themselves hyper-Kähler, but they have proven much more challenging to study. In this talk, I will describe an instance where such an analysis is possible on a four-dimensional manifold. In this case, the moduli space is indeed a hyper-Kähler manifold of dimension 10, deformation equivalent to O'Grady's example.

  The topology of the space of n distinct labeled points in Euclidean space has a long history. Its cohomology is fairly well understood, including as a representation of the symmetric group permuting the n labels. These representations also have mysterious connections with combinatorial notions like descents of permutations, and sometimes "hidden" actions of the symmetric group on n+1 points. We will discuss several results in recent years elucidating some of these connections, including work by and with Marcelo Aguiar, Ayah Almousa, Sarah Brauner, Nick Early, and Sheila Sundaram.
  Note: This talk is part of the activity of the MIUR Excellence Department Project Mat-Mod@TOV (CUP E83C23000330006)

The supersingular Endomorphism Ring problem is the following: given a supersingular elliptic curve, compute all of its endo- morphisms. The presumed hardness of this problem is foundational for isogeny-based cryptography. The One Endomorphism problem only asks to find a single non-scalar endomorphism. We prove that these two problems are equivalent, under probabilistic polynomial time reductions. We prove a number of consequences: on the security of cryptosystems, on the hardness of computing isogenies between supersingular elliptic curves, and on solving the endomorphism ring problem.

We prove the Shafarevich conjecture for a large class of irregular varieties. Our proof relies on the Lawrence-Venkatesh method as used by Lawrence-Sawin, together with the big monodromy criterion from our previous work with Javanpeykar, Lehn and Maculan. This is joint work in progress with Marco Maculan (IMJ Paris).

  We will talk about minimal monomial lifting of cluster algebras. That is sort of a homogenisation technique, whose goal is to identify a cluster algebra structure on some schemes "suitable for lifting", compatibly with a base cluster algebra structure on a given subscheme. We will see how to apply this technique to study some branching problems, in representation theory of complex reductive groups and, time permitting, we will discuss some possible development as the construction of polyhedral models for multiplicities.

  Quasi-invariants are special polynomials associated with a finite reflection group W and a multiplicity function. They appeared in 1990 in the study of Calogero-Moser integrable systems by Chalykh and Veselov, in which case they are the highest symbols of differential operators which form a large commutative ring. Similarly to all the polynomials, quasi-invariants form a free module over invariant polynomials of rank |W|, and they have other good properties. Quasi-invariants form representations of spherical Cherednik algebras as was established by Berest, Etingof and Ginzburg in 2003, which gives a way to establish the freeness property. I am going to explain a more recent application of quasi-invariants to the theory of free multi-arrangements of hyperplanes. In this case one is interested in the module of logarithmic vector fields which is known to be free over polynomials for some arrangements including Coxeter ones. Quasi-invariants can be used to construct elements of this module, and they also lead to new free multi-arrangements in the case of complex reflection groups.
  The talk is based on a joint work with T. Abe, N. Enomoto and M. Yoshinaga.

I will explore some recent results based on the interaction of operator space theory and quantum nonlocality. In particular I will emphasize the connection between large violations of Bell inequalities and certain norms in Banach and operator space categories. Finally, using Grothendieck's inequality, I will derive some interesting consequences for the parallel repetition problem in the context of XOR games.

If X is a projective variety cut out byquadrics, the p^th Syzygy scheme Syz_p(X) is the scheme cut out by quadrics involved in a p^th syzygy of X, and it turns out to capture refined geometrical properties of X in its embedding. We report on joint work in progress with M. Aprodu and E. Sernesi, concerning the second Syzygy scheme Syz_2(C) of a smooth curve in case C is embedded either by the canonical line bundle or by a non-special line bundle L, aiming at a classification of all (C,L) such that Syz_2(C) strictly contains

  A Yetter-Drinfeld module (=YD-module) over a bialgebra H is at the same time module and comodule over H satisfying a compatibility condition. It is well-known that the category of YD-modules (over a finite-dimensional Hopf algebra H) is equivalent to the center of the monoidal category of H-(co)modules as well as the category of modules over Drinfeld doubles of H. Canaepeel, Militaru, and Zhu introduced generalized YD-modules. More precisely, they consider two bialgebras H, K, together with a bicomodule algebra C and bimodule coalgebra over them. A generalized YD-module in their sense, is simultaneously an A-module and a C-comodule with a compatibility condition. Under a finiteness condition, they showed that these modules are exactly modules over a suitable constructed smash product build out of A and C. The aim of this talk is to show how the category of the generalized YD-modules can be obtained as a relative center of the category A-modules, viewed as a bimodule category over categories of H-modules and K-modules. Moreover, we also show how other variations of YD-modules, such as anti-YD-modules, arise as a particular case.
  This talk is based on ongoing work with Joost Vercruysse.

  Many well-known examples of "noncommutative spaces", like Woronowicz' quantum SU(2) or Vaksman-Soibelman odd-dimensional quantum spheres, can be described by C*-algebras associated to directed graphs. More generally, many compact quantum groups and quantum homogeneous spaces, can be described by convolution C*-algebras of "nice" groupoids. C*-algebras associated to combinatorial data (graphs, diagrams, groupoids) allow efficient models to attack key open problems in noncommutative geometry.
  The aim of this talk is to present some basic ideas of noncommutative geometry, using graph and groupoid C*-algebras as examples of "noncommutative spaces that one can see".