Chen-Gounelas-Liedtke recently introduced a powerful regeneration technique, a process opposite to specialization, to prove existence results for rational curves on projective K3 surfaces. In the talk I will present a joint work with G. Mongardi in which we show that, for projective irreducible holomorphic symplectic manifolds, an analogous regeneration principle holds and provides a very flexible tool to prove existence of uniruled divisors, significantly improving known results.

In this talk I will explain how to extract an 'anyon theory' (braided tensor category) from a gapped ground state of an infinite two-dimensional lattice spin system. Just as in the DHR formalism from AQFT, the anyon types correspond to certain superselection sectors of the observable algebra of the spin system. We apply this formalism to Kitaev's quantum double model for finite gauge group G, and find that the anyon types correspond precisely to the representations of the quantum double algebra of G.
The Operator Algebra Seminar schedule is here: https://sites.google.com/view/oastorvergata/home-page?authuser=0

Homotopy theory allows us to study formal moduli problems via their tangent Lie algebras. We apply this general paradigm to Calabi-Yau varieties Z in characteristic p. First, we show that if Z has torsion-free crystalline cohomology and degenerating Hodge-de Rham spectral sequence (and for p=2 a lift to W/4), then its mixed characteristic deformations are unobstructed. This generalises the BTT theorem from characteristic 0 to characteristic p. If Z is ordinary, we show that it moreover admits a canonical (and algebraisable) lift to characteristic zero, thereby extending Serre-Tate theory from abelian varieties to Calabi-Yau varieties. This is joint work with Taelman, and generalises results of Achinger-Zdanowicz, Bogomolov-Tian-Todorov, Deligne-Nygaard, Ekedahl–Shepherd-Barron, Iacono-Manetti, Schröer, Serre-Tate, and Ward.

The Cauchy-Liouville theorem (1844) states that any bounded entire function of a complex variable is necessarily constant. In the realm of PDE's, by a Liouville-type theorem, one usually means a statement asserting the nonexistence of solutions in the whole space (or a suitable unbounded domain). Numerous results of this kind have appeared over the years and many far-reaching applications have arisen, conferring Liouville-type theorems an important role in the theory of PDE's and revealing strong connections with other mathematical areas (calculus of variations, geometry, fluid dynamics, optimal stochastic control). After a brief historical detour (minimal surfaces - Lagrange, Bernstein, de Giorgi, Bombieri,… and regularity theory for linear elliptic systems - Giaquinta, Necas, ...), we will recall the developments of the 1980-2000's on nonlinear elliptic problems, leading to powerful tools for existence and a priori estimates for Dirichlet problems (Gidas, Spruck, Caffarelli, ...), based on the combination of Liouville type theorems and renormalization techniques. In a more recent period, this line of research has also led to much progress in the study of singularities of solutions, both for stationary (elliptic) and evolution PDEs. In particular, in the case of power like nonlinearities, we will recall the equivalence between Liouville type theorems and universal estimates, based on a method of doubling-rescaling (joint work with P. Polacik and P. Quittner, 2007). Then we will present recent developments which show that these renormalization techniques can be applied to nonlinearities without any scale invariance, even asymptotically, with applications to initial and final blowup rates or decay rates in space and/or time.
Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027)

In one dimensional complex dynamics we have an increasingly detailed knowledge about stable components which are periodic and preperiodic. On the other hand, stable components which elude being (pre)periodic (aka wandering domains) also elude our full understanding and are currently an active topic of research. While much of the current research focuses on constructing examples showing a great variety of possibilities, in our work we propose an actual classification of wandering domains according to the behaviour of their internal orbits. This seamlessly leads us to analyzing nonautonomous dynamics for self-maps of the unit disk. For autonomous iteration of inner functions (self-maps of the disk whose radial extension is a self map of the boundary a.e.) there is a remarkable dichotomy due to Aaronson, Doering and Mañé, according to which the internal dynamics of the map determines the dynamical properties of its boundary extension: either (almost all) boundary orbits converge to a single point, or (almost all) boundary orbits are dense. In the nonautonomous setting the situation is more complicated. However, we present a generalization of this dichotomy which is, in a specific sense, optimal. This is joint work with Vasso Evdoridou, Nuria Fagella, Phil Rippon, and Gwyneth Stallard. Parts of this work are still in progress.
Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027)

Circular polygons are closed plane curves formed by concatenating a finite number of circular arcs so that, at the points where two arcs meet, their tangents agree. These curves are strictly convex and C1, but not C2. We study the billiard dynamics in domains bounded by circular polygons. We prove that there is a set accumulating on the boundary of the domain in which the return dynamics is semiconjugate to a transitive shift on infinitely many symbols. Consequently the return dynamics has infinite topological entropy. In addition we give an exponential lower bound on the number of periodic orbits of large period, and we prove the existence of trajectories along which the angle of reflection tends to zero with optimal linear speed. These results are based on joint work with Rafael Ramírez-Ros.
Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027)

  Reductive non-connected groups appear frequently in the study of algebraic groups, for example as centralizers of semisimple elements in non-simply connected semisimple groups. Let G be a non-connected reductive algebraic group over an algebraically closed field of arbitrary characteristic and let D be a connected component of G. We consider the strata in D defined by G. Lusztig as fibers of a map E given in terms of truncated induction of Springer representation. By the definition of the map E, one can see that elements with the same unipotent part and the same centralizer of the semisimple part are in the same stratum. The connected component of the set collecting the elements with these properties are called Jordan classes. In his work, G. Lusztig suggests that the strata are locally closed: in my work I prove this assertion. To prove it, I show that a stratum is a union of the regular part of the closure of Jordan classes. From this result, one can also describe the irreducible components of a stratum in terms of regular closures of Jordan classes.

  The study of the (non)-existence of integral Hopf orders was originally motivated by Kaplansky's sixth conjecture, which is a generalization of Frobenius theorem in the Hopf algebra setting. In fact, Larson proved that a Hopf algebra which admits an integral Hopf order satisfies the conjecture.
  The aim of this talk is to give a partial answer to the following question: "Does a semisimple complex Hopf algebra admit an integral Hopf order?"
  In particular, we will present several families of semisimple Hopf algebras which do not admit an integral Hopf order. These Hopf algebras will be constructed as Drinfeld twists of group algebras.
  This talk is based on a joint work with Giovanna Carnovale and Juan Cuadra and on my Ph.D. thesis.

Circa dieci anni fa, V. Jones introduceva diverse rappresentazioni unitarie dei gruppi di Thompson e vari sottogruppi interessanti sono emersi come stabilizzatori di vettori in queste rappresentazioni. In questo seminario presenterò il lavoro svolto su questo argomento.

We will report on joint works with Paola Frediani, Juan Carlos Naranjo and Gian Pietro Pirola. We study the second fundamental form of the Siegel metric in A_5 restricted to the moduli space of the intermediate jacobians cubic threefolds and the second fundamental form in A_9 restricted to the moduli of the the intermediate jacobians of (2,2) threefolds in P^2xP^2 . In both case there is a natural composition with a multiplication map. For cubic threefold this composition results to be zero, while for (2,2) threefolds it gives a not zero holomorphic section of a bundle.