22/04/24  Seminario  16:00  17:00  1201 Dal Passo  Andrew Clarke  UPC Barcelona  Seminario di Sistemi Dinamici
Chaotic properties of billiards in circular polygons
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 RamírezRos.
Note:
This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (20232027) 
22/04/24  Seminario  14:30  15:30  1201 Dal Passo  Anna Miriam Benini  Università di Parma  Seminario di Sistemi Dinamici
Wandering Domains and Non Autonomous Dynamics on the disk
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 selfmaps of the unit disk. For autonomous iteration of inner functions (selfmaps 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 (20232027) 
19/04/24  Seminario  16:00  17:00  1201 Dal Passo  Elisabetta MASUT  Università di Padova 
Algebra & Representation Theory Seminar (ARTS)
"Nonexistence of integral Hopf orders for certain Hopf algebras"
N.B.: this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)
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.

19/04/24  Seminario  14:30  15:30  1201 Dal Passo  Martina COSTA CESARI  Università di Bologna 
Algebra & Representation Theory Seminar (ARTS)
"Jordan classes and Lusztig strata in nonconnected algebraic groups"
N.B.: this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)
Reductive nonconnected groups appear frequently in the study of algebraic groups, for example as centralizers of semisimple elements in nonsimply connected semisimple groups. Let G be a nonconnected 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.

17/04/24  Seminario  16:00  17:00  1201 Dal Passo  Valeriano Aiello  Sapienza University of Rome 
Operator Algebras Seminar
Colorazioni, sottogruppi del gruppo di Thompson e rappresentazioni
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. 
16/04/24  Seminario  16:00  17:00  1201 Dal Passo  Pietro Majer  Università di Pisa  Seminario di Equazioni Differenziali
On the CWstructure induced by a MorseSmale gradient flow
A classic yet delicate fact of Morse theory states that the unstable manifolds of a MorseSmale gradientflow on a closed manifold M are the open cells of a CWdecomposition of M.
I will describe a selfcontained proof by Abbondandolo and myself. The key tool is a "system of invariant stable foliations", which is analogous to the object introduced by Palis and Smale in their proof of structural stability of Morse Smale diffeomorphisms and flows, but with finer regularity and geometric properties.
[Stable foliations and CW structure induced by a MorseSmale gradient flow, A.Abbondandolo,P.Majer]
https://www.worldscientific.com/doi/10.1142/S1793525321500527
NB:This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C23000330006

16/04/24  Seminario  14:30  16:00  1101 D'Antoni  Elisabetta Colombo  Università degli studi di Milano  On the local geometry of the moduli spaces of cubic threefolds in
A_5 and of (2,2) threefolds in A_9
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. 
16/04/24  Seminario  14:30  15:30  1201 Dal Passo  Michael Barton  Basque Center for Applied Mathematics  Gaussian quadrature rules for univariate splines and their applications to tensorproduct isogeometric analysis
Univariate Gaussian quadrature rules for spline spaces that are frequently used in Galerkin discretizations to build mass and stiffness matrices will be discussed. Their computation is based on the homotopy continuation concept that transforms Gaussian quadrature rules from the so called source space to the target space. Starting with the classical Gaussian quadrature for polynomials, which is an optimal rule for a discontinuous odddegree space, and building the source space as a union of such discontinuous elements, we derive rules for target spline spaces with higher continuity across the elements. We demonstrate the concept by computing numerically Gaussian rules for spline spaces of various degrees, particularly those with nonuniform knot vectors and nonuniform knot multiplicities. We also discuss convergence of the spline rules over finite domains to their asymptotic counterparts, that is, the analogues of the halfpoint rule of Hughes et al., that are exact and Gaussian over the infinite domain. Finally, the application of spline Gaussian rules in the context of isogeometric analysis on subdivision surfaces will be discussed, showing the advantages and limitations of the tensor product Gaussian rules.
This talk is part of the activity of the MUR Excellence Department Project MatMod@TOV (CUP E83C23000330006). 
12/04/24  Seminario  16:00  17:00  1201 Dal Passo  Grant BARKLEY  University of Harvard 
Algebra & Representation Theory Seminar (ARTS)
"Hypercube decompositions and combinatorial invariance for elementary intervals"
N.B.: this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)
The combinatorial invariance conjecture asserts that the KazhdanLusztig (KL) polynomial of an interval [u,v] in Bruhat order can be determined just from the knowledge of the poset isomorphism type of [u,v]. Recent work of Blundell, Buesing, Davies, Velicković, and Williamson posed a conjectural recurrence for KL polynomials depending only on the poset structure of [u,v]. Their formula uses a new combinatorial structure, called a hypercube decomposition, that can be found in any interval of the symmetric group. We give a new, simpler, formula based on hypercube decompositions and prove it holds for "elementary" intervals: an interval [u,v] is elementary if it is isomorphic as a poset to an interval with linearly independent bottom edges. As a result, we prove combinatorial invariance for KazhdanLusztig Rpolynomials of elementary intervals in the symmetric group, generalizing the previously known case of lower intervals.
This is a joint work with Christian Gaetz. 
12/04/24  Seminario  14:30  15:30  1201 Dal Passo  Willem DE GRAAF  Università di Trento 
Algebra & Representation Theory Seminar (ARTS)
"Classifying orbits of complex and real Vinberg representations"
N.B.: this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)
Vinberg representations are representations of algebraic groups that arise from a cyclic grading of a semisimple Lie algebra. In the literature they are mainly known as thetagroups or Vinberg pairs. A distinguishing feature of these representations is that it is possible to classify the orbits of the algebraic group. We sketch how this can be done when the base field is the complex numbers. This mainly uses results of Vinberg of the 70's. Then we describe techniques for classifying the orbits when the base field is the real numbers. This talk is based on joint work with Mikhail Borovoi, Hong Van Le, Heiko Dietrich, Marcos Origlia, Alessio Marrani. 