Dipartimento di Matematica -UTV-

Date	Type	Start	End	Room	Speaker	From	Title
25/02/22	Seminario	14:30	15:30	1201 Dal Passo	Jacopo GANDINI	Università di Bologna	Algebra & Representation Theory Seminar (ARTS) "Fully commutative elements and spherical nilpotent orbits" - in live & streaming mode - ( please click HERE* to attend the talk in streaming )* Abstract Let g be a simple Lie algebra, with a fixed Borel subalgebra b = t+n , and let W be the associated Weyl group. The Steinberg map associates to any element of W a nilpotent orbit in g, which is defined by the corresponding set of inversions. Extending on previous work of Fan and Stembridge, in this talk I will compare two different notions of "smallness", one available in the Weyl group and the other one for nilpotent orbits. N.B.: please click HERE to attend the talk in streaming.
22/02/22	Seminario	16:00	17:00	1201 Dal Passo	Cristian Mendico	Università di Roma	Seminario di Equazioni Differenziali Asymptotic behavior of solutions to Hamilton-Jacobi-Bellmann equations (MS Teams link for the streaming at the end of the abstract) Abstract The analysis of the ergodic behavior of solutions to Hamilton-Jacobi-Bellmann equations has a long history going back to the seminal paper by [Lions, P.-L., Papanicolaou, G. and Varadhan, S.R.S]. Since this work, the subject has grown very fast and when the Hamiltonian is of Tonelli type a large number of results have been proved. A full characterization of the ergodic behavior of solutions to Tonelli Hamilton-Jacobi equations can be found in the celebrated weak KAM theory and Aubry-Mather theory. However, few results are available if the Hamiltonian fails to be Tonelli, i.e., the Hamiltonian is neither strictly convex nor coercive with respect to the momentum variable. In particular, such results cover only some specific structure and so, the general problem is still open. In this talk, I will present some recent results obtained in collaboration with Piermarco Cannarsa and Pierre Cardaliaguet concerning the long time-average behavior of solutions to Hamilton-Jacobi-Bellman equations. We will look, first, to the case of control of acceleration and, then, to sub-Riemannian control systems. Finally, we conclude this talk showing how the previous analysis applies to mean field game systems. MS Teams Link for the streaming Note: This talk is part of the activity of the MIUR Department of Excellence Project MATH@TOV CUP E83C18000100006
22/02/22	Seminario	14:30	15:30	1201 Dal Passo	Roberto Fringuelli	Tor Vergata	Geometry Seminar The automorphism group of the moduli space of principal bundles on a smooth curve Abstract Let G be a complex (connected) reductive group and C be a complex smooth projective curve of genus at least four. It is known that the moduli space of semistable G-bundles over C is a projective variety. The automorphism group of this variety contains the so-called tautological automorphisms: they are induced by the automorphisms of the curve C, outer automorphisms of G and tensorization by Z-torsors, where Z is the center of G. It is a natural question to ask if they generate the entire automorphism group. Kouvidakis and Pantev gave a positive answer when G=SL(n). An alternative proof has been given by Hwang and Ramanan. Later, Biswas, Gomez and Muñoz, after simplifying the proof for G=SL(n), extended the result to the symplectic group Sp(2n). All the proofs rely on the study of the singular fibers of the Hitchin fibration. In this talk, we present a recent work where, by adapting the Biswas-Gomez-Muñoz strategy, we describe the automorphism group of the connected components of the moduli space of semistable G-bundles over C, for any almost-simple group G.
15/02/22	Seminario	16:00	17:00	1201 Dal Passo	Nicolas Augier	CNRS-LAAS, Toulouse	On the use of quasi-static controls and duplication of controls for quantum systems ( MS Teams Link for the streaming ) Abstract In a first part of the talk, I will present some geometric techniques allowing to control quantum systems using slowly-varying controls, in the so-called adiabatic regime. The latter provides strong control results only when the system is driven by at least two controls, which is a strong requirement in practice. The second part of the talk will be dedicated to an averaging approximation (Rotating Wave Approximation) which allows to duplicate controls in this setting. NB:This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C18000100006
15/02/22	Seminario	14:30	15:30	1201 Dal Passo	Luca De Feo	IBM Zurigo	Geometry Seminar Isogeny based cryptography [Click here to attend the talk in Streaming (MS Teams Link)] Abstract Le isogenie sono morfismi di varietà abeliane. La loro teoria algoritmica è sviluppata da oltre 30 anni, motivata in parte dall'algoritmo di Schoof-Elkies-Atkin per il conteggio di punti, algoritmo fondamentale in crittografia ellittica. I progressi algoritmici hanno portato negli ultimi 20 anni allo sviluppo di una nuova branca della crittografia, detta a base d'isogenie. L'oggetto centrale di questa disciplina non è più una curva ellittica isolata, bensì un grafo di curve ellittiche legate da isogenie. I grafi d'isogenie esibiscono diverse strutture combinatorie interessanti (foreste, grafi di Cayeley, grafi espansori), e offrono dei problemi computazionalmente difficili come la ricerca di cammini. Su queste basi, siamo oggi in misura di costruire un vasto spettro di primitive crittografiche: cifratura e firma digitale resistenti agli attacchi quantistici, crittografia a orologeria, sistemi a soglia, ecc. In questo talk, darò un'introduzione alla teoria delle isogenie di curve ellittiche su corpi finiti, e spiegherò come la crittografia è costruita a partire da esse.
11/02/22	Seminario	16:00	17:00	1201 Dal Passo	Corrado DE CONCINI	Accademia Nazionale delle Scienze “Sapienza” Università di Roma	Algebra & Representation Theory Seminar (ARTS) "Paving Springer fibers" - in live & streaming mode - ( please click HERE* to attend the talk in streaming )* Abstract In the paper [De Concini C., Lusztig G., Homology of the zero-set of a nilpotent vector field on a flag manifold, J. Amer. Math. Soc. 1 (1988), no. 1, 15-34] it was proven that the so called Springer fiber B_n for any nilpotent element n in a complex simple Lie algebra g has homological properties that suggest that B_n should have a paving by affine spaces. This last statement was proved to hold in the case in which g is classical, but remained open for exceptional groups in types E₇ and E₈. In a joint project with Maffei we are trying to fill the gap. At this point our efforts have been successful in type E₇ and "almost" in type E₈, where one is reduced to show it only in one case. The goal of the talk is to survey the problem and give an idea on how to show our new results. N.B.: please click HERE to attend the talk in streaming.
11/02/22	Seminario	14:30	15:30	1201 Dal Passo	Domenico FIORENZA	"Sapienza" Università di Roma	Algebra & Representation Theory Seminar (ARTS) "Brackets and products from centres in extension categories" - in live & streaming mode - ( please click HERE* to attend the talk in streaming )* Abstract Building on Retakh's approach to Ext groups through categories of extensions, Schwede reobtained the well-known Gerstenhaber algebra structure on Ext groups over bimodules of associative algebras both from splicing extensions (leading to the cup product) and from a suitable loop in the categories of extensions (leading to the Lie bracket). We show how Schwede's construction admits a vast generalisation to general monoidal categories with coefficients of the Ext groups taken in (weak) left and right monoidal (or Drinfel'd) centres. In case of the category of left modules over bialgebroids and coefficients given by commuting pairs of braided (co)commutative (co)monoids in these categorical centres, we provide an explicit description of the algebraic structure obtained this way, and a complete proof that this leads to a Gerstenhaber algebra is then obtained from an operadic approach. This, in particular, considerably generalises the classical construction given by Gerstenhaber himself. Conjecturally, the algebraic structure we describe should produce a Gerstenhaber algebra for an arbitrary monoidal category enriched over abelian groups, but even the bilinearity of the cup product and of the Lie-type bracket defined by the abstract construction in terms of extension categories remain elusive in this general setting. This is a joint work with Niels Kowalzig, cf. arXiv:2112.11552. N.B.: please click HERE to attend the talk in streaming.
08/02/22	Seminario	16:00	17:00	1201 Dal Passo	Azahara DelaTorre	Università di Roma "La Sapienza"	The fractional Yamabe problem with singularities ( MS Teams Link for the streaming ) Abstract The so called Yamabe problem in Conformal Geometry consists in finding a metric conformal to a given one and which has constant scalar curvature. From the analytic point of view, this problem becomes a semilinear elliptic PDE with critical (for the Sobolev embedding) power non-linearity. If we study the problem in the Euclidean space, allowing the presence of nonzero-dimensional singularities can be transformed into reducing the non-linearity to a Sobolev-subcritical power. A quite recent notion of non-local curvature gives rise to a parallel study which weakens the geometric assumptions giving rise to a non-local semilinear elliptic PDE. In this talk, we will focus on metrics which are singular along nonzero-dimensional singularities. In collaboration with Ao, Chan, Fontelos, González and Wei, we covered the construction of solutions which are singular along (zero and positive dimensional) smooth submanifolds in this fractional setting. This was done through the development of new methods coming from conformal geometry and Scattering theory for the study of non-local ODEs. Due to the limitations of the techniques we used, the particular case of ''maximal’’ dimension for the singularity was not covered. In a recent work, in collaboration with H. Chan, we cover this specific dimension constructing and studying singular solutions of critical dimension. NB:This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C18000100006
08/02/22	Seminario	14:30	15:30	1201 Dal Passo	Paolo Stellari	Università di Milano	Geometry Seminar Stability conditions in the trivial canonical bundle case: Hilbert schemes of points. Abstract The construction of stability conditions on the bounded derived category of coherent sheaves on smooth projective varieties is notoriously a difficult problem, especially when the canonical bundle is trivial. In this talk, I will review some results and techniques related to the latter setting. I will specifically concentrate on the case of Hilbert scheme of points on K3 surfaces and (as a work in progress) on generic abelian varieties of any dimension. This is joint work in progress with C. Li, E. Macri' and X. Zhao.
01/02/22	Seminario	14:30	15:30	1201 Dal Passo	Giovanni Mongardi	Università di Bologna	Geometry Seminar Deforming rational curves Abstract In this talk, we will survey known results on deformations of rational curves inside hyperkähler manifolds, and then provide suitable generalizations for singular hyperkähler varieties. As an application, we will construct uniruled divisors in many hyperkähler varieties. This is joint work with Ch. Lehn and G. Pacienza.