## Notiziario Scientifico

Notiziario dei seminari di carattere matematico
a cura del Dipartimento di Matematica Guido Castelnuovo, Sapienza Università di Roma

Settimana dal 06-12-2021 al 12-12-2021

Lunedì 06 dicembre 2021
Ore 14:30, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
seminario di Analisi Matematica
Asymptotic behavior of solutions to Hamilton-Jacobi-Bellmann equations
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. 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 arising from optimal control problems with control of acceleration, first, and then from optimal control problems of sub-Riemannian type. We will show the existence of a critical constant in both cases but the existence of a critical solution only in case of sub-Riemannian geometry. For this latter case, we also show some results on the Aubry set. We conclude presenting open problems and ideas toward the solution of the general case.

Lunedì 06 dicembre 2021
Ore 17:00, online su Google Meet, Dipartimento di Matematica, Sapienza Università di Roma
Seminario di Dipartimento
Alessio Troiani
Algoritmi Markoviani Paralleli e loro applicazione all'ottimizzazione combinatoria
Presenterò una classe di catene di Markov "parallele" (Probabilistic Cellular Automata) che possono essere utilizzati efficacemente per affrontare problemi di ottimizzazione combinatoria. Di queste catene è possibile controllare la misura stazionaria e, sotto opportune ipotesi è possibile mostrare la convergenza di questa misura alla misura di Gibbs. Per quanto riguarda l'ottimizzazione farò riferimento, in particolare, all'Unconstrained Binary Quadratic Programming descrivendone alcune caratteristiche e mostrando alcuni risultati numerici ottenuti.

Giovedì 09 dicembre 2021
Ore 14:15, Aula M1, Dipartimento di Matematica a Fisica, Università Roma Tre
Seminario di Geometria
Andreas Leopold Knutsen (Bergen)
Moduli of polarized Enriques surfaces
Moduli spaces of polarized Enriques surfaces (parametrizing pairs (S,H), where S is a compact, complex Enriques surface, and H is an ample line bundle on S) have several components, even if one fixes the degree of the polarization. I will present some new results determining the various irreducible components of these spaces and answering a question of Gritsenko and Hulek on the connectedness of the etale double cover from the moduli space of polarized such surfaces to the moduli space of numerically polarized such surfaces. (see https://folk.uib.no/st00895/articolimiei/Knutsen-JMPA2020-II.pdf)
Per informazioni, rivolgersi a: amos.turchet@uniroma3.it

Giovedì 09 dicembre 2021
Ore 14:30, Sala di Consiglio - https://meet.google.com/ads-dekx-bgm, Dipartimento di Matematica, Sapienza Università di Roma
Seminario P(n)
Benedetta Noris (Politecnico di Milano)
Multiplicity of solutions on a Nehari set in an invariant cone
I shall present a recent preprint in collaboration with F. Colasuonno and G. Verzini. For $$1 < p < 2$$ and $$q$$ large, we prove the existence of two positive, nonconstant, radial and radially nondeacreasing solutions of the supercritical equation $$-\Delta_p u+u^{p-1}=u^{q-1}$$ under Neumann boundary conditions, in the unit ball of $$\mathbb R^N$$. We use a variational approach in an invariant cone. We distinguish the two solutions upon their energy: one is a ground state inside a Nehari-type subset of the cone, the other is obtained via a mountain pass argument inside the Nehari set. As a byproduct of our proofs, we detect the limit profile of the low energy solution as $$q\to\infty$$ and show that the constant solution 1 is a local minimum on the Nehari set.
Per informazioni, rivolgersi a: galise@mat.uniroma1.it

Giovedì 09 dicembre 2021
Ore 15:30, Il seminario sarà tenuto in modalità mista, Sala lettura "Maria Corda Costa" (Villa Mirafiori)
Seminari di Ricerca in Didattica e Storia della Matematica
Annalisa Cusi (Sapienza Università di Roma)
Il ruolo chiave delle discussioni di classe nella riflessione sulla costruzione di esempi come strumenti per argomentare: verso lo sviluppo di un modello

Chi è interessato a partecipare a distanza può rivolgersi ad Annalisa Cusi (annalisa.cusi@uniroma1.it)

Venerdì 10 dicembre 2021
Ore 12:00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
Seminario MoMA
Emanuele Rodolà (Sapienza Università di Roma)
In this talk I will present several recent directions of research in the areas of machine learning and applied AI, the so called "geometric deep learning" subfield; common to these topics are the notions of geometry and structure. I'll start by showing how data-driven methods can serve as efficient and robust drop-in replacements for classical axiomatic constructions, even when learning is not strictly necessary, for typical tasks of geometry and signal processing; in turn, I'll show how geometry itself can serve as a valid proxy for better machine learning pipelines. I'll then explore some preliminary steps toward the application of geometric deep learning in other fundamental disciplines, such as mathematical physics and structural biology, and conclude with an overview of some work in progress in our lab as well as some wishful thinking on future perspectives.

Venerdì 10 dicembre 2021
Ore 14:30, Aula "Roberta Dal Passo", Dipartimento di Matematica, Università degli Studi di Roma "Tor Vegata"
Algebra and Representation Theory Seminar
Margherita Paolini (Università de L'Aquila)
Integral forms of affine Lie algebras
Let be g a semisimple finite dimensional Lie algebra and U(g) its universal enveloping algebra. The theory of highest weight representations of g may pass through the description of an integral form of U(g), namely a suitable $$\mathbb{Z}$$-subalgebra of U(g) generated by the divided powers of the Chevalley generators; for this reason it has been studied by several authors (e.g., Chevalley and Cartier). If $$\hat{g}$$ is an affine Lie algebra, in order to extend this approach, the analogous $$\mathbb{Z}$$-subalgebra has been studied by Garland (in the untwisted case) and by Mitzman and by Fisher-Vasta (in the twisted case). Anyhow, the case when $$\hat{g}$$ is of type $$A_{2n}^2$$ still remains obscure. In order to study the representation theory of this algebra we try to find more manageable techniques that will help to get a deeper understanding. The aim of this talk is to present the structure of these integral forms and some related results.
Link per vedere seminario in streaming

Venerdì 10 dicembre 2021
Ore 16:00, Aula "Roberta Dal Passo", Dipartimento di Matematica, Università degli Studi di Roma "Tor Vergata"
Algebra and Representation Theory Seminar
Riccardo Biagioli (Università di Bologna)
Temperley-Lieb algebra and fully commutative elements in affine type C
The Temperley-Lieb algebra is a well studied finite dimensional associative algebra : it can be realized as a diagram algebra and it has a basis indexed by the fully commutative elements in the Coxeter group of type A. A few years ago, Dana Ernst introduced an elegant generalization of such diagrammatic representation for the generalized Temperley-Lieb algebra of affine type C. The proof that such representation is faithful is quite involved and the same author wonders if an easier proof exists. In this talk, we present a new combinatorial way to describe Ernst's algebra homomorphism, from which injectivity and subjectivity follow more easily. Our results are based on a classification of fully commutative elements of type affine C in terms of heaps of pieces, and on certain operations that we define on such heaps. This talk is based on a joint work with Giuliana Fatabbi and Gabriele Calussi. N.B.: this talk is part of the activity of the MIUR Excellence Department Project CUP E83C18000100006
Link per seguire seminario in streaming

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