Notiziario Scientifico

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

Settimana dal 06-03-2023 al 12-03-2023

Lunedì 06 marzo 2023
Ore 14:00, Aula B, Dipartimento di Matematica, Sapienza Università di Roma - ANNULLATO
Mini-corso di dottorato
Benoit Claudon (IRMAR Université de Rennes 1 )
Local fundamental group of klt singularities, after L. Braun
Per informazioni, rivolgersi a: bravi@mat.uniroma1.it

Lunedì 06 marzo 2023
Ore 14:30, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
seminario di Analisi Matematica
Monica Musso (University of Bath)
Leapfrogging for Euler equations
We consider the Euler equations for incompressible fluids in 3-dimension. A classical question that goes back to Helmholtz is to describe the evolution of vorticities with a high concentration around a curve. The work of Da Rios in 1906 states that such a curve must evolve by the so-called "binormal curvature flow". Existence of true solutions whose vorticity is concentrated near a given curve that evolves by this law is a long-standing open question that has only been answered for the special case of a circle travelling with constant speed along its axis, the thin vortex-rings, and of a helical filament, associated to a translating-rotating helix. In this talk I will consider the case of two vortex rings interacting between each other, the so-called leapfrogging. The results are in collaboration with J. Davila (U. of Bath), M. del Pino (U. of Bath) and J. Wei (U. of British Columbia).
Per informazioni, rivolgersi a: azahara.delatorrepedraza@uniroma1.it

Lunedì 06 marzo 2023
Ore 16:00, Aula IV, Dipartimento di Matematica, Sapienza Università di Roma
Seminario Baby A&G
Domenico Fiorenza (Sapienza Università di Roma)
In quanti modi si può scrivere 10011101 come somma di quattro quadrati?
Il 25 dicembre del 1640, tra un biglietto di auguri e l’altro, Marin Mersenne se ne vede recapitare uno che recita: “Un primo p si può scrivere come somma di due quadrati se e solo se p è congruo a 1 modulo 4. Baci, Pierre. p.s. buon Natale e felice 1641”. Avendo una reputazione da mantenere, Pierre de Fermat si guarderà bene dal pubblicare una dimostrazione, ma, a differenza di altri casi, una dimostrazione della sua affermazione può oggi essere ricavata facilmente da chiunque abbia seguito un corso di algebra, usando le proprietà degli interi di Gauss. Passano 130 anni esatti e Giuseppe Ludovico De la Grange Tournier dimostra che ogni primo p, e di conseguenza ogni numero naturale, si può scrivere come la somma di quattro quadrati. Col linguaggio dell’algebra moderna questo risultato si può dimostrare in modo simile a quello di Fermat, passando dai numeri complessi ai quaternioni e lì considerando l’analogo degli interi di Gauss: gli interi di Hurwitz. Ma è nel 1834 che Carl Gustav Jacob Jacobi fa un’entrata da vero villain twittando dal suo account JJOfficial: “a,b,c,d interi, ad-bc=1, 4|c, Im(z)>0, f(z)=(az+b)/(cz+d). 10011101 si può scrivere in 80088816 modi come somma di quattro quadrati”. Come sempre, tutto deve concludersi là dov'è cominciato: nel giugno del 1993 i gruppi modulari \( \Gamma_0(N) \), dei quali quello nel tweet di Jacobi è il caso N=4, porteranno Sharon Stone a scrivere il telegramma "Congratulations, Mr. Wiles". Ma questa è un’altra storia. Link: https://sites.google.com/view/sapienza-baby-ag/home

Lunedì 06 marzo 2023
Ore 16:00, Aula 009, edificio C, Dipartimento di Matematica e Fisica, Università Roma Tre
corso di dottorato
Enrico Arbarello (Università La Sapienza)
Introduction to Bridgeland Theory
Program: t-structures and their hearts. The space of Bridgeland stability conditions on a variety. Stability conditions on K3 surfaces. Construction of stability conditions. The wall and chamber structure. Gieseker stability and Bridgeland stability. The Brill-Noether wall and the Gieseker wall. Mukai’s program for K3-curves. Ample divisors on Bridgeland moduli spaces.

Martedì 07 marzo 2023
Ore 12:00, aula 1B1, palazzina RM002, Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza Università di Roma
Corso di Dottorato
Cornelia Schiebold (Mid Sweden University, Sundsvall, Sweden)
Integrable systems - methods of mathematical physics in interaction
Integrable systems in infinite dimension refer to an area in mathematical physics which is devoted to the study of a certain group of partial differential equations, many of them soliton equations like the classic Korteweg-de Vries equation and the nonlinear Schrodinger equation. One of the striking features is the existence of solutions with particle character, called solitons, remarkable in view of nonlinearity of the governing equations. Methodologically, integrable systems are a meeting point (melting pan) for methods from very diverse parts of mathematics. The main idea of this mini course is to highlight interactions of some of the main approaches to integrable systems, the inverse scattering method and an operator theoretic approach in the first place, and symmetry methods like Backlund transformations, recursion operators and hierarchies to a minor extent. Throughout we will emphasise the recent topic ofnon-commutative integrable systems, like vector- and matrix soliton equations, where many fundamental questions are still open. Notably, the construction of solutions is not interesting only under the mathematical viewpoint, but also under the physical one. Indeed, very important applications of soliton equations are in nonlinear optics, for instance. The lectures are going to be reasonably self-contained. Some familiarity with PDE's and functional analysis is certainly helpful, but not required. An overview on the basic notions used during the course are provided when needed. The course is organised in six lectures (a two hours lecture each week). Needed material as well as references are provided by the Lecturer.

Martedì 07 marzo 2023
Ore 14:00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
Seminario di Probabilità
Markus Fischer (Università di Padova)
On correlated equilibria and mean field games
Mean field games are limit models for symmetric N-player games, as the number of players N tends to infinity. The prelimit models are usually solved in terms of Nash equilibria. A generalization of the notion of Nash equilibrium, due to Robert Aumann (1974, 1987), is that of correlated equilibrium. In a simple discrete setting, we will discuss correlated equilibria for mean field games and their connection with the underlying N-player games. We first consider equilibria in restricted strategies (Markov open-loop), where control actions depend only on time and a player's own state. In this case, N-player correlated equilibria are seen to converge to the mean field game limit and, conversely, correlated mean field game solutions induce approximate N-player correlated equilibria. We then discuss the problem of constructing approximate equilibria when deviating players have access to the aggregate system state. We also give an explicit example of a correlated mean field game solution not of Nash-type. Results (with L. Campi and Federico Cannerozzi) on a related notion of equilibrium in a diffusion-type setting will be mentioned as well. Joint work with Ofelia Bonesini (Imperial College London) and Luciano Campi (University of Milan "La Statale")
Per informazioni, rivolgersi a: giacomo.digesu@uniroma1.it

Martedì 07 marzo 2023
Ore 14:30, Aula Dal Passo, Dipartimento di Matematica, Università di Roma "Tor Vergata"
Seminario di Equazioni Differenziali
Nicolò Forcillo (Università di Roma "Tor Vergata")
The one-phase Stefan problem: perturbative techniques for the free boundary regularity
In Stefan-type problems, free boundaries may not regularize instantaneously. In particular, there exist examples in which Lipschitz free boundaries preserve corners. Nevertheless, in the two-phase Stefan problem, I. Athanasopoulos, L. Caffarelli, and S. Salsa showed that Lipschitz free boundaries in space-time become smooth under a nondegeneracy condition, as well as sufficiently "flat" ones. Their techniques are based on the original work of Caffarelli in the elliptic case. In this talk, we present a more recent approach to investigate the regularity of flat free boundaries for the one-phase Stefan problem. Specifically, it relies on perturbation arguments leading to a linearization of the problem, in the spirit of the elliptic counterpart already developed by D. De Silva. This talk is based on a joint work with D. De Silva and O. Savin.
Per informazioni, rivolgersi a: sorrentino@mat.uniroma2.it

Mercoledì 08 marzo 2023
Ore 14:00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
seminario di Algebra e Geometria
Enrico Fatighenti (Università di Bologna)
Fano varieties of K3 type and their properties
Fano varieties of K3 type are a special class of Fano varieties, which are usually studied for their link with hyperkähler geometry, rationality properties, and much more. In this talk, we will recap some recent results, obtained jointly with Bernardara, Manivel, Mongardi, and Tanturri, that focus on the explicit construction of examples and the study of their Hodge-theoretical properties.
Per informazioni, rivolgersi a: diverio@mat.uniroma1.it

Mercoledì 08 marzo 2023
Ore 16:00, sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
seminario di Fisica Matematica
Rossana Marra (INFN, Università di Roma Tor Vergata)
Ghost effect equations from the Boltzmann theory
The diffusive hydrodynamic limit of the Boltzmann equation in the low Mach number regime is usually described by the incompressible Navier-Stokes-Fourier equations. When the density and temperature at initial time and/or the temperature on the boundary have gradients of order 1 the limiting equations (called “ghost effect equations”) are different and cannot be predicted by the classical fluid theory. Proving the hydrodynamic limit under these conditions has been an open and challenging problem. In this talk I will discuss the rigorous proof of this non standard hydrodynamic behaviour for the stationary Boltzmann equation in a bounded domain with diffuse reflection boundary condition. Work in collaboration with R. Esposito, Y. Guo and Lei Wu

Giovedì 09 marzo 2023
Ore 11:00, Aula 009, edificio C, Dipartimento di Matematica e Fisica, Università Roma Tre
corso di dottorato
Enrico Arbarello (Università La Sapienza)
Introduction to Bridgeland Theory
Program: t-structures and their hearts. The space of Bridgeland stability conditions on a variety. Stability conditions on K3 surfaces. Construction of stability conditions. The wall and chamber structure. Gieseker stability and Bridgeland stability. The Brill-Noether wall and the Gieseker wall. Mukai’s program for K3-curves. Ample divisors on Bridgeland moduli spaces.

Giovedì 09 marzo 2023
Ore 14:15, Aula M1, Dipartimento di Matematica e Fisica
Seminario di Geometria
Renzo Cavalieri (Colorado State)
Pseudo-Stable Hodge Integrals
The moduli space of curves (by which I mean its Deligne-Mumford compactification) is a well studied object in algebraic geometry. Mumford introduced the notion of tautological intersection theory to study a part of the intersection theory which is simple enough to be tractable, but rich enough to be meaningful. Hodge integrals are a class of tautological intersection numbers that arise from intersecting the chern classes of the Hodge bundle and of the cotangent line bundles. In the first part of the talk I will introduce all these concepts and review some "classical" structural results about Hodge integrals. When running the MMP on the moduli space of curves, after the first wall-crossing one sees the moduli space of pseudo stable curves, which is the target of a birational regular morphism from the moduli space of curves. We investigate how the Hodge bundles on either side of this morphism are related, and how, correspondingly, there are very rich combinatorial relation between Hodge integrals and pseudo stable Hodge integrals. This talk is based on joint work with Gallegos, Ross, Wise, Van Over and on some of Matthew Williams' doctoral work.
Per informazioni, rivolgersi a: amos.turchet@uniroma3.it

Giovedì 09 marzo 2023
Ore 14:30, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
P(n)/N(p) : Problemi differenziali nonlineari/Nonlinear differential problems
Alessio Pomponio (Politecnico di Bari)
On the Born-Infeld equation: external charges and nonlinearities
It is well known that, in the classical Maxwell electromagnetic theory, the so called infinite-energy problem appears: the energy of the electrostatic field generated by a pointwise charge is infinite. In the first years of the past century, Born and Infeld proposed to solve such a problem introducing a nonlinear modification of the Maxwell theory that, in the electrostatic case, gives rise to an equation which replaces the Gauss law (or Poisson equation) and that involves the mean curvature operator in the Lorentz-Minkowski metric. The aim of this talk, therefore, is to introduce the Born-Infeld equation, showing its main peculiarities. In the first part, we deal, briefly, with \( \begin{cases} -{\rm div}\left(\frac{\nabla\phi}{\sqrt{1-|\nabla\phi|^2}}\right)=\rho, & x\in\mathbb R^N\\ \lim_{|x|\to\infty}\phi(x)=0, \end{cases}\qquad(1) \) where \( \phi\) is the electric potential and \( \rho \) is an assigned extended charge density. We present some existence, uniqueness and regularity of the solution of (1). In the second part, instead, we deal with \( \begin{cases} -{\rm div}\left(\frac{\nabla\phi}{\sqrt{1-|\nabla\phi|^2}}\right)=g(\phi), & x\in\mathbb R^N\\ \lim_{|x|\to\infty}\phi(x)=0, \end{cases}\qquad(2) \) where \( g\) is a continuous nonlinearity. We present some recent existence results for problem (2) for a large class of nonlinearities by means of variational methods or ODE techniques.
Per informazioni, rivolgersi a: galise@mat.uniroma1.it

Giovedì 09 marzo 2023
Ore 15:00, Aula Picone (anche a distanza, tramite la piattaforma Zoom), Dipartimento di Matematica, Sapienza Università di Roma
Seminari di Ricerca in Didattica della Matematica
Enrico Rogora (Sapienza Università di Roma)
Introdurre la nozione di funzione con l'algebra dei segmenti di Cartesio
Chi è interessato a partecipare a distanza può rivolgersi ad Annalisa Cusi (annalisa.cusi@uniroma1.it)

Venerdì 10 marzo 2023
Ore 11:00, Aula Roberta Dal Passo, Dipartimento di Matematica, Università degli Studi di Roma "Tor Vergata"
corso di dottorato
Martina Lanini (Università di Roma "Tor Vergata") e Marco Trevisiol (Sapienza Università di Roma)
Springer correspondence
This course aims to present one of the first results in geometric representation theory: the Springer Correspondence. The goal of geometric representation theory is to study representations of certain algebraic structures (finite Weyl groups, in our case) via geometric methods. Springer's result (generalised in several ways and still a central tool nowadays) achieves a construction of all representations of a finite Weyl group (e.g. the symmetric group) in terms of the cohomology of a certain variety. The striking point of Springer's construction is that the Weyl group action he discovers is highly non trivial and hidden, since the Weyl group does not act on the variety itself. Some of the arguments we are going to address are: representation theory of the symmetric group via tableaux combinatorics; flag variety and Bruhat decomposition; nilpotent cone, the Springer resolution, Springer fibres; construction of the Weyl group action on the cohomology of the Springer sheaf. The discussion will be more or less detailed, depending on the participants' background and interests.

Venerdì 10 marzo 2023
Ore 14:30, Aula Roberta Dal Passo, Dipartimento di Matematica, Università degli Studi di Roma "Tor Vergata"
Algebra and Representation Theory Seminar
Alexander Puetz (Ruhr-University Bochum)
Desingularizations of Quiver Grassmannians for the Equioriented Cycle Quiver
Originating in the geometric study of quiver representations and in cluster algebra theory, they have been applied extensively in recent years in a Lie-theoretic context, namely as a fruitful source for degenerations of (affine) flag varieties. This approach allows for an application of homological methods from the representation theory of quivers to the study of such degenerate structures. The resulting varieties being typically singular, a construction of natural desingularizations is very desirable. We construct torus equivariant desingularizations of quiver Grassmannians for arbitrary nilpotent representations of an equioriented cycle quiver. This applies to the computation of their torus equivariant cohomology. hr>

Venerdì 10 marzo 2023
Ore 16:00, Aula Picone, Dipartimento di Matematica, Sapienza Università di Roma
Seminari per i docenti a.a. 2022-2023
Andrea Bruno (Università Roma Tre)
La trigonometria mediante rotazioni e numeri complessi

Venerdì 10 marzo 2023
Ore 16:00, Aula Roberta Dal Passo, Dipartimento di Matematica, Università degli Studi di Roma "Tor Vergata"
Algebra and Representation Theory Seminar
Evgeny Feigin (HSE Moscow/Weizmann Institute of Science )
Cauchy identities and representation theory
The celebrated Cauchy identity rewrites a certain infinite product as a sum of products of Schur functions. The identity has a vast number of applications and interpretations; in particular, one can understand the infinite product as the character of polynomial functions on the space of square matrices and the products of Schur functions as characters of tensor products of irreducible gl(n) modules. The classical Cauchy identity has (at least) three natural generalizations: a nonsymmetric version, a q-version and their mixture. The representation theory of the nonsymmetric Cauchy is governed by the Borel subalgebra, the q-Cauchy identity is controlled by the representations of the current algebras and the nonsymmetric q-Cauchy identity has to do with the modules over the Iwahori algebra. We will discuss all these identities and explain the relevant representation theory. Based on joint works with Anton Khoroshkin, Ievgen Makedonskyi and Daniel Orr.

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