Notiziario Scientifico
a cura del Dipartimento di Matematica Guido Castelnuovo, Sapienza Università di Roma
Settimana dal 07-03-2022 al 13-03-2022
Martedì 08 marzo 2022
Ore 11:00, Aula D'Antoni, Dipartimento di Matematica dell'Università di Tor Vergata
Corso di dottorato
Mark Andrea de Cataldo (Stony Brook University)
Cohomological aspects of non abelian Hodge theory for curves in positive characteristic
In these four lectures, I will briefly review the non Abelian Hodge Theorem by Simpson and others for Higgs bundles and flat connections over a compact Riemann surface (curve over the complex numbers). I will then discuss a recent development concerning a cohomological version of this result for curves over fields of positive characteristic. If time allows, I will give cohomological applications to the situation over the complex numbers by lifting results from finite fields.
Il corso può essere seguito usando questo link di teams
Martedì 08 marzo 2022
Ore 15:00, Sala di Consiglio e Zoom ID riunione: 867 4440 2839 Passcode: MDN, Dipartimento di Matematica, Sapienza Università di Roma
Seminario di Modellistica Differenziale Numerica
Andrea Thomann (Mainz University)
An all-speed scheme for isentropic two phase flows
We are interested in the numerical simulation of liquid-gas mixtures, where the sound speed of the liquid phase is consistently faster than the one of the gas phase. If in addition, the material wave is significantly slower than the individual acoustic waves, the system can exhibit three different scales of wave speeds. In these regimes, which are characterized by small, potentially different phase Mach numbers, using an explicit scheme requires a time step that scales with the smallest appearing Mach number. Moreover, the main interest often lies on a sharp resolution of slow dynamics which would allow for a much larger time step. Therefore, we use implicit-explicit (IMEX) time integrators where fast waves are treated implicitly leading to a CFL condition which is restricted only by the local flow velocity. In this talk, we present an all-speed finite volume scheme for isentropic two-phase flows based on a symmetric hyperbolic thermodynamically compatible model given developed by Toro and Romenski (2004). Since the flow regimes can range from compressible for gases to almost incompressible for some liquids, the asymptotic preserving (AP) property together with the correct numerical viscosity are essential. Since the flow regime of the considered two-phase flow model is characterized by two potentially distinct phase Mach numbers, different singular Mach number limits can be obtained which depend on the constitution of the mixture. The AP property of our IMEX scheme is obtained by using a reference solution approach. The consistency with single phase flow, accuracy and the approximation of material waves in different Mach number regimes is illustrated in numerical simulations.
Per informazioni, rivolgersi a: carlini@mat.uniroma1.it
Mercoledì 09 marzo 2022
Ore 14:00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
seminario di Algebra e Geometria
Philippe Eyssidieux (Université Grenoble-Aples)
Orbifold Kähler Groups related to Mapping Class groups
We prove the uniform stability of some smooth DM stacks obtained by further ramifying the moduli stack of stable curves over the Deligne-Mumford boundary divisor and apply this construction to prove that most of the complex projective surfaces proposed by Bogomolov and Katzarkov in 1997 as counter examples to the Shafarevich conjecture on holomorphic convexity do satisfy this statement. Joint work with Louis Funar. arXiv:2112.06726 [math.AG].
Per informazioni, rivolgersi a: diverio@mat.uniroma1.it
Mercoledì 09 marzo 2022
Ore 14:30, Canale Youtube dell'IAC https://www.youtube.com/watch?v=IgelDH-uLbU, Istituto per le Applicazioni del Calcolo, Consiglio Nazionale delle Ricerche
Online Seminars on Artificial Intelligence and Mathematics 2022 https://www.aim.iac.cnr.it/home
Alessandro Rudi (INRIA and École Normale Supérieure, Paris )
Representing non-negative function. With applications in non-convex optimization and beyond
Many problems in applied mathematics are expressed naturally in terms of non-negative functions. While linear models are well suited to represent functions with output in R, being at the same time very expressive and flexible, the situation is different for the case of non-negative functions where the existing models lack one of good properties. In this talk we present a rather flexible and expressive model for non-negative functions. We will show direct applications in probability representation and non-convex optimization. In particular, the model allows to derive an algorithm for non-convex optimization that is adaptive to the degree of differentiability of the objective function and achieves optimal rates of convergence. Finally, we show how to apply the same technique to other interesting problems in applied mathematics that can be easily expressed in terms of inequalities.
Per informazioni, rivolgersi a: roberto.natalini@cnr.it
Giovedì 10 marzo 2022
Ore 11:00, Aula Dal Passo, Dipartimento di Matematica dell'Università di Tor Vergata
Corso di dottorato
Mark Andrea de Cataldo (Stony Brook University)
Cohomological aspects of non abelian Hodge theory for curves in positive characteristic
In these four lectures, I will briefly review the non Abelian Hodge Theorem by Simpson and others for Higgs bundles and flat connections over a compact Riemann surface (curve over the complex numbers). I will then discuss a recent development concerning a cohomological version of this result for curves over fields of positive characteristic. If time allows, I will give cohomological applications to the situation over the complex numbers by lifting results from finite fields.
Il corso può essere seguito usando questo link di teams
Giovedì 10 marzo 2022
Ore 14:00, Aula M3, Dipartimento di Matematica, Università Roma Tre
Seminario di Geometria
Gavril Farkas (Humboldt)
Counting maps with prescribed incidence conditions
The question of computing the number of maps of fixed degree d from a curve to a target variety X and verifying n incidence conditions can be viewed as a counterpart of the problem of determining the Gromov-Witten invariants of X. Using degeneration and Schubert calculus, we solve this problem when the target variety is the projective space of dimension r, and determine these numbers completely for linear series of arbitrary dimension when d is sufficiently large, and for all d when either \(r=1\) or \(n=r+2\). Our formulas generalize and give new proofs of very recent results of Tevelev and of Cela-Pandharipande-Schmitt. Joint work with Carl Lian.
Per informazioni, rivolgersi a: amos.turchet@uniroma3.it
Giovedì 10 marzo 2022
Ore 14:30, Sala di Consiglio - https://meet.google.com/ads-dekx-bgm, Dipartimento di Matematica, Sapienza Università di Roma
P(n)/N(p) : Problemi differenziali nonlineari/Nonlinear differential problems
Qinbo Chen (KTH Royal Institute of Technology)
On nonlinear generalizations of the vanishing discount problem
The vanishing discount problem for Hamilton-Jacobi equations has been studied extensively in recent years. In this talk we discuss the nonlinear generalizations of this problem. Let \( M \) be a compact manifold and \( H(x,p,u): T^*M\times\mathbb{R}\to \mathbb{R }\) be a continuous Hamiltonian which is convex and coercive in \( p \) and is strictly increasing in \( u \). For each parameter \( \lambda>0\) we consider the H-J equation \( H\big(x, d_xu, \lambda\,u(x)\big)=c_0\) and denote its solution by \( u_\lambda\). People are interested in the asymptotic behavior of the family of solutions. By introducing an approximation scheme on selecting Mather measures, I will show that \( u_\lambda\) converges uniformly, as \( \lambda\to 0\), to a specific solution of the critical H-J equation. I also give characterization formulae for the limit solution.
Per informazioni, rivolgersi a: galise@mat.uniroma1.it
Giovedì 10 marzo 2022
Ore 15:00, Aula M3, Dipartimento di Matematica, Università Roma Tre
Seminario di Geometria
Andrea Di Lorenzo (Humboldt)
Integral Chow ring of moduli of stable 1-pointed curves of genus two
Moduli of curves play a prominent role in algebraic geometry. In particular, their rational Chow rings have been the subject of intensive research in the last forty years, since Mumford first investigated the subject. There is also a well defined notion of integral Chow ring for these objects: this is more refined, but also much harder to compute. In this talk I will present the computation of the integral Chow ring of moduli of stable 1-pointed curves of genus two, obtained by using a new approach to this type of questions (joint work with Michele Pernice and Angelo Vistoli).
Per informazioni, rivolgersi a: amos.turchet@uniroma3.it
Giovedì 10 marzo 2022
Ore 15:00, Il seminario sarà tenuto in modalità telematica, Il link per partecipare al seminario sarà condiviso con gli interessati
Seminari di Ricerca in Didattica e Storia della Matematica
Alberto Cogliati (Università di Pisa)
Il teorema fondamentale del calcolo integrale: un percorso storico dalle origini a Darboux
Per informazioni, rivolgersi a: annalisa.cusi@uniroma1.it
Venerdì 11 marzo 2022
Ore 14:30, Aula "Roberta Dal Passo", Dipartimento di Matematica, Università degli Studi di Roma "Tor Vergata"
Algebra and Representation Theory Seminar
Andrea Bianchi (University of Copenhagen)
Symmetric groups, Hurwitz spaces and moduli spaces of surfaces
Let \(d\geq2\), and consider the symmetric group \(S_d\). For \(k\geq0\), the classical Hurwitz space \(\mathrm{hur}_k(S_d)\) parametrises \(d\)-fold branched covers of the complex plane \(\mathbb C\) with precisely k branch points. We introduce an amalgamation of all Hurwitz spaces, for varying \(k\), into a single space \(\mathrm{Hur}(S_d)\). The construction relies on the notion of "partially multiplicative quandle", an algebraic structure slightly weaker than the structure of group, and we will see how to consider \(S_d\) as a PMQ in a convenient way. The main motivation to consider the amalgamated Hurwitz space \(\mathrm{Hur}(S_d)\) is the following. For all \(g\geq0\) and \(n\geq1\), let \(M_{g,n}\) denote the moduli space of Riemann surfaces of genus \(g\) with \(n\) ordered and parametrised boundary components. Our main result ensures that if \(d\) is large enough (with respect to \(g\) and \(n\)), then there exists a connected component of \(\mathrm{Hur}(S_d)\) which is homotopy equivalent to \(M_{g,n}\). Moreover, the space \(\mathrm{Hur}(S_d)\) carries a natural structure of topological monoid, and we will briefly sketch the computation of the stable, rational cohomology of its connected components. The result is very explicit in degrees up to roughly \(d\). Letting \(d\) go to infinity, one can in particular recover the Mumford conjecture on the stable, rational cohomology of moduli spaces of Riemann surfaces, originally proved by Madsen and Weiss.
Link streaming
Venerdì 11 marzo 2022
Ore 16:00, Aula "Roberta Dal Passo", Dipartimento di Matematica, Università degli Studi di Roma "Tor Vergata"
Algebra and Representation Theory Seminar
Rita Fioresi (Università di Bologna)
Generalized Root Systems
In Lie theory we define root systems in several contexts: Lie algebras, superalgebras, affine algebras, etc. There is even more: Kostant defines a more general notion of root systems, by taking roots with respect to a generic toral subalgebra (i.e. not necessarily maximal). All these notion of root systems do not behave well with respect to quotients: the quotient (or projection) of a root systems is not in general a root system. We present here a more general notion of root system, inspired by Kostant, which accomodates all of the above examples and behaves well with respect to quotients and projections. We give a classification theorem for rank 2 generalized root system: there are only 14 of them up to combinatorial equivalence, moreover they are all quotients of Lie algebras root systems. We also prove that root systems of contragredient Lie superalgebras are quotients of root systems of Lie algebras, up to combinatorial equivalence. In the end, we relate our construction with the problem of determining the conjugacy class of two Levi subgroups in a Lie (super)algebra. N.B.: this talk is part of the activity of the MIUR Excellence Department Project CUP E83C18000100006
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