Notiziario Scientifico
a cura del Dipartimento di Matematica Guido Castelnuovo, Sapienza Università di Roma
Settimana dal 22-04-2024 al 28-04-2024
Lunedì 22 aprile 2024
Ore 14:00, Aula A primo piano, Dipartimento di Scienze, Roma Tre, Via della Vasca Navale 84
Seminario di Matematiche Complementari
Inmaculada LIzasoain (Universidad Pública de Navarra - Pamplona, Spagna)
Innovare nei corsi universitari di matematica di base. Il docente come persona e le scelte di contenuto e metodo
Come rapportarsi in quanto docenti universitari all’evoluzione culturale e di contesto nell’insegnamento della matematica nei corsi di base/istituzionali? Partendo dalla propria esperienza nei corsi di laurea in Ingegneria, Magistero e le lauree magistrali per l’insegnamento secondario in Spagna, la prof.ssa Lizasoain proporrà una riflessione sulla componente di incontro fra persone nell’aula, alle volte oscurata dalla riflessione “tecnica” sui contenuti e metodi.
Per informazioni, rivolgersi a: storiadidattica.matematica@uniroma3.it
Lunedì 22 aprile 2024
Ore 14:30, Aula Dal Passo, Dipartimento di Matematica, Università di Roma "Tor Vergata"
Seminario di Sistemi Dinamici
Anna Miriam Benini (Università di Parma)
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 self-maps of the unit disk. For autonomous iteration of inner functions (self-maps 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 (2023-2027)
Per informazioni, rivolgersi a: sorrentino@mat.uniroma2.it
Lunedì 22 aprile 2024
Ore 16:00, Aula Dal Passo, Dipartimento di Matematica, Università di Roma "Tor Vergata"
Seminario di Sistemi Dinamici
Andrew Clarke (UPC Barcelona)
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 Ramírez-Ros. Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027)
Per informazioni, rivolgersi a: sorrentino@mat.uniroma2.it
Lunedì 22 aprile 2024
Ore 16:00, Sala di Consiglio, Dipartimento di Matematica
Seminario di Probabilità
Andrea Agazzi (Università di Pisa)
Random Splitting of Fluid Models: Ergodicity, Convergence and Lyapunov exponents
We consider a family of processes obtained by decomposing the deterministic dynamics associated with some fluid models (e.g. Lorenz 96, 2d Galerkin-Navier-Stokes) into fundamental building blocks - i.e., minimal vector fields preserving some fundamental aspects of the original dynamics - and by sequentially following each vector field for a random amount of time. We characterize some ergodic properties of these stochastic dynamical systems and discuss their convergence to the original deterministic flow in the small noise regime. Finally, we show that the top Lyapunov exponent of these models is positive. This is joint work with Jonathan Mattingly and Omar Melikechi.
Per informazioni, rivolgersi a: silvestri@mat.uniroma1.it
Martedì 23 aprile 2024
Ore 14:30, Aula 1B1, Via scarpa 16, Sapienza, Dipartimento SBAI, Sapienza
Seminario "PDE A Tutto SBAI""
Elide Terraneo (Università di Milano)
Singular solutions to semilinear elliptic equations with exponential nonlinearities in 2-dimensions
By introducing a new classification of the growth rate of exponential functions, singular solutions for \(-\Delta u=f(u) \) in 2-dimensions with exponential nonlinearities are constructed. The strategy is to introduce a model nonlinearity ``close" to $f$, which admits an explicit singular solution. Then, one obtains an approximate singular solution, and concludes by a suitable fixed point argument. Our method covers a wide class of nonlinearities in a unified way, e.g., \(f(u) = u^re^{u^q}\ (q>1,r\in \mathbb{R}), f(u) = e^{u^{q}+u^r}\ ({q>1,\ q/2>r>0}) \) or \(f(u) = e^{e^u} \). As a special case, our result contains a pioneering contribution by Ibrahim--Kikuchi--Nakanishi--Wei for \(u(e^{u^2}-1) \). Joint work with Yohei Fujishima (Shizuoka University, Japan), Norisuke Ioku (Tohoku University, Japan) and Bernhard Ruf (Istituto Lombardo, Italy).
Per informazioni, rivolgersi a: massimo.grossi@uniroma1.it
Martedì 23 aprile 2024
Ore 14:30, aula D'Antoni, Dipartimento di Matematica, Università di Roma Tor Vergata
seminario di Geometria
Lukas Branter (University of Oxford)
Deformations and lifts of Calabi-Yau varieties in characteristic p
Homotopy theory allows us to study formal moduli problems via their tangent Lie algebras. We apply this general paradigm to Calabi-Yau varieties Z in characteristic p. First, we show that if Z has torsion-free crystalline cohomology and degenerating Hodge-de Rham spectral sequence (and for p=2 a lift to W/4), then its mixed characteristic deformations are unobstructed. This generalises the BTT theorem from characteristic 0 to characteristic p. If Z is ordinary, we show that it moreover admits a canonical (and algebraisable) lift to characteristic zero, thereby extending Serre-Tate theory from abelian varieties to Calabi-Yau varieties. This is joint work with Taelman, and generalises results of Achinger-Zdanowicz, Bogomolov-Tian-Todorov, Deligne-Nygaard, Ekedahl–Shepherd-Barron, Iacono-Manetti, Schröer, Serre-Tate, and Ward.
Per informazioni, rivolgersi a: guidomaria.lido@gmail.com
Martedì 23 aprile 2024
Ore 15:00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
Seminario di Modellistica Differenziale Numerica
Jules Berry (IRMAR - INSA Rennes)
Approximation of stable solutions to second order mean field game systems
We introduce a general framework for the study of numerical approximations of a certain class of solutions, called stable solutions, of second order mean-field game systems for which uniqueness of solutions is not guaranteed. To illustrate the approach, we focus on a very simple example of stationary second-order MFG system with local coupling and a quadratic Hamiltonian. We provide sufficient conditions for the stability of solutions and it turns out that stability is a generic property of the MFG. We then re-express the solutions of the system as zeros of a well chosen nonlinear map and establish the fact that stable solutions are regular points of this map. This fact is then used to study the approximation of solutions by finite elements and the local convergence of Newton's method in infinite dimension.
Martedì 23 aprile 2024
Ore 16:00, Aula Dal Passo, Dipartimento di Matematica, Università degli Studi di Roma "Tor Vergata"
Seminario di equazioni differenziali
Philippe Souplet (Université Sorbonne Paris Nord & CNRS)
Liouville-type theorems, singularities and universal estimates for nonlinear elliptic and parabolic problems
The Cauchy-Liouville theorem (1844) states that any bounded entire function of a complex variable is necessarily constant. In the realm of PDE's, by a Liouville-type theorem, one usually means a statement asserting the nonexistence of solutions in the whole space (or a suitable unbounded domain). Numerous results of this kind have appeared over the years and many far-reaching applications have arisen, conferring Liouville-type theorems an important role in the theory of PDE's and revealing strong connections with other mathematical areas (calculus of variations, geometry, fluid dynamics, optimal stochastic control). After a brief historical detour (minimal surfaces - Lagrange, Bernstein, de Giorgi, Bombieri,… and regularity theory for linear elliptic systems - Giaquinta, Necas, ...), we will recall the developments of the 1980-2000's on nonlinear elliptic problems, leading to powerful tools for existence and a priori estimates for Dirichlet problems (Gidas, Spruck, Caffarelli, ...), based on the combination of Liouville type theorems and renormalization techniques. In a more recent period, this line of research has also led to much progress in the study of singularities of solutions, both for stationary (elliptic) and evolution PDEs. In particular, in the case of power like nonlinearities, we will recall the equivalence between Liouville type theorems and universal estimates, based on a method of doubling-rescaling (joint work with P. Polacik and P. Quittner, 2007). Then we will present recent developments which show that these renormalization techniques can be applied to nonlinearities without any scale invariance, even asymptotically, with applications to initial and final blowup rates or decay rates in space and/or time. Nota: Questo seminario rientra tra le attività del progetto MUR "Dipartimenti d'eccellenza" MatMod@TOV (2023-27)
Per informazioni, rivolgersi a: sorrentino@mat.uniroma2.it
Martedì 23 aprile 2024
Ore 17:30, Aula Picone, Dipartimento di Matematica, Sapienza Università di Roma
YAMS - Young Algebraist Meeting in Sapienza
Lorenzo Campioni (Università degli Studi dell'Aquila )
Unrefinable partitions into distinct parts
A partition into distinct parts of a positive integer number is called unrefinable if none of the parts can be written as the sum of smaller integers, without introducing a repetition. Clearly the condition of being unrefinable imposes on the partition a non-trivial limitation on the size of the largest part and on the possible distribution of the parts. We prove an upper bound for the largest part in an unrefinable partition of n, and we call maximal those which reach the bound. We show a complete classification of maximal unrefinable partitions exhibiting a bijection with a suitable partitions into distinct parts, depending on the distance from the successive triangular number. In the last part we see some relations between unrefinable partitions and numerical semigroups and some ideas of future researches.
Per informazioni, rivolgersi a: sabino.ditrani@uniroma1.it
Mercoledì 24 aprile 2024
Ore 14:00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
Seminario di Algebra e Geometria
Marc Troyanov (EPFL (Ecole Polytechnique Fédérale de Lausanne))
Asymptotic Geometry in SOL
SOL is one of Thurston's eight classical homogeneous Riemannian geometries, possibly the most exotic one. To get some insight of this geometry, it might be helpful to visualize the shape of a large spheres in SOL. Clearly, the first challenge is to compute, or at least estimate, the Riemannian distance between two points. In this talk, I will propose a way to circumvent this difficulty by replacing the Riemannian metric with an asymptotically equivalent Finsler metric, inspired by architectural cardboard models. This alternative Finsler metric offers the double advantage of explicit computability of distances, coupled with a rapid convergence that closely aligns with the Riemannian metric, thus simplifying our understanding and representation of SOL geometry. As concrete applications, I will show how to represent the shape of large spheres in SOL and I will compute the volume entropy of this manifold.
Mercoledì 24 aprile 2024
Ore 14:15, online (zoom), registrazione disponibile alla pagina https://indico.gssi.it/event/410/
ciclo Mathematical Challenges in Quantum Mechanics
Alessandro Teta (Università degli Studi di Roma "La Sapienza")
Many-particle systems with contact interactions
Quantum Hamiltonians with contact (or zero-range) interactions are useful models to analyze the behaviour of quantum systems at low energy in different contexts. In this talk we discuss recent mathematical results on the construction of such Hamiltonians for a system of \(N \geq 3\) interacting bosons in dimension three as self-adjoint and lower bounded operators in the appropriate Hilbert space. We will also show the connection with a previous result obtained by Albeverio, Hoegh-Krohn and Streit in 1977 and we will discuss possible applications to the Efimov effect. The talk is based on a series of works in collaboration with G. Basti, C. Cacciapuoti, D. Ferretti, R. Figari, D. Finco and H. Saberbaghi.
Per informazioni, rivolgersi a: monaco@mat.uniroma1.it
Mercoledì 24 aprile 2024
Ore 16:00, Aula Dal Passo, Dipartimento di Matematica, Università di Roma Tor Vergata
Seminario Algebre di Operatori
Alex Bols (ETH Zürich)
The anyon sectors of Kitaev's quantum double models
In this talk I will explain how to extract an `anyon theory' (braided tensor category) from a gapped ground state of an infinite two-dimensional lattice spin system. Just as in the DHR formalism from AQFT, the anyon types correspond to certain superselection sectors of the observable algebra of the spin system. We apply this formalism to Kitaev's quantum double model for finite gauge group G, and find that the anyon types correspond precisely to the representations of the quantum double algebra of G. The Operator Algebra Seminar schedule is here: https://sites.google.com/view/oastorvergata/home-page?authuser=0
Mercoledì 24 aprile 2024
Ore 16:45, sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
seminario di Fisica Matematica
Morris Brooks (Università di Zurigo)
The Fröhlich polaron at strong coupling
In this talk we will review some recent results on the Fröhlich polaron, which is a model for a charged particle coupled to a polarizable medium. We will especially focus on a conjecture due to Landau and Pekar from 1948, claiming that the mass of the charged particle is effectively increased due to its interaction with the environment according to the asymptotic formula \( m_{eff}=\alpha^4 m_{LP} \), where \(\alpha\) is the coupling strength between particle and medium and \( m_{LP}\) is an explicit constant. Notably the recent progress is due to two rather distinct approaches, one working with functional integration (probabilistic/ commutative) and the other one working with functional analytic methods (operator theory/ non-commutative), which are connected by the celebrated Feynman-Kac formula.
Per informazioni, rivolgersi a: basile@mat.uniroma1.it, domenico.monaco@uniroma1.it
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