## Weekly Bulletin (it)

**Notiziario dei seminari di carattere matematico**

a cura del Dipartimento di Matematica

*Guido Castelnuovo*, Sapienza Università di Roma

Settimana dal 20-05-2024 al 26-05-2024

**Lunedì 20 maggio 2024**

Ore 13:30, Sal di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma

Seminario di Dipartimento

Giacomo Di Gesù
*Sharp asymptotics for metastable stochastic processes*

Metastability phenomena show up in the dynamical behavior of a large variety of complex real world systems. From a mathematical point of view the dynamics of such systems may be modeled by means of stochastic differential equations, such as the Langevin equation at small temperature in molecular dynamics. An efficient computation of metastable dynamics over the long time scales of interest is a particularly challenging and important problem for applications.
In this talk I will present some theoretical results providing sharp estimates on key quantities, such as metastable transition times, spectral gaps or exit point distributions. The results concern overdamped Langevin equations in Euclidean domains, discrete space diffusions arising as coarse-grained models in statistical mechanics, and infinite-dimensional systems such as the stochastic Allen-Cahn equation. The proofs are based on a spectral-theoretical approach and employ PDE tools of semiclassical analysis, adapted to the different settings.

**Lunedì 20 maggio 2024**

Ore 14:30, Aula Dal Passo, Dipartimento di Matematica, Università di Roma "Tor Vergata"

Seminario di sistemi dinamici

Stefano Galatolo (Università di Pisa)
*Rare Events and Hitting Time Distribution for Discrete Time Samplings of Stochastic Differential Equations*

We consider a random discrete time system in which the evolution of a stochastic differential equation is sampled at a sequence of discrete times. We set up a functional analytic framework for which we can prove the existence of a spectral gap and estimate the behavior of the leading eigenvalue of the related transfer operator as the system is perturbed by putting a ”hole” in it that corresponds to a rare event. By doing so, we derive the distribution of the hitting times corresponding to the rare event and the extreme value theory associated with it. 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 20 maggio 2024**

Ore 14:30, Sala di Consiglio, Sala di Consiglio

seminario di Analisi Matematica

Armin Schikorra (University of Pittsburgh)
* On s-Stability of \(W^{s,n/s}\)-minimizing maps between spheres in homotopy classes*

We consider maps between spheres \(S^n\) to \(S^\ell\) that minimize the Sobolev-space energy \(W^{s,n/s}\) for some \(s \in (0,1)\) in a given homotopy class. The basic question is: in which homotopy class does a minimizer exist? This is a nontrivial question since the energy under consideration is conformally invariant and bubbles can form. Sacks-Uhlenbeck theory tells us that minimizers exist in a set of homotopy classes that generates the whole homotopy group \(\pi_{n}(S^\ell)\). In some situations explicit examples are known if n/s = 2 or s=1. In our talk we are interested in the stability of the above question in dependence of s. We can show that as s varies locally, the set of homotopy classes in which minimizers exist can be chosen stable. We also discuss that the minimum \(W^{s,n/s}\)-energy in homotopy classes is continuously depending on s. Joint work with K. Mazowiecka (U Warsaw) This seminar is part of the activities of the Dipartimento di Eccellenza CUP B83C23001390001 and it is funded by the European Union – Next Generation EU.

Per informazioni, rivolgersi a: * azahara.delatorrepedraza@uniroma1.it*

**Lunedì 20 maggio 2024**

Ore 16:00, Aula Dal Passo, Dipartimento di Matematica, Università di Roma "Tor Vergata"

Seminario di sistemi dinamici

Fabrizio Bianchi (Università di Pisa)
*Every complex Hénon map satisfies the Central Limit Theorem.*

Hénon maps were introduced by Michel Hénon as a simplified model of the Poincaré section of the Lorenz model. They are among the most studied discrete-time dynamical systems that exhibit chaotic behaviour. Complex Hénon maps have been extensively studied over the last three decades, in parallel with the development of pluripotential theory. I will present a recent result obtained with Tien-Cuong Dinh, where we show that the measure of maximal entropy of every complex Hénon map is exponentially mixing of all orders for Hölder observables. As a consequence of a recent result by Björklund-Gorodnik, the Central Limit Theorem holds for all Hölder observables. 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*

**Martedì 21 maggio 2024**

Ore 14:30, aula d'Antoni, Dipartimento di Matematica, Università di Roma Tor Vergata

seminario di Geometria

Marco D'Addezio (IRMA Strasbourg)
*Edged Crystalline Cohomology*

I will talk about a new cohomology theory for algebraic varieties in positive characteristic, called edged crystalline cohomology. This is a generalisation of crystalline cohomology and depends on the choice of a "decay-function''. Linear decay-functions correspond to integral versions of rigid cohomology, while logarithmic decay-functions produce the conjectured family of log-decay crystalline cohomology theories, parametrised by positive real numbers. During the talk, I will explain the construction of this theory after a brief recall of the classical crystalline and rigid cohomology theories.

Per informazioni, rivolgersi a: * guidomaria.lido@gmail.com*

**Martedì 21 maggio 2024**

Ore 15:00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma

Seminario di Modellistica Differenziale Numerica

Francesca Lourdes Ignoto (Sapienza University of Rome)
*Dissolution of Multiple Variable-in-Shape Drug Particles Using the Level-Set Method*

The dynamics of variable-in-shape drug particles are fundamental to predict the dissolution of drugs in a fluid. In this talk we propose a new approach which consists of describing the dissolution process of a drug particle using the level-set method. In this way, it is possible to recover the particle surface at any time as the zero level set of the solution of an Hamilton-Jacobi equation. This approach extends a previous work of 2022, and it allows us to deal with particles with rectangular shape as well as to simulate the joint dissolution of multiple particles with different shape and size. Firstly, we introduce the problem and the basic functions that come into play in the dissolution process, then we present the new methodology and finally we discuss some numerical simulations in the 2D case.

**Mercoledì 22 maggio 2024**

Ore 14:00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma

Seminario di Algebra e Geometria

George Lusztig (Massachusetts Institute of Technology)
*Decomposing a reductive group into strata*

Let G be a reductive connected group over an algebraically closed field of characteristic p . Of particular importance in the study of G is the set u(G) of unipotent conjugacy classes. It is known that the cardinal of u(G) depends on p. We propose to define another set str(G) which is closely related to u(G) but is independent of p. But while u(G) is a subset of the set of conjugacy classes, str(G) is a quotient of that set. Its elements can be viewed as parts of a partition of G. We also show that the set of conjugacy classes in the Weyl group is partitioned according to str(G). This seminar is part of the activities of the Dipartimento di Eccellenza CUP B83C23001390001 and it is funded by the European Union – Next Generation EU.

**Mercoledì 22 maggio 2024**

Ore 14:00, Aula A, Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre, Via della Vasca Navale 84, e via Teams al seguente link

seminario di Fisica Matematica

Giuseppe Gaeta (Università di Milano)
*RG & PDEs*

I discuss how ideas stemming from Renormalization Group (first presented by Goldenfeld, Martin, Oono & Liu and by Bricmont & Kupiainen – and used by the Japanese school in the analysis of Dynamical Systems, following a seminal paper by Iwasa & Nozaki) can be implemented in the analysis of (nonlinear) PDEs, in particular concerning their asymptotic solutions for long time and/or at large spatial scales. This approach is based on the (Cartan) geometrical theory of PDEs, and on the (asymptotic) symmetry analysis of these.
I will also present some very recent advances concerning a simple computational approach to this problem, for evolution PDEs and a certain class of would-be symmetries.

**Mercoledì 22 maggio 2024**

Ore 14:30, Aula M6, Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre

Seminario di Analisi Matematica

Rafael Lopez Soriano (Universidad de Granada)
*On doubly critical elliptic systems*

We will consider a type of cooperative nonlinear elliptic system in R^N. The interest of this problem is based on the presence of Sobolev or Sobolev-Hardy critical power nonlinearities and a nonlinear coupling, possibly critical, as well as singular potentials of Hardy type. Using variational methods we will focus on the existence of bound and ground states of the underlying energy functional. Finally, for a certain range of parameters, we derive some qualitative properties and a classification criterion. This seminar is based on some works in collaboration with Angel Arroyo (UA), Eduardo Colorado (UC3M), Francesco Esposito (UNICAL) Alejandro Ortega (UNED) and Berardino Sciunzi (UNICAL).

Per informazioni, rivolgersi a: * luca.battaglia@uniroma3.it*

**Mercoledì 22 maggio 2024**

Ore 14:30, Dal Passo, Dipartimento di Matematica, Università Roma Tor Vergata

Seminario RoMaDS

Dario Trevisan (Università di Pisa)
*Gaussian Approximation and Bayesian Posterior Distribution in Random Deep Neural Networks*

We establish novel rates for the Gaussian approximation of randomly initialized deep neural networks with Gaussian parameters and Lipschitz activation functions, in the so-called wide limit, i.e., where the sizes of all hidden layers become large. Using the Wasserstein metric and related functional analytic tools, we demonstrate the distribution of the output of a network and the corresponding Gaussian approximation are at a distance that scales inversely with the width of the network, surpassing previously established rates. Furthermore, we extend our findings to approximate the exact Bayesian posterior distribution of the network when the likelihood is a bounded Lipschitz function of the network output, on a finite training set. This includes common cases, such as the Gaussian likelihood, which is defined as the exponential of the negative mean squared error. Our inequalities thus shed light on the network's Gaussian behavior by quantitatively capturing the distributional convergence results in the wide limit. The exposition will aim to be self-contained, by introducing all the basic concepts related to artificial neural networks and Bayesian statistics to a mathematical audience. Based on arXiv:2203.07379 (joint with A. Basteri) and arXiv:2312.11737. Link streaming

Per informazioni, rivolgersi a: * salvi@mat.uniroma2.it*

**Mercoledì 22 maggio 2024**

Ore 16:00, Aula Dal Passo, Dipartimento di Matematica, Università di Roma Tor Vergata

Seminario Algebre di Operatori

Alessio Ranallo (University of Geneva)
*Low energy spectrum of the XXZ model coupled to a magnetic field*

I will report on recent developments concerning the control of a class of short-range perturbations of the Hamiltonian of an Ising chain. An example covered by our analysis is the celebrated XXZ chain. The talk is based on a joint work with S. Del Vecchio, J. Fröhlich, and A. Pizzo. The Operator Algebra Seminar schedule is here: https://sites.google.com/view/oastorvergata/home-page?authuser=0 Note: This talk is part of the activity of the MUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)

**Mercoledì 22 maggio 2024**

Ore 16:00, Aula M3, Dipartimento di Matematica e Fisica, Università Roma Tre

Junior seminar di Geometria Algebrica

Zakaria Brahimi (Università degli studi Roma Tre)
*Cubic fourfolds: some rational examples*

Projective spaces are the most basic algebraic varieties that we know and rational varieties are those that are the closest possible to being projective spaces. The question of rationality of algebraic varieties has led to the development of a rich and beautiful theory and many questions are still open. Historically, the rationality problem for quadric and cubic surfaces was settled classically over one hundred years ago. In this seminar, we will talk about some rational cubic fourfolds that are known. We look for them inside Hassett Divisors; These divisors are codimension one varieties in the Moduli space of cubic fourfolds. Maggiori informazioni sul sito: https://ricerca.mat.uniroma3.it/seminari/junior_seminars.html

**Mercoledì 22 maggio 2024**

Ore 16:00, sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma

seminari di Fisica Matematica

Abdallah Maichine ( Mohammed V University, Rabat)
*Decomposition of operators commuting with magnetic translations and applications*

For a uniform magnetic field, we provide a decomposition of operators commuting with the associated magnetic translations. As an application, we rewrite the energy per unit surface for an electronic system with 2d symmetry as a function of the magnetic field intensity. We also investigate the ground state of such a system. Besides, we also derive the Thomas-Fermi-like energy density in 2d and 3d. Il seminario si svolgerà all'interno delle attività del progetto PRIN 2022AKRC5P "Interacting Quantum Systems: Topological Phenomena and Effective Theories" finanziato dall’Unione europea – Next Generation EU.

Per informazioni, rivolgersi a: * basile@mat.uniroma1.it, domenico.monaco@uniroma1.it*

**Giovedì 23 maggio 2024**

Ore 13:00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma

Seminario di Modellistica Differenziale Numerica

Vincent Liu (University of Melbourne)
*A Hamilton-Jacobi-Bellman Approach to Ellipsoidal Approximations of Reachable Sets*

Society's ever-increasing integration of autonomous systems in day-to-day life has simultaneously brought forth concerns as to how their safety and reliability can be verified. To this end, reachable sets lend themselves well to this task. These sets describe collections of states that a dynamical system can reach in finite time, which can be used to guarantee goal satisfaction in controller design or to verify that unsafe regions will be avoided. However, general-purpose methods for computing these sets suffer from the curse-of-dimensionality, which typically prohibits their use for systems with more than a small number of states, even if they are linear. In this talk, we derive dynamics for a union and intersection of ellipsoidal sets that, respectively, under-approximate and over-approximate the reachable set for linear time-varying systems subject to an ellipsoidal input constraint and an ellipsoidal terminal (or initial) set. This result arises from the construction of a local viscosity supersolution and subsolution of a Hamilton-Jacobi-Bellman equation for the corresponding reachability problem. The proposed ellipsoidal sets can be generated with polynomial computational complexity in the number of states, making the approximation scheme computationally tractable for continuous-time linear time-varying systems of relatively high dimension.

**Giovedì 23 maggio 2024**

Ore 14:30, Aula 1B1, Dipartimento SBAI, Via Scarpa 10

Seminario "PDE A Tutto SBAI"

Gabrielle Nornberg (Universidad de Chile )
*Fundamental solutions and Liouville results for nonlinear nonlocal operators in cones*

In this talk we discuss the existence of fundamental solutions for a class of nonlinear nonlocal uniformly elliptic operators defined in cones and its application to derive Liouville results for Lane-Emden nonlocal type equations. Joint work with Disson dos Prazeres (Federal University of Sergipe, Brazil) and Alexander Quaas (Federico Santa María Technical University, Chile).

Per informazioni, rivolgersi a: * massimo.grossi@uniroma1.it*

**Venerdì 24 maggio 2024**

Ore 12:00, Dal Passo, Dipartimento di Matematica, Università di Roma "Tor Vergata"

Seminario di Analisi Matematica

Vitaly Moroz (Swansea University)
*Nonlinear elliptic problems with nonlocal interactions*

We present a survey of nonlinear elliptic equations with nonlocal interactions. These equations describe the collective behavior of self-interacting many-body systems at different scales, from atoms and molecules to the formation of stars and galaxies. What sets these models apart from classical nonlinear PDEs is the presence of nonlocal terms in the equations, introduced via Coulomb-type interactions or a fractional Laplacian term, or both. We provide an overview of typical problems with repulsive interactions originating from Density Functional Theory; and Choquard type problems with attractive gravitational interactions. We also outline recent results in problems featuring competing attractive/ repulsive terms, which create particularly complex structures.

NB: This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C23000330006

Per informazioni, rivolgersi a: * molle@mat.uniroma2.it*

**Venerdì 24 maggio 2024**

Ore 14:30, Aula D'Antoni, Dipartimento di Matematica, U Roma Tor Vergata

Joint Topology and Algebra & Representation Theory Seminar

Giovanni Paolini (U Bologna)
*Dual Coxeter groups of rank three*

I will discuss the combinatorics of the noncrossing partition posets associated with Coxeter groups of rank three. In particular, I will describe the techniques used to prove the lattice property and lexicographic shellability. These properties can then be used to solve several problems on the corresponding Artin groups, such as the K(π,1) conjecture, the word problem, the center problem, and the isomorphism between standard and dual Artin groups. This is joint work with Emanuele Delucchi and Mario Salvetti.

**Venerdì 24 maggio 2024**

Ore 15:00, Aula C (anche online, tramite la piattaforma Zoom), Dipartimento di Matematica, Sapienza Università di Roma

Seminari di Ricerca in Didattica della Matematica

Mirko Maracci (Università di Pisa)
*Il ruolo delle teorie nella ricerca in Didattica della Matematica*

Per informazioni, rivolgersi a: * annalisa.cusi@uniroma1.it*

**Venerdì 24 maggio 2024**

Ore 14:30, Aula M2, Dipartimento di Matematica e Fisica, Università Roma Tre

Colloquium di Matematica

Laura DeMarco (Harvard)
*The Mandelbrot set: geometry and arithmetic*

The Mandelbrot set M has been studied for many years and continues to baffle mathematicians. By definition, M is the set of all complex numbers c for which the orbit of 0 is bounded under iteration of f(z) = z^2 + c. Within M, there is a distinguished subset of what we call "PCF" or "special" parameters, where the orbit of 0 is finite. These parameters are special from both a dynamical and - somewhat surprisingly - a number-theoretic point of view. In this talk, we'll explore how the geometry of these PCF parameters is restricted by the number theory. This is joint work with Myrto Mavraki.

**Venerdì 24 maggio 2024**

Ore 16:00, Aula D'Antoni, Dipartimento di Matematica, U Roma Tor Vergata

Algebra and Representation Theory Seminar

Ulrich Kraehmer (TU Dresden)
*The ring of differential operators on a monomial curve is a Hopf algebroid*

The ring of differential operators on a cuspidal curve whose coordinate ring is a numerical semigroup algebra is shown to be a cocommutative and cocomplete left Hopf algebroid. If the semigroup is symmetric so that the curve is Gorenstein, it is a full Hopf algebroid (admits an antipode). Based on joint work with Myriam Mahaman (talk part of the activity of the MIUR Excellence Department Project MATH@TOV CUP CUP E83C23000330006).

Le comunicazioni relative a seminari da includere in questo notiziario devono pervenire esclusivamente
mediante apposita form da compilare online, entro le ore 24 del giovedì precedente la settimana
interessata. Le comunicazioni pervenute in ritardo saranno ignorate.
Per informazioni, rivolgersi all'indirizzo di posta elettronica
*seminari@mat.uniroma1.it*.

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