## Notiziario Scientifico

**Notiziario dei seminari di carattere matematico**

a cura del Dipartimento di Matematica

*G. Castelnuovo*, Sapienza Università di Roma

Settimana dal 29-04-2019 al 05-05-2019

**Lunedì 29 aprile 2019**

Ore 14:00, Aula 211 secondo piano, Dipartimento di Matematica e Fisica Universita' degli Studi Roma Tre Largo San Leonardo Murialdo,1

MINI COURSE PHD IN MATHEMATICS

Amine Asselah (Universite Paris Est Creteil Val-de-Marne )
*Around Random Walks, Interacting or Not *

We plan to explain in six hours how to treat two phenomena involving simple random walks, and using simple estimates about them. The treatment is going to be mathematical even though the problems arise in physics: the actual models are going to be highly idealized, using basically random walks...and we hope to have self-contained lectures (no pre-requisites). 1-The difficulty of building long fingers if random walks move according to internal diffusion limited aggregation which is a celebrated model of erosion. 2-The difficulty of hitting colored lattice sites, if their density is low. This is rather linked with the phenomena of avoiding an acqueous solvant for long hydrophobic polymer. Info: http://www.matfis.uniroma3.it/dottorato/corsi_dottorato_completo.php?dottora to=matematica

**Lunedì 29 aprile 2019**

Ore 14:15, Aula di Consiglio, Dipartimento di Matematica

Seminario di Analisi Matematica

David Lannes (Universita' di Bordeaux)
*Waves Interacting With A Partially Immersed Obstacle In The Boussinesq Regime*

In this talk we shall see the derivation and mathematical analysis of a wave-structure interaction problem which can be reduced to a transmission problem for a Boussinesq system. Initial boundary value problems and transmission problems in dimension d= 1 for 2×2 hyperbolic systems are well understood. However, for many applications, and especially for the description of surface water waves, dispersive perturbations of hyperbolic systems must be considered. We consider here a configuration where the motion of the waves is governed by a Boussinesq system (a dispersive perturbation of the hyperbolic nonlinear shallow water equations), and in the presence of a fixed partially immersed obstacle. We shall insist on the differences and similarities with respect to the standard hyperbolic case, and focus our attention on a new phenomenon, namely, the apparition of a dispersive boundary layer. In order to obtain existence and uniform bounds on the solutions over the relevant time scale, a control of this dispersive boundary layer and of the oscillations in time it generates is necessary. This analysis leads to a new notion of compatibility condition that is shown to coincide with the standard hyperbolic compatibility conditions when the dispersive parameter is set to zero. These phenomena are likely to play a central role in the analysis of initial boundary value problems for dispersive perturbations of hyperbolic systems. Ref: D. Bresch, D. Lannes, G. Métivier, Waves Interacting With A Partially Immersed Obstacle In The Boussinesq Regime, arXiv:1902.04837

**Lunedì 29 aprile 2019**

Ore 14:30, Aula 311, Dipartimento di Matematica e Fisica Largo S. L. Murialdo 1

Final exam Ph.D in Mathematics

Michele Savarese
*Coherent sheaves on primitive multiple curves and their moduli*

**Lunedì 29 aprile 2019**

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

Algebra and Representatino Theory Seminar

Carmelo Antonio Finocchiaro (Università di Catania)
*Spectral spaces of rings and modules and applications*

Let \(K\) be a field and \(D\) be a subring of \(K\). The space \(Zar(K/D)\) of all the valuation domains of \(K\) containing \(D\) as a subring can be endowed with the topology, called the Zariski topology, generated by the sets of the type \(Zar(K/D[x])\) , for every \(x\) in \(K\). In [C. A. Finocchiaro, M. Fontana, K. A. Loper, "The constructible topology on spaces of valuation domains", Trans. Amer. Math. Soc. 365, n. 12 (2013), 6199-6216] it was proved that \(Zar(K/D)\) is a spectral space and a ring whose prime spectrum is homeomorphic to \(Zar(K/D)\) was explicitly provided. In this talk we will introduce a spectral extension of \(Zar(K/D)\) , that is, the space of all \(D\)-submodules of \(K\). More generally, given any ring \(A\), a Zariski-like spectral topology can be given to the space \(S_A(M)\) of \(A\)-submodules of an \(A\)-module \(M\). Some application to flat modules (see [C. A. Finocchiaro, D. Spirito, "Topology, intersection of modules and flat modules", Proc. Amer. Math. Soc. 144 (2016), no. 10, 4125-4133]) will be presented.

**Martedì 30 aprile 2019**

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

Seminario di Analisi Matematica

Nicola Gigli (SISSA)
*Functional analysis and metric geometry*

Aim of the talk is to present some aspects of the important role that functional analysis has in the context of metric geometry. I shall discuss both the case of synthetic description of lower Ricci curvature bounds, where this role is by now well understood, and some potential applications to the world of lower sectional curvature bounds, where it might potentially lead to the solution of long-standing open problems - This talk is part of the activity of the MIUR Excellence Department Project CUP E83C18000100006

**Martedì 30 aprile 2019**

Ore 16:00, 1B1, Pal. RM002, SBAI (Dipartimento di Scienze di Base e Applicate per l'Ingegneria)

Incontri di Algebra e Geometria allo SBAI

Magdalena Boos (Ruhr-Universität Bochum)
*Conjugation on the nilpotent cone*

One of the best known group actions on an affine variety is the conjugation action of GL_n(C) on the nilpotent cone N of complex nilpotent matrices of square size n. In particular interesting is the fact that there are only finitely many orbits which are parametrized by the Jordan canonical form, i.e. by partitions. We generalize this setting by looking at parabolic conjugation on certain subvarieties of N. It is not clear, if the number of orbits is still finite and what the orbits look like. In this talk, we will focus on representation-theoretic methods with which we can examine the actions. Furthermore, we discuss known results and future goals.

**Giovedì 02 maggio 2019**

Ore 14:00, Aula Dal Passo 1201, Dipartimento di Matematica, Università di Roma Tor Vergata

seminario di Analisi Numerica

Michele Benzi (Scuola Normale Superiore, Pisa)
*Solving Differential Equations on Graphs*

There is currently considerable interest in a class of models (known as Quantum Graphs) which can be described in terms of PDEs posed on large and possibly complex graphs. Such models have found applications in quantum chemistry, nanotechnology, solid state physics, neuroscience, network flow, and other areas. Both boundary and initial value problems are of interest in applications, as well as eigenvalue problems. In this talk I will present some methods for solving simple model PDEs on graphs. Discretization of PDEs posed on graphs using linear finite elements and implicit time stepping techniques leads to sparse systems of algebraic equations of huge size which, however, possess favorable properties for iterative solution of the reduced-order system obtained by Schur complement reduction. The use of a preconditioned conjugate gradient method leads to optimal solution complexity in the elliptic case. Some results about diffusion on graphs will also be discussed. This is joint work with Mario Arioli (LUM ``Jean Monnet”, Bari)

**Giovedì 02 maggio 2019**

Ore 14:30, Aula di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma

P(n)/N(p):Problemi differenziali nonlineari/Nonlinear differential problems

Martina Magliocca (Sapienza Università di Roma)
*The regularizing effect of perturbed superlinear gradient terms in elliptic equations*

We want to discuss the regularizing effect induced by superlinear terms in some class of elliptic equations.To give an idea, our model equation is \(-\Delta u=g(u)|\nabla u|^q+f\;\text{in }\Omega,\) being \( \Omega\subset \mathbb{R}^N\) open and bounded,\(N\ge 2\), \(g(u)>0\) and \(q<2\).We assume that the term \(g(u)|\nabla u|^q\)behaves in a superlinear way. Roughly speaking, if \(g(u)\equiv \text{const} \), then we are asking for \(1 < q < 2\). In the more general case \(g(u)\not\equiv \text{const}.\), the \(q\) threshold is influenced by this perturbation term and the superlinear \(q\) range depends on its growth. An important remark on this kind of problems concerns the data assumptions, which have to satisfy well precises compatibility conditions in order to have existence of solutions. We will show that, under certain growth assumptions on \(g(u)\), we can relax the regularity needed on the data w.r.t. the case \(g(u)=\text{const}.\). These results are contained in a ongoing work with S. Segura de León \& M. Latorre Balado.

**Giovedì 02 maggio 2019**

Ore 14:30, Aula 211, Dipartimento di Matematica e Fisica Largo S. L. Murialdo 1

Seminario

Oscar Garcia-Prada (ICMAT, Madrid)
*Higgs bundles and higher Teichmüller spaces*

Some connected components of a moduli space are mundane in the sense that they are distinguished only by obvious topological invariants or have no special characteristics. Others are more alluring and unusual either because they are not detected by primary invariants, or because they have special geometric significance, or both. In this talk, I will describe examples of such special components in moduli spaces of G-Higgs bundles on closed Riemann surfaces or, quivalently, moduli spaces of surface group representations into a semisimple Lie group G. These special components, which occur only for certain groups, generalize Teichmüller space and are the main object of study of higher Teichmüller theory.

**Venerdì 03 maggio 2019**

Ore 12:00, aula di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma

seminario MoMA

Sergio Barbarossa
*Topological Signal Processing*

Signals are used in our everyday life to send and receive information or to extract information from an unknown environment. Typically, signals are defined over a metric space, i.e. time and space. The goal of this talk is to present a set of tools to analyze signals that are defined over a topological (e.g., not necessarily metric) space, i.e. a set of points along with a set of neighbourhood relations for each point. Motivating applications span from gene regulatory networks to social networks, etc. We introduce the Graph Fourier Transform (GFT), derive an uncertainty principle for signals defined over graphs and set the basis for a sampling theory over graphs. We start from signals defined over graphs and then we move to the most general case of signals defined over simplicial complexes. Finally, we illustrate some applications to the recovery of the electromagnetic field from a subset of observations, the inference of the brain functional activity network from electrocorticography (ECoG) signals collected in an epilepsy study and theprediction of data traffic over telecommunication networks.

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