# maths bites trento

Università degli Studi Trento

38123 Povo (TN)

Tel +39 04 61/281508-1625-1701-3898-1980.

dept.math [at] unitn.it

Per partecipare agli eventi (telematici), contattare lo Staff di Dipartimento.

## Prossimo appuntamento

TBA

## Il calendario

TBA

## Eventi passati

### Giovedì 15 ottobre 2020 ore 16:00

**Valter Moretti**, (Università di Trento)

**Il premio nobel per la fisica 2020 (anche) ad un teorema di matematica**

Illustrerò ad un livello divulgativo parte della tecnologia matematica sviluppata da Roger Penrose (ed altri) per formalizzare la teoria della causalità negli spazitempo in termini di geometria semiriemanniana. Particolare enfasi sarà posta sul risultato di Penrose del 1965 noto come Teorema di Singolarità (compagno di un analogo teorema di S. Hawking) che gli è valso il premio Nobel per la fisica 2020. Questo teorema prova come, sotto ipotesi matematiche abbastanza generali per descrivere uno spaziotempo della Relatività Generale di Einstein, lo spaziotempo debba necessariamente sviluppare delle singolarità metriche.

Queste singolarità che sono quindi inevitabili per principio, appaiono in particolare all’interno dei Buchi Neri e nel Big Bang. Il premio Nobel è stato infatti condiviso con Andrea Ghez e Reinhard Genzel per per avere individuato un Buco Nero estremamente massivo all’interno della nostra galassia.

### Giovedì 12 novembre ore 16:00

**Robert Nurnberg**, (Università di Trento)

**Numerics for anisotropic surface diffusion**

In this seminar I will discuss surface diffusion as a proto-type for a geometric evolution equation. The main focus will be on the numerical approximation of this flow, and more generally on numerical approaches for moving interface problems. Here I will consider both front tracking methods and phase field methods. Given that in Materials Science an anisotropic surface energy often plays an important role, I will extend the introduced ideas from the isotropic to the anisotropic setting. Several numerical simulations will be presented, including for extensions of the described techniques to the modelling of snow crystal growth.

### Giovedì 10 dicembre ore 16:00

**Nadir Murru**, (Università di Trento**)**

**Multidimensional continued fractions and p-adic numbers**

The classical theory of continued fractions has been widely studied for centuries for its important properties of good approximation and periodicity: they give the best approximations of real numbers and a periodic

representation for all quadratic irrationals by means of integer sequences. In 1839 Hermite posed to Jacobi the problem of generalizing the construction of continued fractions to higher dimensions. In particular,

he asked for a method of representing algebraic irrationalities by means of periodic sequences that can highlight algebraic properties and possibly provide rational approximations. Perron and Jacobi defined

multidimensional continued fractions for answering this question, however this is still an open problem in number theory. In this talk, we will see some results and approaches to this problem. Moreover, recently

continued fractions have been generalized to p-adic numbers where they present many differences with respect to the real case. We will investigate some results about convergence, approximation and periodicity

of continued fractions in the field of p-adic numbers, as well as their generalization to the multidimensional case.

### Giovedì 18 febbraio ore 16:00

**Nicola Gigli**(SISSA)

**Curvature and Metric Geometry**

Aim of the talk is to give an introduction to the concept of curvature bounds in a non-smooth environment and to the closely related one of distance between metric spaces. I shall survey both classical and more recent results.

### Giovedì 25 marzo 2021 ore 16:00

**Michele Stecconi**(Nantes University)

**What is the degree of a smooth hypersurface?**

The degree of a real algebraic object provides a strong upper bound for its geometrical and topological complexity. By approximating smooth objects with algebraic ones in a controlled way (controlling the degree)

then, it is possible to extend such bounds to hold in a smooth setting. Following this philosophy, in this talk I will define an alternative parameter that controls the geometry of certain smooth objects (the singular loci of

smooth maps) up to isotopy, and provides, in particular, a bound for their Betti numbers. This parameter is obtained as a function of the distance of the object from the set of singular ones in the appropriate functional

space, in analogy with the concept of "condition number". In general, this applies to a very general class of singularities of maps, including the case of "zero set" and "critical points" as the most basic examples.

In this talk, I will focus on the case of smooth closed hypersurfaces in R^{n} to illustrate the main ideas and results. In particular, I will present a quantitative version of Seifert’s Theorem, stating that the hypersurface

is isotopic to an algebraic one (inside a given ball) having degree controlled by an explicit function of the reach of the original one. This talk is based on a recent paper by Antonio Lerario and me, with the same title (see https://arxiv.org/abs/2010.14553).

### Giovedì 15 aprile 2021 ore 16:00

**Paola Antonietti**(POLIMI)

**Numerical models for earthquake ground motion and seismic risk assessment**

Physics-based numerical simulations of earthquake ground motion can be used to better understand the physics of earthquakes, improve the design of site-specific structures, and evaluate the seismic hazard, a key step for the reliable assessment of seismic risk.

The distinguishing features of a numerical method designed for seismic wave propagation are: accuracy, geometric flexibility and parallel scalability.

High-order methods ensure low dissipation and dispersion errors. Geometric flexibility allows complicated geometries and sharp discontinuities of the mechanical properties to be addressed.

Finally, since earthquake models are typically posed on domains that are very large compared to the wavelengths of interest, scalability allows to efficiently solve the resulting algebraic systems featuring several millions of unknowns.

In this talk we present an overview on numerical modelling of the ground motion induced by seismic waves based on employing Discontinuous Galerkin Spectral Element discretizations of the elastodynamics equation. We also present two new approaches to couple the ground motion induced by earthquakes with the induced structural damages of buildings. The first one is based on empirical laws (fragility curves) whereas the second one employs physics-based linear and non-linear differential models.

To validate the first approach we study the seismic damages in the Beijing area as a consequence of ground motion scenarios with magnitude in the range 6.5–7.3 Mw. To validate the second approach we consider the 1999 Mw6 Athens earthquake and study the seismic response of the Acropolis hill and of the Parthenon. Our numerical results have been obtained using the open-source numerical code SPEED (https://speed.mox.polimi.it).

### Giovedì 13 maggio 2021 ore 14:30

**Bernard Hanzon**(University College Cork)

**Attasi nD Systems and Polynomial SystemSolving**

Firstly it will be shown that, using the concept of monomial orderings, the classical 1D theory for linear systems generalizes in a very natural way to (autonomous) Attasi nD-systems, giving the Attasi-Hankel matrix, the Attasi transfer function and Attasi state-space realizations. Secondly we will explain how one can associate an Attasi system to any set of polynomial equations having a finite number of solutions and how Attasi realization theory can be used to (1) describe a commutative matrix solution to the set of polynomial equations (this generalizes the Cayley-Hamilton theorem) and (2) find the (scalar) solutions to the set of polynomial equations by computing the joint eigenvalues and eigenvectors of the commuting matrices. Some remarks will be made about how the (discrete time) Attasi system equations can be solved recursively and how that can be used in large-scale eigensolvers. Using the associated Attasi system to a set of polynomial equations also helps to understand various different approaches to polynomial equation solving and multivariate polynomial and rational function minimization.