## Notiziario Scientifico

Notiziario dei seminari di carattere matematico
a cura del Dipartimento di Matematica G. Castelnuovo, Sapienza Università di Roma

Settimana dal 18-03-2019 al 24-03-2019

Lunedì 18 marzo 2019
Ore 11:00, Aula C, Dipartimento di Matematica, Sapienza Università di Roma
Working Seminar on Calculus of Variations and Gamma-convergence
Gianluca Orlando (TU, Munich)
Does the N-clock model approximate the XY model?
In this seminar we will investigate the relationship between the N-clock model and the XY model (at zero temperature) through a Gamma-convergence analysis as both the number of particles and N diverge. The N-clock model is a two-dimensional nearest neighbors ferromagnetic spin system, in which the values of the spin field are constrained to lie in a set of N equispaced points of the unit circle. For N large enough, it is usually considered as an approximation of the XY model, for which the spin field is allowed to attain all the values of the unit circle. By suitably renormalizing the energy of the N-clock model, we will illustrate how its thermodynamic limit strongly depends on the rate of divergence of N with respect to the number of particles. We shall see that the N-clock model turns out to be a good approximation of the XY model only for N sufficiently large; in other regimes of N, we will show with the aid of cartesian currents that its asymptotic behavior can be described by an energy which may concentrate on geometric objects of different dimensions. The results presented in the talk are based on a work in collaboration with M. Cicalese and M. Ruf.

Lunedì 18 marzo 2019
Ore 14:15, Aula di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
Seminario di Analisi Matematica
Guido De Philippis (SISSA (Trieste))
A uniqueness result for the decomposition of vector fields
First I will review some features of the mathematical modelization of charged droplets. I will then focus on a model, proposed by Muratov and Novaga, which takes into account the regularization effect due the screening of free counterions in the droplet. In particular I will present a partial regularity result for minimizers and I will present some open problems. This is joint work with J. Hirsch e G. Vescovo.

Martedì 19 marzo 2019
Ore 14:30, Aula Dal Passo, Dipartimento di Matematica, Università di Roma Tor Vergata
Marcello Lucia (The City University of New York)
Some results related to Schiffer problem
Motivated by some earlier work by Schiffer, we consider an overdetermined semilinear problem in a two dimensional bounded domain where the Dirichlet data and Neumann boundary conditions are prescribed. In this talk I will provide some conditions that ensure the domain to be a disc. This is a joint work with B. Kawohl.

Martedì 19 marzo 2019
Ore 15:00, Aula 2E, Pal. RM004, Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza Università di Roma
Incontri di Algebra e Geometria allo SBAI
Andrea Vietri (Sapienza Università di Roma)
Graceful polynomials: an introduction and a glance at the graceful tree conjecture.
I will introduce a family of homogeneous polynomials for any given graph, with coefficients (mod 2), one for every degree and with as many variables as the number of vertices. These polynomials are related to graceful labellings: in some cases a graceful polynomial that vanishes (mod 2) could be an efficient tool for proving that the graph is non-graceful, in the same spirit as in a pioneering work by A. Rosa. More generally, graceful polynomials provide necessary conditions for gracefulness.

Giovedì 21 marzo 2019
Ore 11:00, Aula 211, Dipartimento di Matematica e Fisica, Università di Roma Tre, Largo S.L. Murialdo, 1
Mini-corso
Alex Küronya (Goethe-Universität Frankfurt am Main)
Syzygies of Algebraic Varieties
The topic of the course is the study of embeddings of varieties into projective spaces. Given a very ample line bundle L on a projective variety $$X$$, the Kodaira map associated to L gives rise to an embedding of $$X$$ into some projective space, thus realizing $$X$$ as a common zero set of equations in a polynomial ring. Syzygies of the pair $$(X;L)$$ are algebraic invariants of this embedding which describe the higher order relations among the arising system of equations. Following a quick introduction to the basics of the subject, in particular reviewing the necessary material from commutative algebra, we will start focusing on pairs $$(X;L)$$ where the associated syzygy modules are as simple as possible. This is made more precise by the so-called 'property (Np)', which was first considered by Green and Lazarsfeld, and which means that the first $$p + 1$$ syzygies are linear. We will study how verifying property (Np) can be reduced to checking the vanishing of higher cohomology of vector and line bundles, and look at the case of abelian varieties where one obtains a surprisingly uniform answer. In the last part of the course we consider the case of surfaces in more detail and see how one can characterize property (Np) in terms of forbidden subvari- eties. The necessary prerequisites are the basics of graded rings and a working knowledge of positivity, cohomology, and vanishing theorems for projective varieties.

Giovedì 21 marzo 2019
Ore 14:30, Aula 211, Dipartimento di Matematica e Fisica, Università di Roma Tre, Largo S. L. Murialdo 1
Seminario di Geometria
Andrea Fanelli (Versailles, Paris Saclay)
Fano varieties, fibrations and pathologies in positive characteristic.
In this talk, starting from the perspective of characteristic zero, I will discuss some pathologies for Fano varieties in characteristic p>0. I will focus on two aspects: - smoothness for the generic fibre of del Pezzo fibrations in dimension 3; - exotic torsion for the Ne'ron-Severi group of Fano 3-folds. This is a joint project with Stefan Schroer.

Venerdì 22 marzo 2019
Ore 11:00, Aula 211, Dipartimento di Matematica e Fisica, Università di Roma Tre, Largo S.L. Murialdo, 1
Mini-corso
Alex Küronya (Goethe-Universität Frankfurt am Main)
Syzygies of Algebraic Varieties
The topic of the course is the study of embeddings of varieties into projective spaces. Given a very ample line bundle L on a projective variety $$X$$, the Kodaira map associated to L gives rise to an embedding of $$X$$ into some projective space, thus realizing $$X$$ as a common zero set of equations in a polynomial ring. Syzygies of the pair $$(X;L)$$ are algebraic invariants of this embedding which describe the higher order relations among the arising system of equations. Following a quick introduction to the basics of the subject, in particular reviewing the necessary material from commutative algebra, we will start focusing on pairs $$(X;L)$$ where the associated syzygy modules are as simple as possible. This is made more precise by the so-called 'property (Np)', which was first considered by Green and Lazarsfeld, and which means that the first $$p + 1$$ syzygies are linear. We will study how verifying property (Np) can be reduced to checking the vanishing of higher cohomology of vector and line bundles, and look at the case of abelian varieties where one obtains a surprisingly uniform answer. In the last part of the course we consider the case of surfaces in more detail and see how one can characterize property (Np) in terms of forbidden subvari- eties. The necessary prerequisites are the basics of graded rings and a working knowledge of positivity, cohomology, and vanishing theorems for projective varieties.

Venerdì 22 marzo 2019
Ore 16:00, Aula Picone, Dipartimento di Matematica, Sapienza Università di Roma
Seminario per Insegnanti (Piano Lauree Scientifiche)
Benedetto Scoppola*, Emanuela Arnao** (* Universita' di Roma Tor Vergata, ** Liceo Farnesina)
Flussi e riflussi: un percorso didattico sulla teoria delle maree

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