## Notiziario Scientifico

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

Settimana dal 11-02-2019 al 17-02-2019

Lunedì 11 febbraio 2019
Ore 14:15, Aula Consiglio, Dipartimento di Matematica
Seminario di Analisi Matematica
Stefano Bianchini (SISSA)
A uniqueness result for the decomposition of vector fields in $$R^{d+1}$$.
Given a vector field $$\rho (1,{\mathbf b}) \in L1_{\mathrm{loc}}({\mathbb R}^+\times {\mathbb R}^{d},{\mathbb R}^{d+1})$$ such that $${\mathrm{div}}_{t,x} (\rho (1,{\mathbf b}))$$ is a measure, we consider the problem of uniqueness of the representation $$\eta$$ of $$\rho (1,{\mathbf b}) \mathcal L^{d+1}$$ as a superposition of characteristics $$\gamma : (t^-_\gamma,t^+_\gamma) \to {\mathbb R}^d$$, $$\dot \gamma (t)= b(t,\gamma(t))$$. We give conditions in terms of a local structure of the representation $$\eta$$ on suitable sets in order to prove that there is a partition of $${\mathbb R}^{d+1}$$ into disjoint trajectories $$\wp_{\mathfrak a}$$, $${\mathfrak a} \in {\mathfrak A}$$, such that the PDE ${\mathrm{div}}_{t,x} \big( u \rho (1,{\mathbf b}) \big) \in \mathcal M({\mathbb R}^{d+1}), \qquad u \in L^\infty({\mathbb R}^+\times {\mathbb R}^{d}),$ can be disintegrated into a family of ODEs along $$\wp_{\mathfrak a}$$ with measure r.h.s.. The decomposition $$\wp_{\mathfrak a}$$ is essentially unique. We finally show that $${\mathbf b} \in L1_t({\mathrm{BV}}_x)_{\mathrm{loc}}$$ satisfies this local structural assumption and this yields, in particular, the renormalization property for nearly incompressible $${\mathrm{BV}}$$ vector fields.

Martedì 12 febbraio 2019
Ore 14:00, Aula di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
Seminario di Probabilita' e Fisica
Raphael Chetrite (CNRS, Nizza)
On Gibbs-Shannon Entropy
This talk will focus on the question of the physical contents of the Gibbs-Shannon entropy outside equilibrium. It will be based on the article Gavrilov-Chetrite-Bechhoeffer, Direct measurement of weakly nonequilibrium system entropy is consistent with Gibbs-Shannon form, PNAS 2017.

Martedì 12 febbraio 2019
Ore 14:30, Aula 211, Dipartimento di Matematica e Fisica, Università di Roma Tre, Largo San Leonardo Murialdo, 1, Pal. C
Seminario di Fisica Matematica
H. Duminil-Copin (IHES Bures-sur-Yvette and Université de Genève)
Phase transition of Bernoulli percolation via the Gaussian Free Field
In this talk, we prove that Bernoulli percolation on graphs with isoperimetric dimension $$d>4$$ undergoes a non-trivial phase transition (in the sense that $$p_c<1$$). As a corollary, we obtain that the critical point of Bernoulli percolation on infinite quasi-transitive graphs (in particular, Cayley graphs) with super-linear growth is strictly smaller than 1, thus answering a conjecture of Benjamini and Schramm from 1996. The proof relies on a new technique consisting in expressing certain functionals of the Gaussian Free Field (GFF) in terms of connectivity probabilities for a percolation model in a random environment. Then, we integrate out the randomness in the edge-parameters using a multi-scale decomposition of the GFF. We believe that a similar strategy could lead to proofs of the existence of a phase transition for various other models.

Martedì 12 febbraio 2019
Ore 14:30, Aula Dal Passo, Dipartimento di Matematica, Università degli Studi di Roma "Tor Vergata"
Seminario di Equazioni Differenziali
Massimiliano Morini (Università di Parma)
Existence and uniqueness for anisotropic and crystalline mean curvature flows
An existence and uniqueness result, up to fattening, for crystalline mean curvature flows with forcing and arbitrary (convex) mobilities, is proven. This is achieved by introducing a new notion of solution to the corresponding level set formulation. Such solutions satisfy a comparison principle and stability properties with respect to the approximation by suitably regularized problems. The results are valid in any dimension and for arbitrary, possibly unbounded, initial closed sets. As a result of our analysis, we deduce the convergence of a minimizing movement scheme proposed by Almgren, Taylor and Wang (1993), to a unique (up to fattening) “flat flow”.

Martedì 12 febbraio 2019
Ore 15:00, Aula di Consiglio, Dipartimento di Matematica Guido Castelnuovo, Sapienza Università di Roma
Seminario di Modellistica Differenziale Numerica
Sergio Pirozzoli (Dipartimento di Ingegneria Meccanica e Aerospaziale, Sapienza Università di Roma)
Numerical methods for compressible Navier-Stokes equations
We review numerical methods for direct numerical simulation of turbulent compressible flow in the presence of shock waves. Ideal numerical methods should be accurate and free from numerical dissipation in smooth parts of the flow, and at the same time they must robustly capture shock waves without significant Gibbs ringing, which may lead to nonlinear instability. Adapting to these conflicting goals leads to the design of strongly nonlinear numerical schemes that depend on the geometrical properties of the solution. For low-dissipation methods for smooth flows, numerical stability can be based on physical conservation principles for kinetic energy and/or entropy. Shock-capturing requires the addition of artificial dissipation, in more or less explicit form, as a surrogate for physical viscosity, to obtain non-oscillatory transitions. Methods suitable for both smooth and shocked flows will be discussed, and the potential for hybridization is highlighted. Examples of the application of advanced algorithms to DNS/LES of turbulent, compressible flows will be presented.

Mercoledì 13 febbraio 2019
Ore 14:00, Aula di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
Seminario di Algebra e Geometria
Thomas Willwacher (ETH Zurich)
Models for configuration and embedding spaces via Feynman diagrams
Extending ideas of Kontsevich, we show that Feynman diagrams of topological quantum field theories can be used to construct combinatorial models for various spaces of interest in topology, including in particular configuration spaces of points on manifolds and embedding spaces.

Mercoledì 13 febbraio 2019
Ore 16:00, Dal Passo, Dipartimento di Matematica di "Tor Vergata"
Seminario di Algebre di Operatori
Yoshimichi Ueda (Nagoya University)
On Arveson’s boundary theorem
I’ll explain a rather simple proof of Arveson’s boundary theorem by using the notion of Izumi’s non-commutative Poisson boundaries. This talk is based on my joint work with Kei Hasegawa.

Mercoledì 13 febbraio 2019
Ore 17:30, Dal Passo, Dipartimento di Matematica di "Tor Vergata"
Seminario di Algebre di Operatori
Karl-Hermann Neeb (University of Erlangen-Nürnberg)
Finite dimensional endomorphism semigroups of standard subspaces

Giovedì 14 febbraio 2019
Ore 14:30, Aula di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
Seminario P(n)/N(p)
Anna Lisa Amadori (Università di Napoli Parthenope)
Asymptotic profile and Morse index for the Hénon problem
A general formula relates the Morse index of radial solutions to semilinear elliptic problems in the ball with a singular Sturm-Liouville problem. In this talk I will show how this characterization can be used to compute the Morse index at the ends of existence range, in the particular case of the Hénon problem. To this aim another important ingredient is the deep relation which links Hénon and Lane Emden problem, in the radial setting. Similarities and differences with the Lane Emden case will be enlightened. The results have been obtained in collaboration with F. Gladiali (Università di Sassari)

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