## Notiziario Scientifico

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

Settimana dal 27-05-2019 al 02-06-2019

Lunedì 27 maggio 2019
Ore 11:00, Aula 1E, Palazzina RM 004, Dipartimento SBAI, Sapienza Università di Roma
Seminario di Analisi Armonica
Hillel Furstenberg (Hebrew University of Jerusalem)
An equidistribution problem, algebraic functions over finite fields, and Apery-like sequences
A known open problem is the distribution mod 1 of powers of a rational number $$p/q$$ with $$p>q$$. This can be seen as a question regarding the behavior of orbits of "special" points in a cellular automaton. An example of a cellular automaton where this kind of behavior can be determined is where the space consists of sequences from a finite field and the cellular operation is a fixed linear combination of translates. The entries at a fixed position along the orbit turn out to be the coefficient sequence of algebraic functions. A generalization of this construction leads to the notion of a "dirational" sequence: the sequence of diagonal terms in the expansion of a rational function of several variables. These functions are related to modular and hypergeometric functions, and we show that the sequences playing a major role in Apery's proof of the irrationality of $$\zeta(3)$$ are dirational sequences, explaining thereby why the terms are integers or close to integers.

Lunedì 27 maggio 2019
Ore 14:15, Aula di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
Seminario di Analisi Matematica
Eleonora Di Nezza (Sorbonne Université, Paris)
Monge and Ampère : waiting for pluripotential theory
A fundamental problem in Kähler geometry is to look for canonical metrics on a Kähler manifold, where the world "canonical" means that the metric satisfies a curvature condition. Such geometric problem boils down to a complex non-linear PDE of Monge-Ampère type. In the last 50 years the study of the regularity of solutions of such equations has motivated a lot of works. I will give a survey of what is known in this direction (also in singular settings), emphasising how the developments in pluripotential theory were crucial in order to treat (singular) complex Monge-Ampère equations.

Lunedì 27 maggio 2019
Ore 14:30, Aula 1B1, Pal. RM002, Dipartimento SBAI, Sapienza Università di Roma
Incontri di Algebra e Geometria allo SBAI
Paolo Sentinelli (Departamento de Matemáticas, Universidad de Chile)
On a class of representations of right-angled Coxeter groups
We define an action of any right-angled Coxeter group (these groups are also known as "graph groups") on any Coxeter group, whenever is satisfied a condition of universality of a graph constructed with the Coxeter graph of the group on which we want to act. When the graph of the right-angled Coxeter group is a forest with n vertices, and the group on which we want to act is the symmetric group of order n!, the problem of existence of such representations is solved constructively. For generic graphs the existence is conjecturally true.

Lunedì 27 maggio 2019
Ore 14:30, Aula Dal Passo, Dipartimento di Matematica, Università di Roma Tor Vergata
Algebra and Representation Theory Seminar
Margherita Paolini
The integral form of the universal enveloping algebra of twisted affine sl3
In the representation theory of a semisimple Lie algebra L, the subring of the universal enveloping algebra U(L) generated by suitable divided powers arises naturally, thus leading to construct an integral form of U(L). Kostant and Cartier indipendently defined this form and explicitly constructed integral bases when L is finite. Their construction has later been generalized to the untwisted affine case by Garland. An analogous work by Fisher-Vasta extends the construction of the integral form of U(L) to the affine twisted Kac-Moody algebra of rank 1 (type $$A_2^2$$). These works are based on complicated commutation formulas, whose regularity remains hidden; moreover, in the twisted case there are some problems both with the statement and the proof. The aim of this talk is to give a correct description of the integral form of the enveloping algebra of type $$A_2^2$$ , providing explicit and compact commutation relations, so to reach a deeper comprehension and drastic simplification of the problem. This is achieved by means of a careful use of the generating series of families of elements and of the properties of the ring of symmetric functions.

Lunedì 27 maggio 2019
Ore 15:30, Aula 1E, Palazzina RM 004, Dipartimento SBAI, Sapienza Università di Roma
Seminario di Analisi Armonica
Hillel Furstenberg (Hebrew University of Jerusalem)
Affine representations of topological groups
An "affine representation" of a group arises when the group acts on a compact convex set preserving the affine (convex) structure. As in the theory of linear representations, special interest is attached to Irreducible representations, namely, actions for which no proper closed convex subset is invariant. Unlike the linear case such an action may have a non-trivial, non-isomorphic "factor" action. Indeed ALL irreducible representations are derived as factors of a single universal representation. This universal affine representation can be made explicit for Lie groups. We discuss the case of PSL(2,R), or equivalently, the Mobius group of the unit disc. Remarkably this group has a unique non-trivial irreducible action; namely, the action on the space of probability measures on the boundary of the disc. From the irreducibility of this action we will rederive the boundary theory of bounded harmonic functions on the unit disc, with generalizations to more general symmetric spaces.

Lunedì 27 maggio 2019
Ore 16:00, Aula di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
Seminario di Probabilità
Otavio Menezes (Università di Lisbona)
Relative entropy and scaling limits of interacting particle systems
We obtain product approximations to the law of particle systems with exclusion and Glauber dynamics in finite volume, by establishing a bound on the relative entropy between the law of the system and the product measure. As applications of the entropy estimate we obtain the scaling limits of the density fluctuation fields close of equilibrium and bounds on the speed of convergence of the hydrodynamic limit. Joint work with Milton Jara.

Martedì 28 maggio 2019
Ore 11:30, Aula 211, Dipartimento di Matematica e Fisica, Università di Roma Tre, Largo S. L. Murialdo, 1
Seminario
Olivier Ramaré (CNRS / Aix Marseille Université)
On the missing log-factor
The guiding line of this general audience talk is the elementarily proven bound $$|\sum_{n\le x}\mu(n)/n|\le 1$$. The trivial bound for the implied summation is $$\log x+O(1)$$, while the Prime Number Theorem tells us that it is $$o(1)$$. Our starting estimate thus lies in-between, a fact that we explore under different lights.

Martedì 28 maggio 2019
Ore 14:00, Aula Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
seminario di Probabilità e Statistica Matematica
Gerard Letac (IMT, Université Paul Sabatier, Toulouse, France)
Multivariate Reciprocal Inverse Gaussian Distributions: the Surprising Integrals of Supersymmetry
If $$W=(w_{ij})_{1\leq i,j\leq n}$$ is a symmetric matrix with $$w_{ij}\geq 0$$ and zero diagonal, consider the matrix $$M_x=2\mathrm{diag}(x_1,\ldots,x_n)-W$$ and the set $$C_W$$ of $$x\in \mathbb{R}^n$$ such that $$M_x$$ is positive definite. Physicists and probabilists have considered in the last ten years several integrals equivalent to $$\int_{C_W}\exp(-a*M_xa-b^*M_x^{-1}b)(\det M_x)^{q-1}dx.$$ According to the properties of the undirected graph $$G$$ associated to $$W$$ with vertices $$1,\ldots,n$$ and edges $$(i,j)$$ such that $$w_{ij}\neq 0$$, we compute some of these integrals and we study the corresponding exponential families on $$\mathbb{R}^n$$ with many attractive properties.

Martedì 28 maggio 2019
Ore 14:30, Aula di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
seminario di Algebra e Geometria
Charles Favre (CNRS - CMLS)
Hybrid spaces
We shall explain how the construction of a hybrid space by Berkovich enables one to understand the degeneration of natural measures in various contexts in dynamics.

Martedì 28 maggio 2019
Ore 15:00, Aula B, Dipartimento di Matematica, Sapienza Università di Roma
Seminario di Probabilità
Otavio Menezes (University of Lisbon)
Relative entropy and scaling limits of interacting particle systems (lecture I)
The relative entropy method was developed by H.T. Yau in the 90’s to study the hydrodynamics of the Ginzburg-Landau model, and then adapted to several different dynamics. In this course (2 lectures, 2h per lecture) we present the Relative Entropy Method of Yau in the context of general continuous time Markov chains, as well as recent progress in the setting of exclusion and Glauber dynamics.

Martedì 28 maggio 2019
Ore 9:45:, Aula Seminari, Dipartimento SBAI, Sapienza Università di Roma
One Day Workshop on PDEs in honor of Umberto Mosco’s birthday
Programme

• 09:45-10:00 Opening
• 10:00-10:35 V. Vespri, Some questions of regularity for anisotropic parabolic operators.
• 10:35-11:10 C. Sbordone, On BMO-type space of Bourgain-Brezis-Mironescu.
• 11:10-11:30 Coffee
• 11:30-12:05 G. Savaré, Variational convergence of metric gradient flows.
• 12:05-12:40 U. Gianazza, Boundary regularity for the porous medium equation.
• 12:40-13:15 F. Gazzola, Some remarks on the forces exerted by a viscous fluid on a bluff body.
• 13:15-14:45 Lunch
• 14:45-15:20 A. Cianchi, Global gradient regularity of solutions to the p-Laplace system in Campanato type spaces.
• 15:20-15:55 M. Chipot, Nonlocal Issues.
• 15:55-16:10 Coffee
• 16:10-18:30 Round Table on Variational structures and Applied Sciences: the Insight of Umberto Mosco. L. Boccardo, I. Capuzzo Dolcetta, P. Marcellini, U. Mosco
• 18:30-20:00 Cocktail

Mercoledì 29 maggio 2019
Ore 09:00, Aula III, Dipartimento di Matematica "Guido Castelnuovo", Sapienza Università di Roma
Conferenza "Calabi-Yau and Geometry", in onore di Shing-Tung Yau in occasione del suo 70esimo compleanno e del 40esimo anniversario della soluzione della congettura di Calabi.
Programma, giorno I

• 09:00-09:30 Registrazione
• 09:30-10:30 J.-P. Demailly, Université Grenoble Alpes
• 11:00-12:00 J. Jost, Max Planck Institute Leipzig
• 15:00-16:00 S. Cappel, NYU
• 16:30-17:30 L. Göttsche, ICTP
http://www1.mat.uniroma1.it/ricerca/convegni/2019/CYgeometry/index.html

Mercoledì 29 maggio 2019
Ore 10:00, Aula 1B1, Dipartimento SBAI, Sapienza Università di Roma
Incontri di Analisi MaTEmatica
Begoña Barrios Barrera (Universidad de La Laguna)
The sharp exponent in the study of the nonlocal Hénon equation in $$\mathbb{R}^{N}$$.
We will consider the nonlocal Hénon equation $$(-\Delta)^s u= |x|^{\alpha} u^{p},\quad \mathbb{R}^{N},$$ where $$(-\Delta)^s$$ is the fractional Laplacian operator with $$0 < s < 1$$, $$-2s < \alpha$$, $$p > 1$$ and $$N > 2s$$. We prove a Liouville result for positive solutions in the optimal range of the nonlinearity, that is, when $$1 < p < p^{*}_{\alpha, s}:=\frac{N+2\alpha+2s}{N-2s}$$. Moreover, we prove that a kind of bubble solution, that is, a fast decay positive radially symmetric solutions, exists when $$p=p_{\alpha, s}^{*}$$.

Mercoledì 29 maggio 2019
Ore 14:00, Aula B, Dipartimento di Matematica, Sapienza Università di Roma
Seminario di Probabilità
Otavio Menezes (University of Lisbon)
Relative entropy and scaling limits of interacting particle systems (lecture II)
The relative entropy method was developed by H.T. Yau in the 90’s to study the hydrodynamics of the Ginzburg-Landau model, and then adapted to several different dynamics. In this course (2 lectures, 2h for lecture) we present the Relative Entropy Method of Yau in the context of general continuous time Markov chains, as well as recent progress in the setting of exclusion and Glauber dynamics.

Mercoledì 29 maggio 2019
Ore 16:00, Aula Dal Passo, Dipartimento di Matematica, Università di Roma "Tor Vergata"
Colloquium di dipartimento
Albert Fathi (Georgia Institute of Technology (USA))
Singularities of solutions of the Hamilton-Jacobi equation. A toy model: distance to a closed subset
This is a joint work with Piermarco Cannarsa and Wei Cheng. The distance function $$d_F$$ to a closed subset $$F$$ of Euclidean space $${\bf R}^k$$ is given by $$d_F(x)=\inf_{f\in F}\|x-f\|$$. It is a Lipschitz, hence differentiable almost everywhere. We will discuss some topological properties of the set $${\rm Sing}(d_F)$$ of points where $$d_F$$ is not differentiable. More generally, we will discuss properties of the set of singularities of a viscosity solution of the Hamilton-Jacobi equation $$\partial_tU+H(x,\partial_xU)=0,$$ when $$H$$ is a Tonelli Hamiltonian. We will give applications in Riemannian geometry. We will explain during the lecture all notions (beyond common knowledge) necessary to understand it.

Giovedì 30 maggio 2019
Ore 09:30, Aula III, Dipartimento di Matematica "Guido Castelnuovo", Sapienza Università di Roma
Conferenza "Calabi-Yau and Geometry", in onore di Shing-Tung Yau in occasione del suo 70esimo compleanno e del 40esimo anniversario della soluzione della congettura di Calabi.
Programma, giorno II

• 09:30-10:30 R. Berman, Chalmers University
• 11:00-12:00 S. Boucksom, CNRS - CMLS
• 15:00-16:00 R. Hamilton, Columbia University
• 16:30-17:30 D. Lüst, LMU Munich
http://www1.mat.uniroma1.it/ricerca/convegni/2019/CYgeometry/index.html

Giovedì 30 maggio 2019
Ore 14:30, Aula 1200 (Biblioteca Storica), Dipartimento di Matematica, Università di Roma Tor Vergata
Seminario di Geometria Algebrica
Alessandra Sarti (Univ. Poitiers - France)
On non-symplectic automorphisms of K3 surfaces
Automorphisms of K3 surfaces were very much studied in the last years. Depending on the action on the holomorphic two form which can be trivial or not, they are called symplectic or non-symplectic. The aim of the talk is to present recent results in the study of non--symplectic automorphisms of 2-power order. In particular in the case of the order 16, I completely describe the families of K3 surfaces carrying such automorphisms.
[Algebraic Geometry Seminar, in the framework of MIUR Excellence Project 2018-2022 Mat@Tov, CUP: E83C18000100006]

Giovedì 30 maggio 2019
Ore 15:30, Aula 1200 (Biblioteca Storica), Dipartimento di Matematica, Università di Roma Tor Vergata
Seminario di Geometria Algebrica
Samuel Boissiere (Univ. Poitiers - France)
Some families of projective varieties uniformized by the 10-dimensional complex ball
In a famous paper, Allock, Carlson and Toledo described the moduli space of cubic threefolds as the arithmetic quotient of the complementary of a hyperplane arrangement in the 10-dimensional complex ball. I will present an interpretation of this moduli space as the one parametrizing a family of order three nonsymplectic automorphisms on hyperkaehler manifolds deformation equivalent to the Hilbert square of a K3 surface. This is a collaboration with Chiara Camere and Alessandra Sarti.
[Algebraic Geometry Seminar, in the framework of MIUR Excellence Project 2018-2022 Mat@Tov, CUP: E83C18000100006]

Venerdì 31 maggio 2019
Ore 09:30, Aula III, Dipartimento di Matematica "Guido Castelnuovo", Sapienza Università di Roma
Conferenza "Calabi-Yau and Geometry", in onore di Shing-Tung Yau in occasione del suo 70esimo compleanno e del 40esimo anniversario della soluzione della congettura di Calabi.
Programma, giorno III

• 09:30-10:30 J.-P. Bourguignon, IHES
• 11:00-12:00 C. De Lellis, IAS
• 15:00-16:00 W. Ballmann, Max Planck Institute Bonn
• 16:30-17:30 S. Janeczko, Polish Academy of Sciences
http://www1.mat.uniroma1.it/ricerca/convegni/2019/CYgeometry/index.html

Venerdì 31 maggio 2019
Ore 14:30, Aula D'Antoni, Dipartimento di Matematica, Università di Roma Tor Vergata
Colloquium di Dipartimento
Samuel Boissiere (Univ. Poitiers - France)
Involutions on the Hilbert square of a K3 surface
I will use K3 surfaces with special geometry to construct involutions on the Hilbert square of a K3 surface. The existence of such involutions can be proven using the Torelli theorem for hyper-Kaehler manifolds, but it is a challenging problem to produce concrete realizations.
[Department Colloquium (general audience talk) - in the framework of MIUR Excellence Project 2018-2022 Mat@Tov, CUP: E83C18000100006]

Venerdì 31 maggio 2019
Ore 15:30, Aula D'Antoni, Dipartimento di Matematica, Università di Roma Tor Vergata
Colloquium di Dipartimento
Alessandra Sarti (Univ. Poitiers - France)
K3 surfaces and their group of symmetries
Particularly interesting objects in algebraic geometry are K3 surfaces, which are special complex algebraic surfaces. The most easy example of such a surface is the zero set of a homogeneous polynomial of degree four in the three dimensional complex projective space. The name was given by André Weil in 1958 in honour of three famous mathematicians: Kummer, Kähler and Kodaira and in honour of the K2 mountain at Cachemire. Their symmetry group is an important tool to understand their geometry. I will first show some remarkable properties of K3 surfaces and in particular the important role of lattice theory, then I will show some classic and recent results on their symmetry groups.
[Department Colloquium (general audience talk) - in the framework of MIUR Excellence Project 2018-2022 Mat@Tov, CUP: E83C18000100006]

Sabato 1 giugno 2019
Ore 09:30, Aula III, Dipartimento di Matematica "Guido Castelnuovo", Sapienza Università di Roma
Conferenza "Calabi-Yau and Geometry", in onore di Shing-Tung Yau in occasione del suo 70esimo compleanno e del 40esimo anniversario della soluzione della congettura di Calabi.
Programma, giorno IV

• 09:30-10:30 K. O’Grady, SAPIENZA Università di Roma
• 11:00-12:00 J. E. Andersen, Aarhus University
http://www1.mat.uniroma1.it/ricerca/convegni/2019/CYgeometry/index.html

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