seminar

📅 jueves 9 de abril — 12:00
📍 Aula A22, Facultad de Ciencias
👤Conferenciante: Juan Carlos Cantero (Universitat Autònoma de Barcelona)
📖 Título: Skeleton for the one-dimensional aggregation equation
📄 Resumen: For the aggregation equation in $\mathbb{R}$, we consider the evolution of an initial density corresponding to the characteristic function of some set $\Omega_0$. We study the limit measure at the blow up time 1 in the cases $\Omega_0$ is open or compact and we inspect the limit set (skeleton) where this measure is supported. This is a toy model for the general problem in arbitrary dimension. We also review some advances in the two-dimensional case. This is a joint work with Joan Orobitg (UAB).

• IMAG Functional Analysis Seminar
• Title: Representaciones óptimas en productos tensoriales proyectivos y espacios $L_1$
• By Juan Guerrero Viu (Universidad de Zaragoza)
• Abstract: Dados dos espacios de Banach $X$ e $Y$, diremos que un elemento del producto tensorial proyectivo $X\widehat{\otimes}_\pi Y$ alcanza su norma si puede escribirse de forma óptima como una serie de tensores elementales. En esta charla analizaremos qué condiciones geométricas sobre $X$ e $Y$ aseguran que todo tensor alcance su norma, o al menos que los tensores que la alcanzan formen un conjunto denso en $X\widehat{\otimes}_\pi Y$. Además, prestaremos especial atención al caso $X=L_1$, donde describiremos de manera precisa el conjunto de tensores que alcanzan su norma en $L_1\widehat{\otimes}_\pi Y$ cuando $Y$ es estrictamente convexo. Esta charla está basada en trabajos conjuntos con L. C. García-Lirola, T. Procházka y A. Rueda Zoca.

📅 martes 24 de marzo — 11:00

📍 Aula A16, Facultad de Ciencias

👤Conferenciante: Émeric Roulley (Università degli Studi di Milano)

📖 Título: Nonlinear oscillations of electron layers

📄 Resumen: We consider the one-dimensional space periodic Vlasov-Poisson equation. A particular class of solutions, called electron layers, arises when the electron density is given by the characteristic function of a strip in phase space. The evolution of such configurations is governed by a Hamiltonian system consisting of coupled transport equations. We construct, near stationary flat velocity layers, both periodic traveling and quasi-periodic traveling solutions. The periodic traveling solutions are obtained via bifurcation theory, and their phase portrait exhibits a hyperbolic structure. We will also briefly discuss some extensions to the two-component system where the ions are also allowed to have inertia. The construction of quasi-periodic traveling solutions is considerably more delicate and requires a combination of KAM theory, Nash–Moser iteration, and pseudo-differential calculus. In order to exclude resonances and make the KAM scheme applicable, the strip width must be chosen from a suitable set of parameters of almost full measure.

Abstract: Einstein metrics with non-positive scalar curvature are conjectured to be linearly stable. On spin manifolds, the existence of parallel spinors provides a spinorial criterion for stability. In this talk I explain how this mechanism extends to non-spin manifolds using twisted spinors. In particular, we show that Einstein manifolds carrying parallel twisted pure spinors are linearly semistable. This yields stability results for a large class of spaces, including quaternionic Kähler manifolds of dimension greater than 8

Abstract:
I will prove Gromov's conjecture that every 3-manifold of positive scalar curvature contains a short closed geodesic.
The proof uses Min-Max theory of minimal surfaces and a combinatorial version of mean curvature flow.
This is a joint work with Davi Maximo and Regina Rotman.

📅 martes 24 de febrero — 11:00

📍 Aula A13, Facultad de Ciencias

👤Conferenciante: Eduardo Rosinato Longa (Universidade de São Paulo)

📖 Título: Extremal surfaces for the first Dirichlet eigenvalue

📄 Resumen: While many works have investigated extremal domains for the first Dirichlet eigenvalue of the Laplacian, in this talk I will examine the corresponding problem for hypersurfaces, where one varies the immersion rather than the domain inside a fixed ambient manifold. I will present the Euler–Lagrange equation for this variational problem, which links the first eigenfunction to the second fundamental form and leads to both an interior condition and an overdetermined boundary condition. In the minimal case, these equations become particularly rigid and impose strong geometric restrictions. As an application, we show that among compact capillary CMC surfaces in the unit ball, the only extremal examples are totally geodesic discs. The proof combines variational analysis and a Hopf differential argument.

Title: Zoll manifolds with boundary

Speaker: Eduardo Longa Rosinato

Abstract:

Zoll manifolds are Riemannian manifolds all of whose geodesics are closed and have the same length. Beyond the round sphere, nontrivial examples were constructed by Funk and Guillemin, initiating a rich line of research.

In this talk, I introduce a free-boundary analogue of this notion. A compact Riemannian manifold with boundary is said to be Zoll with boundary if every geodesic issuing orthogonally from the boundary returns orthogonally and is nowhere tangent to it.

I will show that such manifolds exhibit strong rigidity: all free-boundary geodesics have the same length and share the same Morse index. Using Morse index theory and algebraic topology, we obtain a complete geometric and topological classification. In particular, when the boundary is connected, the manifold is a tubular neighborhood of a closed embedded submanifold (the “soul”), and the boundary fibers over the soul either as a sphere bundle or as a nontrivial two-fold covering.

This is joint work with Paolo Piccione and Roney Santos.

Seminario de Geometría

Title: Comparaciones tipo Talenti y desigualdades isoperimétricas en variedades Riemannianas

Speaker: Vicente Palmer, Universitat Jaume I, (trabajo en colaboración con V. Gimeno)

More information: abstract

Let M be a properly embedded, connected, complete surface in R^3 with boundary a convex planar curve C, satisfying an elliptic equation H=f(H^2-K), where H and K are the mean and the Gauss curvature respectively – which we will refer to as Weingarten equation. In this talk, we discuss how the symmetries of C may induce symmetries of the whole M. When M is contained in one of the two halfspaces determined by C, we give sufficient conditions for M to inherit the symmetries of C. In particular, when M is vertically cylindrically bounded, we get that M is rotational if C is a circle. In the case in which the Weingarten equation is linear, we give a sufficient condition for such a surface to be contained in a halfspace. Both results are generalizations of results of Rosenberg and Sa Earp, for constant mean curvature surfaces, to the Weingarten setting. In particular, our results also recover and generalize the constant mean curvature case.

In this talk, I will describe a classification result for translating solitons of the mean curvature flow under two natural quantitative/topological assumptions: finite genus and entropy equal to 2. Roughly speaking, these hypotheses place the translator at the borderline between the simplest nontrivial singularity models and genuinely higher–complexity behavior: finite genus controls the global topology, while the entropy bound rigidifies the possible asymptotic geometry and rules out many exotic configurations. I will explain how one combines geometric measure theory and PDE tools for the translator equation to extract strong structural information from the entropy constraint, leading to a precise description of all such examples. Along the way, I will discuss the role of blow-down and compactness arguments, how the entropy pinching interacts with curvature and topology, and what the result says about the landscape of low-entropy translators. This is joint work with E. S. Gama and N. M. Møller.