Superficies mínimas: Introducción a la teoría estructural de Colding y Minicozzi.
Abstract: A shape monotonicity problem asks which shape yields a better value for a given functional. As a model case, we consider eccentric annuli of fixed radii and study the monotonicity of the first Dirichlet eigenvalue with respect to the distance between the centers. We further investigate shape monotonicity problems for eccentric annuli with mixed boundary conditions. To study these problems, we analyze the associated heat kernels and develop probabilistic methods.
In this talk we discuss three aspects. Firstly, we focus on the higher-dimensional Auslander-Reiten theory for infinite-dimensional coalgebras. We introduce the n-transpose of finitely n-copresented comodules and establish the n-Auslander-Reiten formula on n-cluster-tilting subcategories of comodule categories. Secondly, we introduce the Gorenstein transpose via a minimal Gorenstein injective copresentation of a quasi-finite comodule, compare it with the classical transpose, and construct the almost split sequences in terms of the Gorenstein transpose. Finally, we further generalize the classical transpose to the T-transpose via a semidualizing bicomodule T, discuss its homological properties, establish a Foxby equivalence, and characterize T-reflexive comodules.
Abstract: In this talk we investigate the long time dynamics of interacting particle systems subject to singular interactions. We consider a microscopic system of N interacting point particles, where the time evolution of the joint distribution fN(t) is governed by the Liouville equation. Our primary objective is to analyze the system's behavior over extended time intervals, focusing on the stability, the potential chaotic dynamics and the impact of singularities. In particular, we aim to derive reduced models in the regime N >> 1, exploring both the mean-field approximation and configurations far from chaos, where the mean-field approximation no longer holds. These reduced models do not always emerge but in these cases we prove that itis possible to derive uniform bounds in L2, both over time and with respect to the number of particles, on the marginals (fk,N)1 ≤ k ≤ N, irrespective of the initial state’s chaotic nature. Furthermore, we extend previous results by considering a wide range of singular interaction kernels K ∈ W-2/(d+2), d+2 in dimension d ≥ 2, surpassing the traditional Ld regularity barriers. Finally, we address the highly singular case of K ∈ H-1 under a threshold temperature regime, offering new insights into the behavior of such systems.
Given a Riemannian manifold $(M^n, g)$, it is natural to ask whether it admits any closed geodesics and if so, how many are there and how are they distributed along the manifold.
In this talk, I will present the following result: given a closed 2-manifold $M^2$, for a generic (in the Baire sense) Riemannian metric $g$ on $M^2$ there exists an equidistributed sequence $\{\gamma_i\}_{i \in \mathbb{N}}$ of closed geodesics on $(M^2, g)$.
The same question in a higher dimensional ambient manifold $M^n$ turns out to be much harder and is still widely open. However, we can approach the same problem but considering stationary geodesic nets (which are embedded graphs in $(M, g)$ which are stationary with respect to the length functional) instead of closed geodesics.
Together with Yevgeny Liokumovich, we showed that for a Baire-generic metric $g$ on a fixed closed manifold $M^n$, the union of all embedded stationary geodesic nets on $(M^n, g)$ is dense in $M^n$. I will also present the following stronger result: if $n \geq 3$, given a closed $n$-manifold $M^n$, for a Baire-generic metric $g$ on $M^n$ there exists an equidistributed sequence $\{\gamma_i\}_{i \in \mathbb{N}}$ of stationary geodesic nets on $(M^n, g)$. For $n = 3$, this was joint work with Xinze Li.