seminar

SEMINARIO DE JÓVENES INVESTIGADORES

Conferenciante: José Antonio Serrano (Universidad de Granada)

Abstract: En esta presentación comenzaremos definiendo el flujo normal a una subvariedad riemanniana y estudiando la evolución de sus elementos geométricos por dicho flujo. Este estudio tendrá una importante aplicación en teoría de comparación de variedades riemannianas por medio de las curvaturas seccional y de Ricci. De esta comparación obtendremos propiedades analíticas (teoremas de Rauch y Berger), métricas (teoremas de Bishop-Gromov y Toponogov) y globales (teoremas de Klingenberg y Cheeger-Gromoll). Presentaremos una alternativa especialmente intuitiva a la demostración clásica de estos resultados.

SEMINARIO DE ESTUDIANTES DE ANÁLISIS Y GEOMETRÍA

Conferenciante: Tjaša Vrhovnik (Universidad de Granada)

SEMINARIO DE ECUACIONES DIFERENCIALES

Conferenciante: Paolo Cosentino (Università di Roma II 'Tor Vergata')

Abstract: We are concerned with the spectrum of a linearized Liouville-type problem. Among other things, we characterize the case in which the first eigenvalue is zero. To this end, we refine the Alexandrov-Bol inequality suitable for our problem and characterize its equality case. As a consequence, we obtain new pointwise information on the first eigenfunction associated to the zero eigenvalue.

SEMINARIO DE JÓVENES INVESTIGADORES

Conferenciante: Praful Gagrani

Abstract: Chemical reaction networks (CRNs) play a central role in the mathematical modeling of chemistry and biology due to their capacity to capture a wide range of nonlinear phenomena. A CRN is a hypergraph and thus exhibits a rich topological structure. In particular, we will explore a particular feature termed autocatalysis that has many consequences for dynamical organization in CRNs for theoretical and practical reasons. I will define autocatalytic networks and explain the polyhedral geometry induced by the minimal autocatalytic subnetworks (MASs). (For details, see https://arxiv.org/abs/2303.14238.) A CRN, when equipped with the appropriate kinetics, also specifies a transition matrix for the stochastic dynamics of a population state in the positive orthant of the integer lattice. Analogous to the passage of classical mechanics from quantum mechanics, one can show that, to a first order approximation, the stochastic dynamics of the system are governed by solutions of the Hamilton-Jacobi equation. I will outline our functional gradient descent algorithm to estimate these solutions. (For details, see https://doi.org/10.1103/PhysRevE.107.034305.) I will conclude with a perspective on the applications of CRNs and mention open questions in the statistical learning of biochemical systems.

SEMINARIO DE ECUACIONES DIFERENCIALES

Conferenciante: Martina Magliocca (Universidad de Sevilla)

Abstract: Cancer is a very complex phenomenon that involves many different scales and situations.

In this talk, we consider two principal models describing the evolution of a tumor colony and we derive a new asymptotic model for tumor growth. We discuss both the case with free boundary and infinite depth, and the case with two free boundaries.

We focus on the case of a single phase tumor colony taking into account chemotactic effects in an early stage where there is no necrotic inner region. Thus, our models are valid for the case of multilayer avascular tumors with very little access to both nutrients and inhibitors or the case where the amount of nutrients and inhibitors is very similar to the amount consumed by the multilayer tumor cells.

Besides deriving the model, we also prove a well-posedness result in a particular case and discuss some ongoing projects.

This talk is based on joint works with R. Granero-Belinchón and Alejandro Ortega.

SEMINARIO DE ECUACIONES DIFERENCIALES

Conferenciante: Giulio Romani (Università degli Studi dell'Insubria)

Abstract: Systems which couple a Schrödinger and a Poisson equation arise in several physical contexts such as quantum mechanics and electromagnetism. When considered in the whole space and in the limiting case for the Sobolev embeddings, one can deal with exponentially growing nonlinearities, but the application of standard variational tools is spoiled by the fact that the Riesz kernel of the Poisson equation is logarithmic, hence unbounded from above and below.

In this framework, I will present some existence results about local SP-systems in the case of zero mass, and nonlocal SP-systems, by means of a variational approximating procedure for auxiliary Choquard equations, where the logarithmic Riesz kernel is uniformly approximated by polynomial kernels.

This talk is based on joint works with Daniele Cassani (University of Insubria - Varese) and Zhisu Liu (China University of Geosciences - Wuhan)

SEMINARIO DE GEOMETRÍA

Conferenciante: Lilia Mehidi (Universidad de Granada)

Abstract: A compact flat Lorentzian manifold is the quotient of the Minkowski space by a discrete subgroup \(\Gamma\) of the isometry group, acting properly, freely and cocompactly on it. A classical result by Goldman, Fried and Kamishima states that, up to finite index, \(\Gamma\) is a uniform lattice in some connected Lie subgroup of the isometry group, acting properly and cocompactly, generalizing Bieberbach theorem to the Lorentzian signature. Such compact quotients are called "standard". More generally, a compact quotient of a homogeneous space \(G/H\) of a Lie group \(G\) is standard if the fundamental group action extends to a proper cocompact action of a connected Lie subgroup of \(G\). It turns out that looking for standard quotients is an easier problem when studying the existence of compact quotients of homogeneous spaces. This talk is about compact locally homogeneous plane waves. Plane waves can be thought of as a deformation of Minkowski spacetime, they are of great mathematical and physical interests. In this talk, we describe the isometry group of a 1-connected homogeneous non-flat plane wave, and show that compact quotients are “essentially" standard. As an application, we obtain that the parallel flow of a compact plane wave is equicontinuous. This is a joint work with M. Hanounah, I. Kath and A. Zeghib.

SEMINARIO DE GEOMETRÍA

Conferenciante: Raphael Tsiamis (Columbia University)

Abstract: Translators in \(\mathbb{R}^3\) are solitons of the mean curvature flow for embedded 2-surfaces. In the semigraphical case, where the translators are allowed graphical as well as vertical components, Hoffman-Martín-White classified the surfaces into six types. They conjectured the uniqueness of the objects within two families contained in slabs, the "helicoids" and the "pitchforks," for any given width. We present the proof of the conjecture by combining an arc-counting argument motivated by Morse-Radó theory for translators with a rotational application of the maximum principle. We then discuss applications of this result to the classification of semigraphical translators in \(\mathbb{R}^3\) and their limits, related to the work of Hoffman-Martín-White and Gama-Martín-Møller. This is joint work with F. Martín and M. Sáez.

SEMINARIO DE ECUACIONES DIFERENCIALES

Conferenciante: José Antonio Carrillo (University of Oxford)

Resumen: In this talk I will review several results in the modelling of grid cells. The activity generated by an ensemble of neurons is affected by various noise sources. It is a well-recognised challenge to understand the effects of noise on the stability of such networks. We demonstrate that the patterns of activity generated by networks of grid cells emerge from the instability of homogeneous activity for small levels of noise. This is carried out by upscaling a noisy grid cell model to a system of partial differential equations in order to analyse the robustness of network activity patterns with respect to noise. This is rigorously achieved by mean-field type arguments. Inhomogeneous network patterns are numerically understood as branches bifurcating from unstable homogeneous states for small noise levels. We prove that there is a phase transition occurring as the level of noise decreases. Our numerical study also indicates the presence of hysteresis phenomena close to the precise critical noise value. This talk is a summary of four papers/preprints in collaboration with A. Clini, H. Holden, P. Roux and S. Solem.

SEMINARIO DE ESTUDIANTES DE ANÁLISIS Y GEOMETRÍA

Conferenciante: Juan Silverio