seminar

SEMINARIO DE GEOMETRÍA

Conferenciante: Finnur Lárusson (University of Adelaida)

Abstract: The Gauss map of a minimal surface in \(\mathbb R^3\), parametrised as a conformal minimal immersion from an open Riemann surface \(M\) into \(\mathbb R^3\), is a meromorphic function on \(M\). Although the Gauss map has been a central object of interest in the theory of minimal surfaces since the mid-19th century, it was only recently proved by Alarcón, Forstnerič and López, using new complex-analytic methods, that every meromorphic function on \(M\) is a Gauss map. It remains an open problem to usefully characterise those meromorphic functions that are the Gauss map of a complete minimal surface. I will describe recent joint work with Antonio Alarcón, in which we take a new approach to this problem. We investigate the space of meromorphic functions on \(M\) that are the Gauss map of a complete minimal surface from a homotopy-theoretic viewpoint, using a new h-principle as a key tool. My talk will include a brief general introduction to h-principles and their applications.

SEMINARIO DE ECUACIONES DIFERENCIALES

Conferenciante: Víctor Navarro Fernández (Imperial College London)

Abstract: In this work we consider a time-periodic and random version of the Arnold-Beltrami-Childress (ABC) flow. We are concerned with two main subjects. On the one hand, we study the mixing problem of a passive tracer in the three dimensional torus by the action of the ABC vector field. On the other hand, we examine the effect of the ABC flows on the growth of a magnetic field described by the kinematic dynamo equations. To investigate these questions, we analyse the ABC flow as a random dynamical system and examine the ergodic properties of some associated Markov chains. This work settles that the random ABC vector field is an example of a space-time smooth universal exponential mixer in the three dimensional torus, and moreover, that it is an example of a nondissipative kinematic fast dynamo. This is a joint work with M. Coti Zelati (Imperial College London).

SEMINARIO DE ECUACIONES DIFERENCIALES

Conferenciante: Zhilong Xue (Beijing Normal University)

Abstract: In this talk, we will first introduce some recent results about patch solutions for active scaler equation, including two dimensional Euler equation. Then we will present how to establish the existence of simply connected V-states (rotating patches) close to Rankine vortices for active scalar equations with completely monotone kernels. This result allows to unify various results on this topic related to geophysical flows. A key ingredient is a new factorization formula for the spectrum using a universal function which is independent of the model. This function admits sesveral interesting properties allowing to track the spectrum distribution. This talk is based on a joint work with Taoufik Hmidi and Liutang Xue.

Seminario de Ecuaciones Diferenciales: https://www.ugr.es/~sed/

SEMINARIO DE GEOMETRÍA

Conferenciante: Javier Martínez-Aguinaga (Universidad Complutense de Madrid)

Abstract: Is it always possible to park a car in a parking space whose size is exactly the same as the car? Can we do the same with a multi-trailer truck? In this talk, we will show the relationship between these two questions and the theory of distributions on differentiable manifolds. We will review this theory and focus our attention on bracket-generating distributions. Typical examples of this class of distributions are Contact and Engel structures. We will motivate these objects by showing other examples and establish several results about their tangent curves. In particular, we will show that the spaces of regular tangent knots are flexible if the dimension of the manifold is greater than 3. These results are part of a joint work with Álvaro del Pino (Utrecht University).

Conferenciante: Bobomurad Narkuziev (Kimyo International University in Tashkent branch Samarkand, Uzbekistan)

Abstract: In this talk, we examine a time-dependent family of two-dimensional algebras. We investigate the conditions under which any two algebras from this family, formed at different times, are isomorphic. Our findings reveal that the flow comprises of uncountably many pairwise non-isomorphic algebras, including one commutative algebra. The properties of these algebras are described and used for applications in cryptography.

SEMINARIO DE GEOMETRÍA

Conferenciante: Kostya Drach (Universidad de Barcelona)

Abstract: What is the smallest volume a convex body \(K\) in \(\mathbb R^n\) can have for a given surface area? This question is in the reverse direction to the classical isoperimetric problem and, as such, has an obvious answer: the infimum of possible volumes is zero. One way to make this question highly non-trivial is to assume that \(K\) is uniformly convex in the following sense. We say that \(K\) is \(\lambda\)-convex if the principal curvatures at every point of its boundary are bounded below by a given constant \(\lambda>0\) (considered in the barrier sense if the boundary is not smooth). By compactness, any smooth strictly convex body in \(\mathbb R^n\) is \(\lambda\)-convex for some \(\lambda>0\). Another example of a \(\lambda\)-convex body is a finite intersection of balls of radius \(1/\lambda\) (sometimes referred to as ball-polyhedra). Until recently, the reverse isoperimetric problem for \(\lambda\)-convex bodies was solved only in dimension 2. In a recent joint work with Kateryna Tatarko, we resolved the problem also in \(\mathbb R^3\). We showed that the lens, i.e., the intersection of two balls of radius \(1/\lambda\), has the smallest volume among all \(\lambda\)-convex bodies of a given surface area. For \(n>3\), the question is still widely open. I will outline the proof of our result and put it in a more general context of reversing classical inequalities under curvature constraints in various ambient spaces.

Conferenciante: Johan Rene van Dorp (School of Engineering and Applied Science. The George Washington University)

Abstract: A framework of two-sided (TS) densities is presented for asymmetric continuous distributions consisting of two branches each with its own generating density. The framework supports the construction of distributions with positive support and a specified mode. A general expression for the Lorentz curve, depicting income inequality, is derived in terms of these generating densities. The TS beta family of distributions is constructed herein as an instance within that framework. Its generating densities are a half-symmetric beta distribution for its left branch and half Student-t distribution for its right branch. A novel procedure solving for its parameters given a lower quantile, an upper quantile, a modal value and a value for the conditional-value-at-risk (CVaR) with a specified confidence is derived. The procedure shall be demonstrated using publicly available US income distribution data from 2022 by ethnicity by fitting TS beta-t parameters to those data sets. The fitted distributions shall be compared to a fitted Burr XII distributions using the maximum-likelihood estimation (MLE) method. In that process a novel income-inequality metric termed dominance-index is introduced. That dominance-index compares income inequality between two income distributions, whereas the classical Gini-index evaluates income-inequality within a single income distribution.

SEMINARIO DE ECUACIONES DIFERENCIALES

Conferenciante: Laura Baldelli (Universidad de Granada)

Abstract: In this talk, we will disclose the main results contained in a recent paper written jointly with Jarosław Mederski (Institute of Mathematics Polish Academy of Sciences, Warsaw) and Alessio Pomponio (Polytechnic University of Bari, Italy).

Motivated by the fact that physicists are often interested in normalized solutions, we will discuss some recent results concerning existence and nonexistence of normalized solutions to a large class of quasilinear problems, including the well-known Born-Infeld operator.

Our main theorems cover the mass subcritical, critical, and supercritical cases, in the sense of the critical exponents \(2(1+2/N)\), \(q(1+2/N)\). In the mass subcritical cases, we study a global minimization problem and obtain a ground state solution for a \((2,q)\)-type operator which implies the existence of solutions to the Born-Infeld problem. We also deal with the mass supercritical cases, getting an existence result by a mountain pass approach, while in the critical cases, we prove nonexistence results by using asymptotic decays of particular externals, for quasilinear problems involving the \((2,q)\)-type operator.

SEMINARIO DE GEOMETRÍA

Conferenciante: Buse Yalçın (Ankara University)

Abstract: We classify those rotationally invariant translators of the mean curvature flow in the Hyperbolic Einstein Space-time \(\mathbb{H}^n\times_{-1}\mathbb{R}\). Next, we consider a connected, compact space-like translator whose boundary is the boundary of a bounded open domain in a slice. If the domain is invariant by an isometry \(\sigma\) of \(\mathbb{H}^n\), then the traslator is invariant by \(\sigma\times id\). We then characterize one of the rotationally invariant examples.

SEMINARIO DE ECUACIONES DIFERENCIALES

Conferenciante: Prasanta Barik (Universidad de Granada)

Abstract