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Category Archives: Recerca


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SIMBa: Fundamental domains for the Bruhat-Tits tree for $\mathrm{GL}2\left(F{\mathfrak{p}}\right)$

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Dimecres 8 de novembre se celebrarcelebrarà una xerrada del Seminari Informal de Matemàtiques de Barcelona (SIMBa).

Conferenciant: Eloi Torrents Juste.
Universitat: Universitat Autònoma de Barcelona.

Data: Dimecres, 8 de novembre de 2023.
Hora: 13:00, pausa pel cafè; 13:20, xerrada.
Lloc: UAB (aula petita CRM) i Zoom.
Idioma: Anglès.

Títol: Fundamental domains for the Bruhat-Tits tree for \mathrm{GL}_2\left(F_{\mathfrak{p}}\right).

Resum: The computation of fundamental domains of the Bruhat-Tits tree by the action of quaternionic groups allows the computation of harmonic cocycles on it. These are related to automorphic forms and from this fact are derived several applications, as for example the computation of points on Shimura curves and Heegner points on elliptic curves as done by M. Greenberg and later generalized by C. Franc and M. Masdeu.
In this talk we will review these concepts, and we will explain how to apply them in the computation of Heegner points on elliptic curves in cases where the Heegner hypothesis is not satisfied, and therefore the classical arquimedian construction of these points is difficult to compute.


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SIMBa: Overdetermined problems involving the fractional Laplacian

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Dimecres 18 d’octubre se celebrarcelebrarà una xerrada del Seminari Informal de Matemàtiques de Barcelona (SIMBa).

Conferenciant: Jack Thompson.
Universitat: The University of Western Australia.

Data: Dimecres, 18 d’octubre de 2023.
Hora: 13:00, pausa pel cafè; 13:20, xerrada.
Lloc: UB i Zoom.
Idioma: Anglès.

Títol: Overdetermined problems involving the fractional Laplacian.

Resum: Overdetermined problems are a class of partial differential boundary value problem where ‘too many’ boundary conditions are imposed on the solution. Such problems appear in physics, biology, and other applied sciences due to their strong connection with shape optimisation. The mathematical objective is to understand and classify the regions that admit solutions that satisfy all the required properties.
A classical result in this area is Serrin’s problem, which says that a bounded domain in \mathbb{R}^n that admits a function with constant Laplacian, zero Dirichlet data, and constant Neumann data must be a ball. We will consider two generalisations of this problem, both driven by the fractional Laplacian, and give an overview of the current literature and open problems.


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SIMBa: The use of skew braces in Hopf-Galois theory

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Dimecres 4 d’octubre se celebrarcelebrarà una xerrada del Seminari Informal de Matemàtiques de Barcelona (SIMBa).

Conferenciant: Daniel Gil Muñoz.
Universitat: Universitat de Barcelona.

Data: Dimecres, 4 d’octubre de 2023.
Hora: 13:00, pausa pel cafè; 13:20, xerrada.
Lloc: UPC (FME aula 005) i Zoom.
Idioma: Anglès.

Títol: The use of skew braces in Hopf-Galois theory.

Resum: The notion of Galois extension can be rewritten in such a way that it depends only on the Galois group and its action on the top field of the extension. Moreover, the group algebra corresponding to the Galois group can be endowed with a Hopf algebra structure. These facts inspire the notion of Hopf-Galois extension: an extension that admits a Hopf algebra together with a linear action on the top field satisfying analogous conditions. Such a pair is called a Hopf-Galois structure. Consequently, every Galois extension is Hopf-Galois, but not the other way around. This is the beginning of Hopf-Galois theory, a generalization of Galois theory that was introduced in the sixties of the last century and have provided a useful setting to generalize notions and results from Galois theory. More recently, it was found that each Hopf-Galois structure on a Galois extension corresponds to a skew brace: a set endowed with two group structures satisfying a variant of the distributive property. The theory of skew braces was introduced in 2007 and owes its interest to its applications on the study of the Yang-Baxter equation. This is how a connection between Hopf-Galois theory and skew braces was established, leading to a big body of research in the last years. In this talk, we provide an introduction to both of these topics as well as their link, on which we shall view some recent results.


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SIMBa: dues xerrades previstes per al 21 de juny

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Dimecres 21 de juny se celebraran dues xerrades del Seminari Informal de Matemàtiques de Barcelona (SIMBa).

Conferenciant: Meritxell Vila Miñana.
Universitat: Universitat de Barcelona.

Data: Dimecres, 21 de juny, 2023.
Hora: 12:00, pausa pel cafè; 12:20, xerrada.
Lloc: UB (FMI aula B3) i Zoom.
Idioma: Anglès.

Títol: Estimating the dimensionality of complex networks using persistent homology.

Resum: In this work, a new interdisciplinary approach is presented to study the dimensionality of complex networks using techniques from topological data analysis (TDA) through a filtration of graphs by vertex degrees. For each of two real-world complex networks, 30 surrogate graphs were generated in each dimension from 1 to 10, and several TDA descriptors of graphs were compared with the corresponding values for the real networks in order to estimate their latent dimension. Total persistence, Wasserstein distance and scale-space kernel dissimilarity, among other descriptors, yielded consistent outcomes. The results of this study suggest that TDA is sensible to the latent dimension of complex networks, and provide conclusions consistent with those btained in previous studies.

Conferenciant: Teo Gil Moreno de Mora Sardà.
Universitat: Universitat Autònoma de Barcelona, Université Paris-Est Créteil.

Data: Dimecres, 21 de juny, 2023.
Hora: 13:00, pausa pel cafè; 13:20, xerrada.
Lloc: UB (FMI Aula B3) i Zoom.
Idioma: Anglès.

Títol: An isosystolic inequality for Finsler reversible tori and the Busemann-Hausdorff area.

Resum: In 1949, Loewner discovered a much celebrated inequality: the systole of any Riemannian torus of dimension 2 is controlled by its area. The key step in his proof was the reduction to the flat case by means of the conformal representation theorem. The generalization of this inequality to the Finsler framework results in a wide variety of results. In this talk I will survey the different known isosystolic inequalities on two-dimensional Finsler tori involving the two main notions of area in Finsler geometry: the Busemann-Hausdorff area and the Holmes-Thompson area. I will also complete the picture by presenting a new isosystolic inequality on reversible Finsler 2-tori for the Busemann-Hausdorff area, which is obtained again by reduction to the flat case. It is a joint work with Florent Balacheff.

Si voleu proposar una xerrada, ompliu el formulari. Si voleu contactar amb els responsables podeu escriure un missatge a seminari(dot)simba(at)ub(dot)edu.


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SIMBa: Searching for all gene networks capable of pattern formation

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Dijous 24 de msig se celebrarà una nova xerrada del Seminari Informal de Matemàtiques de Barcelona (SIMBa).

Conferenciant: Kevin Martínez Añón.
Universitat: Centre de Recerca Matemàtica (CRM), Universitat Autònoma de Barcelona (UAB).

Data: Dimecres, 24 de maig, 2023.
Hora: 13:00, pausa pel cafè; 13:20, xerrada.
Lloc: UAB i Zoom.
Idioma: Anglès.

Títol: Searching for all gene networks capable of pattern formation.

Resum: Development is the process by which the complex anatomy of multicellular organisms is built in each generation. Not many other natural processes lead to so much complexity in such a short period of time.
Morphological transformations during development are regulated by what we call gene regulatory networks, that is, the specific activatory and inhibitory interactions between gene products at a molecular level. In this talk, under the framework of reaction-diffusion models, we attempt a mathematical approach to these developmental transformations by asking ourselves a very simple question: “which gene regulatory networks are actually capable of pattern formation?” More specifically, we wonder whether the immense number of patterning gene networks that we can think of – especially when the number of genes involved in the network is very large – can all be sorted into some few fundamental classes sharing the same topological features and resulting in similar final patterns.

 

Si voleu proposar una xerrada, ompliu el formulari. Si voleu contactar amb els responsables podeu escriure un missatge a seminari(dot)simba(at)ub(dot)edu.


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SIMBa: Playing with Mixed Boolean-Arithmetic algebra

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Dijous 27 d’abril se celebrarà una nova xerrada del Seminari Informal de Matemàtiques de Barcelona (SIMBa).

Conferenciant: Arnau Gàmez i Montolio.
Universitat: Universitat de Barcelona.

Data: Dimecres, 27 d’abril, 2023.
Hora: 13:00, pausa pel cafè; 13:20, xerrada.
Lloc: UB (FMI aula B1) i Zoom.
Idioma: Anglès.

Títol: Playing with Mixed Boolean-Arithmetic algebra.

Resum: A Mixed Boolean-Arithmetic (MBA) expression is an algebraic expression composed of integer arithmetic operators, e.g. (+,-, \times) and bitwise operators, e.g. (\wedge, \vee, \oplus, \neg). State-of-the-art software protection mechanisms leverage MBA semantics-preserving transformations to obfuscate code. This possibility is motivated by the fact that the combination of operators from these different fields do not interact well together. Moreover, computer algebra systems do not support bitwise operators with symbolic variables (let alone combining them with arithmetic expressions).
In this talk, we introduce the fundamental ideas, constructions and research literature covering MBA algebra, focusing on (and motivated by) applications in program analysis and software protection. We also outline several open problems and research directions aiming to advance the theoretical formalization and tooling development for MBA algebra reasoning and manipulation.

 

Si voleu proposar una xerrada, ompliu el formulari. Si voleu contactar amb els responsables podeu escriure un missatge a seminari(dot)simba(at)ub(dot)edu.


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SIMBa: Splitting of 2D invariant manifolds of a periodically forced 3D volume preserving flow

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Dimecres 29 de març se celebrarà una nova xerrada del Seminari Informal de Matemàtiques de Barcelona (SIMBa).

Conferenciant: Ainoa Murillo López.
Universitat: Universitat de Barcelona.

Data: Dimecres, 29 de març, 2023.
Hora: 13:00, pausa pel cafè; 13:20, xerrada.
Lloc: UB (FMI aula B1) i Zoom.
Idioma: Anglès.

Títol: Splitting of 2D invariant manifolds of a periodically forced 3D volume
preserving flow.

Resum: We shall consider a 3D volume-preserving flow with a 2-dimensional heteroclinic connection to saddle-foci, obtained by truncating the normal form of a generic unfolding of a Hopf-Zero bifurcation. First, we will describe the phase space of the system and then, we will consider the effect of a non-autonomous periodic perturbation on it. This perturbation introduces an additional frequency, which interacts with the intrinsic frequency related to the multiplier of the foci. In particular, our focus will be on the splitting of the 2D stable and unstable invariant manifolds and its quasi-periodic asymptotic behaviour.

 

Si voleu proposar una xerrada, ompliu el formulari. Si voleu contactar amb els responsables podeu escriure un missatge a seminari(dot)simba(at)ub(dot)edu.


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SIMBa: An introduction to Khovanov homology and annular links

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Dimecres 15 de març se celebrarà una nova xerrada del Seminari Informal de Matemàtiques de Barcelona (SIMBa).

Speaker: Sergio García Rodrigo.
Universitat: Universidad Autónoma de Madrid.

Data: Dimecres, 15 de març, 2023.
Hora: 13:00, pausa pel cafè; 13:20, xerrada.
Lloc: UB (FMI aula B1) i Zoom.
Idioma: Anglès.

Títol: An introduction to Khovanov homology and annular links.

Resum: This talk will provide an overview of knot theory, a subfield of algebraic topology that seeks to understand the properties of knots and links through the study of their invariants. Specifically, we focus on two essential invariants in knot theory, namely, the Jones polynomial
and its categorification, the Khovanov homology. Furthermore, we introduce a variation of classical links, known as annular links, which exist in the solid torus instead of \mathbb{R}^3. We then proceed to explore the annular versions of both the Jones polynomial and the Khovanov homology.

Si voleu proposar una xerrada, ompliu el formulari. Si voleu contactar amb els responsables podeu escriure un missatge a seminari(dot)simba(at)ub(dot)edu.


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SIMBa: The Nilpotent Center Problem on Center Manifolds in R^3

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El dimecres 22 de febrer se celebrarà una nova xerrada del Seminari Informal de Matemàtiques de Barcelona (SIMBa).

Speaker: Lucas Queiroz Arakaki.
Universitat: São Paulo State University (UNESP).

Data: Dimecres, 22 de febrer, 2023.
Hora: 13:00, pausa pel cafè; 13:20, xerrada.
Lloc: UAB (aula petita CRM) i Zoom.
Idioma: Català.

Títol: The Nilpotent Center Problem on Center Manifolds in R^3.

Resum: We work with systems of differential equations which have a nilpotent singular point at the origin on a Center Manifold. We study the formal integrability and the center problem for those types of singular points in the monodromic case. Our approach does not require polynomial approximations of the Center Manifold in order to study the center problem.

 

Si voleu proposar una xerrada, ompliu el formulari. Si voleu contactar amb els responsables podeu escriure un missatge a seminari(dot)simba(at)ub(dot)edu.