AnálisisParabolic uniform rectifiability and the Dirichlet problem with Lp data and variable coefficients
Resum: It is a very recent result of Bortz, Hofmann, Martell and Nyström (in combination with earlier works of Hofmann, Lewis and Murray) that the Dirichlet problem for the heat equation is well-posed, with lateral data in Lp, exactly in the domains whose boundary is a (parabolic) Lipschitz graph with the additional property that the graph has a half-time derivative in BMO. They show that this condition, that could possibly look artificial at first glance, is actually optimal.
Continuity of the solution map to some active scalar
equations in Hölder and Zygmund spaces
Resum: In this talk, we explore the continuity properties of the solution map, in Hölder and Zygmund spaces, to a class of nonlinear transport equations in \(\mathbb{R}^n\). The velocity field in these equations is given by the convolution of the density with a kernel that is homogeneous of degree \(-(n-1)\) and smooth away from the origin. This setting encompasses significant models, including the 2D Euler equations and the 3D surface quasi-geostrophic (SQG) equations.
EstadísticaModelització estadística d'esdeveniments extrems en Meteorologia Espacial: Impacte en Infraestructures Crítiques
Resum: La meteorologia espacial constitueix una amenaça per a la seguretat i fiabilitat de les infraestructures crítiques modernes, especialment aquelles que depenen de sistemes de navegació global per satèl·lit (GNSS). En aquest context, un dels fenòmens més preocupants són les fulguracions solars en l'espectre de ràdio (SRBs, Solar Radio Bursts), que poden afectar els senyals GNSS i comprometre la sincronització temporal utilitzada en la gestió de xarxes elèctriques. A mesura que aquestes xarxes evolucionen cap a models intel·ligents amb una dependència creixent de sistemes de temporització precís, la necessitat de comprendre i pronosticar aquests esdeveniments esdevé més crítica que mai.
GeometríaCanonical generalized Kähler structures and the elliptic genus.
Resum: We propose a natural notion of elliptic genus for generalized Kähler manifolds, following work by Heluani and Zabzine on this problem. Our construction requires that the generalized Kähler structure is "canonical", in a precise sense, covering an important class of compact non-Kähler examples, such as compact even dimensional Lie groups. The key technical step is the construction of commuting pairs of the N = 2 superconformal vertex algebra in the chiral de Rham complex of the manifold, endowed with a canonical generalized Kähler structure. Joint work with Andoni de Arriba de la Hera, Luis Alvarez-Consul, and Jethro van Ekeren.
Modelos Estocásticos y DeterministasHitting probabilities for the stochastic heat equation
Resum: We establish a new lower bound for the hitting probabilities associated to the stochastic heat equation. The proof is based on an approximation argument plus an estimate for the density of the supremum of the solution to the stochastic heat equation with additive noise.
Teoría de AnillosNew classes of IYB groups
Resum: In this talk, we shall describe a special structure of a finite solvable group with all Sylow subgroups of nilpotency class at most two. This will be the key to prove that every finite group of odd order with all Sylow subgroups of nilpotency class at most two is an involutive Yang-Baxter group (IYB group for short), i.e. it admits a structure of left brace. We will also prove that every finite solvable group of even order with all Sylow subgroups of nilpotency class at most two and abelian Sylow $2$-subgroups is an IYB group. These results contribute to the open problem asking which finite solvable groups are IYB, in particular they generalize a result of Ben David and Ginosar [BDG] concerned with finite solvable groups with abelian Sylow subgroups.
On the trace simplex of a C*-algebra
Resum: It is a well-known fact that the set of tracial states on a unital C*-algebra is a Choquet simplex (if not empty). I will here present a simple and elementary proof of this fact using the center valued trace on von Neumann algebras and well-known characterizations of Choquet simplices. This part is a joint work with Bruce Blackadar. It is also well-know that the set of all states on a C*-algebra is not a Choquet simplex except if the C*-algebra is abelian. This fact has an interpretation in terms of entanglement in C*-algebras and relates to an old open problem about tensor product of convex compact sets. This latter part is a preliminary report on work in progress.
Jordan homomorphisms
Resum: A Jordan homomorphism between associative rings is an additive map that preserves the Jordan product $xy+yx$. The study of this notion has a rich history, which will be presented in the first part of the talk. In the second part, some new results will be discussed.
Teoría de NúmerosOn the 2-Selmer group of hyperelliptic curves (II)
Resum: In these talks, we are interested in studying the 2-Selmer group of hyperelliptic curves defined over number fields. More precisely, we prove upper and lower bounds on the size of this group in terms of a class group of an algebra associated with the curve. If time permits, we will provide applications to families of curves.
On the 2-Selmer group of hyperelliptic curves (I)
Resum: In these talks, we are interested in studying the 2-Selmer group of hyperelliptic curves defined over number fields. More precisely, we prove upper and lower bounds on the size of this group in terms of a class group of an algebra associated with the curve. If time permits, we will provide applications to families of curves.
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