Noticias en español
Uniform estimates for small volume asymptotics.
RESUMEN: We revisit the problem of studying the impact of a perturbation of the coefficients of an elliptic PDE on a set of small size. We show that the asymptotic structure of the perturbed solution can be described in terms of the spectrum of the Poincaré variational operators defined by the perturbations. This approach turns out to be useful in obtaining estimates which are uniform in the coefficient contrast.
Read MoreIntrinsic ergodicity for a certain class of Derived from Anosov.
RESUMEN: We will talk briefly about some classic examples of Derived from Anosov (DA), that is, homotopic maps to an Anosov diffeomorphism, whose dynamics are partially hyperbolic. We will address some known results related to entropy invariance and the existence (and uniqueness) of measures of maximal entropy for this class of diffeomorphisms. Finally, we will present recent results in collaboration with L. Parra (PUCV) and C. Vásquez (PUCV) for a certain class of DA generated after a Hopf bifurcation, previously introduced by [M. Carvalho’93].
Read MoreColour-bias perfect matchings in hypergraphs.
Abstract: We study conditions under which an r-edge-coloured k-uniform hypergraph has a perfect matching that contains substantially more than n/(kr) monochromatic edges. Our main result solves this problem for perfect matchings under minimum degree conditions, which answers recent questions of Gishboliner, Glock and Sgueglia. This is joint work with Hiêp Hàn, Richard Lang, João Pedro Marciano, Matías Pavez-Signé, Andrew Treglown, and Camila Zárate-Guerén.
Read MoreA strongly polynomial algorithm for the minimum-cost generalized flow problem.
Abstract: We give a strongly polynomial algorithm for minimum cost generalized flow, and hence for optimizing any linear program with at most two non-zero entries per row, or at most two non-zero entries per column. Our result can be viewed as progress towards understanding whether all linear programs can be solved in strongly polynomial time, also referred to as Smale’s 9th problem. Our approach is based on the recent primal-dual interior point method (IPM) due to Allamigeon, Dadush, Loho, Natura and Végh (FOCS ’22). They show that the number of iterations needed by the IPM can...
Read MoreCreación de agentes basados en gráficos de conocimiento con generación de texto estructurado y modelos open-weights
Abstract: Los gráficos de conocimiento son excelentes para representar y almacenar información heterogénea e interconectada de manera estructurada, capturando de manera eficaz relaciones y atributos complejos en diferentes tipos de datos. La generación de texto estructurado permite crear gráficos de conocimiento al proporcionar resultados perfectamente estructurados, lo que lo convierte en un método ideal para extraer información estructurada. De manera similar, la generación de texto estructurado permite la creación de agentes al definir qué herramientas están permitidas y qué entradas de...
Read MoreDifferential-difference equations arising in number theory.
Abstract: In an attempt to find a more intuitive proof of the Prime Number Theorem, Lord Cherwell derived, through heuristic arguments, the equation: f'(x) = -(f(x) f(\sqrt{x})/(2x), where f(x) represents the “density of primes at x”. Through a simple change of variables, the differential equation can be rewritten as the following delay differential equation:h'(u) = -(ln 2)(h(u) + 1)h(u – 1) which marks the first appearance of this type of equation in number theory. In this talk, we present other families of differential equations, both with delay and advance, related to...
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