Homological Invariants of Deformations of Groups and Algebras

Descripción del proyecto

Ampliar los métodos de álgebra homológica para estudiar sistemas dinámicos

El álgebra homológica estudia los functores homológicos y las complejas estructuras algebraicas subyacentes. En ella se extrae la información contenida en complejos en cadena y se presenta en forma de invariantes homológicos de anillos, módulos, espacios topológicos y otros objetos matemáticos. El objetivo del proyecto HIDRA, financiado por las Acciones Marie Skłodowska-Curie, es mejorar aún más los métodos de álgebra homológica con técnicas novedosas basadas en categorías trianguladas, teoría de homotopías y teoría de índices. Se hará hincapié en la conjetura de Baum-Connes, que se relacionará con el cálculo de invariantes K-teóricos y homológicos para sistemas dinámicos destacados. Los resultados del proyecto tendrán repercusiones de calado en las matemáticas puras, la física del estado sólido y la teoría de la información cuántica.

Objetivo

"The pervasive role of algebraic topology in mathematics is proof of the powerful effects that homological invariants produce in the development of the discipline. Extending these techniques beyond the category of topological spaces, in order to include ""quantized"" systems arising from dynamical systems and (quantum) groups, is going to be extremely useful to make fast progress in these fields. The framework of operator algebras and noncommutative geometry is extremely well-suited for these developments and has already been applied with some success. The goal of this proposal is to further develop these homological techniques by supporting them with novel methods based on triangulated categories, homotopy theory, and index theory. The research problems tackled in this Action are deeply related to important topics which attracted a great deal of interest in the mathematical community. For example, we study the celebrated Baum-Connes conjecture (for both groupoids and quantum groups) through a relatively unexplored perspective and relate it to the computation of K-theoretic and homological invariants for notable dynamical systems (e.g. Smale's Axiom A diffeomorphisms). This research will provide mathematicians with both conceptually new approaches and powerful computational tools. Some of these results are relevant not only for pure mathematics, but also for solid-state physics and quantum information theory. This Action will take us one step closer to the solution of significant problems or the formulation of more and more refined research questions. This fellowship will allow V. Proietti to work under the supervision of M. Yamashita (a world-class expert on quantum groups) at the University of Oslo (a leading institution in operator algebras). It will expand the fellow's technical expertise and integrate it with essential management, administrative, and dissemination skills which will help V. Proietti reach a position of professional maturity."