Miercoles 7 de octubre, 16hs. Aula Magna

Título: «Spherical transforms and spectral analysis of differential operators»

Resumen: I will introduce two different topics fitting inside in the title.

Given a connected Lie group $G$ of polynomial volume growth and a self-adjoint sublaplacian on it, it is possible to introduce a sort of Plancherel measure on the nonnegative half-line which allows to define a spherical-like transform associated with the operator. I will present results obtained by my student Leonardo Tolomeo in his undergraduate thesis.
The second topic is an insight into the spherical transform of a given $K$-type on a nilpotent Gelfand pair $(N,K)$. The results are joint work with Amit Samanta.

Viernes 4 de septiembre, 11hs. Aula 10

Título: «Difference equation for the Heckman-Opdam hypergeometric function and its confluent Whittaker limit«

Resumen: We will discuss explicit difference equations for the Heckman-Opdam hypergeometric function associated with root systems (a generalization of the Gauss hypergeometric function to various variables). Our method exploits the fact that for discrete spectral values on a (translated) cone of dominant weights the Heckman-Opdam hypergeometric function truncates in terms of Heckman-Opdam Jacobi polynomials. This permits us to derive/prove the desired difference equations in two steps: first for the discrete spectral values by performing a $q\to 1$ degeneration of a recently found Pieri formula for the celebrated Macdonald polynomials, and then for arbitrary spectral values upon invoking an analytic continuation argument borrowed from Rösler (based on known growth estimates for the Heckman-Opdam hypergeometric function that enable one to apply Carlson’s theorem).
If time permits we will also mention analogous difference equation for the class-one Whittaker function diagonalizing the open quantum Toda chain associated with reduced root systems.
Based on joint work with Jan Felipe van Diejen (Universidad de Talca).

Jueves 30 de julio, 16hs. Aula 15

Título: «El principio de transferencia de Lefschetz”

Resumen: El llamado principio de transferencia de Lefschetz afirma que la geometría algebraica sobre el cuerpo de los números complejos es la misma que la de cualquier otro cuerpo algebraicamente cerrado de característica 0. La teoría de modelos explica y permite generalizar este principio heurístico, uno de los ejemplos mas sencillos de interacción no trivial entre lógica y otras áreas de las matemáticas. Discutiremos principios de transferencia más generales que ilustran el papel de la teoría de modelos como instrumento matemático en álgebra, análisis y teoría de números.

Lunes 22 de junio, 20hs. Aula Magna

Título: «Tango y Ciencia: Un encuentro científico-musical.» (VER PDF)