Programa

Cursillos

Milton Espinoza – Universidad de La Serena

Special values of generalized Dirichlet L-functions.

This course aims to introduce avatars of Dirichlet’s L functions in higher dimensions, in addition to describing a method to explicitly calculate their special values in non-positive integers (and also positive when the respective functional equation allows it). A delicate interaction of ideas of an algebraic, analytical, geometric and topological nature is necessary to address the multiplicity of dimensions. The epiphany took place in the late 1970’s and we owe it to the Japanese mathematician Takuro Shintani, who finally discovered how to generalize the first Riemann demonstration of the analytical continuation of the zeta function to certain Dirichlet series of several variables. The main results that we will use from number theory will be presented throughout the course.

Felipe Gonçalves – IMPA / University of Texas at Austin

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Expositores

Jonathan Bober – University of Bristol

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Emanuel Carneiro – The Abdus Salam International Centre for Theoretical Physics

Fractional derivatives and the equidistribution of Galois orbits

In this talk I will discuss effective versions of the celebrated Bilu’s equidistribution theorem for Galois orbits of sequences of points of small height, identifying the qualitative dependence of the convergence in terms of the regularity of the test functions considered. I will present a general Fourier analysis framework that extends previous results in the literature. This is based in a joint work with Mithun Das (ICTP).

Andrés Chirre – Pontificia Universidad Católica del Perú

Bounds for the partial sums of the Möbius function and extremal functions

One of my favorite quotations in Mathematics is from Titchmarsh, who remarked: «The finer theory of the partial sums of the Möbius function is extremely obscure, and the results are not nearly so precise as the corresponding ones in the prime number problem.« In this talk, we will show how certain extremal functions in Fourier analysis can be employed to obtain good bounds for the partial sums of the Möbius function.

Alexander Dobner – University of Michigan

Alternative random models of the zeros of the Riemann zeta function

There is a great deal of numerical evidence suggesting that the zeros of the Riemann zeta function «look like» the eigenvalues of a random unitary matrix. This empirical connection between number theory and random matrix theory has enabled number theorists to make precise conjectures about the Riemann zeta function. However, there is still a large gap between what has been conjectured and what is known rigorously. To test the limitations of our current knowledge, it is interesting to ask whether there are other random models of the zeta zeros that are consistent with what is known rigorously. One such model called ACUE was recently proposed by Tao and independently by Lagarias and Rodgers. I will discuss this strange ’alternative’ model of the zeta zeros, and give some new results about it.

Cristian González-Riquelme – Centro de Recerca Matemática

Maximizers of extension inequalities for quadratic surfaces in finite fields

Strichartz estimates are important inequalities in PDEs. There is a correspondence between them and Fourier extension inequalities for quadratic manifolds. On the other hand, since the work of Mockenhaupt and Tao, much effort has been made in order to establish the finite field analogues for these Fourier extension inequalities. In this case, the arithmetic structure of these spaces plays a major role. In this talk, we present optimal versions of these inequalities in this setup. This is based on joint works with Diogo Oliveira e Silva and Tolibjon Ismoilov.

 

Oleksiy Klurman – University of Bristol

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Nicolás Valenzuela – Universidad de Chile

An Introduction to Physics Informed Neural Networks and Their Application to Nonlinear Dispersive Equations

In recent years, Deep Learning (DL) techniques have emerged as powerful tools for approximating solutions to certain partial differential equations (PDEs). Most applications to date focus on bounded domains, due the capability of traditional numerical methods. In this talk, we introduce a novel approach that combines Physics Informed Neural Networks (PINNs) -a recently developed DL framework- with stability theory to approximate solutions of nonlinear dispersive equations posed on unbounded domains. We specifically explore the effectiveness of PINNs in addressing the nonlinear Schrödinger and generalized Korteweg-de Vries (gKdV) equations, highlighting how this method can be employed in the absence of boundary conditions.

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Hanne Van Den Bosch – Universidad de Chile

Spectral flow methods for edge states at soft walls

The goal of this work is to understand the appearance of edge states in models from solid state physics. I will give a general introduction to the concept of spectral flow and its properties that allow us to estimate when these edge modes appear.
The talk is based on joint work with David Gontier.

 

Jan Felipe van Diejen – Universidad de Talca

Elementary hypergeometric functions, the dynamics of zeros, and KdV solitons

It is well-known that the one-dimensional stationary Schrödinger equation with a Pöschl-Teller potential can be solved exactly by means of Gauss’ hypergeometric series. For special values of the coupling constants such that the potential becomes reflectioness, this hypergeometric series factorizes essentially in terms of a plane wave and a polynomial in the spectral parameter. We point out that the positions of the zeros of this polynomial satisfy an integrable Hamiltonian system of differential equations in the spatial variable. Integration of the pertinent Hamiltonian dynamics gives rise to detailed insight into the zeros of the reflectionless hypergeometric series. Moreover, the Pöschl-Teller potential can be reconstructed from the positions of the zeros in question. By acting with a time-flow generated by a second integral of the Hamiltonian system for the zeros, the KdV solitons are recovered.

Jeanine Van Order – Pontifícia Universidade Católica do Rio de Janeiro

New approaches to the shifted convolution problem for GL(n)

I will survey the shifted convolution problem for GL(n) L-function coefficients, then describe an approach showing the analytic continuation of the underlying Dirichlet series using variation of vectors in Kirillov models.
This is based on work in progress with Dorian Goldfeld.

 

Luis Lomelí – Pontifícia Universidade Católica de Valparaíso 

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