COLOQUIO DE MATEMÁTICAS 2017-II
ENTRADA LIBRE
 
Se ofrecerán café, aromática y galletas antes del Coloquio a las 3.30 p.m. frente al Departamento de Matemáticas (edificio H), si el clima lo permite.
Coloquios
anteriores:
Jueves 10 de Agosto del 2017
ML606
4.00 p.m.
Ben Weinkove - Northwestern University, U.S.A.
What are the metrics on an algebraic variety?

- Resumen


An algebraic variety is the set of solutions of a system of polynomial equations. It can be regarded as a geometric object, and we would like to introduce geometric notions like distance and curvature. To do so we need the notion of a metric. However, there are many different metrics we could choose. Infinitely many. What are they? Do they themselves form an interesting space? I will discuss this in the context of complex projective varieties.

Invitado por:

Florent Schaffhauser
Jueves 10 de Agosto del 2017
ML606
5.00 p.m.
Pablo Cúbides - Université de Caen, France
Geometría no-arquimediana y espacios de Berkovich

- Resumen


Los números p-ádicos, introducidos por Hensel a finales del siglo XIX, han jugado un papel crucial en la teoría de números moderna. Así como los números reales, los números p-údicos tienen una topología natural asociada a una norma, en este caso, la norma p-ádica. Sin embargo, a diferencia del caso real (y complejo), la topología es totalmente discontinua. Esta diferencia hace que muchos resultados clásicos de la geometría real y compleja no tengan una contraparte en este contexto. En particular, la teoría de las funciones holomorfas sobre dichos campos requiere un tratamiento especial. En esta charla hablaré de una de las alternativas de atacar este problema: los espacios de Berkovich. Si el tiempo alcanza, discutiré algunos resultados obtenidos por Hrushovski y Loeser sobre dichos espacios utilisando métodos modelo-teóricos. .

Invitado por:

Florent Schaffhauser
Jueves 17 de Agosto del 2017
ML606
4.00 p.m.
Victoria Hoskins - Freie Universität Berlin, Germany
Group quotients and moduli problem in algebraic geometry

- Resumen


Moduli problems are classification problems in algebraic geometry whose solutions are given by moduli spaces. These moduli spaces are often constructed as a quotient of a group action via geometric invariant theory. In this talk, I will explain how to construct quotients in algebraic geometry, where our primary motivation is to construct moduli spaces as such quotients. I will take an example-based approach to the theory and, in particular, focus on the example of moduli of vector bundles over a smooth projective algebraic curve. If there is time, I will also explain some techniques for studying the geometry of these moduli spaces.

Invitado por:

Florent Schaffhauser
Jueves 17 de Agosto del 2017
ML606
5.00 p.m.
Benjamin Linowitz - Oberlin College, U.S.A.
You cannot hear the shape of a hyperbolic drum.

- Resumen


In 1966 Marc Kac famously asked if one could hear the shape of a drum. In more technical language, Kac was asking whether two planar domains could be isospectral (i.e., have the same spectrum with respect to the Laplace operator) but not isometric. In 1980 Vigneras used orders in quaternion algebras in order to prove the existence of hyperbolic surfaces which were isospectral but not isometric. Thus one cannot hear the shape of a hyperbolic drum. The example appearing in Vigneras' original paper was a pair of manifolds of genus 100801. We will exhibit substantially simpler examples: a pair of genus 6 manifolds and a pair of orbifolds whose underlying surface has genus 0. This is joint work with John Voight.

Invitado por:

Guillermo Mantilla
Viernes 18 de Agosto del 2017 - Coloquio extraordinario
AU201
2.00 p.m.
Diego Cifuentes - M.I.T., U.S.A.
Sampling Algebraic Varieties for Sum of Squares Programs

- Resumen


We study sum of squares (SOS) relaxations to optimize polynomial functions over a set $\mathcal{V}\cap \mathbb{R}^n$, where $\mathcal{V}$ is a complex algebraic variety. We propose a new methodology that, rather than relying on some algebraic description, represents $\mathcal{V}$ with a generic set of complex samples. This approach depends only on the geometry of $\mathcal{V}$, avoiding representation issues such as multiplicity and choice of generators. It also takes advantage of the dependencies in the coordinate ring of $\mathcal{V}$ to reduce the size of the corresponding semidefinite program (SDP). In addition, the input can be given as a straight-line program. Our methods are particularly appealing for varieties which are easy to sample from but for which the defining equations are complicated, such as $SO(n)$, Grassmannians or rank~$k$ tensors. Nonetheless, for arbitrary varieties we can obtain the samples by using the tools of numerical algebraic geometry. In this way we connect the areas of SOS optimization and numerical algebraic geometry.

Invitado por:

Mauricio Velasco
Miércoles 23 de Agosto del 2017 - Coloquio extraordinario
AU103
2.00 p.m.
Fabrice Gamboa - Université de Toulouse, France
Large deviations and sum rules

- Resumen


In this talk, we will discuss some of our recent results concerning the proof of sum rules that uses large deviations properties of spectral measures built on random matrices. Roughly speaking, a sum rule is an identity relating the Kullback information to a function of coefficients involved in the construction of orthogonal polynomials. One of the most famous sum rule is the Szego Verblunsky Theorem.

Invitado por:

Mauricio Velasco
Miércoles 23 de Agosto del 2017 - Coloquio extraordinario
AU103
3.00 p.m.
Thierry Klein - Université de Toulouse, France
Fréchet means and PCA in the Wasserstein space.

- Resumen


In the first part of this talk, we will present the notion of Fréchet means in the Wasserstein space. In particular we will explain how to define a natural notion of mean when the mathematical objects considered are probability measures. In the second part of this talk, we will explain how we can use the pseudo Riemannian structure of the Wasserstein space to perform a geodesic PCA of a collection of probability measures.

Invitado por:

Mauricio Velasco
Jueves 24 de Agosto del 2017
ML606
4.00 p.m.
Andrés Jaramillo - Institut de Mathématiques de Jussieu - Paris Rive Gauche, France
Title

- Resumen


TBA.

Invitado por:

Florent Schaffhauser
Jueves 24 de Agosto del 2017
ML606
5.00 p.m.
Daniel Massart - Université de Montpellier II, France
Intersección algebraica de curvas cerradas en superficies orientables

- Resumen


Veremos cómo definir la intersección algebraica int(alpha, beta) de dos curvas cerradas alpha y beta en una superficie orientable X. Veremos que la intersección algebraica induce una forma bilineal anti-simétrica en el primer grupo de homología de X, un objeto algebraico que contiene información topológica sobre X. Luego, considerando la longitud como un costo y la intersección como un beneficio, trataremos de optimizar la cantidad Int(alpha,beta)/(longitud(alpha)longitud(beta)). Nos enfocaremos en las superficies mas homogéneas: las cuya geometría local es euclidiana o hiperbólica.

Invitado por:

Florent Schaffhauser
Jueves 31 de Agosto del 2017
ML606
4.00 p.m.
TBA - TBA
Title

- Resumen


TBA.

Invitado por:

Michael Hoegele
Jueves 7 de Septiembre del 2017
ML606
4.00 p.m.
TBA - TBA
Title

- Resumen


TBA.

Invitado por:

Michael Hoegele
Jueves 14 de Septiembre del 2017
ML606
5.00 p.m.
Carolina Benedetti - Universidad de los Andes
Title

- Resumen


TBA.

Invitado por:

Michael Hoegele
Jueves 21 de Septiembre del 2017
ML606
4.00 p.m.
Guillermo Mantilla-Soler - Universidad de los Andes
Title

- Resumen


TBA.

Invitado por:

Michael Hoegele
Jueves 12 de Octubre del 2017
ML606
4.00 p.m.
TBA - TBA
Title

- Resumen


TBA.

Invitado por:

Michael Hoegele
Jueves 19 de Octubre del 2017
ML606
4.00 p.m.
TBA - TBA
Title

- Resumen


TBA.

Invitado por:

Michael Hoegele
Jueves 26 de Octubre del 2017
ML606
4.00 p.m.
TBA - TBA
Title

- Resumen


TBA.

Invitado por:

Michael Hoegele
Jueves 2 de Noviembre del 2017
ML606
4.00 p.m.
TBA - TBA
Title

- Resumen


TBA.

Invitado por:

Michael Hoegele
Jueves 9 de Noviembre del 2017
ML606
4.00 p.m.
TBA - TBA
Title

- Resumen


TBA.

Invitado por:

Michael Hoegele
Jueves 16 de Noviembre del 2017
ML606
4.00 p.m.
Andrés Ángel - Universidad de los Andes
Title

- Resumen


TBA.

Invitado por:

Michael Hoegele