Favor contactarme si está interesado en trabajar en uno de los siguientes temas o en temas conexos:
- Uniformización de Schottky (pregrado)
- Grupos fundamentales orbifold (maestría)
- Fibrados vectoriales en curvas algebraicas (maestría)
- Álgebras de Clifford y grupos de Spin, David Jaramillo Duque (pregrado)
- Teoría de Galois de álgebras separables sobre un cuerpo, Santiago Cortés Gómez (pregrado)
- El teorema de no encaje afín de Gromov, Alirio Calderón (pregrado; co-dirección con Carlos Antonio Julio Arrieta, Universidad Distrital Francisco José de Caldas)
- El grupo modular de una superficie compacta orientable, Simón Soto Ochoa (pregrado)
- Teoría de Morse equivariante en geometría simpléctica, Alejandro Rivera (pasantía ENS Lyon)
- Fibrados de Higgs y reducción hiperkähleriana en dimensión infinita, Camilo Vargas Contreras (maestría)
- Uniformización de curvas algebraicas reales, Andrés Jaramillo Puentes (maestría)
- Cocientes GIT y cocientes simplécticos: el teorema de Kempf-Ness, Ramón Urquijo Novella (maestría)
- Superficies de Hurwitz, Sergio Pedraza Rodríguez (pregrado)
In this undergraduate thesis, Santiago studies étale and separable algebras over a field and explores their Galois theory. The classical Galois correspondence is formulated in categorical language. The goal is to provide the necessary background in order to later see the category of étale algebras as a first example of Galois category in the sense of Grothendieck. (pdf)Santiago Cortés Gómez (pre-grado): Teoría de Galois de álgebras separables sobre un cuerpo
In this undergraduate thesis, Alirio presents a proof of Gromov’s affine non-squeezing theorem, which is the first step towards the notion of (affine) symplectic capacity. (pdf)Alirio Calderón (pre-grado): El teorema de no encaje afín de Gromov
In this undergraduate thesis, Simón studies the definition and basic properties of the modular group of a compact orientable surface, the goal being to understand a proof of a classical result known as Lickorish’s theorem: the modular group of a closed orientable surface of genus g>0 is generated by 3g-1 twists de Dehn associated to explicit curves on the surface. (pdf)Simón Soto Ochoa (pre-grado): El grupo modular de una superficie compacta orientable
In this undergraduate thesis, Alejandro studies the momentum map of a Hamiltonian torus action on a compact symplectic manifold from the point of view of Morse-Bott theory, the goal being to achieve an understanding of Atiyah’s proof that the image of such a momentum map is a convex polytope, namely the convex hull of the images of the fixed points of this action. (pdf)Alejandro Rivera (pasantía ENS Lyon, pre-grado): Teoría de Morse equivariante en geometría simpléctica
In this master thesis, Camilo studies the formalism of Kähler and hyperkähler reduction in order to understand the construction of the moduli space of semistable Higgs vector bundles of fixed rank and degree from that point of view. The hyperkähler momentum map conditions in this context are known as the Hitchin equations and provide a natural differential-geometric framework in which to formulate the algebro-geometric condition of stabilty for Higgs vector bundles. (pdf)Camilo Vargas Contreras (maestría): Fibrados de Higgs y reducción hiperkähleriana en dimensión infinita
In this master thesis, Ramón studies the relationship between GIT quotients and symplectic quotients in the case of smooth, affine, complex algebraic varieties acted upon linearly by a reductive algebraic group G. The goal of the thesis is to understand a proof of the Kempf-Ness theorem in this case, which states that the poly-stable points of such an action are precisely the G-translates of points that go to zero under the K-momentum map and that the orbit of any such point contains a unique K-orbit, where K is a given maximal compact sub-group of G. This implies the existence of a homeomorphism between the GIT quotient under G and the symplectic quotient under K. (pdf)Ramón Urquijo Novella (maestría): Cocientes GIT y cocientes simplécticos, el teorema de Kempf-Ness
In this master thesis, Andrés studies the (discrete) fundamental group of a real algebraic curve (X,s) and its action on the complex analytic universal covering space of the curve. Special attention is dedicated to the hyperbolic case, for which the universal covering space is the Poincaré upper-half plane H. This has a canonical real structure t and the group of analytic and anti-analytic automorphisms of H can be identified with PGL(2,R). This readily gives a uniformization result for hyperbolic real curves: there is a discrete, faithful representation of the real fundamental group into PGL(2,R), extending the discrete, faithful representation of the ordinary fundamental group inside PSL(2,R), such that the quotient real curve obtained from (H,t) is the original real curve (X,s). (este trabajo fue publicado en la revista Lecturas Matemáticas, Vol 33 (2) 2012 107-131, publicada por la Sociedad Colombiana de Matemáticas, pdf)Andrés Jaramillo Puentes (maestría): Uniformización de curvas reales
In this undergraduate thesis, Sergio studies the automorphism group of a compact connected Riemann surface of genus g > 1. The goal is to understand a proof of the Hurwitz theorem, which states that this group is finite and of order no greater than 84(g-1). A Riemann surface of genus g > 1 with exactly 84(g-1) automorphisms is then known as a Hurwitz surface, and Sergio studies, as an example, the plane projective curve of equation x^3 y + y^3 z + z^3 x =0, known as the Klein quartic. It is shown that this (smooth) curve of genus 3 has automorphism group isomorphic to PSL(2,F_7), which has order 168 = 84(3-1). (pdf)Sergio Pedraza Rodríguez (pre-grado): Superficies de Hurwitz