Mathematical Doctorands Seminar of the Luxembourg University
2013/2014

General Information

Organisers

Talks

April 10, 2014 Jérémy Toulisse, Teichmüller space, quantization and invariants of surface diffeomorphisms(II)
April 3, 2014 Jérémy Toulisse, Teichmüller space, quantization and invariants of surface diffeomorphisms(I)
Abstract: Available as a pdf-file.
March 27, 2014 Tiffany Covolo, Categorical approach to the graded determinant
Abstract: I will report on the recent joint work with Jean-Philippe Michel. I will present how a well-known isomorphism of categories - between the categories of graded-commutative algebras, also called color algebras, and the categories of (graded) super-commutative algebras -- gives an intuitive way to transport the notions of trace and determinant to the graded side. I will then investigate the notions thus obtained, showing the power but also the limitations of this method, illustrated in the particular case of matrices over the (graded-commutative) algebra of quaternions.
March 6, 2014 Gilles Becker, Cuspidal functions on quotients of SL(2, R) (II)
February 27, 2014 Gilles Becker, Cuspidal functions on quotients of SL(2, R) (I)
Abstract: In this mini-course, we deal with cuspidal functions on quotients of G := SL(2, R) of the form Γ\G, where Γ is a discrete subgroup of G with finite covolume (e.g. Γ = SL(2, Z)). Roughly speaking, cuspidal functions are locally integrable functions on Γ\G with vanishing "constant term". Our main goal is to show a theorem of Gel'fand, Graev and Piatetski-Shapiro. It is contained in the book "Representation theory and automorphic functions" published in the Sixties and says that the space °L^2(Γ\G) of cuspidal functions in L^2(Γ\G) can be decomposed as a Hilbert direct sum of closed irreducible G-invariant subspaces with finite multiplicities. If Γ\G is compact, then the theorem is almost a consequence of Ascoli's theorem and it simply says that L^2(Γ\G) admits a decomposition as above.
February 20, 2014 Dr. Erlend Grong, Geometry and stochastic processes(II)
Abstract: Last time we looked at the differences between Riemannian geometry and sub-Riemannian geometry.
This week, we will look at stochastic processes ("random variables evolving in time"). We will first look at the case real valued processes, in particular Brownian motion and martingales in general. We will then look at manifold valued processes, in particular diffusions of second order operators. We end with some application to Riemannian and sub-Riemannian geometry.
February 13, 2014 Dr. Erlend Grong, Geometry and stochastic processes(I)
Abstract: A stochastic processes can roughly be described as a random variable evolving in time. I will first do a "quick and dirty" introduction to stochastic processes with values on the real line and stochastic integrals. Much of this theory can then easily be generalized to Euclidean space, however, to get a "good" theory for manifolds is a real hassle. I will explain why these difficulties occur on manifolds, and what we can do to get around them. In the end, to justify all of the hassle, I will give some examples of how one can use stochastic processes to get results in Riemannian geometry (and perhaps even sub-Riemannian geometry, if there is such a thing). It helps to know what Riemannian manifolds and random variables are, but otherwise, very little prior knowledge in probability theory or geometry is needed.
November 28, 2013 Tiffany Covolo, Closed monoidal functors (II)
November 21, 2013 Tiffany Covolo, Closed monoidal functors (I)
Abstract: I will present the classical notions of monoidal functor between monoidal categories and closed functor between closed category, and then discuss the combination of these two notions in the case of closed monoidal categories. As an application, I will shortly outline how this notion was useful in my recent work.
November 14, 2013 Dr. Matteo Tommasini, Introduction to (algebraic) Deligne-Mumford stacks (V)
November 7, 2013 Dr. Matteo Tommasini, Introduction to (algebraic) Deligne-Mumford stacks (IV)
October 31, 2013 Dr. Matteo Tommasini, Introduction to (algebraic) Deligne-Mumford stacks (III)
October 24, 2013 Dr. Matteo Tommasini, Introduction to (algebraic) Deligne-Mumford stacks (II)
October 17, 2013 Dr. Matteo Tommasini, Introduction to (algebraic) Deligne-Mumford stacks (I)
Abstract: In this mini-course of 3-4 lectures I will go through all the basis of (algebraic) Deligne-Mumford stacks. Some prerequisites of algebraic geometry and category theory are welcome but not strictly necessary for understanding the relevant ideas of the theory.
October 10, 2013 Hector Castejon-Diaz, Introduction to Toric Varieties (III)
September 30, 2013 Hector Castejon-Diaz, Introduction to Toric Varieties (II)
September 23, 2013 Hector Castejon-Diaz, Introduction to Toric Varieties (I)
Abstract: In this mini-course of 2-3 sessions, I will explain what are Toric Varieties, how to construct them and some properties which will show the audience why they are interesting to study. Some notions of geometry (definition of smooth manifold, algebraic variety, line bundle, spectrum of a ring, etc.) are recommended to understand the motivation, but not completely necessary to follow the course.

Mathematics Research Unit