EPFL MATHEMATICS COLLOQUIUM

  • Prof. Alexander Mednykh

    Sobolev Institute, Novosibirsk State University
    Thursday, January 26, 2012
    17.15 - CM5

    Geometry of Polyhedra in the Hyperbolic and Spherical Spaces

    Abstract

    The calculation of volume of polyhedron is very old and difficult problem. Probably, the first result in this direction belongs to Tartaglia (1499 -- 1557) who found the volume of an Euclidean tetrahedron. Nowadays this formula is more known as Caley-Menger determinant. Recently it was shown by I. Kh. Sabitov (1996) that the volume of any Euclidean polyhedron is a root of algebraic equation whose coefficients are functions depending of combinatorial type and lengths of polyhedra. In hyperbolic and spherical spaces the situation is much more complicated. Gauss used the word "der Dschungel" in relation with volume calculation in non-Euclidean geometry. In spite of this, Janos Boyai, Nicolay Lobachevsky and Ludwig Schlafli obtained very beautiful formulae for non-Euclidean volume of a biorthogonal tetrahedron (orthoscheme). The volume of the Lambert cube and some other polyhedra were calculated by R. Kellerhals (1989), D. A. Derevnin, A. D. Mednykh (2002), A. D. Mednykh, J. Parker, A. Yu. Vesnin (2004), E. Molnar, J. Szirmai (2005) and others. The volume of hyperbolic polyhedra with at least one vertex at infinity was found by E. B. Vinberg (1992). The general formula for volume of tetrahedron remained to be unknown for a long time. A few years ago Y. Choi, H. Kim (1999), J. Murakami, U. Yano (2005) and A. Ushijima (2006) were succeeded in finding of a such formula. D. A. Derevnin, A. D. Mednykh (2005) suggested an elementary integral formula for the volume of hyperbolic tetrahedron. We note that the volume formula for symmetric tetrahedra whose opposite dihedral angles are mutually equal is rather simple. For the first time this phenomena was discovered by Lobachevsky for ideal hyperbolic tetrahedra, which is automatically symmetric. The respective result in quite elegant form was presented by J. Milnor (1982). For general case of symmetric tetrahedron the volume was given by D. A. Derevnin, A. D. Mednykh and M. G. Pashkevich (2004). Surprisedly, but a hundred years ago, in 1906 an essential advance in volume calculation for non-Euclidean tetrahedra was achieved by Italian mathematician Gaetano Sforza. It came to light during discussion of the author with Jose Maria Montesinos-Amilibia at the conference in El Burgo de Osma (Spain), August 2006. The aim of this lecture is to give a survey of the above mentioned results

  • Prof. Tamas Szamuely

    Alfred Renyi Institute of Mathematics, Hungarian Academy of Sciences
    Thursday, February 23, 2012
    17.15 - CM4

    Galois Theory: Past and Present

    Abstract

    In October 2011 the mathematical world celebrated the 200th anniversary of the birth of Evariste Galois. In today's lecture I shall explain what Galois himself discovered in the theory of algebraic equations and also how his ideas are still with us in present-day mathematics.

  • Prof. Don Zagier

    Max Planck Institute for Mathematics and Collège de France
    Thursday, March 8, 2012
    17.15 - CM5

    Multiple Zeta Values and Their Applications

    Abstract

    TBA

  • Prof. Vera Sos

    Alfred Renyi Institute of Mathematics, Hungarian Academy of Sciences
    Wednesday, March 28, 2012
    17.15 - MA11

    Large graphs and Convergence of Dense Graph Sequences

    Abstract

    Many areas in mathematics, computer science, biology, physics, etc., study properties of very large (deterministic or random) graphs or increasing graph sequences. Very often these graphs are not completely known. The properties we are interested in are mostly different from the questions we investigate in traditional graph theory (for exactly given or defined graphs). The questions that first arise are:
    -when are two large graphs similar, close to each other ,
    -which are the main characteristics of large graphs,
    -which and how local and global properties are related to each other,
    -how large graphs can be approximated by "small" ,"simple" graphs?
    For different classes of graphs (like for dense graphs, for sparse graphs, for hypergraphs etc.) the nature and difficulty of the problems are very different. Here we consider dense graphs. We define the notion of convergence of graph sequences. This, and the method based on it provide to study large graphs and to get e.g. some answers for the questions formulated above. The results and problems are related, e.g., to Szemeredi's Regularity Lemma, to random graphs, to parameter and property testing and also to other mathematical fields like algebra , analysis, probability. We will speak on some joint work with Ch.Borgs, J.Chayes, L.Lovasz, and K.Vesztergombi and also on some results of Lovasz and B.Szegedy, as an outline of a part of a wide ongoing project which developed in the last few years in the center with L.Lovasz. (The talk only assumes a basic familiarity with graph theory, and is essentially self-contained.)

  • Prof. Iain Gordon

    University of Edinburgh
    Thursday, June 14, 2012
    17.15 - CM4

    Title TBA

    Abstract

    TBA

Mathematics Colloquium

The EPFL Mathematics Colloquium features lectures intended for all mathematicians. The Colloquium convenes only a few times each semester, usually Thursdays at 17h15.

For questions or comments, please contact the organiser, Prof. Victor Panaretos.

Bernoulli Lectures

Further to the Mathematics Colloquium, the Bernoulli Centre organises a series of Bernoulli Lectures every semester, featuring mathematicians participating in the semester-long research themes of the Centre.

This semester, the Bernoulli Centre hosts the Programme on Stochastic Analysis and Applications (mathematics contact: Prof. Robert Dalang).

Upcoming Lectures include:

  • Prof. René Carmona, Princeton [15/3/2012, 17:15]
  • Prof. Hans Föllmer, Humbolt [24/5/2012, 17:15]
  • Prof. Etienne Pardoux, Provence [28/6/2012, 17:15]
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    Past Mathematics Colloquia

    Fall 2011

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