Past Colloquia

26 February 2010
16:30
Professor Pierre Cartier (IHES)
Abstract
We shall report on the use of algebraic geometry for the calculation of Feynman amplitudes (work of Bloch, Brown, Esnault and Kreimer). Or how to combine Grothendieck's motives with high energy physics in an unexpected way, radically distinct from string theory.
22 January 2010
16:30
Professor Don Zagier
Abstract
Many problems from combinatorics, number theory, quantum field theory and topology lead to power series of a special kind called q-hypergeometric series. Sometimes, like in the famous Rogers-Ramanujan identities, these q-series turn out to be modular functions or modular forms. A beautiful conjecture of W. Nahm, inspired by quantum theory, relates this phenomenon to algebraic K-theory. In a different direction, quantum invariants of knots and 3-manifolds also sometimes seem to have modular or near-modular properties, leading to new objects called "quantum modular forms".
27 November 2009
16:30
Professor Alessio Corti
Abstract
A key birational invariant of a compact complex manifold is its "canonical ring." The ring of modular forms in one or more variables is an example of a canonical ring. Recent developments in higher dimensional algebraic geometry imply that the canonical ring is always finitely generated:this is a long-awaited major foundational result in algebraic geometry. In this talk I define all the terms and discuss the result, some applications, and a recent remarkable direct proof by Lazic.
8 May 2009
16:30
Professor Steven N. Evans
Abstract
<span lang="EN-GB"> <p> A common question in evolutionary biology is whether evolutionary processes leave some sort of signature in the shape of the phylogenetic tree of a collection of present day species. </p> <p> Similarly, computer scientists wonder if the current structure of a network that has grown over time reveals something about the dynamics of that growth. </p> <p> Motivated by such questions, it is natural to seek to construct``statistics'' that somehow summarise the shape of trees and more general graphs, and to determine the behaviour of these quantities when the graphs are generated by specific mechanisms. </p> <p> The eigenvalues of the adjacency and Laplacian matrices of a graph are obvious candidates for such descriptors. </p> <p> I will discuss how relatively simple techniques from linear algebra and probability may be used to understand the eigenvalues of a very broad class of large random trees. These methods differ from those that have been used thusfar to study other classes of large random matrices such as those appearing in compact Lie groups, operator algebras, physics, number theory, and communications engineering. </p> <p> This is joint work with Shankar Bhamidi (U. of British Columbia) and Arnab Sen (U.C. Berkeley). </p> <p> &nbsp; </p> </span>
6 March 2009
16:30
Professor Bao Chau Ngo
Abstract
Coefficients of the characteristic polynomial are generators of the ring of polynomial functions on the space of matrices which are invariant under the conjugation. This was generalized by Chevalley to general reductive groups. By looking closely on the centralisers, one is lead to a very natural 2-category attached to Chevalley characteristic morphism. This abstract, but yet elementary, construction helps one to understand the symmetries of the fibres of the Hitchin fibration, as well as those of affine Springer fibers.<br /> <br /> We will also explain how these groups of symmetries are related to the notion of endoscopic groups, which was introduced by Langlands in his stabilisation of the trace formula. We will also briefly explain how the symmetry groups help one to acquire a rather good understanding of the cohomology of the Hitchin fibration and eventually the proof of the fundamental lemma in Langlands' program.
6 February 2009
16:30
Professor Ivar Ekeland
Abstract
In classical economic theory, one discounts future gains or losses at a constant rate: one pound in t years is worth exp(-rt) pounds today. There are now very good reasons to consider non-constant discount rates. This gives rise to a problem of time-inconsistency: a policy which is optimal today will no longer be optimal tomorrow. The concept of optimality then no longer is useful. We introduce instead a concept of equilibrium solution, and characterize it by a non-local variant of the Hamilton-Jacobi equation. We then solve the classical Ramsey model of endogenous growth in this framework, using the central manifold theorem<br /> <br />
7 November 2008
16:30
Abstract
A random environment (in Z^d) is a collection of (random) transition probabilities, indexed by sites. Perform now a random walk using these transitions. This model is easy to describe, yet presents significant challenges to analysis. In particular, even elementary questions concerning long term behavior, such as the existence of a law of large numbers, are open. I will review in this talk the model, its history, and recent advance, focusing on examples of unexpected behavior.
6 June 2008
16:30
Prof. Michael Harris
Abstract
Let E be an elliptic curve defined by a cubic equation with rational coefficients. <br />The Sato-Tate Conjecture is a statistical assertion about the variation of the number of points of E over finite fields. I review some of the main steps in my proof of this conjecture with Clozel, Shepherd-Barron, and Taylor, in the case when E has non-integral j-invariant. Emphasis will be placed on the steps involving moduli spaces of certain Calabi-Yau hypersurfaces with level structure.<br /><br />If one admits a version of the stable trace formula that should soon be available, the same techniques imply that, when E and E' are two elliptic curves that are not isogenous, then the numbers of their points over finite fields are statistically independent. For reasons that have everything to do with the current limits to our understanding of the Langlands program, the analogous conjectures for three or more non-isogenous elliptic curves are entirely out of reach.<br /><br />

Pages