Past Colloquia

12 November 2010
16:30
Professor Luis Caffarelli
Abstract
Anomalous ( non local) diffusion processes appear in many subjects: phase transition, fracture dynamics, game theory I will describe some of the issues involved, and in particular, existence and regularity for some non local versions of the p Laplacian, of non variational nature, that appear in non local tug of war.
22 October 2010
16:30
Nicola Fusco
Abstract
<p>The isoperimetric inequality is a fundamental tool in many geometric and analytical issues, beside being the starting point for a great variety of other important inequalities.</p> <p>We shall present some recent results dealing with the quantitative version of this inequality, an old question raised by Bonnesen at the beginning of last century. Applications of the sharp quantitative isoperimetric inequality to other classic inequalities and to eigenvalue problems will be also discussed.</p>
11 June 2010
16:30
Professor Jacob Lurie
Abstract
Let L be a positive definite lattice. There are only finitely many positive definite lattices L' which are isomorphic to L modulo N for every N > 0: in fact, there is a formula for the number of such lattices, called the Siegel mass formula. In this talk, I'll review the Siegel mass formula and how it can be deduced from a conjecture of Weil on volumes of adelic points of algebraic groups. This conjecture was proven for number fields by Kottwitz, building on earlier work of Langlands and Lai. I will conclude by sketching joint work (in progress) with Dennis Gaitsgory, which uses topological ideas to attack Weil's conjecture in the case of function fields.
14 May 2010
16:30
Professor Artur Avila
Abstract
Since the work of Feigenbaum and Coullet-Tresser on universality in the period doubling bifurcation, it is been understood that crucial features of unimodal (one-dimensional) dynamics depend on the behavior of a renormalization (and infinite dimensional) dynamical system. While the initial analysis of renormalization was mostly focused on the proof of existence of hyperbolic fixed points, Sullivan was the first to address more global aspects, starting a program to prove that the renormalization operator has a uniformly hyperbolic (hence chaotic) attractor. Key to this program is the proof of exponential convergence of renormalization along suitable ``deformation classes'' of the complexified dynamical system. Subsequent works of McMullen and Lyubich have addressed many important cases, mostly by showing that some fine geometric characteristics of the complex dynamics imply exponential convergence. We will describe recent work (joint with Lyubich) which moves the focus to the abstract analysis of holomorphic iteration in deformation spaces. It shows that exponential convergence does follow from rougher aspects of the complex dynamics (corresponding to precompactness features of the renormalization dynamics), which enables us to conclude exponential convergence in all cases.
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>

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