Colloquia (past)
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Mon, 29/04 16:30 |
George Papanicolaou (Stanford University) |
Colloquia  |
L2 |
| The quantification and management of risk in financial marketsis at the center of modern financial mathematics. But until recently, riskassessment models did not consider the effects of inter-connectedness offinancial agents and the way risk diversification impacts the stability ofmarkets. I will give an introduction to these problems and discuss theimplications of some mathematical models for dealing with them. | |||
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Fri, 22/02 16:30 |
Professor Anand Pillay (University of Leeds) |
Colloquia |
L1 |
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There are many recent points of contact of model theory and other |
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Fri, 09/11/2012 16:30 |
Randall J. LeVeque (Applied Mathematics Department University of Washington) |
Colloquia |
L2 |
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Many geophysical flows over topography can be modeled by two-dimensional |
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Fri, 08/06/2012 16:30 |
Bruce Kleiner (NYU) |
Colloquia |
L2 |
| A map betweem metric spaces is a bilipschitz homeomorphism if it is Lipschitz and has a Lipschitz inverse; a map is a bilipschitz embedding if it is a bilipschitz homeomorphism onto its image. Given metric spaces X and Y, one may ask if there is a bilipschitz embedding X—>Y, and if so, one may try to find an embedding with minimal distortion, or at least estimate the best bilipschitz constant. Such bilipschitz embedding problems arise in various areas of mathematics, including geometric group theory, Banach space geometry, and geometric analysis; in the last 10 years they have also attracted a lot of attention in theoretical computer science. The lecture will be a survey bilipschitz embedding in Banach spaces from the viewpoint of geometric analysis. | |||
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Fri, 04/05/2012 16:30 |
Professor Steven Strogatz (Cornell University) |
Colloquia |
L2 |
| Consider a fully-connected social network of people, companies,or countries, modeled as an undirected complete graph with real numbers onits edges. Positive edges link friends; negative edges link enemies.I'll discuss two simple models of how the edge weights of such networksmight evolve over time, as they seek a balanced state in which "the enemy ofmy enemy is my friend." The mathematical techniques involve elementaryideas from linear algebra, random graphs, statistical physics, anddifferential equations. Some motivating examples from internationalrelations and social psychology will also be discussed. This is joint workwith Seth Marvel, Jon Kleinberg, and Bobby Kleinberg. | |||
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Fri, 02/03/2012 16:30 |
Stephan Luckhaus |
Colloquia |
L2 |
| What is a phase transition? The first thing that comes to mind is boiling and freezing of water. The material clearly changes its behaviour without any chemical reaction. One way to arrive at a mathematical model is to associate different material behavior, ie., constitutive laws, to different phases. This is a continuum physics viewpoint, and when a law for the switching between phases is specified, we arrive at pde problems. The oldest paper on such a problem by Clapeyron and Lame is nearly 200 years old; it is basically on what has later been called the Stefan problem for the heat equation. The law for switching is given e.g. by the melting temperature. This can be taken to be a phenomenological law or thermodynamically justified as an equilibrium condition. The theory does not explain delayed switching (undercooling) and it does not give insight in structural differences between the phases. To some extent the first can be explained with the help of a free energy associated with the interface between different phases. This was proposed by Gibbs, is relevant on small space scales, and leads to mean curvature equations for the interface – the so-called Gibbs Thompson condition. The equations do not by themselves lead to a unique evolution. Indeed to close the resulting pde’s with a reasonable switching or nucleation law is an open problem. Based on atomistic concepts, making use of surface energy in a purely phenomenological way, Becker and Döring developed a model for nucleation as a kinetic theory for size distributions of nuclei. The internal structure of each phase is still not considered in this ansatz. An easier problem concerns solid-solid phase transitions. The theory is furthest developped in the context of equilibrium statistical mechanics on lattices, starting with the Ising model for ferromagnets. In this context phases correspond to (extremal) equilibrium Gibbs measures in infinite volume. Interfacial free energy appears as a finite volume correction to free energy. The drawback is that the theory is still basically equilibrium and isothermal. There is no satisfactory theory of metastable states and of local kinetic energy in this framework. | |||
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Fri, 10/02/2012 16:30 |
Professor Karen Vogtmann (Cornell University) |
Colloquia |
L2 |
| Free groups, free abelian groups and fundamental groups of closed orientable surfaces are the most basic and well-understood examples of infinite discrete groups. The automorphism groups of these groups, in contrast, are some of the most complex and intriguing groups in all of mathematics. I will give some general comments about geometric group theory and then describe the basic geometric object, called Outer space, associated to automorphism groups of free groups. This Colloquium talk is the first of a series of three lectures given by Professor Vogtmann, who is the European Mathematical Society Lecturer. In this series of three lectures, she will discuss groups of automorphisms of free groups, while drawing analogies with the general linear group over the integers and surface mapping class groups. She will explain modern techniques for studying automorphism groups of free groups, which include a mixture of topological, algebraic and geometric methods. | |||
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Fri, 04/11/2011 16:30 |
Professor John W.M Bush (Massachusetts Institute of Technology) |
Colloquia |
L2 |
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Yves Couder and co-workers have recently reported the results of a startling series of experiments in which droplets bouncing on a fluid surface exhibit several dynamical features previously thought to be peculiar to the microscopic realm. In an attempt to develop a connection between the fluid and quantum systems, we explore the Madelung transformation, whereby Schrodinger's equation is recast in a hydrodynamic form. New experiments are presented, and indicate the potential value of this hydrodynamic approach to both visualizing and understanding quantum mechanics.
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Fri, 24/06/2011 16:30 |
Professor Sir Vaughan Jones (University of California) |
Colloquia |
L2 |
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Voiculescu showed how the large N limit of the expected value of the trace of a word on n independent hermitian NxN matrices gives a well known von Neumann algebra. In joint work with Guionnet and Shlyakhtenko it was shown that this idea makes sense in the context of very general planar algebras where one works directly in the large N limit. This allowed us to define matrix models with a non-integral number of random matrices. I will present this work and some of the subsequent work, together with future hopes for the theory.
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Fri, 03/06/2011 16:30 |
Prof Graeme Segal (Oxford) |
Colloquia |
L2 |
| Graeme Segal shall describe some of Dan Quillen’s work, focusing on his amazingly productive period around 1970, when he not only invented algebraic K-theory in the form we know it today, but also opened up several other lines of research which are still in the front line of mathematical activity. The aim of the talk will be to give an idea of some of the mathematical influences which shaped him, of his mathematical perspective, and also of his style and his way of approaching mathematical problems. | |||
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Fri, 04/03/2011 16:30 |
Prof Arkani-Hamed |
Colloquia |
L2 |
| "Scattering amplitudes in gauge theories and gravity have extraordinary properties that are completely invisible in the textbook formulation of quantum field theory using Feynman diagrams. In this usual approach, space-time locality and quantum-mechanical unitarity are made manifest at the cost of introducing huge gauge redundancies in our description of physics. As a consequence, apart from the very simplest processes, Feynman diagram calculations are enormously complicated, while the final results turn out to be amazingly simple, exhibiting hidden infinite-dimensional symmetries. This strongly suggests the existence of a new formulation of quantum field theory where locality and unitarity are derived concepts, while other physical principles are made more manifest. The past few years have seen rapid advances towards uncovering this new picture, especially for the maximally supersymmetric gauge theory in four dimensions. These developments have interwoven and exposed connections between a remarkable collection of ideas from string theory, twistor theory and integrable systems, as well as a number of new mathematical structures in algebraic geometry. In this talk I will review the current state of this subject and describe a number of ongoing directions of research." | |||
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Fri, 28/01/2011 16:30 |
Professor Camillo De Lellis. |
Colloquia |
L2 |
| There are nontrivial solutions of the incompressible Euler equations which are compactly supported in space and time. If they were to model the motion of a real fluid, we would see it suddenly start moving after staying at rest for a while, without any action by an external force. There are C1 isometric embeddings of a fixed flat rectangle in arbitrarily small balls of the three dimensional space. You should therefore be able to put a fairly large piece of paper in a pocket of your jacket without folding it or crumpling it. I will discuss the corresponding mathematical theorems, point out some surprising relations and give evidences that, maybe, they are not merely a mathematical game. | |||
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Fri, 12/11/2010 16:30 |
Professor Luis Caffarelli |
Colloquia |
L2 |
| 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. | |||
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Fri, 22/10/2010 16:30 |
Nicola Fusco |
Colloquia |
L2 |
| 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. 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. | |||
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Fri, 11/06/2010 16:30 |
Professor Jacob Lurie (Harvard University) |
Colloquia |
L2 |
| 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. | |||
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Fri, 14/05/2010 16:30 |
Professor Artur Avila (IMPA) |
Colloquia |
L2 |
| 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. | |||
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Fri, 26/02/2010 16:30 |
Professor Pierre Cartier (IHES) (IHES) |
Colloquia |
L2 |
| 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. | |||
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Fri, 22/01/2010 16:30 |
Professor Don Zagier (Max-Planck Institut) |
Colloquia |
L2 |
| 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". | |||
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Fri, 27/11/2009 16:30 |
Professor Alessio Corti (London) |
Colloquia |
L2 |
| 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. | |||
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Fri, 23/10/2009 16:30 |
Professor Norm Zabusky (Weizmann Institute) |
Colloquia |
L2 |
| An overview of the early history of the soliton (1960-1970) and equipartition in nonlinear 1D lattices : From Fermi-Pasta-Ulam to Korteweg de Vries, to Nonlinear Schrodinger*…., and recent developments . | |||
