Bayesian melding is difficult when the model takes a long time to run. I will describe how one can use Bayesian model averaging to make calibrated inference using only a few model runs, and apply it to probabilistic weather forecasting.
 

I will talk about a family of conformally invariant differential equations and the curvature invariants they define. The invariants can be used to understand the conformal structure in low dimensions.
 

Most of us as children saw those paper or cardboard cutouts, which we could call "foldouts," whose edges glue to form (boundaries of) 3-dimensional convex polyhedra. Just how did anyone figure out how to make them? Given a 3-dimensional convex polyhedron, does there always exist a foldout in the plane? What about higher dimensions? These questions have surprising answers, depending on the precise meaning of "foldout." One method is to treat boundaries of polyhedra like Riemannian manifolds. Algorithmic concerns then raise
fundamental issues of computational complexity for the combinatorics of geodesics on polyhedra.  This talk is on joint work with Igor Pak.
 

Suppose you are given an algebraic curve C defined, let us say, as the locus of zeroes of a polynomial f (x,y) in two variables with rational coefficients. Suppose you are told that C has at least one rational point, i.e., there is a pair of rational numbers (a,b) such that f (a,b) = 0. How likely is it that C will have infinitely many rational points? We present new data and old conjectures about this question. This is joint work with Barry Mazur and Mark Watkins.
 

A household is modelled as a set of individuals with different utilities, sharing a common budget constraint, and trying to reach an efficient allocation of resources. Two basic question then arise. Does the model have testable consequences ? Can one identify the parameters from observable data? This has been the subject of an ongoing work program with P.A. Chiappori, and I will present some of our results. References can be found at the URL: http://www.pims.math.ca/~ekeland/
 

Applications in computer vision and medical imaging have driven the study of geometry on various infinite dimensional manifolds: esp. the group of diffeomorphisms, the "Chow" manifolds of all submanifolds in a fixed ambient space. What are geodesics like in these spaces and what are the sectional curvatures in the most natural metrics? Although this program is work in progress, some clear themes have emerged. There is a dichotomy between weak metrics producing positively curved spaces which wrap up infinitely tightly in high-frequency dimensions and stronger metrics which negatively curve the space. In the case of the manifold of simple closed plane curves, Teichmuller theory gives one of nicest global descriptions. Work in this area (jointly and separately by Peter Michor, Michael Miller, Eitan Sharon and others) will be outlined and illustrated.
 

This is a survey talk. We will discuss a number of results connected by a common theme: we need more "coordinate-type" functions than the dimension. Such "excessive" coordinate systems can be thought as embeddings in some higher-dimensional vector space equipped with an additional structure.
 

In many cases two-dimensional inverse problems have turned out to be more difficult then their higher dimensional counterparts. This is specially the case with inverse boundary value problems and with inverse scattering problems with fixed energy. In the talk we recall some open problems in the field as well as demonstrate some recent development. In particular, we will discuss the solution of the two-dimensional Calderón problem, the detection of singularities in Schrödinger scattering and finally consider a 2D inverse problem for random Markov potentials. The research is done jointly with K. Astala, M. Lassas, P.Ola, V. Serov and E. Saksman.
 

We shall describe the concept and the calculus of anti-selfdual (ASD) Lagrangians and their derived vector fields which seem inherent to many partial differential equations and evolutionary systems. They are natural extensions of gradients of convex functions--hence of self-adjoint positive operators--which usually drive dissipative systems, but also provide representations for the superposition of such gradients with skew-symmetric operators which normally generate unitary flows. They yield variational formulations and resolutions for large classes of non-potential boundary value problems and initial-value parabolic equations. Solutions are minima of newly devised energy functionals, however, and just like the self (and antiself) dual equations of quantum field theory (e.g. Yang-Mills) the equations associated to such minima are not derived from the fact they are critical points of the functional I, but because they are also zeroes of suitably derived Lagrangians. The approach has many advantages: it solves variationally many equations and systems that cannot be obtained as Euler-Lagrange equations of action functionals, since they can involve non self-adjoint or other non-potential operators; it also associates variational principles to variational inequalities, and to various dissipative initial-value first order parabolic problems. These equations can therefore be analyzed with the full range of methods--computational or not--that are available for variational settings. Most remarkable are the permanence properties that ASD Lagrangians possess making their calculus relatively manageable and their domain of applications quite broad.

Much of this ongoing work has been done with your own Leo Tzou and with a UBC postdoc Abbas Moameni.
 

This is an introduction to joint work with Julia Pevtsova, partially in collaboration with Jon Carlson and Andrei Suslin. We are interested in actions of a group G (or similar structure) on a vector space over a field of characteristic p > 0. I shall begin by discussing the fully understood
example of G = Z/p. I shall then discuss what we do and do not know in the case G = Z/p x Z/p. Finally, I will explain how the formalism extends to a very general context which presents many challenges for the future.
 

Mirror symmetry is a mysterious duality between pairs of manifolds (or orbifolds) which are Calabi-Yau, that is, compact Kahler with trivial canonical bundle. It is supposed to interchange geometric structures on these orbifolds (such as their spaces of deformations) in a complicated fashion. A geometric interpretation of mirror symmetry has been proposed by Strominger, Yau and Zaslow. This appears to link mirror symmetry to several of the other great dualities of mathematics, such as Langlands duality and the Fourier transform. We will survey the subject, then discuss certain very simple examples, namely quotients of tori by finite group actions, in which the conditions of SYZ are easily satisfied and the desired identity of Hodge numbers holds.
 

I will describe some recent work on the understanding of the space of constant mean curvature surfaces (and hypersurfaces) in Riemannian manifolds. In particular, I will emphasize the role of minimal submanifolds in the compactification of this space. I will also mention the similarities between this study and the study of some semilinear elliptic equations or some dynamical systems.
 

I will explain how techniques from quantum N-body scattering shed light on scattering theory on (arbitrary rank) symmetric spaces of non-compact type. In particular, this connection explains which features of symmetric spaces are "typical" in a larger context, and which are due to the special algebraic features.
 

I will present a celebrated theorem of Kostant and Rallis and its implications in invariant theory. Next, I will describe a program to generalize the result, and report some success in this direction for the classical instances of the problem.

Specifically, I will present joint work with Roger Howe and Eng Chye Tan. Our efforts exploit classical branching rules to obtain a formula for the stable graded multiplicity in a space of harmonic polynomials associated to a symmetric space. The results are combinatorial in nature as they involve the ubiquitous Littlewood-Richardson coefficients.
 

Networks are all around us, from the World Wide Web to the gene networks inside our cells. In many cases, these networks change over time, as nodes and edges are added, deleted, or rewired. The math problems raised by such evolving networks are of interest not just to graph theorists and computer scientists, but also to people working in dynamical systems and statistical physics.

In this talk, I'll discuss what may be the simplest model of a randomly growing network. At each time step, a new node is added; then, with probability δ, two nodes are chosen uniformly at random and joined by an undirected edge. This process is repeated for t time steps. In the limit of large t, the resulting graph displays surprisingly rich characteristics. In particular, a giant connected component emerges in an infinite-order phase transition at δ = 1/8.

No knowledge of graph theory is needed to follow this talk; the main ideas come from elementary probability and differential equations.

This is joint work with Duncan Callaway, John Hopcroft, Jon Kleinberg, and Mark Newman.