Friday, January 19

13:45: Tea and Coffee

14:15: Opening Addresses: Felix Otto; Matthias Winiger; Armin B. Cremers; Peter Finger; Wolfgang Lück; Friedrich Hirzebruch

15:15: Tea and Coffee

16:00: Michael Hopkins (Harvard): Classical and quantum invariants in algebraic topology

Abstract: Over the past couple of decades, the mathematical demands of theoretical physics have engendered a re-evaluation of many fundamental mathematical questions. In this talk I will describe the issues that led, more than one hundred years ago, to the apparatus of "algebraic topology," in which sophisticated, qualitative measurements of shape are made. The modern take on these old questions is leading to a new conception of the most basic aspects of geometry. I will describe this new geometry, and some examples that, in retrospect, are representative of its new features.

17:00: Tea and Coffee

17:30: László Lovász (Budapest): Very large graphs

Abstract: There are many areas of science where huge networks (graphs) are central objects of study: the internet, the brain, various social networks, VLSI, statistical physics, etc. To study these graphs, new paradigms are needed: What are meaningful questions to ask? When are two huge graphs "similar"? How to scale down these graphs without changing their fundamental structure? How to generate random examples with the desired properties? A reasonably complete answer can be given in the case when the huge graphs are dense (in the more diffcult case of sparse graphs there are only partial results). Among others, a "limit" of growing graph sequences can be defined in the form of a 2-variable measuable function, which is often nice and smooth, and whose analytic properties reflect important graph-theoretic properties of the members of the sequence. For example, Szemeredi's Regularity Lemma is equivalent to a compactness result in this setting. The talk will survey joint work with Christian Borgs, Jennifer Chayes, Balazs Szegedy, Vera Sos and Katalin Vesztergombi.

18:30: Food and Music

Saturday, January 20

10:00: Elliott Lieb (Princeton): The Stability of Matter and Quantum Electrodynamics

Abstract: Ordinary matter is held together with electromagnetic forces, and the dynamical laws governing the constituents (electrons and nuclei) are those of quantum mechanics, i.e., Schroedinger's equation. These laws, found a century ago, were able to account for the fact that electrons do not fall into the nuclei and thus atoms are quite robust. It was only in 1967 that Dyson and Lenard were able to show that matter in bulk was also stable and that two stones had a volume twice that of one stone. Simple as this may sound, the conclusion is not at all obvious and hangs by a thread – namely Pauli's "exclusion principle" (which states that two electrons cannot be in the same state or that Schroedinger's equation has to be restricted to antisymmetric functions). In the ensuing three decades much was accomplished to clarify, simplify and extend this result. We now understand that matter can, indeed, be unstable when relativistic effects and magnetic fields are taken into account – unless the electron's charge is small enough (which it is, fortunately). These delicate and non-intuitive conclusions will be summarized. The requisite mathematical apparatus needed for these results is itself interesting. Finally, we can now hope to begin an analysis of the half-century old question about the ultimate theory of ordinary matter, called quantum electrodynamics (QED). This is an experimentally successful theory, but one without a decent mathematical foundation. Some recent, very preliminary steps to resolve the problems of QED will be presented.

11:00: Tea and Coffee

11:30: Andreu Mas-Colell (Barcelona): Adaptive Play in Game

Abstract: Adaptive behaviour in repeated interacting social systems modelled as games will be considered. In particular, fictitious play and regret matching will be presented as instances of adaptive play. Convergence results of sample play to the static equilibrium notion of correlated equilibrium will be presented. The role of mathematical approachability theorems à la Blackwell will be highlighted.

12:30: Lunch Break

14:00: Wendelin Werner (Paris): Macroscopic randomness

Abstract: Probability theory and statistical physics are concerned with the understanding of big systems that are built out of a large number of random microscopic inputs. I will discuss some examples in order to illustrate the different types of possible behaviours of such large systems, focusing in particular on two-dimensional models, that turn out to be closely related to various branches of mathematics.

15:00: Tea and Coffee

15:30: Douglas Arnold (Minneapolis): The geometrical basis of numerical stability

Abstract: The accuracy of a numerical solution to a partial differential equation depends on the consistency and stability of the discretization method. While consistency is usually elementary to establish, stability of numerical methods can be subtle, and for some key PDE problems the development of stable methods is extremely challenging. The finite element exterior calculus is a new theoretical approach to the design and understanding of discretizations for a variety of systems of partial differential equations. This approach obtains stability by developing discretizations which are compatible with the geometrical and topological structures which underlie well-posedness of the PDE problem being solved. In this talk we will illustrate some of the challenges surrounding the development of stable algorithms, and then introduce the basic structures and constructions of finite element exterior calculus, and discuss some of its applications.