Abstract: The derived category of coherent sheaves almost certainly plays a role of fundamental importance in string theory on an algebraic variety. We review the various ideas that have led to this belief and outline the resulting consequences for geometry and physics.

A limit of a suitably-chosen string theory compactified on an elliptic Calabi-Yau threefold is believed to be equivalent to a Yang-Mills field theory with gravity in six dimensions. The Lie group associated to this Yang-Mills theory is encoded in the geometry of the Calabi-Yau space --- the Cartan subalgebra is generated by ruled surfaces and the weights of representations of fields appear as rational curves living inside the threefold. Cancellation of anomalies in the six-dimensional field theory predicts rather peculiar constraints on the configurations of such ruled surfaces and rational curves.

Dynamic singularities are ubiquitous. They arise in mathematical models of phenomena as grand as star formation or as familiar as the breakup of a thread of honey as it is being added to tea. Drop breakup allows one to study dynamics close to a singularity in a simple context which is also accessible to experiments. Recent works have revealed that a viscous liquid drop close to breakup looks self-similar---the drop profile looks the same if the length scales are rescaled appropriately. A new numerical strategy is developed to capture the drop breakup dynamics and show good agreement with experimental measurements. Surprisingly, the presence of even small amounts of viscous dissipation in the surrounding can dramatically alter the self-similar profile. In particular, when no exterior viscous dissipation is present, the thread profile is symmetric about the point of pinch-off. When small amounts of exterior viscous dissipation are present, the thread profile becomes severely asymmetric. An understanding of the final breakup process is crucial in elucidating the mechanisms underlying the formation of satellite drops, an issue relevant to the development of ink-jet printing technologies and emulsification processes.

Strominger, Yau and Zaslow have proposed that Calabi-Yau mirror partners should be families of special Lagrangian tori over the same base which are dual to each other. I will exhibit some striking evidence for this proposal: pairs of Calabi-Yau orbifolds satisfying the requirements of SYZ for which the expected equality of stringy Hodge numbers can be completely verified. This is in a sense the first enumerative evidence supporting SYZ. The construction I will present is the first of several suggesting a general principle: for any construction using a compact Lie group to construct a Calabi-Yau, mirror partners arise from applying the construction to Langlands dual groups.

Stochastic PDEs have become important models for many phenomenon. Nonetheless, many fundamental questions about their behavior remain poorly understood. Often such SPDE contain different processes active at different scales. Not only does such structure give rise to beautiful mathematics and phenomenon, but I submit that it also contains the key to answering many seemingly unrelated questions. Questions such as ergodicity and Mixing. Given a stochastically forced dissipative PDE such as the 2D Navier Stokes equations, the Ginzburg-Landau equations, or a reaction diffusion equation; is the system Ergodic ? If so, at what rate does the system equilibrate ? Is the convergence qualitatively different at different physical scales ? Answers to these an similar questions are basic assumptions of many physical theories such as theories of turbulence. I will try both to convince you why these questions are interesting and explain how to address them. The analysis will suggest strategies to explore other properties of these SPDEs as well as numerical methods. In particular, I will show that the stochastically forced 2D Navier Stokes equations converges exponentially to a unique invariant measure. I will discuss under what minimal conditions one should expect ergodic behavior. The central ideas will be illustrated with a simple model systems. Along the way I will explain how to exploit the different scales in the problem and how to overcome the fact that the problem is an extremely degenerate diffusion on an infinite dimensional function space. The analysis points to a class of operators in between STRICTLY ELLIPTIC and HYPOELLIPTIC operators which I call EFFECTIVELY ELLIPTIC. The techniques use a representation of the process on a finite dimensional space with memory. I will also touch on a novel coupling construction used to prove exponential convergence to equilibrium.

One of the most basic objects in probability theory is the simple random walk, which one can think of as a random map from Z to Z mapping adjacent points to adjacent points. A similar theory for random maps from Z² to Z had until recently remained elusive to mathematicians, despite being known (non-rigorously) to physicists. In this talk we discuss some natural families of random maps from Z² to Z. We can explicitly compute both the local and the large-scale behavior of these maps. In particular we construct a "scaling limit" for these maps, in a similar sense in which Brownian motion is a scaling limit for the simple random walk. The results are in accord with physics.

A new method will be reviewed for analyzing boundary value problems for linear and integrable nonlinear PDEs. This method involves the following:

1. Given a PDE for q(x), x in Rⁿ, construct a closed (n-1)-differential form W(x,k), k in C^n-1.
2. Given a convex domain D contained in Rⁿ, perform the spectral analysis of W
3. Given appropriate boundary conditions, analyze the global relation
   

For linear PDEs, the relation of this method with the Ehrenpreis-Palamodov principle, as well as the relations with applied techniques such as the Weiner-Hopf technique will be discussed.

Associated with a dynamical system, which evolves single initial conditions, the Frobenius-Perron operator evolves ensemble densities of initial conditions. Including a brief tutorial on the topic, we will present our new applications of this global and statistical point of view:

1. The inverse Frobenius-Perron problem (IFPP) is a global open-loop strategy to control chaos by constructing a "nearby" dynamical system with desirable invariant density. We reduce the question of stabilizing an arbitrary invariant density to the question of a hyperplane intersecting a unit hyperbox; several controllability theorems follow. Applications will be described.
2. Well-known models have been found to exhibit new and interesting dynamics under the addition of stochastic perturbations. Using the Frobenius-Perron operator for stochastic dynamical systems, we develop new tools designed to predict the effects of noise and to pinpoint stochastic transport regions in phase space. As an example, we study a model from population dynamics for which chaos-like behavior can be induced, as the standard deviation of the noise is increased. We identify how stochastic perturbations destabilize two attracting orbits, effectively completing a heteroclinic orbit, to create chaos-like behavior. Other physical applications will also be discussed.

Abstract: One of the major themes of modern differential geometry is the relationship between the curvature and topology of Riemannian manifolds. In this lecture, I will describe some links between curvature and SMOOTH topological invariants --- i.e. invariants which can distinguish between different smooth structures on a given topological manifold. The specific invariants I will discuss are the Seiberg-Witten invariants of 4-manifolds, and I will describe the impact these have on the existence problem for Einstein metrics and some related Riemannian variational problems.

The combinatorial structure of a d-dimensional simple convex polytope -- as given, for example, by the set of the (d-1)-regular subgraphs of facets -- can be reconstructed from its abstract graph [Blind & Mani 1988, Kalai 1988]. However, no polynomial/efficient algorithm is known for this task, although a polynomially checkable certificate for the correct reconstruction exists [Kaibel & Koerner 2000]. A much stronger certificate would be given by the following characterization of the facet subgraphs, conjectured by M. Perles: ``The facet subgraphs of a simple d-polytope are exactly all the (d-1)-regular, connected, induced, non-separating subgraphs'' [Perles 1970,1984]. We give examples for the validity of Perles conjecture: In particular, it holds for the duals of cyclic polytopes, and for the duals of stacked polytopes. On the other hand, we observe that for any counterexample, the boundary of the (simplicial) dual polytope P^* contains a 2-complex without a free edge, and without 2-dimensional homology. Examples of such complexes are known; we use a simple modification of ``Bing's house'' (two walls removed) to construct explicit 4-dimensional counterexamples to Perles' conjecture.

The combinatorial laplacian is a method for computing the rational homology of an algebraic or simplicial complex. Although the method is quite old, it has recently been applied with success to a number of problems in algebraic combinatorics. We will survey the method and describe some of these recent applications.

This lecture, which assumes only a basic abstract algebra background, provides an introduction to the Galois theory of linear ordinary differential equations. At present, no algorithm exists for computing the Galois group of an arbitrary equation. We will give a survey of recent work on some special classes of equations, including results to be included in the speaker's PhD dissertation.

The p-power torsion of a family of elliptic curves in characteristic p defines a local system on the base space. Igusa shows that the associated monodromy representation is maximal, in the sense that the image of the fundamental group is the full automorphism group of a fiber. I will discuss variants of this situation which can make the group in question bigger (higher-dimensional abelian varieties), smaller (with large endomorphism rings), or simply different (and no physical p-torsion). I hope to interpret these results as statements about certain differential equations.

A celebrated theorem of Carleson shows that the Fourier series of an L^2 function converges pointwise almost everywhere. At the heart of this work lies an L^2 estimate for a particular type of maximal singular integral operator, which has since become known as a Carleson operator. In the past 40 years, a number of important results have been proved for generalizations of the original Carleson operator. In this talk we will introduce the Carleson operator and survey its generalizations, and then describe new joint work with Po Lam Yung on Carleson operators with a certain type of polynomial phase that also incorporate the behavior of Radon transforms.