On a compact 4-manifold, every self-dual connection and every anti-self-dual connection minimizes the Yang-Mills energy. In this talk, I will answer the converse question for compact homogeneous 4-manifolds. I will also survey related stability results in other dimensions.

Let $\Omega$ and $\tilde{\Omega}$ be domains in the hyperbolic plane with smooth boundary. Assume that both domains are uniformly convex, and have the same area. We show that there exists an area-preserving, orientation-preserving diffeomorphism $f: \Omega \to \tilde{\Omega}$ such that the graph of $f$ is a minimal surface in $\mathbb{H}^2 \times \mathbb{H}^2$. Moreover, we show that the set of all such diffeomorphisms is parametrized by the circle.

Our numerical computation group has studied several problems in computational finance over the past decade. One that we've looked at recently is the pricing of "collateralized debt obligations" (CDOs). The market for CDOs has grown rapidly to over US$1 trillion annually in 2006, since the appearance of JP Morgan's Bistro deal, the first synthetic CDO, in December 1997. Much of the turmoil in the financial markets recently has been due to such credit derivatives. As this suggests, there are still many open problems associated with the pricing and hedging of these complex financial instruments. I will talk briefly about some work that we have done recently in this area.

We consider idealized stochastic models for a network of pulse-coupled oscillators where there is randomness both in input and in network architecture. We describe the various types of dynamics which arise in this system, analyze scalings which arise in the infinite-network limit, and study the various "finite-size" effects as perturbations of these limits. Most notably, the networks we consider can simultaneously support both synchronous and asynchronous modes of behavior and will switch stochastically between these modes due to "rare events". We also relate the analysis of certain scaling limits of this network to classical graph-theoretical results involving the size of components in the Erdos-Renyi random graph. This work is joint with Charles Peskin and Joel Spencer.

A Calabi-Yau orbifold is locally modeled on C^n/G with G a finite subgroup of SU(n). If the singularity is isolated, then the crepant resolution (if it exists) is an ALE manifold, for which index-type results are well known. However, most of the time the singularity is not isolated, and for the corresponding crepant resolution there is no index theorem so far. In this talk, I present the first step towards obtaining such a result: I will introduce the class of iterated cone-edge singular manifolds and the corresponding quasi-asymptotically conical spaces (of which orbifolds and their resolutions of singularities are examples), and build-up the general set-up for studying Fredholm properties of geometrical elliptic operators on these spaces. This is joint work with Rafe Mazzeo.

A Zoll metric is a metric whose geodesics are all circles of equal length. In this talk, we will first review the definition of the twistor correspondence of LeBrun and Mason for Zoll metrics on the sphere $S^{2}$. It associates to a Zoll metric on $S^{2}$ a family of holomorphic disks in $CP_{2}$ with boundary in a totally real submanifold $P\subset CP_{2}$. For a fixed $P\subset CP_{2}$, we will indicate how one can show that such a family is unique whenever it exists, implying that the twistor correspondence of LeBrun and Mason is in some sense injective. One of the key ingredients in the proof will be the blow-up and blow-down constructions in the sense of Melrose.

A bihermitian structure is a Riemannian metric compatible with two distinct orthogonal complex structures. In the mathematical literature this notion appeared in 90's in the study of the curvature of conformal 4-manifolds. However, bihermitian metrics were already studied in the physics literature in the 80's, as a building bloc of what Gates, Hull and Rocek call `the target space for a (2,2) super-symmetric sigma model'. There has been a great deal of interest in bihermitian geometry more recently, motivated by its link with the notion of generalized Kaehler geometry, introduced by Gualtieri and Hitchin. In this talk I will explain some main features of 4-dimensional bihermitian manifolds, as developed in the 90's, and report on recent classification results that I obtained with M. Gualtieri and G. Dloussky.

The simplest model of vortex filament motion in an ideal fluid leads to an integrable nonlinear evolution equation, known as the localized induction approximation or the vortex filament flow, closely related to the cubic focusing nonlinear Schroedinger equation. For closed finite-gap solutions of this flow, certain geometric and topological features of the evolving curves appear to be correlated with the algebro-geometric data used to construct them. In this talk, I will briefly discuss this construction, and some low-genus examples (in particular, Kirchhoff elastic rod centerlines) where this correlation is well understood. I will mainly discuss recent joint work with Annalisa Calini, describing how to generate a family of closed finite-gap solutions of increasingly higher genus via a sequence of deformations of the multiply covered circle. We prove that every step in this sequence corresponds to constructing a cable on previous filament; moreover, the cable's knot type (which is invariant under the evolution) can be read off from the deformation sequence.

The first step in finding a spacetime solution to the Einstein gravitational field equations via the inital value formulation is to construct initial data which satisfy the Einstein constraint equations. There are three ways of carrying out this construction which have been found to be useful: the conformal and conformal thin sandwich methods, the gluing techniques, and the quasi-spherical approaches. We describe each of these, we discuss their advantages and disadvantages, we outline some of their recent successful applications, and we present some of the outstanding questions remaining to be solved from each of these perspectives.

For each K¨ahler class on a compact K¨ahler manifold there is a lower bound of the Calabi functional, which we call the ``potential energy''. Fixing the volume and letting the K¨ahler classes vary, the energy defines a functional which may be studied in it?s own right. Any critical point of the energy functional is then a K¨ahler class whose extremal K¨ahler metrics (if any) are so-called strongly extremal metrics. We take the well-studied case of Hirzebruch surfaces and generalize it in two different directions; along the dimension of the base and along the genus of the base. In the latter situation we are able to give a very concrete description of the corresponding dynamical system (as defined first by S. Simanca and L. Stelling). The talk is based on work in progress with Santiago Simanca.

Kapustin and Witten introduced interesting boundary value problems for magnetic monopoles on a Riemann surface times an interval. They described the moduli space of such solutions in terms of Hecke modifications of holomorphic bundles over the Riemann surface. I will explain this and prove existence and uniqueness for such monopoles.

In this talk I will introduce some new ideas to the existence theory for a class of non-variational existence problems arising naturally in geometry and analysis. I will discuss some applications (and potential applications) to positive mass-type and Penrose-type theorems in general relativity.

The link between meromorphic connections on a Riemann surface and their monodromy is a very classical one, indeed so classical that it was the subject of one of Hilbert?s problems. The deformation theory of these connections, and when these deformations preserve the monodromy, is almost equally ancient. I will give an overview of some results in the area, some ancient, and some quite recent.

In this talk, I will examine the geometry of moduli spaces of stable bundles on Hopf and Kodaira surfaces, which are compact complex surfaces that do not admit Kaehler metrics. In particular, I will show that these moduli admit interesting geometric structures such as hypercomplex structures and strong HKT-metrics, in the case of Hopf surfaces, as well as algebraic integrable systems.

The k-dilation of a map measures how much the map stretches k-dimensional volumes. The 1-dilation is the usual Lipschitz constant. We consider the problem of finding the smallest k-dilation among all degree 1 maps from one rectangle to another rectangle. (These are n-dimensional rectangles.) In general the linear map is far from optimal.

A quasi-local mass has been a long sought after quantity in general relativity. A recent candidate has been the Liu-Yau mass. One can show that the Liu-Yau mass of any two-surface is the maximum of the Brown-York energy for that two-surface. This means that it has significant disadvantages as a mass. It is much better interpreted as an energy and I will show one way of doing so. The Liu-Yau mass is especially interesting in spherical geometries, where mass and energy are indistinguishable. For a spherical two-surface, it equals the minimum of the amount of energy at rest that one needs to put inside the two-surface to generate the given surface geometry. Thus it gives interesting information about the interior, something no other mass or energy function does.

An important special case of the general construction of black holes translates into a problem in Riemannian geometry, since a totally geodesic slice of spacetime is an asymptotically flat Riemannian manifold with nonnegative scalar curvature, and the restriction of the event horizon to the slice is the apparent horizon in the slice. In this talk we show how to construct new examples of Riemannian manifolds with nonspherical apparent horizon, in dimensions four and above. More precisely, for any $n,m\ge 1$, we construct asymptotically flat, scalar flat Riemannian manifolds with apparent horizon that is a smooth outermost minimal hypersurface with topology $S^n\times S^{m+1}$.

Two metrics are geodesically equivalent if they have the same (unparameterized) geodesics. During my talk I describe geodesically equivalent metrics on closed manifolds (which is an answer to Beltrami's question) and explain the proof of Lichnerowicz-Obata conjecture (which is an answer on the infinitesimal version of the Beltrami question known as Schouten problem).

A number of classical integrable systems, for example harmonic maps of the plane to a compact Lie group or symmetric space, can be transformed into a \{\\em linear\} flow on a complex torus. This torus is the Jacobian of an algebraic curve, called the spectral curve. Recently several authors have produced an analogous one-dimensional analytic variety for conformal 2-tori in $S4$ (which are not in general integrable!) using the geometry of the quaternions. It is hoped that this new development will lead to progress on the Willmore conjecture for reasons that I will explain. However this variety is at present quite mysterious; very little is known about it. I will discuss the simplest case, namely constant mean curvature tori in $\mathbb{R}3$. I will demonstrate that in this case the variety is not at all mysterious and interpret its points geometrically in terms of transformations generalising the classical transform of Darboux. This is joint work with Katrin Leschke and Franz Pedit.

In this talk I will review the construction of the Lie algebra associated to the mapping class group of a (possibly decorated) surface and explain how this generalizes the Lie algebra associated to the pure braid group. I will also explain the analogue of the KZ-equation in the mapping class group case. In the second talk I will discuss filtrations of this Lie algebra associated to curve systems on the surface and their relation to handlebody groups.

In this talk I will review the construction of the Lie algebra associated to the mapping class group of a (possibly decorated) surface and explain how this generalizes the Lie algebra associated to the pure braid group. I will also explain the analogue of the KZ-equation in the mapping class group case. In the second talk I will discuss filtrations of this Lie algebra associated to curve systems on the surface and their relation to handlebody groups.

The cosmic censorship conjecture roughly states that singularities in the evolution of spacetime are always hidden from the outside world by event horizons. As a test for this conjecture Penrose proposed the inequality M >= (A/16pi)^1/2, relating the total ADM mass M of a spacetime to the area A of an event horizon. For time symmetric initial data sets of Einstein's equations this inequality has been confirmed, independently by Huisken and Ilmanen (for one black hole) and by Bray (for multiple black holes). The purpose of this talk is to show how the time symmetric proofs can be generalized to apply to general initial data, assuming existence for a canonical degenerate elliptic system of equations. This is joint work with Hubert Bray.

I will discuss some examples of minimal graphs with jump discontinuities in their boundaries. Robert Huff and I constructed these examples in response to a question of John Urbas: Is it possible for a minimal graph over a smooth annular domain to have an isolated jump discontinuity on the inner boundary component? I will also give a brief overview of the boundary consistency problem for Di Giorgi's generalized solutions of the minimal surface equation and discuss this question in that context. The construction of the examples uses the Weierstrass representation and the developing map introduced by Huff in the study of capillary problems.

The Yamabe invariant of a closed manifold appears naturally when studying the total scalar curvature functional on the space of Riemannian metrics on the manifold. Computations are difficult, in particular in the positive case (when the manifold admits metrics of positive scalar curvarture, and there is no unicity of metrics of constant scalar curvature on a conformal class). In this talk I will review a little of what is known about the computation of the invariant and discuss some recent joint work with K. Akutagawa and L. Florit on the Yamabe constants of Riemannian products.

One of the central questions in Geometric Analysis is the interplay between the curvature of the manifold and the spectrum of an operator. In this talk, we will be considering the Hodge Laplacian on differential forms of any order $k$ in the Banach Space $L^p$. In particular, under sufficient curvature conditions, it will be demonstrated that the $L^p\,$ spectrum is independent of $p$ for $1\!\leq\!p\!\leq\! \infty.$ The underlying space is a $C^{\infty}$-smooth non-compact manifold $M^n$ with a lower bound on its Ricci Curvature and the Weitzenb\"ock Tensor. The further assumption on subexponential growth of the manifold is also necessary. We will see that in the case of Hyperbolic space the $L^p$ spectrum does in fact depend on $p.$ As an application, we will show that the spectrum of the Laplacian on one-forms has no gaps on certain manifolds with a pole and on manifolds that are in a warped product form. This will be done under weaker curvature restrictions than what have been used previously; it will be achieved by finding the $L^1$ spectrum of the Laplacian.

A central theme in Riemannian geometry is understanding the relationships between the curvature and the topology of a Riemannian manifold. Positive isotropic curvature (PIC) is a natural and much studied curvature condition which includes manifolds with pointwise quarter-pinched sectional curvatures and manifolds with positive curvature operator. We will give a survey of results on PIC and discuss recent joint work with J. Wolfson on fundamental groups of manifolds of PIC. The techniques used involve minimal surfaces.

We discuss an approach to the proof of positivity of mass without spin assumption, for asymptotically hyperbolic Riemannian manifolds, based on the general methodology of Schoen and Yau. Our approach makes use of the "BPS brane action" introduced by Witten and Yau in their work on the AdS/CFT correspondence, and takes hints from work of Lohkamp. This is joint work with Lars Andersson and Mingliang Cai.