Abstract: The Calabi conjecture, proved by Yau thirty years ago, says that on a compact Kahler manifold one can find a unique Kahler metric in every Kahler class with prescribed volume form. Donaldson recently conjectured that this theorem can be extended to symplectic forms with a compatible almost complex structure in 4 dimensions, and gave possible applications to the symplectic topology of 4-manifolds. I will discuss Donaldson's conjecture and some recents developments (joint work with B. Weinkove and partly with S.-T. Yau).

Motivated by the need to formulate and solve Plateau type problems in higher dimensions and codimensions, normal and integral currents were introduced by Federer and Fleming around 1960; their work was, to some extent a generalization of earlier work by DeGeorgi in codimension one as well as the work of Reifenberg in arbitrary codimensions. Since then a great deal of work has been done on the Plateau problem and related variational problems. This work has always been based on geometric measure theory. The so-called closure theorem for integral currents and the boundary rectifiability theorem are essential ingredients in all of this work; these theorems depend on the Besicovitch-Federer structure theory for set of finite Hausdorff measure in Euclidean space. More recently, in the work of Ambrosio and others, a useful theory of Sobolev spaces for functions with values in an arbitrary metric space has been developed and applied to a variety of problems. Ambrosio and Kirchheim have developed a theory of currents in metric spaces in which they are able to give geometrically appealing proofs of generalizations of the aforementioned closure and rectifiability theorems using some ideas of Almgren and DeGiorgi and avoiding the use of the Besicovitch-Federer structure theory. In this talk I will describe how they do it.

We formulate the Nahm transform producing self-dual Hermitian connections on Asymptotically Locally Flat hyperkaehler manifolds. Using this formulation we describe the moduli spaces of such connections and explicitly compute their asymptotic metrics.

In the talk we will discuss curvature conditions under which we can guarantee the existence of a smooth solution to the Ricci flow and the mean curvature flow equation. These are improvements of Hamilton's and Husiken's results on extending the Ricci flow and the mean curvature flow, under conditions that the norm of Riemannian curvature and the norm of the second fundamental form are uniformly bounded along the flow in finite time, respectively.

In this talk, we will investigate existing Legendrian knot invariants and discuss new connections between the theory of generating families and the Chekanov-Eliashberg differential graded algebra (CE-DGA). The geometric origins of the CE-DGA are Floer theoretic in nature and come out of the Symplectic Field Theory developed by Eliashberg and Hofer. On the other hand, Legendrian invariants derived from the study of 1-parameter families of smooth functions (called generating families) are Morse theoretic in nature. In the last decade, connections have been found between the Legendrian invariants derived using these two methods. In this talk, I will try to provide a clearer picture of the relationship between generating families and the CE-DGA.

I will tell two stories. The first is the story of static spacetimes with black hole boundaries and the attempt to classify them. The second is the story of the Penrose inequality. I will then weave these two stories together in the setting of negative curvature. This last part is a report on joint work-in-progress with A. Neves.

In the early 1980s Atiyah and Bott described a new approach to studying the cohomology of the moduli space of stable bundles: the equivariant Morse theory of the Yang-Mills functional. There are many other interesting moduli spaces that fit into a similar framework, however the catch is that the total space is singular, and it is not obvious how to construct the Morse theory of the appropriate functional. In this talk I will describe how to get around these difficulties for the moduli space of stable pairs, for which we prove a Kirwan surjectivity theorem and give a Morse-theoretic interpretation of the change in cohomology due to a flip. This builds upon earlier work with George Daskalopoulos, Jonathan Weitsman and Richard Wentworth for rank 2 Higgs bundles.

On a Fano manifold M we study the supremum of the possible t such that there is a Kähler metric in c_1(M) with Ricci curvature bounded below by t. We relate this to Aubin's continuity method for finding Kähler-Einstein metrics and we give bounds on it for certain manifolds.

I will discuss the question: When does a vector bundle admit a holomorphic structure? I will explore applications of Yang-Mills theory and geometric quantization to this problem.

It is known that, on closed manifolds, Einstein metrics of negative scalar curvature are critical points of the usual volume functional constrained to the space of metrics of constant scalar curvature. In this talk, I will discuss how this variational characterization of Einstein metrics can be localized to compact manifolds with boundary. I will present the critical point equation and focus on geometric properties of its general solutions. In particular, when a solution has zero scalar curvature and its boundary can be isometrically embedded into the Euclidean space, I will show that the volume of this critical metric is always greater than or equal to the Euclidean volume enclosed by the image of the isometric embedding and two volumes are the same if and only if the critical metric is isometric to the Euclidean metric on a standard round ball. This is a joint work with Prof. Luen-Fai Tam.

Homological Mirror Symmetry proposes an equivalence between the derived category of coherent sheaves on a complex manifold and the (derived) Fukaya category of the mirror symplectic manifold. It is natural to consider the behaviour of these categories and equivalences under deformations of the underlying spaces.

In this talk I will describe Toda's results on deformations of the category Coh(X) of coherent sheaves on a complex manifold X. They come from deformations of X as a complex manifold, non-commutative deformations, and gerby deformations. These can all be interpreted as deformations of X as a generalized complex manifold; in some instances it is possible to deform X to a symplectic manifold. Toda also described how to deform Fourier-Mukai equivalences, and I will present some examples coming from mirror SYZ fibrations.

A smooth knot in a contact 3-manifold is called Legendrian if it is always tangent to the contact planes. In this talk, I will discuss Legendrian knots in R^3 and the solid torus where knots can be conveniently viewed using their `front projections'. In particular, I will describe how certain decompositions of front projections known as `normal rulings' (introduced by Fuchs and Chekanov-Pushkar) can be used to give combinatorial descriptions for parts of the HOMFLY-PT and Kauffman polynomials. I will conclude by discussing recent generalizations to Legendrian solid torus links. It is usual to identify the `HOMFLY-PT skein module' of the solid torus with the ring of symmetric functions. In this context, normal rulings can be used to give a knot theory description of the standard scalar product determined by taking the Schur functions to form an orthonormal basis.

We study the Seiberg-Witten equations on a 4-manifold X with the homology of S1 x S3. The count of such solutions, called the Seiberg-Witten invariant, depends on choices of Riemannian metric and perturbation of the equations to make a smooth moduli space. A similar issue is resolved in dimension 3 by relating the jumps in the Seiberg-Witten invariant to the spectral flow of the Dirac operator. In dimension 4, we use Taubes' theory of periodic-end operators to relate the jumps in the Seiberg-Witten invariant to the index theory of the Dirac operator on the infinite cyclic cover of X. This circle of ideas has applications to classification of smooth manifolds homeomorphic to S1 x S3, to questions about positive scalar curvature, and to homology cobordisms. This is joint work with Tomasz Mrowka and Nikolai Saveliev.

To each three-component link in the 3-dimensional sphere we associate a characteristic map from the 3-torus to the 2-sphere, and establish a correspondence between the pairwise and Milnor triple linking numbers of the link and the Pontryagin invariants that classify its characteristic map up to homotopy. This can be viewed as a natural extension of the familiar fact that the linking number of a two- component link is the degree of its associated Gauss map from the 2- torus to the 2-sphere. When the pairwise linking numbers are all zero, we give an integral formula for the triple linking number analogous to the Gauss integral for the pairwise linking numbers. The integrand in this formula is geometrically natural in the sense that it is invariant under orientation-preserving rigid motions of the 3-sphere.

We discuss recent and ongoing work, joint with A. Little, on estimating the intrinsic dimensionality of data sets assumed to be sampled from a low-dimensional manifold in high dimensions and perturbed by high-dimensional noise. This work is motivated by several applications, including machine learning and dynamical systems, and by the limitations of existing algorithms. Our approach is based on a simple tool such as principal component analysis, used in a multiscale fashion, a strategy which has its roots in geometric measure theory. The theoretical analysis of the algorithm uses tools from random matrix theory and exploits concentration of measure phenomena in high-dimensions. The talk will have a tutorial flavour: no previous knowledge of what mentioned above will be required, and several toy examples to build intuition about some measure-geometric phenomena in high-dimensions will be presented.

The Fundamental Theorem of Algebra first rigorously proved by Gauss states that each complex polynomial of degree n has precisely n complex roots. In recent years various extensions of this celebrated result have been considered. We shall discuss the extension of the FTA to harmonic polynomials of degree n. In particular, the 2003 theorem of D. Khavinson and G. Swiatek that shows that the harmonic polynomial z − p(z); deg p = n > 1 has at most 3n − 2 zeros as was conjectured in the early 90's by T. Sheil-Small and A. Wilmshurst. More recently L. Geyer was able to show that the result is sharp for all n.

In 2004 G. Neumann and D. Khavinson showed that the maximal number of zeros of rational harmonic functions z − r(z); deg r = n > 1 is 5n − 5. It turned out that this result conrfimed several consecutive conjectures made by astrophysicists S. Mao, A. Petters, H. Witt and, in its final form, the conjecture of S. H. Rhie that were dealing with the estimate of the maximal number of images of a star if the light from it is deflected by n co-planar masses. The first non-trivial case of one mass was already investigated by A. Einstein around 1912.

We shall also discuss the problem of gravitational lensing of a point source of light, e.g., a star, by an elliptic galaxy, more precisely the problem of the maximal number of images that one can observe. Under some more or less "natural" assumptions on the mass distribution within the galaxy one can prove that the number of visible images can never be more than four in some cases and six in the other. Interestingly, the former situation can actually occur and has been observed by astronomers. Still there are much more open questions than there are answers.

A couple of years ago Kronheimer and I revisited our work on instantons with codimension two singularities. This leads to knot invariants associated to each partial flag manifold. There invariants are related to Khovanov homology for the case of $\IP1$ and Khovanov-Rozansky homology fo $\IP^n$. We have recently understood how to use the $\IP1$ case to prove that Khovanov homology detects the unknot.

We will discuss the existence and uniqueness of the foliation by stable spheres with constant mean curvature for asymptotically flat manifolds satisfying the Regge-Teitelboim condition at infinity. This work generalizes the earlier results of Huisken/Yau, Ye, and Metzger. We will also discuss the concept of the center of mass in general relativity.

TBA

TBA

We present recent results that show that for any portion of a compact manifold that admits a bi-Lipschitz parametrization by a Euclidean ball one may find a well-chosen set of eigenfunctions of the Laplacian that gives a bi-Lipschitz parametrization almost as good as the best possible. A similar, and in some respect stronger result holds by replacing eigenfunctions with heat kernels. These constructions are motivated by applications to the analysis of the geometry of data sets embedded in high-dimensional spaces, that are assumed to lie on, or close to, a low-dimensional manifold. This is joint work with P.W. Jones and R. Schul.

Heegaard Floer homology, a kind of (3+1)-dimensional field theory, associates chain complexes to 3-manifolds and chain maps to 4-manifolds with boundary. These complexes and maps are defined by counting holomorphic curves, and are hard to compute. Bordered Floer homology extends Heegaard Floer theory one dimension lower, assigning algebras to surfaces and differential modules to 3-manifolds with (parameterized) boundary. After introducing the bordered Floer framework, we will illustrate its construction in a toy case where it is explicit and combinatorial: planar grid diagrams. This is joint work with Peter Ozsvath and Dylan Thurston.

We consider the problem of constructing systems of hyperbolic conservation laws in one space dimension with prescribed geometry in state space: the eigenvectors of the Jacobian of the flux are given. This is formulated as a system of algebraic-differential equations whose solution space is analyzed using Darboux and Cartan-K\"ahler theorems. It turns out that already the case with three equations is fairly complex. We give a complete list of possible scenarios for the general systems of two and three equations and for rich systems (i.e. when the given eigenvector fields are pairwise in involution) of arbitrary size. As an application we characterize conservative systems with the same eigencurves as compressible gas dynamics.

This is joint work with Kris Jenssen (Penn State University)

Let M be an algebraic complex surface equipped with a singular foliation F. We assume that F leaves invariant a closed current on M or, equivalently, that F possesses a transversely invariant measure. The purpose of this talk is two-fold. First we want to classify the pairs (M, F) as above, a problem that is usually regarded as a step towards developing a suitable Ergodic Theory for these foliations. On the other hand we want to explain the connection of this problem with the Kobayashi hyperbolicity of general type surfaces. In particular we shall sketch a new proof of McQuillan's theorem proving the Green-Griffiths conjecture for general type surfaces having positive Segre class.

A 'correspondence' between two manifolds is a submanifold in the product. This generalizes the notion of a map (whose graph is a correspondence) ... and is of little use in general since the composition of correspondences, though naturally defined, can be highly singular.

Lagrangian correspondences between symplectic manifolds however are highly useful (and will be defined carefully). They were introduced by Weinstein in an attempt to build a symplectic category that has morphisms between any pair of symplectic manifolds (not just symplectomorphic pairs).

In joint work with Chris Woodward we define such a cateory, in which all Lagrangian correspondences are composable morphisms. We extend it to a 2-category by constructing a Floer homology for generalized Lagrangian correspondences. One of the applications is a general prescription for constructing topological invariants. We consider e.g. 3-manifolds or links as morphisms (cobordisms or tangles) in a topological category. In order to obtain a topological invariant from our generalized Floer homology, it suffices to

(i) decompose morphisms into simple morphisms (e.g. by cutting between critical levels of a Morse function)

(ii) associate to the objects and simple morphisms smooth symplectic manifolds and Lagrangian correspondences between them (e.g. using moduli spaces of bundles or representations)

(iii) check that the moves between different decompositions are associated to 'good' geometric composition of Lagrangian correspondences

Let Y be a hypersurface in a 2n-dimensional holomorphic symplectic manifold X. The restriction $\sigma|_Y$ of the holomorphic symplectic form induces a rank one foliation on Y. If this "characteristic foliation" has compact leaves, then the space of leaves Y/F is a holomorphic symplectic manifold of dimension 2n-2. This construction also works when Y is a coisotropic submanifold of higher codimension, and is known as "coisotropic reduction". In this talk we will consider when the characteristic foliation has compact leaves, and look at some applications of coisotropic reduction.

The combinatorial complexity of the union of a set of geometric objects is the total number of faces of all dimensions that lie on the boundary of the union. We review some recent results on the complexity of the union of geometric objects in 2d and 3D satisfying various natural conditions and on computing the union. We then discuss the critical roles they play in computing an epsilon net and a hitting set of a set system.

In this talk we will discuss a proof of the path-connectedness of the space of positive scalar curvature metrics on the three-sphere. The proof uses the Ricci flow with surgery and the connected sum construction of Gromov and Lawson. The work of Perelman on Hamilton's Ricci flow is fundamental. If time permits we will also discuss an application to general relativity.