"Differentiation, bi-Lipschitz nonembedding and embedding"

Rademacher's theorem asserts the almost everywhere differentiability of Lipschitz functions  f: Rⁿ  ®  Rⁿ .  We will review a theory which provides an extension of Rademacher's theorem to PI spaces, i.e. metric measure spaces (X, m) satisfying a doubling condition and a Poincaré inequality in the sense of upper gradients. An application yields a large class of PI spaces X which do not bi-Lipschitz embed in any finite dimensional  R^k . In recent joint work with Bruce Kleiner, we extended the differentiation and nonembedding results to infinite dimensional Banach space targets, which are separable and dual to some Banach space. The separable space L¹  is not a dual space and differentiability of  L¹  valued Lipschitz functions, R ®  L¹  is well-known to fail. Indeed, members of a class PI spaces including so-called Laakso spaces, to which the separable dual space discussion applies, do bi-Lipschitz embed in  L¹ .  But for certain special PI spaces, including the Heisenberg group with its Carnot-Caratheodory metric, by means of a novel differentiation theory, we prove bi-Lipschitz nonembedding in  L¹ .
 

"Fitting a smooth function to data"

Fix positive integers m,n, and suppose we are given N points in Rⁿ⁺¹ .
How can we compute a C^m function  F:Rⁿ ®  R  whose graph passes through (or close to) all the given points, with the C^m norm of F nearly as small as possible? How small can we take the C^m norm of such an F? What happens if we are allowed to discard a few of the N points as "outliers"? How many computer operations are needed for these problems? What if we fit the N points to an imbedded hypersurface instead of a graph? Joint work with Bo'az Klartag.
 

"On the analytic and geometric foundations of symplectic field theory"

Symplectic field theory may be viewed as a theory of invariants for symplectic cobordisms.  These invariants are constructed from the solution spaces of a countable number of nonlinear Cauchy-Riemann type problems.  All these problems show a lack of compactness and transversality for any choice of geometric data defining the  elliptic problem. On the other hand these problems are all interdependent, which is the source for the rich structure of the invariants obtained.  The talk addresses the underlying geometric and analytic issues in the construction of SFT.
 

"Reeb vector fields and open book decompositions"

According to a theorem of Giroux, there is a 1-1 correspondence between isotopy classes of contact structures and equivalence classes of open book decompositions.  We prove that any contact structure  (M, x)  (in dimension 3) which is supported by an open book with periodic monodromy satisfies the Weinstein conjecture, namely any Reeb vector field  R  of (M, x)  admits a closed orbit.  The approach is to study holomorphic curves in the symplectization of  (M, x)  for a particularly nice Reeb vector field  R ,  when x is universally tight with universal cover  R³ .   In such a case we show that the contact homology is cylindrical and nonzero. This is joint work with Vincent Colin.
 

"The Dynamics Theorem for embedded minimal surfaces"

First I will give a quick survey of recent classification theorems for classical minimal surfaces.  These results include the uniqueness of the catenoid, the helicoid, Scherk's singly and doubly periodic minimal surfaces, Riemann's family of minimal surfaces and the Topological Classification Theorem for properly embedded minimal surfaces in R³ .  Next I will present and prove some recent results on minimal laminations of 3-manifolds.  These results include the Minimal Lamination Closure Theorem, the Lamination Metric Theorem, the Local Removable Singularity Theorem, and the C^1,1 Regularity Theorem for a
Colding-Minicozzi lamination and its converse.  As a consequence, I will then sketch the proofs of the following central theorems.

1. The Quadratic Decay of Curvature Theorem, which states that a complete embedded minimal surface in  R³  has quadratic decay of curvature (in terms of the distance R from the origin) if and only if it has finite total curvature

2. The Dynamics Theorem for a properly embedded minimal surface M in R³. This theorem demonstrates that if  M  does not have finite total curvature, then its space  D(M)  of nonflat divergent dilation limits always contains a nonempty minimal dilation invariant subset, in the sense of dynamics.

3. The Bounded Topology Theorem, which gives bounds on the topology of a complete embedded minimal surface of finite topology in terms of a bound on its genus.  In particular, a complete embedded minimal surface of finite index of stability and fixed genus must have a bound on its index that depends only on its genus.
 

"Asymptotic geometry of the mapping class group"

"Manifolds with Density"

Riemannian manifolds with a positive, smooth density function, the smooth case
of the "mm spaces" of Gromov, have long appeared on an ad hoc basis in mathematics. In freshman calculus one studies volumes and surfaces of revolution by considering planar regions and curves with density 2 p r.  Gauss space ... Euclidean space with Gaussian density ~ exp(—x²/2) ... is of particular interest to probabilists.  The grand goal is to generalize Riemannian geometry to the larger category of manifolds with density. I'll mention some generalizations of Gauss-Bonnet (by undergraduates) and of isoperimetric theorems in which Gauss space replaces the sphere as the natural model. One almost unexplored topic is properties and examples of minimal surfaces in Gauss space.
 

"Link Floer homology and the Thurston norm"

The aim of this talk is to introduce Floer homology groups for links and study their relationships with the Thurston norm of the link complement.  This is a joint work with Peter Ozsvath.