Department of Mathematics

Three outstanding mathematicians joined our department this academic year.  They are at the cutting edge of their fields and they are passionate about their work. Each of them has provided us with a description of their research which by necessity is highly technical, but it gives us some impression of the difficulty and also of the beauty of their endeavor. Their comments are quoted below.

Marco Gualtieri, Assistant Professor (differential geometry and mathematical physics).

Marco got his PhD in 2003 at Oxford University UK under the supervision of Nigel Hitchin. His last position was at MIT.

"My work focuses primarily on the development of generalized complex geometry. This is a relatively new subject in differential geometry, originating in 2001 with the work of Hitchin on geometries defined by differential forms of mixed degree. It has the particularly interesting feature that it interpolates between two very classical areas in geometry: complex algebraic geometry on the one hand, and symplectic geometry
on the other hand. As such, it has bearing on some of the most intriguing geometrical problems of the last few decades, namely the suggestion by physicists that a duality of quantum field theories leads to a "mirror symmetry" between complex and symplectic geometry.

Examples of generalized complex manifolds include complex and symplectic manifolds; these are at opposite extremes of the spectrum of possibilities. Because of this fact, there are many connections between the subject and existing work on complex and symplectic geometry. More intriguing is the fact that complex and symplectic methods often apply, with subtle modifications, to the study of the intermediate cases. Unlike symplectic or complex geometry, the local behaviour of a generalized complex manifold is not uniform. Indeed, its local structure is characterized by a Poisson bracket, whose rank at any given point affects the local geometry. Using a combination of tools from algebraic and Poisson geometry, we have recently discovered examples of generalized complex 4-manifolds which admit neither complex nor symplectic structures. This opens up the question of what obstructions there are to the existence of generalized complex structures, just as Seiberg-Witten invariants obstruct the existence of symplectic, and in some cases complex structures.

Just as in the theory of Kähler manifolds, we may introduce a Riemannian metric compatible with a given generalized complex structure; the result is a generalized Kähler manifold. These have the interesting feature that the underlying manifold becomes a complex manifold in two distinct ways, each of which is compatible with the Riemannian metric. Such bi-Hermitian metrics are of interest to Riemannian geometers but also to physicists, who discovered that quantum field theories constructed from such spaces carry a N = (2, 2) supersymmetry. Much of my work recently has been directed towards constructions of new examples of these Riemannian structures.

One intriguing and urgent problem in the field of generalized complex geometry is that of constructing an algebraic category which represents the geometry. In the case of complex geometry, this is simply the category of coherent sheaves. In the case of symplectic geometry, this is the Fukaya category, which describes the Floer-theoretic intersection of Lagrangian submanifolds. We expect that a similar category should exist for a large class of generalized complex manifolds. Developing such a category would have profound consequences: it would suggest not only a mirror symmetry between complex and symplectic manifolds but a full theory of morphisms between categories associated to complex and symplectic manifolds".

Larry Guth, Assistant Professor (geometry and functional analysis)

Larry comes to us from Stanford University.

"I do work in quantitative topology - trying to combine geometric estimates and ideas from topology.

 The first example I want to talk about begins with ideas from topology and develops into some geometric estimates.  In the early 1900's, topologists wanted to define the "dimension" of various topological spaces.  Uryson suggested a definition that involved comparing an arbitrary metric space to a polyhedron.  He said that a metric space X has dimension at most n if - for any small number d - we can map X to an n-dimensional polyhedron in such a way that every fiber of the map has diameter less than d.  To show that his definition agrees with the usual one, he proved that any map from the unit n-cube to a polyhedron of dimension n-1 contains a fiber of diameter at least 1.  Notice that this is a geometric estimate.

 Fifty years later, Gromov revisited Uryson's idea from the perspective of geometry.  He defined the Uryson n-width of a metric space X as the smallest W so that X can map to an n-dimensional polyhedron with fibers of diameter less than W.  According to Uryson's definition, a space has dimension at most n if its Uryson n-width vanishes.  According to Uryson's estimate, the (n-1)-width of the unit cube is at least 1.  Rather than trying to define dimension, Gromov viewed the Uryson n-width as a way of describing the "shape" of a metric space such as a Riemannian manifold.  It provides a quantitative, precise language for saying that a Riemannian manifold is "thin".

 Gromov conjectured that a Riemannian n-manifold of volume 1 has Uryson width at most C(n).  In the important paper "Filling Riemannian
 manifolds", he proved a result which is slightly weaker than this. Building on his work, I was recently able to prove Gromov's conjecture.

The second example I want to give begins with geometry.  We consider a map F between Riemannian manifolds.  I'm interested in surface areas, so we choose a dimension k, and we measure how much F stretches the areas of k-dimensional surfaces.  We say that the k-dilation of F is at most D if F maps each k-dimensional surface of volume V in the domain to a k-dimensional surface of volume at most DV.

 Here is a typical problem about k-dilation.  Fix two n-dimensional ellipsoids, E and E', and try to find the minimal k-dilation of any
 diffeomorphism from E to E'.  One might at first expect the linear diffeomorphism to deliver the smallest k-dilation, but it turns out that
 this is badly false when 1 < k < n.  For some pairs of ellipsoids, there are funny non-linear maps with k-dilations far smaller than the linear map's.  I've used a number of tools from topology to give lower bounds for k-dilation, including rational homotopy invariants of various orders, cup powers, and Steenrod squares".

Joel Kamnitzer, Assistant Professor (representation theory, algebraic geometry, combinatorial representation theory, homology).

Joel got his PhD at the University of California, Berkeley. He held postdoctoral positions at MIT and UC Berkeley. He was also a visiting Assistant Professor at UC Berkeley and a member of MSRI.

"My work concerns complex reductive groups and their representation theory.  This is a well studied field dating back to the 19th century, but it continues to a source of interesting problems and structures.  I focus on exploring connections between representation theory and algebraic geometry and combinatorics.

One problem concerns the notion of canonical bases.  One would like to find nice bases for representations that interact well with the action of the group and allow one to answer natural enumerative questions, such as tensor product multiplicities.

I have been studying such bases via a geometric approach coming from the geometric Langlands problem.  In this approach, we obtain a natural basis labelled by certain algebraic varieties.  We then study geometric properties of these varieties in order to learn more about our basis.

Once one has a canonical basis, it becomes very interesting to study its combinatorics and this leads to a beautiful combinatorial structure called a crystal.  I also have worked on the combinatorics of these crystals.

Recently, I have also been working in the field of knot homology.  Here, we have been trying to define homological knot invariants associated to complex reductive groups and their representations.  Again, we have been using a geometric approach.  We associate an algebraic variety to each representation and then study categories associated to these varieties".