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XVIth Annual Geometry Festival

Northeastern University, Boston, MA
Friday April 20 - Sunday April 22, 2001

ABSTRACTS OF TALKS
Robert Bryant,   Rigidity and quasi-rigidity of extremal cycles in Hermitian symmetric spaces

Let M be a compact Hermitian symmetric space and let W be a compact, non-empty complex subvariety of M of codimension p. There exists a nontrivial holomorphic exterior differential system I on M with the property that any compact complex subvariety $V\subset M$ of dimension p that satisfies $[V]\cap[W]=0$ is necessarily an integral variety of I.

The system Iis almost never involutive. However, its p-dimensional integral varieties (when they exist) can sometimes be described explicitly by taking advantage of this non-involutivity. In this talk, several of these ideals I will be analyzed, particularly in the case where M is a Grassmannian, and the results applied to prove various results about the rigidity of algebraic subvarieties with certain specified homology classes.

These rigidity results have implications for the classification of holomorphic bundles over compact Kähler manifolds that are generated by their global sections but for which one or more of the Schur-Chern classes vanish, and this will be explained in the talk.

Tobias Colding,   Embedded minimal surfaces in 3-manifolds

 

Boris Dubrovin,   Normal forms of integrable PDEs and Frobenius manifolds

The well-known observation that goes back to Riemann identifies a significant part of the theory of quasilinear systems of the first order evolutionary PDEs

\begin{displaymath}{\partial u^i\over \partial t} = \sum_{j=1}^n A^i_j(u) {\partial x}, ~~u=\left( u^1(x,t), \dots, u^n(x,t)\right)\eqno(*)\end{displaymath}

as a subject of differential geometry. According to this observation, the dependent variables u=(u1,...,un) can be considered as coordinates of a point of an n-dimensional manifold $M\ni u$. The system (*) can be treated as a flow on the loop space ${\cal L}(M)$ when dealing with periodic boundary conditions $u^i(x+2\pi, t) = u^i(x,t)$ (actually, one can forget about boundary conditions by standing on the point of view of the formal calculus of variations). The classification of the flows of the form (*) on ${\cal L}(M)$ reduces to a differential-geometric classification of sections Aij(u) of $TM\otimes T^*M$.

As it has been understood much later by S.P. Novikov and the author, also the generic Hamiltonian structures of the flows (*) on ${\cal L}(M)$ are labelled by finite-dimensional geometrical objects, namely, by flat Riemannian or pseudo-Riemannian metrics on M.

In the talk we will address the problem of classification of integrable systems on ${\cal L}(M)$ of the form (*) and their integrable deformations in the class of evolutionary PDEs. The following two features of integrability are adopted as the basis for the classification program developed recently by Youjin Zhang and the author: 1) bihamiltonian structure and 2) existence of a $\tau$-function. An appropriate extension of the group of diffeomorphisms of M acting on the loop space ${\cal L}(M)$ is involved in the classification of deformations.

The first classification result says that, under certain genericity assumption, hierarchies of integrable systems of the form (*) on ${\cal L}(M)$ are labelled by Frobenius manifold structures on M or by their degenerations. Frobenius manifolds were introduced by the author in the beginning of 90s as the geometric setup of the so-called equations of associativity discovered by physicists E.Witten, R.Dijkgraaf, E. and H. Verlinde. For mathematicians it is best known appearance of Frobenius manifolds in the theory of the genus zero Gromov-Witten invariants of compact symplectic manifolds, although Frobenius manifolds appear also in other branches of mathematics.

Remarkably, the relationship between integrable hierarchies and Gromov-Witten invariants persists also in the classification of integrable deformations of systems (*), but the Gromov-Witten invariants of higher genera become involved. At the moment this conjectural relationship remains to look rather misterious. However, we found some evidences supporting this conjecture. In particular, we proved that the leading order deformation is always described in terms of the theory of elliptic Gromov-Witten invariants. In certain particular cases we were able to go beyond the leading order to reproduce the known identities for the genus 2 Gromov-Witten invariants starting from integrable hierarchies.

Open problems. To classify integrable systems on ${\cal L}(M)$, dim M = n+m, m=2 m1, written, in some coordinates u1, ..., un, v1, ..., vm in the form

\begin{displaymath}\begin{aligned}{\partial u^i\over \partial t} & &\sum_{j=1}{l=1}^m D_l^k(u,v) {\partial v^l\over \partial x}.\end{aligned}\end{displaymath}

Classify also their integrable deformations.

Remark. A Hamiltonian structure of such a system induces on M a structure of symplectic foliation with a n dimensional base $N\ni (u^1, \dots, u^n)$ with the symplectic leaves of the dimension m=2 m1 and also a flat Riemannian or pseudo-Riemannian metric on N. The particular case m=0 is considered in the talk; in the opposite case n=0 the Poisson bracket on ${\cal L}(M)$ is determined by a symplectic structure on M. For other values (n,m) the classification problem of such Poisson structures remains open.

John Lott,   Heat equation methods in noncommutative geometry

I will first describe Connes' index theorem for etale groupoids, which includes his foliation index theorem as a special case. I'll then present a heat equation proof of Connes' theorem, along the lines of the McKean-Singer approach to the Atiyah-Singer index theorem. The proof, which is joint work with A. Gorokhovsky, uses superconnections in the context of noncommutative geometry, something which I will also explain. As an application of the methods, I'll describe a proof of the homotopy invariance of higher signatures for manifolds-with-boundary, when the fundamental group is Gromov hyperbolic (joint work with E. Leichtnam and P. Piazza).

Dusa McDuff,   Seminorms on the Hamiltonian group and the nonsqueezing theorem

Let $\phi_t^H, t\in [0,1],$ be the path in the Hamiltonian group generated by the functions $H_t, t\in [0,1],$ on the closed symplectic manifold $(M, \omega)$. Assuming that Ht has zero mean $\int_M H_t \omega^n$, we can define the negative and positive parts of its length by setting ${\mathcal L}^-(H_t) = \int_0^1 -\min_{x\in M} H(x,t)\,dt,\quad {\mathcal L}^+(H_t) = \int_0^1 \max_{x\in M} H(x,t)\,dt.$ Accordingly, we define seminorms $\rho^\pm$ and $\rho$ on the Hamiltonian group ${\rm Ham}(M, \omega)$ by taking $\rho^\pm(\phi)$to be the minimum of ${\mathcal L}^\pm(H_t)$ over all Hamiltonians Ht with time 1 map $\phi$. Similarly, $\rho(\phi)$ is the minimum of ${\mathcal L}^-(H_t) + {\mathcal L}^+(H_t)$ over all such paths. Thus $\rho(\phi)$ is the well known Hofer norm: it is invariant under conjugation and taking inverses as well as being nondegenerate.

The one sided seminorms $\rho^-$ and $\rho^+$ are much less well understood, though they have a very natural geometric interpretation. This talk will concentrate mostly on their sum $\rho^- + \rho^+$. Because this is invariant under taking inverses, for each manifold $(M, \omega)$it must either be identically zero or be nondegenerate. We will describe geometric arguments (using Gromov-Witten invariants on suitable Hamiltonian fibrations over S2) that show that it is nondegenerate in certain cases, for example if $(M, \omega)$ is a projective space or is weakly exact. We also show why short $\rho$-minimizing paths always exist in ${\rm Ham}(M, \omega)$.

Open Problems:

  1. Does ${\rm Ham}(M, \omega)$ always have infinite diameter with respect to the Hofer norm $\rho$?
  2. Is $\rho^- + \rho^+$ a norm on all closed $(M, \omega)?$
  3. Is $\rho^-$ ever nondegenerate? (This is not even known for the case M = S2!)
  4. Find new ways to get lower bounds for the seminorms.
  5. If the Hamiltonian flow of $H: M\to {\bf R}$ is not periodic, does $\rho(\phi_t^H) \to \infty$ as $t\to \infty$?
Rick Schoen,   Variational approaches to the construction of minimal lagrangian submanifolds

In this talk we will describe a natural variational approach to the construction of minimal lagrangian submanifolds. These include special lagrangian submanifolds which play an important role in the geometry of Calabi-Yau manifolds. The idea is to construct least volume submanifolds in suitable classes of lagrangian submanifolds. Subtleties arise connected with stability issues for homology classes and (non)orientability of the competing submanifolds.

Shing-Tung Yau,   Mirror symmetry

 

Back to the XVIth
East Coast Geometry Festival
Web page:  Alexandru I. Suciu
Comments to:  alexsuciu@neu.edu 		Started: February 23, 2001.   Updated: April 11, 2001
URL: http://www.math.neu.edu/geomfest/abstracts.html