Analysis-Geometry Seminar 1996-7

Archived Talks from 1996-7:

• September 27, 1996
  Speaker: Shunhui Zhu (Dartmouth)
  Title: Constant Mean Curvature Hypersurfaces in Euclidean and Minkowski Space

• October 4, 1996
  Speaker: Mikhail Shubin (Northeastern)
  Title: L²-Holomorphic Functions on Noncompact Manifolds
  Abstract: A generalization of the classical Oka-Grauert theory to a non-compact situation will be discussed. Consider a complex manifold M with a non-empty strongly pseudo-convex boundary and a free holomorphic action of a group G such that the quotient M/G is compact. (The classical case corresponds to the case of trivial group G.) Then type II von Neumann algebra technique allows us to prove that the space of L²-holomorphic functions is infinite-dimensional provided G is unimodular (in particular, discrete). An example is given where G is a solvable (but not unimodular) 3-dimensional Lie group, M has the complex dimension 2, M/G=[0,1], and there is no non-trivial L²-holomorphic functions on M. (This is a joint work with M.Gromov and G.Henkin.)

• October 11, 1996
  Speaker: Bong Lian (Brandeis)
  Title: The Large Radius Limit Problem in Mirror Symmetry
  Abstract: Mirror Symmetry purports to yield precise information about the enumerative geometry of a Calabi-Yau 3-fold by studying the variation of Hodge structure, and its degeneration, of another Calabi-Yau 3-fold. We will consider the problem of constructing and describing this degeneration for Calabi-Yau hypersurfaces in a Fano toric variety.

• October 18, 1996
  Speaker: George Kamberov (University of Massachusetts, Amherst)
  Title: Quaternions, Dirac Spinors, and Prescribing Mean Curvature
  Abstract: A fundamental question in geometry is to what extent one can use mean curvature data to determine the shape of a surface in Euclidean three space. We are interested in the uniqueness and as well as in the existence aspects of this question.
      In the first part of the talk we will present our results on the uniqueness problem suggested by Bonnet: show that for most Riemannian surfaces there is at most one, up to a rigid motion, immersion with a given mean curvature function, and describe the exceptional surfaces for which the uniqueness fails. The corresponding existence problem is overdetermined and so the existence research is centered on the problem: given a conformal structure and a function H, is there a conformal immersion with mean curvature H?
      In the second part of the talk we outline a new existence theory for conformal immersions. This is joint work with U. Pinkal and F. Pedit. The central idea in the new existence theory is to use quaternionic and spinor calculus to define square roots of basic geometric objects, e.g., area elements, differentials of maps, and then determine the equations satisfied the square root of the differential of a conformal immersion. From an analytic point of view the main advantage of the new approach is that the existence problem for prescribing mean curvature is reduced to a first order linear elliptic system. Our theory suggests a new paradigm: the data used to determine the immersion are, the conformal structure, the regular homotopy class of the immersion, and the mean curvature half-density, i.e., the mean curvature appropriately weighted by the metric. These data determine the immersion via a generalized Weierstrass-Kenmotsu formula.

• October 25, 1996
  Speaker: John Lott (University of Michigan)
  Title: The Zero-in-the-Spectrum Question
  Abstract: The zero-in-the-spectrum question asks whether it is true that for any complete Riemannian manifold, there is some number p such that zero lies in the spectrum of the p-form Laplacian. I will explain how this question arose and its relation to questions in topology and combinatorial group theory.

• Tuesday, October 29, 1996 (Note Special Day!)
  Speaker: Walter Strauss (Brown)
  Title: Breathers as Harmonic Maps
  Abstract: The classical breather solution of the sine-Gordon equation is almost unique. Others have been searched for but never found. However, the breather can be viewed as a harmonic map from periodic 1+1 Minkowski space into the 2-sphere, also called a wave map. This map is a solution of a certain nonlinear hyperbolic PDE. It is homoclinic, in the sense that it converges to the same limit as t goes to + or - infinity. It can also be viewed as an unstable orbit of a very simple steady-state map. Are there any other homoclinic wave maps? If so, they would be the long-sought breathers!

• November 8, 1996
  Speaker: Ross Niebergall (University of Northern British Columbia)
  Title: Tight Immersions of Highly Connected Manifolds
  Abstract: An immersion of a compact manifold is tight if it admits the minimal total absolute curvature over all immersions of the manifold. The most significant result in the study of minimal total absolute curvature immersions is the theorem of Chern and Lashof, which characterizes minimal total absolute curvature immersions, and tight immersions, of spheres into a Euclidean space. In this talk we examine tight immersions of highly connected manifolds; i.e., 2k-dimensional manifolds that are (k-1)-connected by not k-connected.

• November 15, 1996
  Speaker: Eckhard Meinrenken (M.I.T.)
  Title: Quantization and Reduction
  Abstract: Let G be a compact Lie group, M a compact almost complex G-manifold and L a complex G-line bundle. The equivariant index of the corresponding Dirac operator, twisted by the line bundle L, is a (virtual) representation of G known as its Riemann-Roch number RR(M,L). Even though the equivariant index theorem gives an explicit formula for the character of this representation, it is in general difficult to find the decomposition into irreducible components. In this talk, I will describe a situation arising in the context of symplectic geometry in which such a description is possible and is quite beautiful: The "quantization commutes with reduction theorem".

• 1:30-2:30 pm Tuesday, November 19, 1996 (Note Special Day and Time!)
  Speaker: Xi Du (Northeastern University)
  Title: Isometric Immersions of Space Forms in Space Forms
  Abstract: We will discuss a rational loop group action on such immersions which includs some known transformations (i.e. Backlund). We will also talk about how to apply the Adler-Kostant-Symes theorem to obtain such immersions.

• November 22, 1996
  Speaker: Tang Zizhou (Academia Sinica, Beijing)
  Title: Harmonic Maps between Spheres
  Abstract: A basic problem in Riemannian geometry posed by Eels and Sampson is the following: when does a homotopy class of maps between compact Riemannian manifolds admit a harmonic representative? By harmonic join, Smith showed that every element of the m-th homotopy group of S^m admits a harmonic representative for m 8. By using orthogonal multiplications, we show that for almost all odd m, there exists a map f from S^m to itself with degree 2 which is harmonic.

• January 24, 1997
  Speaker: Ryszard Nest (Copenhagen)
  Title: Quantum Riemann Surfaces
  Abstract: We construct continuous fields of C*-algebras deforming smooth functions on a Riemann surface and related to the C* algebra of the fundamental group, and study their properties.

• January 31, 1997 (Joint with the Geometry-Algebra-Singularities Seminar)
  Speaker: Joseph Landsberg (U. North Carolina)
  Title: Dual varieties, systems of quadrics of constant rank, symmetric degeneracy loci and local differential geometry

• February 7, 1997
  Speaker: Christopher King (Northeastern University)
  Title: Phase boundaries for a lattice flux line model.
  Abstract: We consider a statistical model of non-intersecting self-avoiding random walks (flux lines) on the lattice. The model is non-isotropic, and in the thermodynamic limit exhibits at least three distinct phases. We locate the boundaries of these phases, and describe the bulk correlations of the flux lines.

• February 14, 1997
  Speaker: Naum Zobin (Ohio State University)
  Title: Whitney's Problem on Extendability in Planar Domains
  Abstract: We consider the space of functions with bounded k-th derivatives in a general domain in Rⁿ. Is every such function extendable to a function of the same class defined on the whole Rⁿ? H. Whitney showed in 1934 that the equivalence of the geodesic metric in this domain to the Euclidean one is sufficient for such extendability. There was an old conjecture (going back to H. Whitney) that this equivalence is also necessary for extendability. We disprove this conjecture in all dimensions starting from 2. The counterexamples are infinitely connected domains in R². It is possible to construct counterexamples in Rⁿ, n>2, homeomorphic to balls, so no topological restrictions can help in dimensions 3 and higher. As for dimension 2, we prove that, nevertheless, the Whitney's Conjecture is true for bounded finitely connected domains.
  In this talk we are going to concentrate on the proof of the Whitney's Conjecture for finitely connected planar domains.

• 1:30-2:30 pm, February 18, 1997   (Note Special Time and Day!)
  Speaker: Leon Ehrenpreis (Temple University)
  Title: Hypergeometric Functions
  Abstract: The usual works on hypergeometric functons seem to involve a morass of unrelated formulae. In this lecture we shall introduce the concept of conformal group of partial differential equations which serves as a unifying force.
      Analogues for q-hypergeometric functions will be discussed.

• February 21, 1997
  Speaker: Boris Fedosov (Moscow, visiting Courant Institute of Math. Sciences)
  Title: On the index of elliptic operators on a cone and a wedge.
  Abstract: Elliptic operators of zero order on a cone and a wedge are considered in the framework of Schulze's algebra of pseudodifferential operators on manifolds with singularities. Index formulas for such operators are obtained in terms of their operator-valued symbols. The relations to previous results of Plamenevski-Rozenblum and Luke are discussed.
  

• 3:00-4:00 pm, February 28, 1997   (Note Special Time!)
  Speaker: Leonid Friedlander (Univ. of Arizona, Tuscon)
  Title: Glueing formulae for the analytic and Reidemeister torsions
  Abstract: Let (M₁,d₊M₁,d_-M₁) and (M₂,d₊M₂,d_-M₂) be bordisms, and assume that d₊M₁ is diffeomorphic to d_-M₂. Then one can glue them together to obtain a bordism (M,d_-M₁,d₊M₂). I will discuss how the L²-analytic and L²-Reidemeister torsion of M is related to corresponding torsions of M₁ and M₂. The talk is based on the joint work with D.Burghelea and Th.Kappeler.

• March 21
  Speaker: Nikolai Nadirashvili (ETH, Zürich)
  Title: Recent Advances in Isoperimetric Inequalities
  Abstract: We discuss some recent results concerning isoperimetric inequalities for the eigenvalue problems on euclidean domains and manifolds.

• March 28
  Speaker: Vladimir Tulovsky (St.John University, New York)
  Title: Generalized Fourier Transform and Maslov's Canonical Operator
  Abstract: An attempt to develop a unified approach to PDE with constant and variable coefficients will be presented. The idea of the approach is to construct a complete system of eigenfunctions using Maslov's canonical operator.

• April 4
  Speaker:Alexandr Demidov (Moscow State University)
  Title: Some applications of the Helmholtz -- Kirchhoff method
  Abstract: We will discuss proofs of existence and non-existence of equilibrium forms of steady plasma in a given topological class. Inverse equilibrium problems, high-frequency asymptotics and Stokes --Leibenson problem for the Hele-Shaw flow will be briefly reviewed.

• 2:30-3:30 pm, April 11 (Note Special Time!)
  Speaker:Robert McOwen (Northeastern University)
  Title: Asymptotics and the Singular Yamabe Problem
  Abstract: An attempt to asymptotically describe the unique solution to the singular Yamabe problem, when the singular set is a submmmanifold with boundary and low codimension, leads to a linear analysis involving function spaces with two weights.
  

• 2:00-3:00 pm, May 2
  Speaker:Hyam Rubinstein (University of Melbourne, Australia)
  Title: Recent developments in decision problems for 3-manifolds
  

• May 23: Double Header!
  
  • 2:00-3:00 pm
    Speaker:Robin Forman (Rice University)
    Title: The Combinatorial Laplace Operator and Riemannian Geometry
    Abstract: Let M be a simplicial complex, or perhaps a more general cell complex. For each p there is a combinatorial Laplace operator which maps the space of p-cochains on M to itself. This operator can be thought of as a finite dimensional approximation of the Laplace operator acting on p-forms on a Riemannian manifold. In fact, we will show that the combinatorial Laplace operator surprisingly captures some of the subtle properties of the Riemannian Laplace operator. Along the way, we will show how one can define combinatorial analogues of Ricci curvature and the Witten Laplacian, which behave much like the corresponding Riemannian notions. We hope that this project will be the beginning of a more complete theory of combinatorial Riemmanian geometry.
    
  • 3:30-4:30 pm
    Speaker:Helen Moore (Bowdoin College)
    Title: Minimal Submanifolds with Finite Total Curvature
    Abstract: I will discuss the theory of minimal surfaces in R^3 with finite total Gauss curvature, as well as the analogous theory for minimal submanifolds with finite total scalar curvature. I am currently interested in existence results for minimal submanifolds, and the behavior of their Gauss maps, and will discuss my work on these problems.