Department of Mathematics
Northeastern University
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 Sm 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 Sm 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.
Last modified June 4, 1997
Comments or corrections to:
mcowen@neu.edu