Archived Talks from 1999-2000:

• May 19, 2000
  Speaker: Martin Bridgeman (Boston College)
  Title: Convex Hull boundaries of Hyperbolic 3-Manifolds
  Abstract: We define the average bending of a geodesic on the boundary of the convex hull of a quasi-fuchsian group and prove that it is universally bounded by a number K. We use this to prove that 1+K is a universal bound on the lipschitz constant for the map from the intrinsic hyperbolic structure on the convex hull boundary to the hyperbolic structure on the domain of discontinuity facing it. We further prove that the length of the bending lamination of the convex hull of a quasi-fuchsian group is bounded by K (pi)² times the euler characteristic of the underlying surface.

• May 12, 2000
  Speaker: Walter Strauss (Brown University)
  Title: Electromagnetic instabilities in a plasma
  Abstract: Given an evolution PDE that possesses equilibria, a fundamental question is: which ones are stable and which are unstable? In a plasma (using a kinetic formulation and ignoring collisions) there are many equilibrium solutions. The answer to the question depends on what kinds of perturbations are allowed, purely electrostatic or electromagnetic. Y. Guo and I have found a surprising result that some equilibria can be electro- statically stable, yet unstable when a magetic field is introduced. This partly solves a 40-year-old problem of Penrose. (No knowledge of physics will be required to understand this talk.)

• May 5, 2000
  Speaker: Andreas Kollross (University of Augsburg)
  Title: Polar Actions on Symmetric Spaces
  Abstract: An isometric action of a Lie group on a Riemannian manifold is called polar if there exists a submanifold which intersects all orbits orthogonally. Such a submanifold is then called a section. If the sections are flat, then the action is called hyperpolar. Examples of hyperpolar actions are given by isotropy actions of symmetric spaces. We give a classification of hyperpolar actions on irreducible compact symmetric spaces. This includes the classification of cohomogeneity one actions on these spaces.
  Recently, a classification of polar actions on rank one symmetric spaces has been given by Podesta and Thorbergsson. However, it remains an interesting open problem if there exist polar actions on irreducible symmetric spaces of higher rank with non-flat sections. We will explain some partial results in this direction.

• April 28, 2000
  Speaker: Megan Kerr (Wellesley College)
  Title: A deformation of noncompact Einstein solvmanfolds
  Abstract: A Riemannian manifold is said to be Einstein if it has constant Ricci curvature. We construct new examples of homogeneous Einstein manifolds of negative Ricci curvature. We restrict our attention to the case of Riemannian solvmanifolds, since every known example of a homogeneous Einstein manifold of negative Ricci curvature is a solvmanifold, that is, a solvable Lie group with a left-invariant metric. Beginning with a given Einstein solvmanifold, in order to construct nearby Einstein spaces, one must modify not the metric but rather the Lie bracket structure on the associated metric Lie algebra. We know from the work of Jens Heber that the moduli space of Einstein solvmanifolds near a given one may have very large dimension. Complementing Heber's existence results, we give an explicit description of a continuous family of Einstein manifolds with a positive dimensional parameter space, including a continuous subfamily of manifolds with negative sectional curvature.

• April 21, 2000
  Speaker: Barbara Keyfitz (University of Houston, visiting Brown)
  Title: Solving Hyperbolic Problems with Elliptic Methods
  Abstract: Although the theory of hyperbolic conservation laws in a single space variable (and time) has advanced considerably in the last few years, there has not been comparable progress in the theory of systems of conservation laws in several dimensions. Smooth solutions do not exist for all time, but the nature of weak solutions of these systems remains a mystery. This despite the fact that applications of conservation laws are important, and much of computational fluid dynamics involves simulation of hyperbolic systems.
  Recently, with Suncica Canic, I have been studying self-similar solutions of two-dimensional systems. We have obtained some interesting reductions, and have solved (that is, proved existence of solutions of) a prototype problem: one case of regular reflection of weak shocks. The governing equations change type, in a manner similar to steady transonic flow. One feature of our approach is that it leads to quasilinear degenerate elliptic equations with novel properties.
  In this talk, I will give some history of the approaches to solving conservation laws, describe the self-similar reduction, summarize our attack on the reduced problem and outline the case we have succeeded in solving.

• April 14, 2000
  Speaker: Huai-Dong Cao (Texas A&M)
  Title: On Kaehler-Ricci Solitons
  Abstract: Kaehler-Ricci solitons arise from the study of limiting behavior of Hamilton's Ricci flow on compact Kaehler manifolds. They are natural extension of Kaehler-Einstein metrics. In this talk we'll survey some of the recent developments on the subject and present a structure theorem of complete gradient Kaehler-Ricci solitons with positive Ricci curvature.

• April 7, 2000
  Speaker: Tatiana Suslina (St.Petersburg)
  Title: Absolute continuity of spectrum of the Schroedinger operator with delta-like potential supported by a periodic graph.
  Abstract: In L₂(R^d) we consider the Schroedinger operator H=(igrad+A(x))² +V(x) + s(x)D_S(x). Here S is a periodic system of curves on R², D_S is a delta-function supported by S, V(x) and A(x) are periodic functions on R² and s(x) is a periodic function on S. We prove the absolute continuity of the spectrum of operator H. This is a joint work with M.Birman and R.Shterenberg.

• March 31, 2000 (Regular Meeting 2-3 pm)
  Speaker: Alexander Turbiner (UNAM, Mexico)
  Title: Exactly-solvable Schroedinger equations and representation theory
  Abstract: A notion of an "exactly-solvable spectral problem" is introduced. It is shown that all A-B-C-D Perelomov-Olshanetsky quantum integrable multidimensional Hamiltonians (rational and trigonometric) emerge from a single quadratic polynomial in generators of the sl(n)-algebra. The coefficients of the polynomial keep memory about A-B-C-D origin of the problem we study. Corresponding spectral problems are exactly-solvable. Perturbations of above systems admitting an algebraic treatment are introduced.

• March 31, 2000: SPECIAL MEETING 4-5 pm
  Speaker: Alexander Gorban (Krasnoyarsk, visiting Northeastern)
  Title: Method of invariant manifolds for dissipative systems
  Abstract: The goal of nonequilibrium statistical physics is understanding how a system with many degrees of freedom becomes one with a small number of degrees of freedom. The method of invariant manifolds gives a new approach to the problem of a reduced description for dissipative systems. It is a "dissipative version" of the Kolmogorov-Arnold-Moser theory (KAM). Slow invariant manifolds are constructed by iterative Newton-type procedures of successive approximations. This method has been successful in several concrete problems, for example in the derivation of higher-order hydrodynamics from the Boltzmann equations. A related preprint is available on http://mystic.math.neu.edu/gorban/

• March 17, 2000
  Speaker: Izabella Laba (Princeton)
  Title: Recent progress on the Kakeya conjecture
  Abstract: A Besicovitch set is a subset of R^n which contains a unit line segment in each direction; the Kakeya conjecture states that such sets must have Minkowski and Hausdorff dimension n. This turns out to be relevant to a number of open problems in analysis, including the restriction conjecture and the Bochner-Riesz conjecture. In this talk I will report on some new partial results on the Kakeya conjecture: in particular, a combinatorial approach to the problem due to Bourgain (1998) and the recent work of Katz, Tao, and myself will be discussed.

• March 10, 2000
  Speaker: Natan Krugljak (Yaroslavl, Russia, visiting University of Illinois, Chicago)
  Title: On some recent results in Real Interpolation
  Abstract: Lions-Peetre reiteration formula and K-divisibility for finite tuples of Banach spaces will be discussed.

• March 3, 2000
  Speaker: Robert Brooks (Technion-Israel Institute of Technology)
  Title: Isoscattering
  Abstract: Two geometrically finite hyperbolic manifolds are said to be isoscattering if they have the same poles of the scattering operator. Being isoscattering is in a sense the infinite-volume analogue of being isospectral. We construct two hyperbolic manifolds which are isoscattering, and which have the same (conformally equivalent) boundaries at infinity, but which are nonetheless not isometric. (This is joint work with Ruth Gornet and Peter Perry)

• February 25, 2000
  Speaker: Thomas Puettmann (University of Pennsylvania)
  Title: Pinching Constants of Homogeneous Spaces with Positive Curvature
  Abstract: Pinching constants measure how much the geometry of a compact Riemannian manifold with positive sectional curvature deviates from the geometry of a standard sphere. The various sphere theorems are examples where pinching constants appear. In this talk we will speak about the optimal pinching constants for the homogeneous spaces of positive sectional curvature and how they can be determined by geometrical and computational methods.

• February 11, 2000
  Speaker: Walter Craig (Brown University)
  Title: Traveling water waves
  Abstract: This talk describes an existence theorem for spatially periodic traveling wave solutions for gravity and capillary-gravity water waves in two dimensions, and capillary-gravity water waves in three dimensions. This is a problem in bifurcation theory, yielding curves in the two dimensional case and bifurcation surfaces in the three dimensional case. In order to address the presence of resonances, the proof is based on a variational formulation and a topological argument, which is related to the resonant Lyapunov center theorem.

• January 28, 2000
  Speaker: Rob Kusner (UMass, Amherst)
  Title: Classifying complete embedded constant mean curvature surfaces
  Abstract: Soap bubbles and fluid interfaces are physical systems modeled by embedded constant mean curvature (cmc) surfaces. The round sphere and cylinder are two well-known examples in a family of complete embedded cmc surfaces of revolution, the unduloids, which can be described via an ordinary differential equation. Any cmc surfaces beyond this family require analysis of partial differential equations closely related to the minimal surface equation in the three sphere. In recent work (joint with K. G. Brauckmann and J. M. Sullivan) we classify all the complete embedded cmc surfaces of trousers (thrice-punctured sphere) topology. These "triunduloids" can serve as building blocks for complete, embedded cmc surfaces of all other topological types. (If time permits, we will also briefly discuss a loop-group approach to these problems, and show a short GANG video by N. Schmitt illustrating these surfaces and their moduli.)

• January 14, 2000
  Speaker: Maxim Braverman (Northeastern Univ)
  Title: Index theory for equivariant Dirac operators on a non-compact manifold
  Abstract: I will present a new generalized version of an index of a Dirac operator on a complete Riemannian manifold endowed with an action of a compact Lie group G. I show that this index is an invariant of a non-compact cobordism of the type considered by V. Guillemin, V. Ginzburg and Y. Karshon. Combined with the fact that any G-manifold is cobordant to the normal bundle to the fixed point set, it leads to a new proof of the Atiyah-Segal-Singer equivariant index theorem. I also obtain a generalization of this theorem to non-compact manifolds. If the time permits, I will discuss applications to geometric quantization.

• December 3, 1999
  Speaker: Franz Pedit (UMass, Amherst)
  Title: Quaternionic holomorphic geometry and the Willmore problem
  Abstract: I will explain the notion of quaternionic holomorphic vector bundles over Riemann surfaces, their invariants and basic properties: Riemann-Roch, Pluecker relations etc. The basic objects of this theory are conformal maps into the 4-sphere, which should be seen as the analogs of holomorphic maps into the Riemann sphere in the complex case. Quaternionic holomorphic structures carry an energy which has to do with the Willmore functional on immersed surfaces. I will conclude by indicating a few applications to conformal surface theory, e.g. how the quaternionic Pluecker relations give lower bounds on the Willmore energy of an immersed surface.

• November 19, 1999
  Speaker: Michael Gruber (MIT)
  Title: Kadison property and band spectrum for magnetic Schrödinger operators
  Abstract: The spectrum of a periodic Schrödinger operators has band structure, as ordinary Bloch (or Floquet) theory shows. Adding a constant magnetic field to get a magnetic Schrödinger operator in general breaks the symmetry with respect to translations, but it is now periodic w.r.t. to "magnetic" translations which in general do not form a group. We describe how to use these translations to set up a non-commutative version of the ordinary Bloch-theory, and we relate band structure to a property of the C*-algebra generated by the magnetic translations, the Kadison property.

• November 5, 1999
  Speaker: Chris King (Northeastern University)
  Title: Minimal entropy conjectures for stochastic maps
  Abstract: A review of recent work on classification of stochastic maps on C². Includes a conjecture on the additivity of minimal entropy for such maps.

• October 29, 1999
  Speaker: Chuu-Lian Terng (Northeastern University)
  Title: Integrable systems, loop groups, and Virasoro algebra
  Abstract: This talk concerns some joint work with K. Uhlenbeck. Many integrable systems naturally arise from loop group factorizations, for example, KdV, SGE, NLS, n-wave equation, and isothermic surfaces in R³. Explicit solutions of these equations can be constructed from rational loops using residue calculus. We identify the properties of loops for rapidly decay solutions, algebraic geometry solutions, and local analytic solutions. Symmetries of these equations will also be discussed.

• October 22, 1999
  Speaker: Luchezar Stoyanov (Univ. of Western Australia)
  Title: On inverse scattering by obstacles
  Abstract: Given an obstacle K in Rⁿ (n > 2 and odd) with smooth boundary, it is known that the singularities of the scattering kernel are related to the so called scattering length spectrum -- the set of sojourn (travelling) times of scattering rays in the exterior of the obstacle. So, the problem of finding geometric information about the obstacle from the singularities of the scattering kernel can be stated in a simple geometric form (which makes sense in any dimension). Some properties of obstacles that can be recovered from the scattering length spectrum (SLS) will be discussed. It turns out that in some special classes of obstacles the SLS provides enough information to completely recover the obstacle.

• October 15, 1999
  Speaker: Mikhail Shubin (Northeastern)
  Title: Discreteness of spectrum for magnetic Schrödinger operators
  Abstract: The spectrum of a magnetic Schrödinger operator can be discrete due to the growth of the electric potential alone, but also due to a regular growth of magnetic field because of non-commutativity of magnetic translations which can be used in an uncertainty principle type argument. I will explain new conditions of the discreteness of spectrum which combine influence of electric and magnetic fields by introduction of effective potentials. (These new results are joint with V.Kondrat'ev.)

• October 8, 1999
  Speaker: Robert McOwen (Northeastern)
  Title: Singular Sturm-Liouville Theory on Manifolds
  Abstract: In this talk we investigate Schrödinger operators L on a compact Riemannian manifold (M,g), where the potential function a(x) is assumed to be continuous, but not necessarily bounded, outside of some closed set F of measure zero in M. Under certain geometric hypotheses on F and growth conditions on a(x) as x approaches F, we prove that the Dirichlet extension of L is bounded from below with discrete spectrum; in many cases, a(x) is allowed to approach negative infinity as x approaches F. We also consider conditions on F and a(x) under which the Sturm-Liouville theory of L is "singular'" in that no boundary conditions are needed to specify the eigenvalues and eigenfunctions of L; in particular, this occurs when the domain of L does not depend on boundary conditions, for example, when L is essentially self-adjoint or more generally "essentially Dirichlet" (a new property that we define). The behavior of L on weighted Sobolev spaces is discussed; this is required in the analysis of essential self-adjointness. In most of the talk we assume that F is a k-dimensional submanifold without boundary, but in the last section we generalize our results to stratified sets.

• October 1, 1999
  Speaker: Misha Shubin (Northeastern)
  Title: Discreteness of spectrum for Schrödinger operators
  Abstract: In 1953 A. Molchanov gave a necessary and sufficient condition for the discreteness of spectrum (or, equivalently, compactness of the resolvent) for the Schrödinger operator with a semibounded below potential in Rⁿ. This condition is formulated in terms of the potential and uses Wiener capacity.
  In the talk the Molchanov criterion and its recent generalization to manifolds of bounded geometry (which is due to V. Kondrat'ev and the speaker) will be explained together with all necessary preliminaries on capacities.