Functional Analysis/Operator Algebras Seminar, Spring 1999

Upcoming talks

• Date: 4/16/99
  • Speaker: Henrik Shahgholian, Royal Institute of Technology, Stockholm
  • Title: On the porosity of free boundaries in degenerate variational inequalities
  • Abstract: In this talk we consider a certain degenerate variational problem with zero constraint. The exact growth of the solution near the free boundary will be established. As a consequence of this we show that the free boundary is porous and therefore its Hausdorff dimension is less than $N$ and hence it is of Lebesgue measure zero.
    
    Keywords: Obstacle problem, nonhomogeneous $p$-Laplace equation, free boundary, porosity.

• Date: 4/22/99, Mathematics Colloquium, SH 6635, 3:30-4:30pm
  • Speaker: Hans Wenzl, UCSD
  • Title: Braids, Categories and Invariants
  • Abstract: Classically, symmetric groups were used to describe the decomposition of tensor products of representations of the general linear group. A similar and more general interplay exists between braid groups and quantum groups. This way one obtains an elementary construction of fusion categories (for classical Lie types), which in turn can be used to construct invariants of links and 3-manifolds, and subfactors.

• Date: 4/23/99
  • Speaker: Hans Wenzl, UCSD
  • Title: Tensor categories for exceptional Lie groups
  • Abstract: While tensor categories related to classical Lie groups have been well studied, the ones for exceptional Lie groups still are quite mysterious. Inspired by Vogel's work on finite type link invariants, Deligne proposed to study the 5 exceptional Lie groups simultaneously in an `exceptional series'. We describe his evidence for such a series, and how one can obtain additional information from braid groups by extending this approach to the quantum setting.

• Date: 4/30/99
  • Speaker: Dan Voiculescu, UC Berkeley
  • Title: Topics in free entropy : liberation and mutual free information

• Date: 5/3/99, Advancement to Candidacy, 1:00-3:00 pm, SH 4607 !!
  • Speaker: Anne Louise Svendsen
  • Title: Analytical properties of the standard invariant for subfactors

• Date: 5/7/99
  • Speaker: Joerg Eschmeier, Universitaet Saarbruecken
  • Title: Invariant subspaces for commuting contractions
  • Abstract: A classical result of Brown, Chevreau and Pearcy shows that each contraction on a Hilbert space with dominating spectrum in the open unit disc possesses a non-trivial closed invariant subspace. As an application of the Scott Brown technique we prove corresponding invariant-subspace results for commuting systems of contractions and for spherical contractions. We indicate that subnormal n-tuples with rich spectrum in the unit ball or unit polydisc are reflexive.

• Date: 5/14/99
  • no meeting

• Date: 5/21/99
  • Speaker: Sorin Popa, UCLA
  • Title: Universal constructions of subfactors

• Date: 5/28/99
  • Speaker: Peyman Milanfar, SRI - Stanford
  • Title: Connections Between Inverse Problems, Moments, and Array Signal Processing
  • Abstract: Much of what we learn about the world is measured indirectly. Inverse problems are termed as such because they are concerned with solving for the "cause" of a measured physical "effect". Inverse problems arise in many diverse areas of science and engineering, including geophysical exploration, astronomy, medicine, and computational vision. In this talk, I will explore a class of inverse problems and show how the respective measurements can be distilled into moments of the underlying quantities we seek to reconstruct. Hence, these measurements can directly yield geometric information about the underlying object(s) of interest. In turn, I will discuss some deep connections between the problem of inverting a shape or function from its moments and two apparently disparate areas: 1) numerical quadrature in the plane, and 2) array signal processing.
    
    As a result of these observations, I will demonstrate, in two different contexts -- computed tomography and gravimetric inversion -- how the shape of a region may be reconstructed from X-ray attenuation or gravitational field measurements respectively, using stable and efficient numerical algorithms akin to those used in array signal processing. In this way, both new physical interpretations and new algorithms for these inverse problems are derived.

• Date: 6/4/99
  • Speaker: Bruce Reznick, UIUC
  • Title: Cones of polynomials in several variables
  • Abstract: We will discuss some closed convex cones of real homogeneous polynomials in several variables (e.g. the cone of psd forms, the cone of sums of squares, the cone of sums of powers of linear forms), as well as what appears to be the "right" inner product to place on them. Topics touched on will include Hilbert's 17th Problem, Moment Problems and the still-open question of determining which real polynomials in a single variable can be written as a sum of fourth powers of polynomials. Partial result: there exist real $(a_k,b_k,c_k)$ so that $t^8 + ct^4 + 1 = \sum_k(a_kt^2 + b_kt + c_k)^4$ if and only if $c \ge -14/9$.