Classical Analysis in One and Several Complex Variables

Celebrating the work of John Wermer

Friday, October 23 - Sunday, October 25, 2009

University of North Carolina, Chapel Hill


One hour lectures by:

  • Jean-Pierre Rosay, University of Wisconsin

    • TITLE: `Uniqueness in Rough Almost Complex Structures and Differential Inequalities.'

      ABSTRACT: The equation of J-holomorphicity leads to differential inequalities. Questions under investigation are classical ones: Do vanishing on an open set, or vanishing to infinite order, or having a non-isolated zero imply vanishing? Do these properties depend on the equations of J-holomorphicity or are they mere consequences of differential inequalities? Among the surprising facts: results for vector valued maps and for functions are different and results of uniqueness (that are unexpected in view of the failure of uniqueness, even for O.D.E.'s) are still true for non Lipschitz continuous structures.

  • Eugene Poletsky, Syracuse University

    • TITLE: Wermer's example and a quest for an analytic disc replacement.

      ABSTRACT: Wermer's example of a compact set whose non-trivial polynomial hull contains no analytic discs reappears any time when the existence of such a disc is conjectured. This raises a question: What kind of structure is suited to replace analytic discs.My talk will contain the review of old results and, also, recent results in this direction.

  • Thomas Ransford, Université Laval

    • TITLE: Computation of capacity

  • Mike Jury, University of Florida

    • TITLE: Schur class functions on the ball in C^n.

Half hour lectures by:

  • James Brennan, University of Kentucky

    • TITLE: Analytic capacity and some problems in approximation theory
      ABSTRACT: Let X be a compact subset of the complex plane and let dA denote two-dimensional Lebesgue (or area) measure. For each p, 1 < p < \infty, let Hp(X; dA) and Rp(X; dA) be the closed subspaces of Lp(X; dA) that are spanned by the polynomials and the rational functions having no poles on X, respectively. Likewise, C(X) is the usual space of continuous functions on X endowed with the uniform norm, and R(X) is the closed subspace of C(X) generated by the rational functions.  The following two questions date from the late 1960's: If R(X) not equal C(X) can it happen that (1) Rp(X; dA) = Lp(X; dA) for all p? (2) Hp(X; dA) = Lp(X; dA) for all p? The answer to the first has been known for more than forty years, whereas the second has only recently been settled. It is my intention to indicate the connection between these two problems, the presence of non-peak points in R(X), and the existence of certain types of representing measures. The results depend in an essential way on the Vitushkin scheme for approximation, and on the more recent work of Tolsa on the semiadditivity of analytic capacity. They represent, in part, joint work with Erin Militzer. Department of Mathematics, University of Kentucky, Lexington, KY 40506 E-mail address: brennan@ms.uky.edu

  • Andrew Browder, Brown University

  • Brian Cole, Brown University

  • Erlend Fornaess-Wold, University of Oslo, Norway

  • Alexander Izzo, Bowling Green State University

    • TITLE: The impact of a paper of Wermer.

  • Norm Levenberg, Indiana University

    • TITLE: Recent Results in Pluripotential Theory

  • Dmitry Khavinson, University of South Florida

    • TITLE: On a Uniqueness Property of Harmonic Functions

      ABSTRACT: We shall discuss the problem of uniqueness for functions u harmonic in a domain G  in R^n and vanishing on some parts of the intersection  V
      (not necessarily connected) of   G with a line m. The question originated more than a decade ago with N. Nadirashvili (private communication).  For example, let G be a spherical shell, i.e., the  region between two concentric spheres, and m is a line through the origin. Does u vanish on both segments along which m intersects G if it does so on one of them? To illustrate the depth of the question note that if you let G to be the annulus with a sector cut out, the function u= arg z in the plane does vanish on the positive part of the real axis, but not on the whole intersection. What happens if G is a spherical shell but m does NOT pass through the center? What if we replace harmonic functions by polyharmonic functions, or, more generally, solutions of analytic elliptic equations, or even worse, by linear combinations of Riesz potentials that satisfy no PDE altogether? The answers are by no means obvious and, in some cases, may be judged as surprising.

  • Romain Dujardin,(tentative) Paris 7

    • TITLE: Wermer examples and currents

      ABSTRACT:  I'll show how the classical example of J. Wermer of a polynomial hull without analytic structure allows to construct pathological, almost smooth, solutions to the homogeneous complex Monge-Ampère equation in the unit ball of C2. reference : http://arxiv.org/abs/0904.4179

  • Michael Stessin, (tentative) University of Albany

    • TITLE: Spectra of Toeplitz operators and regularity of optimal recovery problems

Schedule:  All talks will be in Phillips 332.  The registration and teas will take place in the adjacent lounge, Phillips 330.

Friday -
 
1:30-2:00
 A. Browder
2:30-3:00
 D. Khavinson
3:00-3:30
 Tea
3:30-4:30
 B. Lawson
5:30-7:30
 Banquet

Saturday -

9:00-9:30
 J. Brennan
10:00-10:30
 R. Dujardin
11:00-12:00
 T. Ransford
1:30-2:20
 J. P. Rosay
2:30-3:00
 M. Stessin
3:10-3:40
 E. Fornaess-Wold
3:50-4:20
 B. Cole
4:30-5:20
 M. Jury
8:00
 Beer and Chips

Beer and Chips at J. Cima's home.
507 Weaver Mine Trail (map)
 Sunday -
 
9:00-9:30
 N. Levenberg
10:00-10:30
 A. Izzo
11:00-12:00
 E. Poletsky


Banquet:  There will be a banquet to be held at 411 West (411 West Franklin St., 919-967-2782).  It will begin at 5:30.  The cost is roughly $45.00 per person.  Map

Lodging:  There are a block of rooms available at a group rate at the Holiday Inn in Chapel Hill.  Please call 1-888-452-5765 or 919-929-2172.  The rooms are $102.00/night (including tax) for either king size bed or two double beds.  The hotel has a shuttle service that will bring guests from the hotel to campus.  Please inquire at the front desk.

Note that there is a home football game on Thursday, October 22.  So it may be difficult to get rooms on that evening.

Some info on Chapel Hill can be found here, including information on the free bus system and a restaurant guide.  The airport is RDU (Raleigh-Durham International Airport).  It is about 20-25 miles from town.  There are vans from the airport which cost about $40.00 one way.  Transportation can also be provided by Chapel Hill-Durham Executive Express Taxi 919-233-6798.  The weather in Chapel Hill in October is usually pleasant with mild temperatures and perhaps a bit of rain.

Organizers:  Joseph Cima, N. Levenberg, J. Anderson

Some local information:  Campus maps, Visiting the UNC math