Scheduled talks at Analysis Seminar - fall 2008

See the deparmental seminar calendar for additional information.

Sep. 03         Organizational Meeting

Sep. 09         No seminar

Sep. 15         Ilya Spitkovsky (College of William and Mary)
        On the current state of the factorization problem for almost periodic
        matrix functions

Abstract: Factorization of almost periodic (AP) matrix functions arises
naturally in a variety of problems, both theoretical and applied, and
for many of them the matrix in question is 2-by-2 and triangular. Even in
this setting the factorability properties remain a mystery, in striking
difference with both the scalar almost periodic case and with purely
periodic matrix case.  We will give a survey of currently available
existential results regarding the factorization of general AP matrix
functions, as well as available approaches to constructive factorization
of the specific matrices mentioned above.

Sep. 22         John Wermer (Brown U.)
        Analytic disks and projective hulls

Abstract: An analytic disk F in a complex manifold X is defined to be
an analytic map of the unit disk into X.  In the 1980s, E. Poletsky
characterized the polynomial hull of a compact set in C^n in terms of
certain analytic disks.  Larusson and Sigurdsson expanded on Poletsky's
method for some problems in complex potential theory.  In this talk,
we shall give applications by Blaine Lawson and the speaker of this work
to the problem of characterizing projective hulls in C^n.

Sep. 29         Sergei Treil (Brown U.)
        H^1 and dyadic H^1

Abstract: I will present a simple proof of the fact that the average
over all dyadic lattices of the dyadic $H^1$-norm of a function gives
an equivalent $H^1$-norm.  The proof works for both one-parameter and
multi-parameter Hardy spaces.

The results of such type are known, and, by duality, such results are
equivalent to the ``BMO from dyadic BMO'' statements.

The main idea of treating square function as a Calderon-Zygmund operator
is a commonplace in harmonic analysis; the main observation, on which the
paper is based, is that one can treat the random dyadic square function
this way.  After that, all is proved by using the standard and well-known
results about Calderon-Zygmund operators in the Hilbert-space-valued
setting.

Oct. 10         John Anderson (Holy Cross)
        Special Friday Seminar: 4 pm, Kassar 105

        Products of H^1 and BMO functions

Abstract: In 1971 Charles Fefferman identified the dual of the Hardy
space H^1 with the space BMO of functions of bounded mean oscillation.
The dual pairing is somewhat subtle, because the product of an H^1
function and a BMO function need not be integrable.  In a recent paper
Bonami, Iwaniec, Jones and Zinsmeister (BIJZ) give meaning to this product
as a distribution.  In the setting of the unit disk, they identify the
product of an H^1 function and an analytic BMO function with a function
in a  certain Hardy-Orlicz space.  I will give an exposition of this
and other results in the BIJZ paper.

Oct. 13         No seminar (Columbus Day)

Oct. 20         Brett Wick (U. South Carolina)

Oct. 27         Todd Kemp (MIT)

Nov. 03         Michael Bateman (Indiana U.)

Nov. 10         Lillian Pierce (Princeton U.)

Nov. 17         Justin Holmer (Brown U.)