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Series: Analysis Seminar

There is a large literature to study the behavior of the image curves f(\partial {\mathbb D}) of analytic functions f on the unit disc {\mathbb D}. Our interest is on the class of analytic functions f for which the image curves f(\partial {\mathbb D}) form infinitely many (fractal) loops. We formulated this as the Cantor boundary behavior (CBB). We develop a general theory of this property in connection with the analytic topology, the distribution of the zeros of f'(z) and the mean growth rate of f'(z) near the boundary. Among the many examples, we showed that the lacunary series such as the complex Weierstrass functions have the CBB, also the Cauchy transform F(z) of the canonical Hausdorff measure on the Sierspinski gasket, which is the original motivation of this investigation raised by Strichartz.

Series: Analysis Seminar

This talk is about a random Schroedinger operator describing the
dynamics
of an electron in a randomly deformed lattice. The periodic displacement
configurations
which minimize the bottom of the spectrum are characterized. This leads to
an
amusing problem about minimizing eigenvalues of a Neumann Schroedinger
operator
with respect to the position of the potential.
While this conﬁguration is essentially unique for dimension greater than
one, there are inﬁnitely many different
minimizing conﬁgurations in the one-dimensional case.
This is joint work with Jeff Baker, Frederic Klopp, Shu Nakamura and Guenter
Stolz.

Series: Analysis Seminar

Calderon's algebra can be thought of as a world whichincludes singular integral operators and operators of multiplicationwith functions which grow at most linearly (more precisely, whose firstderivatives are bounded).The goal of the talk is to address and discuss in detail the followingnatural question: "Can one meaningfully extend it to include operatorsof multiplication with functions having polynomial growth as well ?".

Series: Analysis Seminar

I will discuss linear extremal problems in the Bergman spaces $A^p$ ofthe unit disc and a theorem of Ryabykh about regularity of thesolutions to these problems. I will also discuss extensions I havefound of Ryabykh's theorem in the case where $p$ is an even integer.The proofs of these extensions involve Littlewood-Paley theory and abasic characterization of extremal functions.

Series: Analysis Seminar

I will survey recent results about the convergence of the Wash-Fourier series near L1. Joint work with Michael Lacey.

Series: Analysis Seminar

Let C[-1, 1] be the space of continuous functions on [-1, 1], and denote by \Delta^2
the set of convex functions f \in C[-1, 1]. Also, let E_n(f) and En^{(2)}_n(f) denote
the degrees of best unconstrained and convex approximation of f \in \Delta^2 by
algebraic polynomials of degree < n, respectively.
Clearly, E_n(f) \le E^{(2)}_n (f), and Lorentz and Zeller proved that the
opposite inequality E^{(2)}_n(f) \le CE_n(f) is invalid even with the constant
C = C(f) which depends on the function f \in \Delta^2.
We prove, for every \alpha > 0 and function f \in \Delta^2, that
sup{n^\alpha E^{(2)}_n(f) : n \ge 1} \le c(\alpha)sup{n^\alpha E_n(f): n \ge 1},
where c(\alpha) is a constant depending only on \alpha.
Validity of similar results for the class of piecewise convex functions having
s convexity changes inside (-1,1) is also investigated. It turns out that there
are substantial differences between the cases s \le 1 and s \ge 2.

Series: Analysis Seminar

The Seiberg-Witten equations, introduced by Edward Witten in 1994, are a first-order semilinear geometric PDE that have led to manyimportant developments in low-dimensional topology. In this talk,we study these equations on cylindrical 4-manifolds with boundary, which we supplement with (Lagrangian) boundary conditions that have a natural Morse-Floer theoretic interpretation. These boundary conditions, however, are nonlinear and nonlocal, and so the resulting PDE is highlyunusual and nontrivial. After motivating and describing the underlying geometry for the Seiberg-Witten equations with Lagrangian boundary conditions, we discuss some of the intricate analysis involved in establishing elliptic regularity for these equations, including tools from the pseudodifferential analysis ofelliptic boundary value problems and nonlinear functional analysis.

Series: Analysis Seminar

When Calderón studied his commutators, in connection with the Cauchy
integral on Lipschitz curves, he ran into the bilinear Hilbert
transform by dropping an average in his first commutator. He posed the
question whether this new operator satisfied any L^p estimates. Lacey
and Thiele showed a wide range of L^p estimates in two papers from 1997
and 1999. By dropping two averages in the second Calderón commutator
one bumps into the trilinear Hilbert transform. Finding L^p estimates
for this operator is still an open question.
In my talk I will discuss L^p estimates for a singular integral
operator motivated by Calderón's second commutator by dropping one
average instead of two. I will motivate this operator from a historical
perspective and give some comments on potential applications to partial
differential equations motivated by recent results on the water wave
problem.

Series: Analysis Seminar

The quest for a suitable geometric description of major analyticproperties of sets has largely motivated the development of GeometricMeasure Theory in the XXth theory. In particular, the 1880 Painlev\'eproblem and the closely related conjecture of Vitushkin remained amongthe central open questions in the field. As it turns out, their higherdimensional versions come down to the famous conjecture of G. Davidrelating the boundedness of the Riesz transform and rectifiability. Upto date, it remains unresolved in all dimensions higher than 2.However, we have recently showed with A. Volberg that boundedness ofthe square function associated to the Riesz transform indeed impliesrectifiability of the underlying set. Hence, in particular,boundedness of the singular operators obtained via truncations of theRiesz kernel is sufficient for rectifiability. I will discuss thisresult, the major methods involved, and the connections with the G.David conjecture.

Series: Analysis Seminar

In this talk, we will discuss the theory of Hardy spacesassociated with a number of different multiparamter structures andboundedness of singular integral operators on such spaces. Thesemultiparameter structures include those arising from the Zygmunddilations, Marcinkiewcz multiplier. Duality and interpolation theoremsare also discussed. These are joint works with Y. Han, E. Sawyer.