We can consider the Szegö kernel S(z,z) on the boundary of model domains W = {(z₁,z₂) Î C² | - Áz₂ > b(Âz₁)}. If b is convex, the only singularities of S(z,z) are on the diagonal z=z. When the function b is a certain non-convex function, we show that near certain points, there are singularities off the diagonal.

We will then discuss some recent work on a related problem, the [`(¶)]_b problem for the model domain considered above.

This is joint work with Jennifer Halfpap.

Refinements of the Fourier restriction estimate have been used by Bourgain and others to prove mass concentration occurs in finite time blowup solutions of mass critical nonlinear Schrodinger equations. I'll revisit this argument and remark on certain quantifications and extensions.

For b < 1, the fractional Carleson condition for a measure m on the upper half-space Rⁿ⁺¹₊:

We show that the time evolution of the magnetic Schrodinger operator

This is a joint work with M. Goldberg and W. Schlag.

Fourier integral operators (FIOs) with singularities are used in inverse scattering theory. In such problems, artifacts can appear when inversion is attempted and one would like to understand precisely and eliminate them as far as possible. The operators considered here arise in linearized seismic imaging and synthetic aperture radar. The singularities which appear in this problems are folds, submersion with folds and cross caps. One would like to understand the composition of such operators since in general, the composition of two FIOs is not a FIO. We will establish a composition calculus for FIOs associated to folding canonical relations, working away from the fold points.

A geometric resolution of singularities algorithm for real-analytic functions is described. This method is elementary in its statement and proof, extensively using explicit coordinate systems. Each coordinate change used in the resolution procedure is one-to-one on its domain, and is of one of a few explicit canonical forms. Applications to classical analysis are given, including a theorem regarding the existence of critical integrability exponents.

We show how earlier work on counterexamples for the Calderón problem (i.e., whether a conductivity function or tensor s(x) on a domain W is determined by the boundary values of the solutions to Ñ·(sÑ)u = 0) can be extended to obtain rigorous results concerning invisibility (or "cloaking") for solutions of the Helmholtz and Maxwell equations.

This is joint work with Yaroslav Kurylev, Matti Lassas and Gunther Uhlmann.

We investigate the mapping properties of directional maximal operators on the plane with smooth densities. If vectors of all lengths in a given set of directions are taken in defining the maximal operator, then the boundedness of the operator depends on the order of the stationary points of the density function. In contrast, if the set of directions is limited to sums of diadic directions, and the density function has only finitely many stationary points, each of finite order, the maximal operator is bounded on all L^p for p > 1.

This is joint work with M. Roginskaya.

We investigate asyptotic behavior of the generalized eigenfunctions of discrete Schrodinger operator with potentials ¶b(n) Î l^p(Z), 1 < p < 2. Our approach is based on the work of Christ and Kiselev in the continuous case. The main new step is a development of discrete version of multilinear operator analysis and maximal function estimate.

Joint work with A. Kiselev.

We will show the global uniqueness for the Calderon's inverse problem for a piecewise linear, or more generally piecewise smooth across polyhedra, conductivity in R³ by using the geometry of Faddeev Green's function.

We consider Haar functions in the unit cube in three dimensions, normalized in L^¥. The question at hand is a `non-trivial' lower bound on the L^¥ norm of the sum

Joint work with Dmitry Bilyk.

A theorem of Sárközy and Furstenberg states that if A Í Z with positive upper Banach density, then the set of quadratic return times {d | AÇ(A+d²) ¹ Æ} is non-empty. Using Fourier analysis we give a new proof of the fact that the set of all quadratic return times is syndetic and obtain uniform lower bounds for the density of these return times.

This is joint work with Ákos Magyar.

A maximal theorem is proved for averages taken over suitable discrete sub-varieties of nilpotent Lie groups.

We show that an obstacle inside a known inhomogeneous medium can be determined from the scattering amplitude at one frequency. Moreover, we show that if the scattering operators for two obstacles are comparable at one non-resonant frequency, then the obstacles must coincide. The proof gives a new reconstruction procedure; it is based on an extension of the factorization method, which I'll review.

This is joint work with Lassi Paivarinta and Ari Teirila.

In joint work with Jennifer Halfpap and Stephen Wainger, we obtain estimates for the Bergman and Szegö kernels in domains of the form

A (d,k) set is a subset of R^d containing a translate of every k-dimensional plane. We use mixed-norm estimates for the x-ray transform to obtain improved dimension estimates for (d,k) sets.

A variant of the square function has applications to boundary value problems and to characterizing A^¥ weights.

In the 1950's Buck raised the question of whether or not there exists an infinite subset Z = {z_j}_j=1^¥ of D that is interpolating for H^¥(D). In 1958 Carleson gave an affirmative answer and characterized all such interpolating sequences in the disk. In 1961 Shapiro and Shields demonstrated the equivalence of this problem with certain Hilbert space analogues involving l²(m) where m = å_j k_zj(z_j)^-1d_zj and k_z is the reproducing kernel. Suppose that Z = {z_j}_j=1^¥ Ì D is separated and let R be the restriction map Rf = {f(z_j)}_j=1^¥. Then the results of Carleson and Shapiro and Shields show the following four conditions are equivalent: R maps H^¥(D) onto l^¥(Z); R maps H²(D) onto l²(m); R maps H²(D) into l²(m); m( T(I) ) £ C|I| for all arcs I Ì T. In 1995 Marshall and Sundberg extended this theorem to the Dirichlet space D(D) and its multiplier algebra M_D(D), but without the second condition. In fact, as observed by Bishop in 1995, even when the measure m is finite, R maps D(D) onto l²(m) for sequences Z more general than those for which R maps D(D) into l²(m). We give two results toward resolving the open question of when R maps D(D) onto l²(m): the first is a geometric characterization of such Z in the case m is a finite measure, and the second shows there are such Z with m an infinite measure, thus answering a question of Bishop.

We shall report recent joint work with Jong-Guk Bak and Daniel Oberlin on Fourier restriction for curves in R^d and related oscillatory integral operators.

We present a comprehensive study of the existence, blow up and regularity properties of solutions of the Burgers equation with fractional dissipation. We prove existence of the finite time blow up for the power of Laplacian a < 1/2, and global existence as well as analyticity of solution for a ³ 1/2. We also discuss solutions with very rough initial data.

In this talk I will give an overview of the global well-posedness results for the L² critical NLS below the H¹ norm. The recent result of Tao-Visan-Zhang proves global well-posedness in L² for radial data in higher dimensions d ³ 3. This problem is still open in lower dimensions and for nonradial data. In this talk I will present some recent results of global well-posedness that I obtained with De Silva, Pavlovic and Tzirakis for any data in H^s_d for a certain 0 < s_d < 1.

We present estimates for Bochner-Riesz means with respect to the R² ×R² cylinder variant { x, max{x₁² + x₂², x₃² + x₄²} = 1 }. The operator S^l to be studied is given by