Abstracts for some talks (Spring Semester 2009) Tuesday, January 6, 3:00 p.m., VV B139 Betsy Stovall (UCLA) Quasi-extremals for convolution with surface measure on the sphere Let $T$ be the operator which maps functions on $\mathbb{R}^d$ to their convolution with surface measure on the unit sphere. Then a classical result from harmonic analysis states that there exists a constant $C$ so that for any functions $f,g \in L^{(d+1)/d}(\mathbb{R}^d)$, \[ |\langle Tf, g \rangle | \leq C \|f\|_{L^{(d+1)/d}} \|g\|_{L^{(d+1)/d}}. \] But what happens if we try to reverse this inequality? Can we describe the pairs of functions $f,g \in L^{(d+1)/d}(\mathbb{R}^d)$ satisfying \[ | \langle Tf,g \rangle | \geq \varepsilon \|f\|_{L^{(d+1)/d}} \|g\|_{L^{(d+1)/d}}? \] In this talk, we will give a partial answer to this question, sketch its proof, and highlight the differences between convolution with surface measure on the sphere and convolution with surface measure on the paraboloid. Friday, January 9, 2:00 p.m., VV B139 Alexander Fish (Ohio State) Rigidity of measures on the torus invariant under the action of a "large" multiplicative semigroup Abstract: We prove that if a Borel probability measure on the circle group is invariant under the action of a "large" multiplicative semigroup (lower logarithmic density is positive) and the action of the whole semigroup is ergodic then the measure is either Lebesgue or has finite support. Tuesday, January 20, 4:00 p.m., VV B139 Maxim Zinchenko (Caltech) A Criterion for Purely Absolutely Continuous Measures on Homogeneous Sets Abstract: Let $E$ be a homogeneous subset of $\mathbb{R}$ in the sense of Carleson. Let $\mu$ be a finite positive measure on $\mathbb{R}$ and $H_\mu(x)$ its Hilbert transform. We prove that $\lim_{t\to\infty} t|E\cap\{x : |H_\mu(x)|>t\}|=0$ if and only if $\mu_s(E)=0$, where $\mu_s$ is the singular part of $\mu$. This is joint work with Alexei Poltoratski and Barry Simon. Tuesday, February 24, 4:00 p.m., VV B139 Jae-Seong Cho (Purdue) Sharp Subelliptic Estimates for the dbar-Neumann Problem on a special class of domains in C^n Abstract : Subelliptic Estimates guarantee solvability and regularity of the dbar-Neumann problem, the problem of finding a solution to the Kohn-Laplacian equation. After presenting basic concepts and previous results, we will discuss how to obtain a good estimate on a special class of pseudoconvex domains. Tuesday, February 24, 4:00 p.m., VV B139 Alexander Barvinok (University of Michigan) Approximating norms by polynomials and convex bodies by ellipsoids Abstract: I'll describe a general construction which allows one to approximate a given norm in an n-dimensional vector V space by a an expression of the type p^{1/2k}, where p: V --> R is a polynomial of degree 2k. The approximation is within a factor of c(n, k)={n+k-1 choose k-1}^{1/2k} and it is not clear whether the approximation is optimal except in the case of k=1 when p is a quadratic form and p^{1/2} is a Euclidean norm. It is also not clear whether polynomial p can be chosen in such a way that p^{1/2k} is itself a norm (again, apart from the classical case of k=1). I'll also describe some special cases where the approximated norm possesses a large symmetry group, in which case p^{1/2k} is a variation of an L^2k norm. Tuesday, March 3, 4:00 p.m., VV B139 Nets Katz (Indiana University) Discrete analogs of the Kakeya problem Abstract: (joint with L. Guth) Using the algebraic method of Dvir, we solve the Joints problem, proving that any set of $N$ lines in $R^3$ describe at most $O(N^{{3 \over 2}})$ joints, where a joint is a point at which three of the lines intersect in a noncoplanar fashion. We solve a related problem of Bourgain. Wednesday, Match 11 Elizabeth Wulcan (Michigan) Sparse effective Nullstellensaetze via residue currents ABSTRACT: Residue currents are generalizations of one complex variable residues and can be thought of as currents representing ideals of holomorphic functions or polynomials. I will discuss how residue currents on toric varieties can be used to obtain certain effective versions of Hilbert's Nullstellensatz; this is work in progress. Tuesday, March 24, 4:00 p.m., VV B139 Sergei Treil (Brown University) Abstract: I will present a simple proof of the fact that the average over all dyadic lattices of the dyadic $H1$-norm of a function gives an equivalent $H1$-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 everything 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. Tuesday, March 31, 4:00 p.m., VV B139 Christian Remling (University of Oklahoma) The absolutely continuous spectrum of Jacobi matrices Abstract: I would like to discuss general structural properties of the absolutely continuous spectrum of difference operators in one space dimension. It turns out that there are universal building blocks with very peculiar properties that must be used to generate any kind of absolutely continuous spectrum. I'll report on this, and, if time permits, also subsequent attempts, joint with Alexei Poltoratski, to analyze these in more detail. Tuesday, April 21,4:00 p.m., VV B139 Ciprian Demeter (Indiana University) Title: Multilinear processes with fractional rank Abstract: We investigate averages from Additive Combinatorics (relevant to detecting patterns, such as arithmetic progressions), and Harmonic Analysis, which exhibit fractional rank. A relation between rank and dimension is established, that guarantees that a given process has Fourier complexity. The argument is a mixture of Combinatorics and multi-scale Analysis. (joint with Christoph Thiele and Malabika Pramanik). Tuesday, April 28, 4:00 p.m., VV B139 Markus Keel (University of Minnesota) Title: Resonant decompositions and almost conservation laws for dispersive PDE in higher dimensions. Abstract: This is joint work with J. Colliander, G. Staffilani, H. Takaoka, and T. Tao, on the global well-posedness of the defocusing cubic Nonlinear Schroedinger Equation in two space dimensions from relatively rough initial data. The proof relies on an almost conserved quantity constructed using multilinear correction terms. The main new difficulty is to control the contribution of resonant interactions to these correction terms, which is significant due to the multidimensional setting of the problem.