Abstracts for some talks ( Fall Semester 2009) Tuesday, September 8, 4:00 p.m.: Carlos Kenig (University of Chicago) Compact radial solutions to energy-supercritical nonlinear wave equations, with applications. Abstract: We will discuss a decay estimate for radial solutions to energy-supercritical nonlinear wave equations in R^3, which exist for all time and are precompact in the critical Sobolev space, modulo scalings. An application to scattering of bounded (in the critical space) radial solutions, for the defocusing equation, will be explained. This is joint work with F.Merle. Tuesday, September 15, 4:00 p.m., VV B139 VV. Christoph Thiele (UCLA) A conjecture about Calderon Zygmund decompositions for multiple frequencies and application to an extension of a Lemma of Bourgain. COLLOQUIUM: Friday, September 18, 4:00 p.m. B 239 VV. Variations and Applications of Carleson's theorem Abstract: Carleson's theorem of the 1960s states that a Fourier series with square summable coefficients converges almost everywhere. In the past ten years this theorem has been much studied and generalized, and its variants have applications to singular integrals along vector fields and to theorems in ergodic theory. We will present the theorem and elaborate on its variants and applications. Monday, September 21, 2;30 p.m. Alexander Eremenko (Purdue) Title: Meromorphic functions with linearly distributed values and Julia sets of rational functions. Abstract: We prove that if the Julia set of a rational function is contained in a smooth curve, then it is a subset of a circle. Tuesday, September 29, 4:00 p.m., B139 VV Alexei Poltoratski (Texas A M) will speak about Title: Entire Functions and Gap Theorems. Abstract: In my talk I will discuss solutions to two problems of classical analysis obtained using an approach recently developed in our joint papers with Nikolai Makarov. A sequence of real numbers is called a Polya sequence if any entire function of exponential type zero that is bounded on that sequence is a constant. The first problem that I will discuss is an old problem by Polya and Levinson that asks for a description of such sequences. This part is based on joint work with my student Mishko Mitkovski. The second problem is the Beurling's gap problem. If $X$ is a closed set on the real line, denote by $G_X$ the supremum of the size of the gap in the support of the Fourier transform of $\mu$, taken over all non-trivial complex measures $\mu$ supported on $X$. I will present a formula for $G_X$ in terms of metric characteristics of $X$. Tuesday, October 6, 4:00 p.m., B139 VV Keith Rogers (Madrid) Mass concentration for the 2d cubic nonelliptic Schrödinger equation Abstract: We consider the cubic nonelliptic Schr\"odinger equation, $$i\partial_t u+(\partial^2_{x}-\partial^2_{y})u=|u|^2u,$$ with initial datum in $L^2(\mathbb{R}^2)$. We show that if the solution blows up in finite time, then there is a mass concentration phenomenon near the blow-up time. The key ingredient is a refinement of the Strichartz inequality on the saddle. This is joint work with Ana Vargas. Tuesday, October 13, 4:00 p.m., B139 VV Alexander Koldobsky (Missouri). Title: Positive definite functions and multidimensional versions of random variables Abstract: We say that a random vector $X=(X_1,...,X_n)$ in $\R^n$ is an $n$-dimensional version of a random variable $Y$ if for any $a\in \R^n$ the random variables $\sum a_iX_i$ and $\gamma(a) Y$ are identically distributed, where $\gamma:\R^n\to [0,\infty)$ is called the standard of $X.$ An old problem is to characterize those functions $\gamma$ that can appear as the standard of an $n$-dimensional version. In this talk we prove the conjecture of Lisitsky that every standard must be the norm of a space that embeds in $L_0.$ This result is almost optimal, as the norm of any finite dimensional subspace of $L_p$ with $p\in (0,2]$ is the standard of an $n$-dimensional version ($p$-stable random vector) by the classical result of P.L\`evy. An equivalent formulation is that if a function of the form $f(\|\cdot\|_K)$ is positive definite on $\R^n,$ where $K$ is an origin symmetric star body in $\R^n$ and $f:\R\to \R$ is an even continuous function, then either the space $(\R^n,\|\cdot\|_K)$ embeds in $L_0$ or $f$ is a constant function. Combined with known facts about embedding in $L_0,$ this result leads to several generalizations of the solution of Schoenberg's problem on positive definite functions. Monday, October 19, 2:25 p.m., B113 VV Igor Verbitsky (Missouri) Dyadic models of some linear, quasilinear and Hessian equations Recent results on linear, quasilinear and fully nonlinear elliptic PDE will be discussed. This includes bilateral global estimates of Green's functions, criteria of solvability, and characterizations of removable singularities. Equations involving fractional Schr\"odinger,p-Laplacian or k-Hessian operators, and singular source terms and data will be considered. Analysis of solutions is based on certain dyadic models of these equations. Tuesday, October 27, 4:00 p.m., B139 VV Loukas Grafakos, University of Missouri Bilinear Fourier Integral Operators Abstract: We develop the framework for the study of bilinear Fourier integral operators and we present the first results in this area. We focus on the local L^2 theory, which may be perceived as analogous to the L^2 theory in the linear case, although we also discuss results in the cases where the target spaces are Banach or quasi-Banach Lebesgue spaces. This study of this topic is motivated by the need to understand the mapping properties of the restriction of classical linear Fourier integral operators on R^{2n} along the diagonal {(x,x): x in R^n}. All results are joint with Marco Peloso. Tuesday, November 3, 4:00 p.m., B139 VV. Wanke Yin (Wuhan University) A Bishop surface with a vanishing Bishop invariant Abstract: Bishop surfaces are generically embedded surfaces in the complex Euclidean space of dimension two. The surfaces have been playing important roles in the study of several complex variables. In this talk, I will focus on the equivalence problem for such surfaces, as well as its connection with classical dynamics and hyperbolic geometry. I will discuss a joint work by Huang and myself which solves an open problem asked by Moser and Webster more than two decades ago. Tuesday, November 10, 4:00 p.m., B139 VV. Vladimir Sverak (Minnesota) Minimal initial data for potential Navier-Stokes singularities Abstract: Assuming some smooth data lead to a singularity for the 3d Navier-Stokes equations, we show that there are also initial data with minimal $\dot H^{1/2}$ -- norm which will produce a singularity. The set of such initial data is compact modulo the symmetries of the equation. (Joint work with Walter Rusin). Thursday, November 19, 2:25 p.m., B325 VV Allan Greenleaf (Rochester) Microlocal analysis of some singular Lagrangian manifolds Abstract: I will describe classes of generalized Fourier integral operators that one can associate (canonical relations) with singularities, the so-called open umbrellas of Arnold and Givental. The primary motivation for this work comes from an inverse problem in seismology, but there are also simple examples arising from averaging operators for certain curves and surfaces. This is joint work with R. Felea. Tuesday, December 1, 4:00 p.m., Adam Coffman (IUPUI) A nonlinear differential inequality and counterexamples for Holder continuous almost complex structures Abstract: We consider a second order quasilinear partial differential inequality for real valued functions on the unit ball and find conditions under which there is a lower bound for the supremum of nonnegative solutions that do not vanish at the origin. As a consequence, for complex valued functions f(z) satisfying df/dzbar=|f|^alpha, 0<\alpha<1, and f(0) not =0, there is also a lower bound for sup|f| on the unit disk. For each alpha, we construct a manifold with an alpha-Holder continuous almost complex structure where the Kobayashi-Royden pseudonorm is not upper semicontinuous, generalizing an example of Ivashkovich, Pinchuk, and Rosay. Tuesday, December 8, 4:00 p.m., B139 VV. Benoit Pausader(Brown University) The Mass critical fourth order Schrödinger equation Abstract: We will discuss global existence and scattering for solutions to the L^2-critical fourth order Schroedinger equation of finite mass. In high dimensions, the global existence holds for every initial data in L^2 in the defocusing case. If time permit, we will discuss the case of the small dimensions and the focusing case. Math department Colloquium: Gigliola Staffilani (MIT) Friday, December 11, 4;00 p.m., vv B239. Title: On dispersive equations and their importance in mathematics. Abstract: Dispersive equations, like the Schr\"odinger equation for example, have been used to model several wave phenomena with the distinct property that if no boundary conditions are imposed then in time the wave spreads out spatially. In the last fifteen years this field has seen an incredible amount of new and sophisticated results proved with the aid of mathematics coming from different fields: Fourier analysis, differential and symplectic geometry, analytic number theory, and now also probability and a bit of dynamical systems. In this talk it is my intention to present few simple, but still representative examples in which one can see how these different kinds of mathematics are used in this context.