Abstracts for some talks (Spring Semester 2011) Tuesday, January 25, 4:00 p.m. VV B139 Sanghyuk Lee (Seoul National University) Pointwise convergence along curves for solutions of the Schr\"odinger equation. Abstract: A classical problem regarding pointwise behavior of the Schr\"odinger equation is to determine the optimal exponent $s$ for which $\lim_{t\to 0} e^{it\Delta}f(x)=f(x) \text{ a.e. }x\in \mathbb R^d,$ whenever $f\in H^s(\mathbb R^d)$. This problem was settled for $d=1$ but still remains open for the higher dimensions even though some progresses have been made for $d=2$. In this talk we consider pointwise convergence along variable curves and prove that the convergence problem along a class of curves are equivalent to the one along vertical lines. Via lens transform this yields the convergence results for the Hermite Schr\"odinger equation including some class of Schr\"odinger equations with quadratic potentials. We also show that the time-space estimates for theses equations are equivalent. This is a joint work with Keith Rogers. Tuesday, February 8, 4:00 p.m., VV B139 Serguei Denissov (UW) The generalized capacity and harmonic measure: applications to the multidimensional scattering Abstract: we develop a special potential theory to study some subtle questions in multidimensional scattering. In particular, we will prove analogs of the classical Hall's lemma and Carleman inequalities. This talk is based on the paper with S. Kupin. Tuesday, February 15, 4:00 p.m., VV B139 Alexander Kiselev (UW) Nonlocal maximum principles for active scalars Abstract: Active scalars appear in many problems of fluid dynamics. The most common examples of active scalar equations are 2D Euler, Burgers, and 2D surface quasi-geostrophic equations. Many questions about regularity and properties of solutions of these equations remain open. We develop the recently introduced idea of nonlocal maximum principle, formulating a more general criterion and providing new applications. The most interesting application introduced the idea of a nonlocal maximum principle, formulating a more general criterion and providing new applications. The most interesting application is finite time regularization of weak solutions in the supercritical regime. Tuesday, February 22, 4:00 p.m., VV B139 Benjamin Jaye (University of Missouri) TBA Tuesday, March 1, 4:00 p.m., VV B139 Alexander Fish (UW) Ergodic Plunnecke inequalities Abstract: We extend classical Plunnecke inequalities from additive number theory to ergodic setting. As a corollary of ergodic Plunnecke inequalities via Furstenberg's correspondence principle for sumsets we obtain a new number theoretic result. Joint work with M.Bjorklund. Saturday, March 5, Events honoring Professor Stephen Wainger Elias M. Stein (Princeton University) The mathematical adventures of Stephen Wainger Alexandru Ionescu (Princeton University and UW) On the global regularity of energy critical dispersive equations in non-euclidean backgrounds Abstract: I will discuss some recent work with Benoit Pausader and Gigliola Staffilani on global well-posedness of certain energy critical dispersive models on manifolds. We will consider two examples: the defocusing energy critical NLS on the hyperbolic space $H^3$ and the semiperiodic defocusing energy critical NLS on $R*T^3$. Akos Magyar (University of British Columbia) Discrete analogues in harmonic analysis and some applications. Abstract: Discrete analogues of problems in harmonic analysis arise when the underlying Euclidean space is replaced by the set of integer points. In such problems properties of exponential exponential sums play a crucial role, hence they are more of arithmetical nature and in turn have implications in other fields, such as number theory, ergodic theory and arithmetic Ramsey theory. We will discuss some problems and recent results in this area. Jim Wright (University of Edinburgh) Maximal functions along convex curves in the plane: L^2 estimates Abstract: A basic result in this topic states that the maximal function is bounded on L^2 if the so-called h functional has the doubling property. We explore to what extent L^2 boundedness holds under weaker conditions. Tuesday, March 15, 4:00 p.m., VV B139 Joint Analysis-PDE seminar Francois Hamel (Marseille) Optimization of eigenvalues of non-symmetric elliptic operators Abstract: The talk is concerned with various optimization results for the principal eigenvalues of general second-order elliptic operators in divergence form with Dirichlet boundary condition in bounded domains of R^n. To each operator in a given domain, we can associate a radially symmetric operator in the ball with the same Lebesgue measure, with a smaller principal eigenvalue. The constraints on the coefficients are of integral, pointwise or geometric types. In particular, we generalize the Rayleigh-Faber-Krahn inequality for the principal eigenvalue of the Dirichlet Laplacian. The proofs use a new symmetrization technique, different from the Schwarz symmetrization. This talk is based on a joint work with N. Nadirashvili and E. Russ. Tuesday, March 22, 4:00 p.m., VV B139. David Damanik (Rice University) Title: Uniform localization Abstract: We discuss the concept of localization for the Schr\"odinger equation, in both the time-dependent and the time-independent setting. After surveying some of the work in this area, we discuss a recent result, obtained in joint work with Zheng Gan, that illustrates the phenomenon of uniform localization, again in both the time-dependent and the time-independent setting. Tuesday, March 29, 4:00 p.m., VV B139. [Tent.] Marius Junge (University of Illinois at Urbana-Champaign) Fourier multipliers and cocycles Abstract: We use techniques from operator algebras and Tao Mei's BMO space to prove results on Fourier multipliers. Primarily we want to prove results on L_p boundedness of Fourier multipliers for noncommmutative versions of T^m, the multidimensional torus. With this (mildly) noncommutative approach we also obtain new results for multipliers on classical spaces. Tuesday, April 12, 4:00 p.m., VV B139. Jim Brennan (University of Kentucky) Analytic capacity and some problems in approximation theory The abstract is uploaded at http://www.math.wisc.edu/~seeger/abstracts/brennan11.pdf TBA Tuesday, April 19, 4:00 p.m., VV B139. Alexander Nagel (UW) Diagonal estimates for Bergman kernels in monomial type domains Tuesday, April 26, 4:00 p.m., VV B139. Dong Li (Iowa) Global smooth solutions for the 2D Euler-Poisson system Abstract: The Euler-Poisson system is a fundamental two-fluid model to describe the dynamics of the plasma consisting of compressible electrons and a uniform ion background. By using the dispersive Klein-Gordon effect, Guo first constructed a global smooth irrotational solution in the three dimensional case. It has been conjectured that same results should hold in the two-dimensional case. The main difficulty in 2D comes from the slow dispersion of the linear flow and certain nonlocal resonant obstructions in the nonlinearity. I will discuss a new method to overcome these difficulties and construct smooth global solutions for the 2D Euler-Poisson system. April 30 - May 1, 2011 Madison Springtime Analysis and PDE Workshop Tuesday, May 3, 4:00 p.m., VV B139. Malabika Pramanik (University of British Columbia) Title: A multi-dimensional resolution of singularities with applications to analysis Abstract: The structure of the zero set of a multivariate polynomial is a topic of wide interest, in view of its ubiquity in problems of analysis, algebra and geometry. The study of such sets, known in algebraic geometry literature as resolution of singularities, originated in the pioneering work of Jung, Abhyankar and Hironaka and has seen substantial recent advances, albeit in an algebraic setting. In this talk, I will discuss a few situations in analysis where the study of polynomial zero sets play a critical role, and discuss prior work in this analytical framework in two dimensions. Our main result (joint with Tristan Collins and Allan Greenleaf) is a formulation of an algorithm for resolving singularities of a multivariate real-analytic function with a view to applying it to a class of problems in harmonic analysis.