Abstracts for some talks (Spring Semester 2012) Tuesday, January 31, 4:00 p.m., VV B139. Alexandru Ionescu (Princeton University) On the global stability of the constant equilibrium solution of the Euler-Poisson system Abstract: The Euler-Poisson system is a model used to describe the dynamics of a two-fluid plasma. I will discuss some recent work on the question of global existence of solutions of this system, in the case of small perturbations of a constant background. Friday, February 3, 2012, 4:00 p.m., VV B239. Colloquium Akos Magyar (University of British Columbia) On prime solutons to linear and quadratic equations Abstract: The classical results of Vinogradov and Hua establishes prime solutions of linear and diagonal quadratic equations in sufficiently many variables. In the linear case there has been a remarkable progress over the past few years by introducing ideas from additive combinatorics. We will discuss some of the key ideas, as well as their use to obtain multidimensional extensions of the theorem of Green and Tao on arithmetic progressions in the primes. We will also discuss some new results on prime solutions to non-diagonal quadratic equations of sufficiently large rank. Most of this is joint work with B. Cook. Joint PDE/Analysis seminar Monday, February 6, B115, 3:30 p.m. Yao Yao (UCLA) Degenerate diffusion with nonlocal aggregation: behavior of solutions Abstract: Thee Patlak-Keller-Segel (PKS) equation models the collective motion of cells which are attracted by a self-emitted chemical substance. While the global well-posedness and finite-time blow up criteria are well known, the asymptotic behavior of solutions is not completely clear. In this talk I will present some results on the asymptotic behavior of solutions when there is global existence. The key tools used in the paper are maximum-principle type arguments as well as estimates on mass concentration of solutions. This is a joint work with Inwon Kim. Tuesday, February 21, 4:00 p.m., VV B139. Malabika Pramanik (University of British Columbia) Maximal operators and differentiation on sparse sets Friday, February 24, 2012 Colloquium Malabika Pramanik (University of British Columbia) Analysis on sparse sets http://www.math.wisc.edu/~seeger/abstracts/malabika2012.pdf Tuesday, February 28, 4:00 p.m., VV B139. Dmitri Ryabogin (Kent State University) On problems of Bonnesen and Klee. Abstract: This is joint work with Fedor Nazarov and Artem Zvavitch. We will discuss some results related to the questions of Bonnesen about unique determination of convex (non-symmetric) bodies given the volumes of their maximal sections and projections. We will show that if $d\ge 4$ is even, then one can find two essentially different convex bodies such that the volumes of their maximal sections, central sections, and projections coincide for all directions. Tuesday, March 6, 2012, 4:00 p.m., VV B139. Han Peters (KdVI, University of Amsterdam) Cusps and Complex Dynamics Sufficiently small perturbations of singular holomorphic mappings from one to two complex dimensions have the property that their images intersect the original image. After discussing how to prove this, we show how this result can be used to obtain new results in two dimensional complex dynamics. This is joint work with Misha Lyubich. Friday, March 16, 2012, 4:00 p.m., VV B239. Colloquium Burak Erdogan (University of Illinois - Urbana-Champaign) Smoothing for the KdV equation and Zakharov system on the torus Tuesday, March 20, 4:00 p.m., VV B139. Stephen Wainger (UW) Discrete singular operators on discrete subgroups of nilpotent Lie groups Monday, March 26, 4:00 p.m., VV B115, PDE seminar. Vlad Vicol (University of Chicago) Shape dependent maximum principles and applications Abstract: We present a non-linear lower bound for the fractional Laplacian, when evaluated at extrema of a function. Applications to the global well-posedness of active scalar equations arising in fluid dynamics are discussed. This is joint work with P. Constantin. Friday, March 30, 2012, 4:00 p.m., VV B239. Colloquium Wilhelm Schlag (University of Chicago) Invariant manifolds and dispersive Hamiltonian equations Abstract: We will review recent work on the role that center-stable manifolds play in the study of dispersive unstable evolution equations. More precisely, by means of the radial cubic nonlinear Klein-Gordon equation we shall exhibit a mechanism in which the ground state soliton generates a center-stable manifold which separates a region of data leading to finite time blowup from another where solutions scatter to a free wave in forward time. This is joint work with Kenji Nakanishi from Kyoto University, Japan. Monday, April 9, 3:30 p.m., VV B115 Joint PDE/Analysis Seminar Charles Smart (MIT) Title: PDE methods for the Abelian sandpile Abstract: The Abelian sandpile growth model is a deterministic diffusion process for chips placed on the $d$-dimensional integer lattice. One of the most striking features of the sandpile is that it appears to produce terminal configurations converging to a peculiar lattice. One of the most striking features of the sandpile is that it appears to produce terminal configurations converging to a peculiar fractal limit when begun from increasingly large stacks of chips at the origin. This behavior defied explanation for many years until viscosity solution theory offered a new perspective. This is joint work with Lionel Levine and Wesley Pegden. Monday, April 16, 3:30 p.m., VV B115. PDE Seminar Jiahong Wu (Oklahoma State) The 2D Boussinesq equations with partial dissipation Abstract: The Boussinesq equations concerned here model geophysical flows such as atmospheric fronts and ocean circulations. Mathematically the 2D Boussinesq equations serve as a lower-dimensional model of the 3D hydrodynamics equations. In fact, the 2D Boussinesq equations retain some key features of the 3D Euler and the Navier-Stokes equations such as the vortex stretching mechanism. The global regularity problem on the 2D Boussinesq equations with partial dissipation has attracted considerable attention in the last few years. In this talk we will summarize recent results on various cases of partial dissipation, present the work of Cao and Wu on the 2D Boussinesq equations with vertical dissipation and vertical thermal diffusion, and explain the work of Chae and Wu on the critical Boussinesq equations with a logarithmically singular velocity. Tuesday, April 17, 4:00 p.m., VV B139. Ilya Kossovskiy (University of Western Ontario) Analytic Continuation of Holomorphic Mappings From Non-Minimal Hypersurfaces Abstract: The classical result of H. Poincare states that a local biholomorphic mapping from an open piece of the 3-sphere in $\mathbb{C}^2$ onto another open piece extends analytically to a global holomorphic automorphism of the sphere. This theorem was generalized by H. Alexander to the case of a sphere in $\mathbb{C}^n$ ($n\geq 2$), then later by S. Pinchuk for the case of strictly pseudoconvex hypersurface in the preimage and a sphere in the image, and finally by R. Shafikov and D. Hill for the case of an essentially finite hypersurface in the preimage and a quadric in the image. In this joint work with R. Shafikov we consider the - essentially new - case when a hypersurface $M$ in the preimage contains a complex hypersurface. We demonstrate that the above extension results fail in this case, and prove the following analytic continuation phenomenon: a local biholomorphic mapping from $M$ onto a non-degenerate hyperquadric in $\mathbb{CP}^n$ extends to a punctured neighborhood of the complex hypersurface, as a multiple-valued local biholomorphic mapping. We also establish an interesting interaction between non-minimal spherical real hypersurfaces and linear differential equations with an isolated singular point. Tuesday, May 8, 4:00 p.m., VV B139. Sergey Denisov Analysis of the instability in two-dimensional fluids Abstract: For the two-dimensional Euler equation describing incompressible nonviscous fluid, we obtain lower bounds for the growth of Sobolev norms. We also explain the merging mechanism for the dynamics of centrally symmetric vortex patches.