for Harmonic Analysis and Differential Equations Seminar events the year of Friday, September 14, 2018.
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1:00 pm in 347 Altgeld Hall,Tuesday, March 13, 2018
Harmonic Analysis and Differential Equations Seminar
Low Regularity Global Existence for the Periodic Zakharov System
Erin Compaan (MIT)
Abstract: In this talk, we present a low-regularity global existence result for the periodic Zakharov system. This is a dispersive model for the motion of ionized plasma. Its dynamics have been extensively studied, and existence of solutions is in known for data in the Sobolev space $H^\frac12 \times L^2$. We present a global existence result which holds for even rougher data, in a class of Fourier Lebesgue spaces. It is obtained by combining the high-low decomposition method of Bourgain with an almost-conserved energy result of Kishimoto. Combining these two tools allows us to obtain a low-regularity result which was out of reach of either method alone.
Submitted by tzirakis
1:00 pm in 347 Altgeld Hall,Tuesday, September 11, 2018
Unlabeled distance geometry problem
Ivan Dokmanic [email] (Illinois - Electrical and Computer Engineering)
Abstract: The famous distance geometry problem (DGP) asks to reconstruct the geometry of a point set from a subset of interpoint distances. In the unlabeled DGP the goal is the same, alas without knowing which distances belong to which pairs of points. Both problems are of practical importance: the DGP models sensor network localization, clock synchronization, and molecular geometry reconstruction from NMR data, while the unlabeled DGP models room geometry reconstruction from echoes, positioning by multipath, and nanostructure determination by powder diffraction. The unlabeled DGP in 1D is known as the turnpike reconstruction problem, and it was one of the first techniques used to reconstruct genomes. The mathematics of the unlabeled DGP is nowhere near as well-understood as that of the DGP. I will introduce the unlabeled DGP, explain how it arises in various applications, discuss connections with phase retrieval and explain our approach based on empirical measure matching. Along the way I will point out numerous theoretical and algorithmic aspects of the problem that we do not understand but we wish we did.
Submitted by laugesen
1:00 pm in 347 Altgeld Hall,Tuesday, October 2, 2018
From Neumann to Steklov via Robin: the Weinberger way
Richard Laugesen [email] (Illinois Math)
Abstract: Lord Rayleigh asserted in 1877, and Faber and Krahn proved fifty years later, that "If the area of a membrane be given, there must evidently be some form of boundary for which the pitch (of the principal tone) is the gravest possible, and this form can be no other than the circle." In modern terminology, Rayleigh was claiming that among all planar domains of given area, the one that minimizes the first eigenvalue of the Laplacian (under Dirichlet boundary conditions) is the disk. In terms of heat flow in 3 dimensions, that means a room of given volume whose boundary is maintained at temperature zero will cool off slowest when it is a ball. What about a room whose boundary is perfectly insulated (Neumann boundary conditions)? Szego and Weinberger discovered in the 1950s that the room's temperature will equilibrate fastest when it is spherical - which is admittedly an unlikely shape for a room, except for fans of Futuro Flying Saucer homes. Insulation is never perfect, as every homeowner knows, and so we are led to the Robin boundary condition for partial insulation, and to recent progress and open problems on "isoperimetric type" eigenvalue inequalities that extend the work of Rayleigh-Faber-Krahn and Szego-Weinberger.
The Ricci iteration on homogeneous spaces
Artem Pulemotov (University of Queensland)
Abstract: The Ricci iteration is a discrete analogue of the Ricci flow. Introduced in 2007, it has been studied extensively on Kähler manifolds, providing a new approach to uniformisation. In the talk, we will define the Ricci iteration on compact homogeneous spaces and discuss a number of existence, convergence and relative compactness results. This is largely based on joint work with Timothy Buttsworth (Queensland), Yanir Rubinstein (Maryland) and Wolfgang Ziller (Penn).
Submitted by palbin
1:00 pm in 347 Altgeld Hall,Tuesday, December 4, 2018
Unconditional uniqueness for the derivative nonlinear Schrodinger equation
Razvan Mosincat (The University of Edinburgh)
Abstract: We consider the initial-value problem for the derivative nonlinear Schrödinger equation (DNLS) on the real line. We implement an infinite iteration of normal form reductions (namely, integration by parts in time) and reformulate a gauge-equivalent equation in terms of an infinite series of multilinear terms. This allows us to show the unconditional uniqueness of solutions to DNLS in an almost end-point space. This is joint work with Haewon Yoon (National Taiwan University).