Department of

Mathematics


Seminar Calendar
for events the day of Thursday, August 31, 2017.

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Thursday, August 31, 2017

11:00 am in 241 Altgeld Hall,Thursday, August 31, 2017

Polynomial Roth type theorems in Finite Fields

Dong Dong (Illinois Math)

Abstract: Recently, Bourgain and Chang established a nonlinear Roth theorem in finite fields: any set (in a finite field) with not-too-small density contains many nontrivial triplets $x$, $x+y$, $x+y^2$. The key step in Bourgain-Chang's proof is a $1/10$-decay estimate of some bilinear form. We slightly improve the estimate to a $1/8$-decay (and thus a better lower bound for the density is obtained). Our method is also valid for 3-term polynomial progressions $x$, $x+P(y)$, $x+Q(y)$. Besides discrete Fourier analysis, algebraic geometry (theorems of Deligne and Katz) is used. This is a joint work with Xiaochun Li and Will Sawin.

4:00 pm in 245 Altgeld Hall,Thursday, August 31, 2017

Symplectic non-squeezing for the cubic nonlinear Schrodinger equation on the plane

Monica Visan (UCLA)

Abstract: A famous theorem of Gromov states that a finite dimensional Hamiltonian flow cannot squeeze a ball inside a cylinder of lesser radius, despite the fact that the ball has finite volume and the cylinder has infinite volume. We will discuss an infinite-dimensional analogue of Gromov's result, in infinite volume. Specifically, we prove that the flow of the cubic NLS in two dimensions cannot squeeze a ball in $L^2$ into a cylinder of lesser radius. This is joint work with R. Killip and X. Zhang.