Thursday, February 2, 2023
02/02/2023 - 4:00pm
Lattices in SL(n,R) (for n at least 3) are known to exhibit various rigidity properties relative to linear representations and similar rigidity phenomena is expected for actions on manifolds. For instance, it is know there are no actions on manifolds of dimension below (n-1). In this talk, I talk about work in progress to understand actions on manifolds of dimension (n-1) and dimension n. Especially in dimension n, I’ll discuss how dynamical properties (positive topological entropy) significantly constrains the action.
02/02/2023 - 4:15pm
I will present recent work which proves a sharp L^7 square function estimate for the moment curve (t , t^2, t^3) in R^3 using ideas from decoupling theory. In the context of restriction theory, which concerns functions with specialized (curved) Fourier support, this is the only known sharp square function estimate with a non-even L^p exponent (p=7). The basic set-up is to consider a function f with Fourier support in a small neighborhood of the moment curve. Then partition the neighborhood into box-like subsets and form a square function in the Fourier projections of f onto these box-like regions. We will use a combination of recent tools including the “high-low” method and wave envelope estimates to bound f in L^7 by the square function of f in L^7.