The default time for the analysis seminar this semester is Thursdays at 11:15-12:10pm in French 2109.
If you are interested in giving a talk contact Gareth Speight (Gareth.Speight<at>uc.edu).
To find out more about our department click here.
Michael Goldberg (University of Cincinnati)
Thursday, September 3, 11:15-12:10, Seminar Room (French 4206)
Tools for Proving Sharp Dispersive Estimates
Solutions of the free Schrodinger equation in $R^n$ satisfy an $L^2$ conservation law while higher-power $L^p$ norms decrease over time. This combination is possible if the solution evolves toward smaller pointwise values over a larger volume of support. The $L^p$ bounds with time decay are known as dispersive estimates. They occur because wave packets with different wavelength travel with different group velocity and eventually separate from one another.
The same dispersive estimates are often true for the Schrodinger equation with a short-range potential, once bound states are projected away. It is a useful shorthand to think of "short-range" as decaying faster than $O(|x|^{-2})$. The relevant comparison is that the Laplacian has exactly the same scaling properties as multiplying a function pointwise by $\frac{1}{|x|^2}$.
In the talk we will explore functional analysis tools that make it possible to prove Schrodinger dispersive estimates with a potential at the scaling limit of being short-range.
Oleg Asipchuk (University of Cincinnati)
Thursday, September 10, 11:15-12:10, French 2109
A sparse sampling conjecture
Sinai Robins conjectured that two convex, centrally symmetric bodies of positive measure that are not multi-tilers must agree up to a rigid motion whenever the Fourier transforms of their indicator functions agree on (\mathbb{Z}^d).
In my talk, I will present two counterexamples that disprove the conjecture. I will also discuss a related positive result obtained in collaboration with Sinai Robins.
Michal Wojciechowski (Mathematical Institute of the Polish Academy of Sciences)
Thursday September 17, French 2109, 4-5pm, French 2109
Aleksandr Logunov (Massachusetts Institute of Technology)
Friday, September 18th, 3:25-4:30, FRENCH-W 4221