The default time for the analysis seminar this semester is Thursdays at 11:15-12:10pm.
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
Isomorphic classification of $BV$ and Sobolev spaces on simply connected planar domains.
Let $\Omega\subset\mathbb{R}^2$ be any bounded simply connected domain, and let $Q=(0,1)^2$. We prove, without any regularity assumptions on $\partial\Omega$, that both the homogeneous and nonhomogeneous $BV$ spaces on $\Omega$ are isomorphic to the corresponding spaces on the square $Q$. The same holds for the Sobolev spaces $W^{1,p}$ and their homogeneous versions for every $1\leq p<2$. As a separate ingredient, we prove a general result for countable locally finite weighted trees: the space defined by the natural norm combining weighted vertex values and weighted edge differences is isomorphic to $\ell^p$ for every $1\leq p\leq 2$. (This is a part of joint work with M. Derezinski and. F. Nazarov)
Aleksandr Logunov (Massachusetts Institute of Technology)
Friday, September 18th, 3:35-4:30, FRENCH-W 4221
Real zeroes of linear combinations of radicals of polynomials.
We will talk about the number of real roots of the expressions of the form
\[
f(x) = \sum_{k=1}^{n} c_k \bigl(P_k(x)\bigr)^{\alpha_k},
\]
where $c_k, \alpha_k \in \mathbb{R}$ and each $P_k$ is a real polynomial of degree at most $d$ that is non-negative on an interval $I\subset \mathbb{R}$. We will describe an elementary ODE method that improves previously known exponential upper bounds for the number of roots on $I$ to bounds that are polynomial in $n$, linear in $d$, and independent of the exponents $\alpha_k$. After that we will discuss problems in between PDE and real algebraic geometry.
Based on a joint work with Gal Binyamini, Avner Kiro, Dmitry Novikov, Dmitrii Zakharov.
Kyle Hammer
Thursday, September 24, 11:15-12:10, Seminar Room (French 4206)
Computation of Critical Energies in the 1-D Anderson-Bernoulli Polymer Model
For the one-dimensional discrete Schrödinger operators with ergodic random potential, the spectral type of the operator is almost entirely dependent on the Lyapunov Exponent \gamma(E) of the associated transfer matrices to the problem. It is known that absolutely continuous spectrum can only live when E is a critical energy, that is, a zero of \gamma. Through some classical estimates, one can show the spectrum is pure point whenever \gamma is positive. So, the zeros of the Lyapunov exponent have a very unique local spectral behavior. It is known for single site potentials, the Lyapunov Exponent is always positive, giving Anderson localization everywhere in the spectrum.
To force \gamma to have zeros, we will need to introduce potentials with correlated blocks called "polymers", which introduce a finite set of critical energies. We will discuss an extension of a technique of Germinet and De Bievre (2000) where they explicitly constructed a model with simple critical energies for the Dimer model (polymer size = 2). We will discuss the proof that extends their result for any polymer size and give a formula for the roots of \gamma in this extended case and compare the result numerically with work of Hislop and Nakano (2025). If time permits, we will discuss an open problem that these results naturally pose.