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 Pavel Zatitskii (zatitspl<at>ucmail.uc.edu) or 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 (University of Cincinnati)
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.
Hassan Babaei (University of Cincinnati)
Thursday, October 1, 4:00-5:00, French 2109
Determining wavenumbers and Kolmogorov-like dissipation scales for 3D Hall and electron magnetohydrodynamics (MHD) equations
Kolmogorov’s theory of turbulence predicts that, in three-dimensional fluid flows, most of the energy is concentrated in Fourier modes with wavenumbers below a certain threshold, known as the Kolmogorov dissipation wavenumber. At scales beyond this threshold, viscous effects become dominant. Similar dissipation scales have been identified through numerical studies and simulations involving Hall and electron magnetohydrodynamics (MHD), which are important models for understanding turbulence in astrophysical plasmas.
In this work, we use Littlewood–Paley decomposition techniques to develop a rigorous mathematical counterpart to these physically observed scales. Specifically, we establish the existence of time-dependent determining wavenumbers for weak solutions of 3D Hall- and electron-MHD equations. Although these wavenumbers may become unbounded if a solution develops a singularity, we show that their time averages are bounded above by the Kolmogorov-like dissipation numbers predicted by phenomenological studies of plasma turbulence. We also establish a uniform bound on the magnetic determining wavenumber under suitable Besov regularity assumptions. These results connect the determining wavenumber framework with the physical dissipation scales suggested by phenomenological theories of Hall and electron MHD turbulence.
Kyle Roy (Kent State University)
Thursday, October 8, 11:15-12:10, French 2109
Convex bodies homothetic to their floating bodies
In a recent paper, \textbf{(Alfonseca, Nazarov, Ryabogin, Stancu, Yaskin, Rev. Un. Mat. Argentina 69 (2026), 93–108, MR5022112)} showed that origin symmetric convex bodies which are dilations of their floating bodies in $\mathbb{R}^2$ and close to the unit ball must be ellipses. Additionally, they showed that there exist asymmetric convex bodies which are not ellipses but are dilations of their floating bodies. We will discuss the important ideas they used, and how they were adapted to prove similar results in higher dimensions.
Alex Misiats (Virginia Commonwealth University) - Colloquium
Thursday October 15
Mike Roysdon (The Ohio State University)
Thursday October 22 at 11:15-12:10 in the Seminar Room (French 4206)
Functional Santaló Inequalities in Polish Spaces
The classical Blashke-Santaló inequality from convex geometry gives a way to measure of how far a finite dimensional normed space is from being Euclidean by examining the product of the volumes of the associated unit balls of each space; this produces an isoperimetric-type inequality that is invariant under non-degenerate linear transforms. Functional versions of this inequality have shown great utility in the literature, dubbed functional Santaló inequalities, and have found applications in analysis, pdes, optimal transport, and probability theory.
In this talk, we will develop a transference principle, that provides a sharp equivalence between geometric Blaschke-Santaló -type inequalities and their functional counterparts in Polish spaces. We will examine several examples of such inequalities, arising from isoperimetric problems on the matrix space, spherical space, Gauss space (and more generally RCD(k,\infty) spaces), and on the discrete cube. If time permits, we will discuss applications of this principle to optimal transport theory.
Based on a joint work with Dylan Langharst and Andreas Malliaris.
Mohammad Sadegh M. Moakhar (University of Cincinnati)
Thursday October 29 at 11:15-12:10 in the Seminar Room (French 4206)
SAT Actions of Discrete Quantum Groups on von Neumann Algebras
We introduce a natural generalization of the notion of strongly approximately transitive (SAT) states for actions of locally compact quantum groups. In the case of discrete quantum groups of Kac type, we show that the existence of unique stationary SAT states entails rigidity results concerning injective extensions of quantum group von Neumann algebras.
https://www.ams.org/journals/proc/2024-152-10/S0002-9939-2024-16882-9/S0002-9939-2024-16882-9.pdf