The default time for the analysis seminar this semester is Wednesday at 2:30-3:30pm in the Seminar Room (French Hall 4206).
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.
Michal Wojciechowski (Mathematical Institute of the Polish Academy of Sciences)
Wednesday September 17 at 2:30-3:30pm in the Seminar Room
On the Mityagin-Mirkhil-DeLeeuw theorem with differential constraints
The classical result of the authors mentioned in the title says that the uniform norm of Q(D)f is bounded by the uniform norms of Q_j(D)f, j=1,2,...,k for functions f of compact support if and only if Q is in the span of Q_j's (here Q and Q_j's are homogeneous polynomials of the same degree). In the talk I will present the analogs of this result with additional constraint that the estimate holds only for f being a solution of the differential equation P(D)f=0.
This is joint work with Eduard Curca.
Michael Penrod (University of Cincinnati)
Wednesday September 24 at 2:30-3:30pm in the Seminar Room
The Christ-Goldberg Maximal Operator in Variable Lebesgue Spaces
Mike Roysdon (University of Cincinnati)
Wednesday October 1 at 2:30-3:30pm in the Seminar Room
Comparison Problems for Radon Transforms.
We will discuss comparison problems for L^p norms of pairs of functions based on information exhibited from their Radon transforms. We will discuss how these problems are connected to the celebrated Busmann-Petty problem and Bourgain slicing problem from convex geometry. Based on joint work with Koldobsky and Zvavitch.
Oleg Asipchuk (University of Cincinnati)
October 8 at 2:30-3:30pm in the Seminar Room
Methods of construction of exponential bases on planar domains
The problem of proving (or disproving) the existence of exponential bases on measurable sets of the plane is still largely unsolved. An exponential basis is a set of exponential functions in the form of $\{e^{2\pi i \langle\lambda,\vec x\rangle}\}_{\lambda\in\Lambda}$, where $\Lambda$ is a discrete set of $\mathbb{R}^2$. The main examples of domains where exponential bases exist are shapes that tile the plane by translations, polygons with central symmetry, and unions of rectangles with sides parallel to the axes. We still know nothing about the existence of exponential bases on triangles, irregular polygons, and disks. In my talk, I will introduce an approach for constructing exponential bases on specific types of polygons using approximation by dyadic squares. I will also discuss examples and explore generalizations of the results.
Mike Roysdon (University of Cincinnati)
Wednesday October 15 at 2:30-3:30pm in the Seminar Room
Functional Liftings of Restricted Geometric Inequalities
The equivalence between the dimenion free version of the celebrated Brunn-Minkowski inequality and its functional counterpart, the Prekopa-Leindler inequality, is well known in the literature. Among the applications of these inequalities is that the standard Guassian probability measure on \R^n is log-concave with respect to Minkowski summation and the Gaussian concentration phenomena. In 2010, it was conjectured by Gardner and Zvavitch that, by restricting one's considerations to origin symmetric convex sets rather than general Borel measurable sets, the Gaussian measure is, in fact, (1/n)-concave. This conjecture was verified in 2020 by Eskenazis and Moschidis.
Inspired by the above, in this talk, we consider the term "sup-convolutions," and show that functional inequalities (aka PL) that enjoy an interpretation as sup-convolution inequalities can be deduced from the special case of indicator functions corresponding to a geometric inequality (aka BM). As consequences of our discussion we derive a Borell-Brascamp Lieb inequality for the Gaussian Brunn-Minkowski inequality and give a functional analog of the log-Brunn-Minkowski conjecture (2010) if time permits.
Joint work with Malliaris, Melbourne and Roberto.
Dmitry Ryabogin (Kent State University)
Wednesday October 22 at 2:30-3:30pm in the Seminar Room
On the homothety conjecture with the homothety coefficient $1/2$.
Let $K=-K$ be an origin-symmetric plane convex body and let $1/2 K=K_{\delta}$ for some $\delta\in (0,1/2)$; $K_{\delta}$ is a body of flotation of $K$.
We will discuss a recent proof of Ayala-Figueroa, Jer\'onimo-Castro and Jimenez-Lopez, who showed that under the above conditions $K$ must be an ellipsoid.
Karl Liechty (DePaul University)
Wednesday October 29 at 2:30-3:30pm in the Seminar Room
The six-vertex model with domain wall and related boundary conditions via orthogonal polynomials.
The six-vertex model is a well-known model in 2d statistical mechanics which, for certain boundary conditions, admits a determinantal formula for its partition function. I’ll discuss recent and current work on the asymptotic analysis of orthogonal polynomials arising from these determinantal formulas to obtain limit theorems for the model. I’ll discuss joint work with Vadim Gorin, Jimmy He, and Umid Ahmadali.
Michael Goldberg (University of Cincinnati)
Wednesday November 12 at 2:30-3:30pm in the Seminar Room
How L^1 inversion lemmas prove scaling-sharp Schroedinger dispersive bounds: A survey
The Riemann-Lebesgue Lemma says that the Fourier transform of an integrable function is continuous. Its converse (continuous functions have integrable Fourier transform) is false, and there's no good characterization of these functions in terms of smoothness alone. There is a nice relationship between functions in this class, however. In 1935, Norbert Wiener proved that any time f(x) has integrable Fourier transform, its reciprocal 1/f(x) does too as long as there is no division by zero.
The Schroedinger equation in R^n is considered a "dispersive" PDE because energy is conserved, yet the solution decreases over time in other norms due to cancellation between waves of different frequencies. When the Hamiltonian is a translation-invariant operator, such as the Laplacian and its powers, the time-decay rate can be found by direct evaluation of oscillatory integrals. We would like to show the same decay rates hold if the Hamiltonian is perturbed by a scalar multiplier V(x) that is sufficiently localized (a.k.a. a "short range potential").
The dilation scaling of the Laplacian suggests a necessary condition for V(x) to be short range. Sufficiency has been proved in some settings, and the proof has relied on Wiener's theorem (or similar results) every single time. I'll give an overview of how the argument works and where it's worked so far.
Jan Lang (The Ohio State University)
Tuesday November 18 at 2:30-3:30pm in the Seminar Room
Quality of non-compactness for Sobolev embeddings and some other maps
In this talk, we investigate the structure of certain non-compact linear operators and embeddings between function spaces.
Plenty of research has been devoted to the study of compact operators and the “quality” of their compactness. By means of approximation, Kolmogorov, and entropy numbers, one can describe the degree of compactness through approximation via finite-dimensional structures or covering rates. However, these techniques fail to yield insight into the structural properties of non-compact mappings. We employ more refined concepts, like strict singularity, finite strict singularity, and Bernstein numbers, to provide an adequate characterization of the degree of non-compactness and describe “quality” of non-compactness.
We focus in particular on embeddings between Sobolev and Besov spaces, as well as on Fourier and Hilbert transforms, and present recent results in understanding “quality” of non-compactness of these operators through the framework of Bernstein numbers and the notions of maximal non-compactness and strictly singular operators. Our results show that Bernstein numbers, although initially introduced and subsequently neglected, are indispensable for revealing subtle structural distinctions among non-compact mappings. In particular, we establish conditions under which embeddings are strictly singular, finitely strictly singular, or fail to exhibit strict singularity, thereby enriching the classical theory of functional analysis.
The talk is based on a series of joint papers with Chian-Yeong Chuah, Liding Yao, David E. Edmunds, and others.
Michael Goldberg (University of Cincinnati)
Wednesday December 3 at 2:30-3:30pm in the Seminar Room (Tentative)
How L^1 inversion lemmas prove scaling-sharp Schroedinger dispersive bounds: A survey (Part 2)