2023-2024
Fall 2023
The standard time for the analysis seminar will be Thursday at 3:35-4:30PM in the French Hall Seminar Room (French Hall 4206). If a seminar conflicts with a colloquium, it will be postponed to the next week or alternate to Friday at 3:35-4:30PM
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.
Ilmari Kangasniemi (University of Cincinnati)
Thursday September 7 at 3:35-4:30PM in the French Hall Seminar Room
On Quasiregular Values
A quasiregular (QR) map is a Sobolev map f: ℝⁿ → ℝⁿ satisfying the distortion inequality |Df(x)|ⁿ ≤ K det(Df(x)) at almost every x, where K ≥ 1 is a constant. QR maps form a higher-dimensional class of maps with many similar geometric properties as single-variable holomorphic maps. In this talk, we consider a generalization of the distortion inequality of the form |Df(x)|ⁿ ≤ K det(Df(x)) + Σ(x)|f(x) - y|ⁿ, where Σ is a real-valued function and y ∈ ℝⁿ is a fixed point. Our recent results show that under various Lᵖ-integrability assumptions on Σ, this condition can be used to prove single-value counterparts to many fundamental results of QR-maps. The list of generalized results includes the QR-versions of the open mapping theorem, Liouville theorem, and Picard theorem. Joint work with Jani Onninen.
Andrew Lorent (University of Cincinnati)
Thursday September 14 at 3:35-4:30PM in the French Hall Seminar Room
On rigidity for differential inclusions into nowhere Elliptic curves in 2x2 matrices
After surveying results on differential inclusions we will state a regularity/rigidity result for differential inclusions into a class of nowhere Elliptic connected closed curves in 2x2 matrices. This results generalizes our previous result for the special case of the nowhere Elliptic closed curve (that arises in the study of the Aviles Giga conjecture) to a somewhat general class of nowhere Elliptic curves. The methods of the proof are from the theory of the Aviles Giga functional and from scalar conservation laws. The main proof ideas will motivated and sketched in board outline.
(Department Colloquia on September 21 and 28)
Gamal Mograby (University of Cincinnati)
Thursday October 5 at 3:35-4:30PM in Room W Charlton 119
Jacobi operators on graphs: Applications to almost Mathieu operators and Grover’s quantum walk
Jacobi operators can be viewed as the discrete analog of Sturm-Liouville operators and appear in various applications. This talk introduces an approach to studying such operators from a graph theoretical perspective. We present a class of Jacobi operators ("piecewise centrosymmetric Jacobi operators") defined on certain substitution graphs. We show that their spectral analysis can be explicitly related to the spectral theory of graph Laplacians using certain orthogonal polynomials. Depending on time, we will discuss applications to almost Mathieu operators and Grover’s quantum walk on graphs.
Feng Li (University of Cincinnati)
Thursday October 12 at 3:35-4:30PM in the French Hall Seminar Room
Critical boundary condition of Holder extension for Harmonic functions - from local to nonlocal
The well-known Wiener criterion gives a critical condition of the continuity up to the boundary for local and nonlocal Laplacian operators. In this talk, we will give some sufficient and necessary boundary conditions (by uniformly positive boundary capacity) of the Holder extension for both local and nonlocal Laplacian operators, in which, the local case was solved by Hiroaki Aikawa and Nages, (for sufficient condition, see also Maz'ya in 1967, and Gariepy and Ziemer in 1983), and our recent work solved the fractional case (p=2), but the more general case is still open.
(Taft Lecture on October 19)
Chun Ho (University of Cincinnati)
Thursday October 26 at 3:35-4:30PM in the French Hall Seminar Room
Real Hardy Spaces and Local Hardy Spaces: A Very (Very) Short Introduction
In this talk, I would like to introduce the definitions of real Hardy spaces and local Hardy spaces, as well as some properties, including other characterizations of the spaces, their dual spaces, and Hardy's inequalities. Some parts of the talk are based on joint work with Dafni, Picon, and Vasconcelos.
Alim Sukhtayev (Miami University)
Thursday Nov 2 at 3:35-4:30PM in 60W Charlton 119
Spectral decomposition and decay to grossly determined solutions for a simplified BGK model.
For a simplified 1D BGK model we show that $H^1$ solutions decay exponentially in $L^2$ to a subclass of the class of grossly determined solutions as defined by Truesdell and Muncaster in the context of Boltzmann's equation. In the process, we determine the spectrum and generalized eigenfunctions of the associated non-selfadjoint linearized operator and derive the associated generalized Fourier transform and Parseval's identity. Notably, our analysis makes use of rigged space techniques originating from quantum mechanics, as adapted by Ljance and others to the nonselfadjoint case.
This is joint work with Kevin Zumbrun.
Jani Onninen (Syracuse University) - Part of Geometric Analysis Reading Seminar
Wednesday Nov 8 at 3:30-4:30PM in the French Hall Seminar Room
Existence of Sobolev homeomorphisms
In nonlinear elasticity theory, the class of acceptable deformations is typically a space of Sobolev homeomorphisms. Whether a given boundary map extends to such a homeomorphism is therefore a fundamental question to study in this field. We share some recent developments regarding this subject. The main question we will consider is the Sobolev Jordan-Schönflies problem of extending a boundary map between planar Jordan domains as a Sobolev homeomorphism. This is based on a joint work with Aleksis Koski.
Chris Gartland (UC San Diego)
Thursday Nov 9 at 3:35-4:30PM in the French Hall Seminar Room
Stochastic Embeddings of Hyperbolic Metric Spaces
This talk is based on ongoing work of the speaker. We will discuss the stochastic embeddability of snowflakes of doubling metric spaces into ultrametric spaces and the induced stochastic embeddings of their hyperbolic fillings into trees. As an application, we obtain that finitely generated Gromov hyperbolic groups admit proper, uniformly Lipschitz affine actions on $L^1$.
Seung-Yeon Ryoo (Caltech)
Friday Nov 17 at 3:35-4:30PM in 60W Charlton 119 - will bring cookies!
Quantitative nonembeddability of nilpotent groups into uniformly convex spaces
Computing the bilipschitz distortion of a given metric space into a given Banach space is a recurring theme in metric geometry. We compute the bilipschitz distortion up to constant factors when the given metric space is a ball in a finitely generated group of polynomial growth that is not virtually abelian and when the target Banach space is an $L^p$ $(1<p<\infty)$ space, to be $(\log n)^{1/p}$ when $p\ge 2$, and $\sqrt{\log n}$ when $p< 2$, where $n$ is the radius of the ball. We will discuss quantitative nonembeddability into $L^1$, which has potential applications to theoretical computer science.
(Break for Thanksgiving)
Matthew Romney (Stevens Institute of Technology)
Friday Dec 1 at 3:35-4:30PM in 60W Charlton 119
Uniformization of metric surfaces
The classical uniformization theorem, conjectured by Klein in 1882 and finally proved independently by Poincaré and Koebe in 1907, states that every simple connected Riemann surface is conformally equivalent to either the disk, the plane, or the 2-sphere. In this talk, we consider generalizations of this theorem to metric spaces, with an emphasis on metric surfaces. We show that any surface can be parametrized by a constant curvature surface under a weakly quasiconformal map, assuming only that the surface has locally finite Hausdorff 2-measure. As an application, we can recover earlier results due to Bonk-Kleiner (for Ahlfors 2-regular surface), Rajala (for surfaces with controlled conformal modulus) and Lytchak-Wenger (for surfaces satisfying a quadratic isoperimetric inequality). This talk is based on joint work with D. Ntalampekos.