2019-2020

University of Cincinnati: Analysis and PDE Seminar (2019-2020)

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.

Spring 2020

The analysis seminar will usually be on Wednesday at 1:25-2:20 in CGD 214.

Jeff Lindquist

Wednesday February 5 at 1:25-2:20 in CGD 214

Carathéodory-type extension theorems with respect to prime end boundaries

Using a notion of prime ends developed by Adamovicz, Björn, Björn, and Shanmugalingam, we prove a Carathéodory-type extension of BQS-homeomorphisms between two domains in proper, locally path-connected metric spaces as homeomorphisms between their prime end closures. We also give a Carathéodory-type extension of geometric quasiconformal mappings between two such domains provided the two domains are both Ahlfors Q-regular and support a Q-Poincaré inequality when equipped with their respective Mazurkiewicz metrics. (Joint work with Nageswari Shanmugalingam and Joshua Kline at the University of Cincinnati).

Deniz Bilman

Wednesday February 12 at 1:25-2:20 in CGD 214

Toda lattice and its Hamiltonian Perturbations

Doubly-infinite Toda lattice is a discrete-space nonlinear evolution equation which is also a completely integrable Hamiltonian system. It admits soliton solutions and exhibits soliton resolution in the long-time regime. The resolution is established by the direct and inverse scattering transforms for doubly-infinite Jacobi matrices, which are well-known to linearize the Toda flow. This framework also yields a powerful numerical method to compute the solutions for arbitrarily large times without time-stepping (joint work with T. Trogdon). On the other hand, there is the common belief, usually referred to as “the solitary wave resolution conjecture”, that (in the absence of finite time blow-up) generic solutions of nonlinear Hamiltonian partial differential equations can be decomposed at large times into a sum of solitary waves plus a dispersive tail (i.e., radiation). After giving an overview of what’s stated above, we will present results and challenges from a study aiming at this conjecture by considering Hamiltonian perturbations of the Toda lattice (joint work with I. Nenciu).

David Herron

Wednesday February 19 at 1:25-2:20 in CGD 214

You Can't See a Fat Annulus

Each plane domain $\Omega$ with two (or more) boundary points supports many non-Euclidean distances. Of special interest are the hyperbolic distance $h$ and quasihyperbolic distance $k$ and it is well known that the two metric spaces $(\Omega,h), (\Omega,k)$ can be quite different.

Nonetheless, it is fairly simple to establish a Rigidity Theorem that says that these metric spaces are either quite similar (bilipschitz equivalent) or quite different (not QS equivalent).

After a brief review, we'll discuss this, and then other results which reveal that the underlying geometries are in fact "always" quite similar.

David Herron

Wednesday February 26 at 1:25-2:20 in CGD 214

Using Uniformity

1st a question for all: Did you _see_ the fat annulus used right at the end of Wed's talk? :-)

Poincare inequalities, quasihyperbolic distance, and uniform spaces are without doubt three of the most important players in metric space analysis. After a brief intro about Gromov hyperbolicity, I'll discuss some of the ideas behind the proof of what I call the QS theorem:

For any Gromov hyperbolic plain domain Omega, the conformal gauges on the Gromov boundaries of (Omega, h) and (Omega, k) are naturally QS equivalent.

There will be lots of interplay between quasihyperbolic, hyperbolic, spherical, Euclidean, and uniform geometry.

Clark Butler (Princeton University)

Wednesday March 4 at 1:25-2:20 in CGD 214

Unbounded uniformizations of Gromov hyperbolic spaces

In a fundamental work Bonk, Heinonen, and Koskela established a conformal correspondence between Gromov hyperbolic spaces and bounded uniform spaces (satisfying certain additional hypotheses) that generalized the classical conformal correspondence between the Euclidean unit disk and the hyperbolic plane. We prove a similar conformal correspondence between Gromov hyperbolic spaces and unbounded uniform spaces that extends the correspondence between the Euclidean upper half plane and the hyperbolic plane. Our uniformization procedure is particularly well-adapted to the study of hyperbolic fillings of unbounded metric spaces.

David Minda

Wednesday March 11 at 1:25-2:20 in CGD 214

Normal Families of Mobius Maps in the Extended Complex Plane

Abstract available here

The seminar is temporarily suspended at least until in-person classes resume.

Tentative:

David Minda (Part 2)

Rajinder Mavi

Fall 2019

Michael Goldberg

Thursday December 5 at 12:20-1:15 in CGD 608

Strichartz estimates for higher-dimensional Schr\"odinger operators with lower-dimensional potentials

We consider Schr\"odinger operators H = −∆ +μ on R^n, n≥3, whose potential is a compactly supported d-dimensional measure for some d < n. We verify that H is self-adjoint provided the codimension n−d is less than 2, and we show thatStrichartz inequalities hold for the propagator e^{itH}P_{ac}(H) provided the codimension is less than 1 +1/(n−1). As an example, the results hold for potentials which are the product of a surface measure and any bounded function.

This is joint work with Burak Erdogan (Illinois) and William Green (Rose-Hulman).

Bingyu Zhang

Thursday November 21 at 12:20-1:15 in CGD 608

Well-posedness and Critical Index Set of the Cauchy Problem for the Coupled KdV-KdV Systems

Abstract available here.

Senping Luo

Wednesday November 13 at 3:30-4:30 in CGD 608

On homogeneous stable solutions to the polyharmonic equations

A unified framework is provided to classify the homogeneous stable solutions of the arbitrary polyharmonic Lane-Emden equations and Caffarelli-Silvestre extension equations. Our method is achieved by introducing the differential operator and the corresponding symmetric functions, the properties of the symmetric functions are carefully discussed. The connection between the symmetric functions and homogeneous stable solutions are uncovered by the generic stability inequalities.

Christopher Gartland (University of Illinois at Urbana Champaign)

Thursday November 7 at 12:20-1:15 in CGD 608

The Ribe Program and Markov Convexity of Model Filiform Groups

The Ribe program is the research program concerned with generalizing local properties of Banach spaces to biLipschitz invariant properties of metric spaces. Among such generalizations that have been found is the notion of Markov p-convexity, proven by Mendel-Naor to generalize uniform p-convexity. One of the first important spaces for which this invariant has been calculated is the Heisenberg group, proven by Li to be Markov p-convex for every p >= 4 and not Markov p-convex for any p < 4. In this talk, we'll start with background on the Ribe program and applications to metric space embedding theory, and then introduce the model filiform groups - a class of Carnot groups containing the Heisenberg group - and explain how to use random walks on graphs to compute their Markov convexities.

Tom Hill

Thursday October 31 at 12:20-1:15 in CGD 608

New Method for Dispersive Estimates of One-Dimensional Schrodinger Equations

Abstract available here.

Leonid Slavin

Thursday October 17 at 12:20-1:15 in CGD 608

An A_2 Refinement of the Helson-Szego Theorem

Abstract available here.

Tehri Moisala (University of Jyvaskyla, Finland)

Tuesday October 8 at 12:20-1:15 in CGD 608

Infinite-dimensional Carnot groups

I will introduce a class of spaces that can be seen as an infinite-dimensional analogue of Carnot groups and, on the other hand, as a noncommutative generalization of Banach spaces. We will see examples of these spaces as well as a geometric characterization. I will also show that any Lipschitz map with infinite-dimensional Carnot group domain has a point of Gâteaux differentiability (joint work with Enrico Le Donne and Sean Li) and that these spaces naturally appear as direct limits of classical Carnot groups (in preparation, with Enrico Pasqualetto).

Andrew Lorent

Thursday October 3 at 12:20-1:15 in CGD 608

On the Rank-$1$ convex hull of a set arising from a hyperbolic system of Lagrangian elasticity

There has recently been a lot of progress on a number of outstanding problems in PDE by reformulating the PDE as a differential inclusion. For example recently the decades old problem of the Onsager conjecture has been resolved in this way. An approach to studing entropy solutions of conservation laws via differential inclusion was suggested for a model system arising from Lagrangian elasticity by B. Kirchheim; S. Muller; V. Sverak. In "Studying nonlinear pde by geometry in matrix space" Geometric analysis and nonlinear partial differential equations, 347--395, Springer, Berlin, 2003. They asked a series of questions, P1,..P4. In joint work with G. Peng we previously answered P4 as to the structure of "polyconvex hull" of the set that forms the differential inclusion ascociated to the system. More recently we have studied P1, P2 that asks about the Rank-$1$ convex hull. This is the harder question and the one more relivant for the construction of non-trivial differential inclusions. G. Peng and I have shown no Tartar square can be embedded into the set https://arxiv.org/abs/1909.05938. I will give a board introduction to these notions and questions.

Gareth Speight

Thursday September 26 at 12:20-1:15 in CGD 608

Whitney Extension and Lusin Approximation for Horizontal Curves in the Heisenberg Group

The Whitney extension problem characterizes when a mapping defined on a compact subset can be extended to a smooth map defined on the whole space. This talk will describe a version of this for horizontal curves in the Heisenberg group and a recent application to obtain Lusin type C^m approximations of horizontal curves. Based on joint work with Capolli, Pinamonti and Zimmerman.

Angela Wu (Indiana University in Bloomington)

Tuesday September 17 at 12:20-1:15 in CGD 608

Ahlfors Regular Conformal Dimensions of Visual Spheres and Their Weak Tangents

Given an expanding Thurston map $f: \mathbb{S}^2 \to \mathbb{S}^2$, one can construct a metric on the 2-sphere, called the visual sphere of $f$, so that the map $f$ locally expands distance by a fixed expansion factor $\Lambda > 1$. Given a weak tangent $T$ of the visual sphere, we show that the Ahlfors regular conformal dimension of $T$ and the original visual sphere are the same, and the Ahlfors regular conformal dimension of $T$ can be attained if and only if the Ahlfors regular conformal dimension of the visual sphere can be attained.

Clark Butler (Princeton University)

Wednesday September 4 at 3:30-4:30 in CGD 217

Characterizing symmetric spaces by their Lyapunov spectra

We associate to each closed geodesic of a closed, negatively curved Riemannian manifold Y its Lyapunov spectrum, which is a list of real numbers measuring the exponential growth rate of the singular values of the derivative of the geodesic flow of Y along that geodesic. We show that if Y is homotopy equivalent to a negatively curved locally symmetric space X of dimension at least three and the Lyapunov spectra of the closed geodesics of Y match those of X then Y is isometric to X. The Mostow rigidity theorem follows as a corollary. The proof involves the use of several tools drawn from smooth dynamics and quasiconformal mapping theory. In this talk I will highlight the tools used from quasiconformal mapping theory, which are a theorem of Tyson on when the conformal dimension of a metric space is attained and an absolute continuity on lines result of Balogh-Koskela-Rogovin for quasiconformal maps between Ahlfors Q-regular spaces with the domain being a Carnot group.