University of Cincinnati: Analysis and PDE Seminar (2018-2019)

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 2019

Wednesday April 24, 3:30-4:30PM, French Hall Seminar Room

Andrea Pinamonti (University of Trento)

$\Gamma$-convergence for integral functionals depending on vector fields.

The aim of this talk is to present a recent result of $\Gamma$-convergece for functionals depending on Lipschitz vector fields. In particular, we will prove that, under some assumptions, any sequence of integral functionals depending on Lipschitz vector fields, $\Gamma-$ converges and the $\Gamma$-limit is an integral functional. The talk is based on a joint work with A. Maione and F. Serra Cassano.

This talk is part of a Taft Research Seminar on Non-Smooth Analysis.

Thursday April 18, 4-5PM, 60W Charlton 240

Andrea Pinamonti (University of Trento)

An introduction to Sub-Riemannian geometry

The aim of this talk is to introduce some basic ideas in sub-Riemannian geometry. We will start by introducing sub-Riemannian manifolds and we characterize their tangent space using the celebrated Mitchell's theorem. This will allow us to introduce and study Carnot groups. We will conclude by describing some challenging open problems in sub-Riemannian geometry.

This talk is part of a Taft Research Seminar on Non-Smooth Analysis.

Wednesday April 17th, 4-5pm, Swift 808 (Colloquium)

Tatiana Toro (University of Washington)

The geometry of measures

In the 1920's Besicovitch studied linearly measurable sets in the plane, that is sets with locally finite "length". The basic question he addressed was whether the infinitesimal properties of the "length" of a set E in the plane yield geometric information about E itself. This simple question marks the beginning of the study of the geometry of measures and the associated field, Geometric Measure Theory.

In this talk I will present several examples of measures that arise naturally in different contexts, for example PDEs and free boundary regularity problems. I will discuss how the infinitesimal properties of a measure yield a great deal of information about the measure and its support. In turn this sheds light on the original problem which gave rise to the measure in question.

Tuesday April 16th, 4-5PM, Rec Center 3240 (Taft Lecture)

Tatiana Toro (University of Washington)

The world through mathematical lenses: rough vs smooth

This lecture will introduce non-experts to some of the tools used in mathematics to distinguish between rough and smooth objects. Then it will explore some of their wide-ranging applications. One of the goals is to illustrate how the development of deep mathematical concepts is often motivated by real life observations.

Wednesday April 10th, 3:30-4:30PM, French Hall Seminar Room

Andrea Pinamonti (University of Trento)

Geometric aspects of P-capacitary potentials

The aim of this talk is to provide monotonicity formulas for solutions to the p-Laplace equation defined in the exterior of a convex domain. A number of analytic and geometric consequences are derived as well as new characterizations of rotationally symmetric solutions and domains. The talk is based on a joint work with L. Mazzieri and M. Fogagnolo.

This talk is part of a Taft Research Seminar on Non-Smooth Analysis.

Thursday April 4th, 4pm, Room 220, 60W Charlton (Colloquium)

Krystal Taylor (Ohio State University)

The geometry of sets from the perspective of Fourier analysis and projection theory

In this talk, we consider a circle of geometric problems from the lens of harmonic analysis, geometric measure theory, and number theory. This includes understanding the existence of finite point configurations within fractal sets, obtaining estimates in lattice point counting problems, and determining the dimension, measure, and interior of Euclidean sets of the form A+B and AB (sum and product sets).

The common theme is decomposing intricate objects into simpler components using Fourier transforms and projection theorems. The Fourier transform decomposes a function or measure into frequencies which are often easier to analyze. Orthogonal projections offer a means to view higher dimensional objects in terms of lower dimensional information. This talk will be accessible to a wide mathematical audience, and relevant background information is given.

Thursday March 14, 11AM-12PM, Rec Center 3250

Paul Hagelstein (Baylor University)

Solyanik Estimates in Harmonic Analysis

Abstract available here.

Wednesday March 13, 3:30-4:30PM, French Hall Seminar Room

Anders Bjorn (Linkoping University)

p-harmonic Green functions and their local integrability

Abstract available here.

This talk is part of a Taft Research Seminar on Non-Smooth Analysis.

Thursday March 7, 12:20-1:20PM, WCharlton 250

Anders Bjorn (Linkoping University)

Uniform continuity and some other parts from the history of analysis in the 19th century

Abstract available here.

This talk will be much much less technical and might appeal to a wider audience.

This talk is part of a Taft Research Seminar on Non-Smooth Analysis.

Wednesday March 6, 3:30-4:30PM, French Hall Seminar Room

Anders Bjorn (Linkoping University)

Boundary regularity for the porous medium equation

Abstract available here.

This talk is part of a Taft Research Seminar on Non-Smooth Analysis.

Wednesday February 27, 3:30-4:30PM, French Hall Seminar Room

Jana Bjorn (Linkoping University)

The Dirichlet problem and boundary regularity for nonlinear parabolic equations

Abstract available here.

This talk is part of a Taft Research Seminar on Non-Smooth Analysis.

Monday February 25, 4-5PM, 277 West Charlton

Fabrice Baudoin (University of Connecticut)

Heat semigroup based Besov spaces and BV functions

In abstract Dirichlet spaces, we develop a theory of Besov spaces which is based on the heat semigroup. This approach offers a new perspective on BV and Korevaar-Schoen classes in settings including sub-Riemannian manifolds or fractal spaces. The key assumption on the underlying space is a weak Bakry-\'Emery type curvature assumption which is weaker than the RCD(0,\infty) condition extensively studied in the last few years.

The talk will based on joint works with Patricia Alonso-Ruiz, Li Chen, Luke Rogers, Nageswari Shanmugalingam and Alexander Teplyaev.

Thursday February 21, 12:20-1:20PM, WCharlton 250

Jana Bjorn (Linkoping University)

Sphericalization and p-harmonic functions on unbounded domains in Ahlfors regular metric spaces

We use sphericalization of unbounded metric spaces to transform p-harmonic functions on unbounded domains to p-harmonic functions on bounded ones, for which the theory is much more developed and there are plenty of methods and results. In particular, we consider the Dirichlet problem in unbounded domains, with a particular emphasis on boundary regularity at infinity. As a byproduct, we obtain a result on the p-harmonic measure in R^n.

This talk is part of a Taft Research Seminar on Non-Smooth Analysis.

Wednesday February 20, 3:30-4:30PM, French Hall Seminar Room

Jana Bjorn (Linkoping University)

Good and bad news about quasiminimizers (part 2)

Abstract available here.

This talk is part of a Taft Research Seminar on Non-Smooth Analysis.

Friday February 15, 1:25-2:25PM, Rec Center 3230

Xining Li

Sphericalization and flattening with applications in quasimetric measure space

In this talk, we prove that the Ahlfors regular measure and doubling measure can be preserved under sphericalization and flattening in quasimetric spaces, which gives a generalization of a recent of Wang and Zhou. And we show that the Loewner condition is quasimobius invariant between two Ahlfors Q-regular spaces.

Wednesday February 13, 3:30-4:30PM, French Hall Seminar Room

Jana Bjorn (Linkoping University)

Good and bad news about quasiminimizers (part 1)

Abstract available here.

This talk is part of a Taft Research Seminar on Non-Smooth Analysis.

Thursday January 31, 1-2PM, West Charlton 273: Sylvester Eriksson-Bique (UCLA)

What happens if the Sierpinski carpet attains its conformal dimension?

It is a long standing open question what the value of the conformal dimension of the Sierpinski carpet is, and whether or not it is attained. In this talk, in addition to defining all the notions just mentioned, I will discuss what special properties such a minimizer would have. So far, no space satisfying all such properties is not known to exist, but a number of examples exist that almost satisfy them. This raises the question, if it is at all possible for such a space to exist. These questions have some connections to geometric group theory and a conjecture from Bruce Kleiner on the so called combinatorial Loewner property.

Thursday January 24, 12:20-1:20PM, 60W Charlton 227: John Dever (Bowling Green State University)

Space and Time Scaling Exponents on Fractals and Compact Metric Spaces

We discuss two scaling exponents that may be defined on any compact metric space: the local Hausdorff dimension and the local walk dimension. Intuitively, the local Hausdorff dimension determines how the volume of metric balls scales with the radius of the ball, and the local walk dimension determines how fast a diffusion process leaves a ball on average relative to its radius. Spaces that have walk dimension not equal to 2 exhibit what is called anomalous diffusion. We give examples of fractals that have continuously variable Hausdorff dimension and continuously variable walk dimension. Moreover, we explain how these dimensions may be useful in approximating diffusion on a compact metric space.

Fall 2018

Tuesday November 27th, 12:30-1:30pm, French Hall Seminar Room: Nageswari Shanmugalingam (University of Cincinnati)

A notion of prime ends for domains in metric spaces, and associated Dirichlet problem

Caratheodory's notion of prime ends was developed with the goal of extending conformal mappings between two simply connected planar domains to their respective boundaries. For non-simply connected planar domains and for domains in higher dimensional Euclidean domain. Caratheodory's notion is not as productive, and so in the metric setting it is not applicable either. We will discuss an alternate notion of prime ends that seems to be geared towards setting up Dirichlet problems that respect the shape of the domain. The talk is based on joint work with Tomasz Adamowicz, Anders Bjorn, Jana Bjorn, and Dewey Estep.

Colloquium: Wednesday November 14th, 4-5pm, Location TBA: David Freeman (University of Cincinnati, Blue Ash)

Toward a Bi-Lipschitz Characterization of Invertible Homogeneous Metric Spaces

We study the geometry of certain metric spaces that are homogeneous with respect to uniformly bi-Lipschitz self-homeomorphisms and quasi-invertible in a sense that generalizes inversion in a Euclidean sphere. In particular, we provide new characterizations of such spaces.

Tuesday November 13th, 12:30-1:30pm, French Hall Seminar Room: Leonid Slavin (University of Cincinnati)

The BMO-BLO norm of the maximal operator on alpha-trees

Tuesday November 6th, 12:30-1:30pm, French Hall Seminar Room: Michael Goldberg (University of Cincinnati)

The Stein-Tomas Theorem, now with derivatives

The Stein-Tomas restriction theorem shows that the Fourier transform of a function in L^p(R^n) has trace values on any curved surface, even though it may only be defined almost everywhere. We prove a generalization for trace values for the gradient of the Fourier transform, and for derivatives of an associated bilinear form. This is joint work with Dmitriy Stolyarov.

Tuesday October 30th, 12:30-1:30pm, French Hall Seminar Room: Rebekah Jones (University of Cincinnati)

Title: Modulus of sets of finite perimeter and quasiconformal maps between metric spaces of globally $Q$-bounded geometry

Abstract: In Euclidean space, it is well-known that quasiconformal maps quasi-preserve the $n$-modulus of curves. In 1973, Kelly also showed that the $n/(n-1)$-modulus of surfaces is quasi-preserved. We generalize this result to the setting of Ahlfors $Q$-regular metric spaces supporting a 1-Poincar\'e inequality. In fact, we consider a larger class of surfaces so our results are new even in Euclidean space. This talk is based on joint work with Panu Lahti and Nageswari Shanmugalingam.

Monday October 29th, 11am-12pm, French Hall Seminar Room: Angela Wu (UCLA)

Title: A metric sphere not a quasisphere but for which every weak tangent is Euclidean

Abstract: For all $n \geq 2$, there exists a doubling linearly locally contractible metric space $X$ that is topologically a $n$-sphere such that every weak tangent of $X$ is isometric to $\R^n$, but $X$ is not quasisymmetrically equivalent to the standard $n$-sphere. We discussed the $2$-dimensional example, with emphasis on how $2$-modulus helps in the study of quasisymmetric equivalence.

Tuesday October 23rd, 12:30-1:30pm, French Hall Seminar Room: Jeffrey Lindquist (University of Cincinnati)

Title: Branched Quasisymmetric Mappings and Extensions (Continued)

Tuesday October 16th, 12:30-1:30pm, French Hall Seminar Room: Jeffrey Lindquist (University of Cincinnati)

Title: Branched Quasisymmetric Mappings and Extensions

Abstract: We investigate a class of maps called branched quasisymmetric mappings (BQS maps) between Ahlfors regular metric spaces. BQS maps are generalizations of quasisymmetric maps that may not be locally injective. We define vertical quasi-isometric mappings (VQI maps) between fixed hyperbolic fillings of bounded turning Ahlfors regular metric spaces. We relate the classes of BQS maps and VQI maps under some mild assumptions. We connect BQS mappings to quasiregular mappings in the compact Riemannian setting. (Joint work with Pekka Pankka from the University of Helsinki)

Monday October 8th, 11am-12pm: French Hall Seminar Room: Scott Zimmerman (University of Connecticut)

Title: The Traveling Salesman Theorem in Carnot groups

Abstract: Peter Jones proved his famous Traveling Salesman Theorem in the plane in 1990. His result classified those sets in the plane which are contained in a rectifiable curve via the boundedness of a certain Carleson integral. The methods introduced by Jones have seen applications in harmonic analysis and geometric measure theory, and versions of his theorem have since been proven in the settings of Hilbert spaces, the Heisenberg group, and the graph inverse limits of Cheeger and Kleiner. I will present recent work with V. Chousionis and S. Li in which we proved one direction of the Traveling Salesman Theorem for rectifiable curves in any Carnot group. A Carnot group is a type of nilpotent Lie group whose abelian members are precisely Euclidean spaces, and these groups have been the focus of much recent study in geometric measure theory. As an application, I will show that this theorem may be used to prove uniform boundedness of the singular integral operator associated with a certain non-negative kernel on any set contained in a rectifiable curve.

Tuesday September 25, 12:30-1:30pm, French Hall Seminar Room: Gareth Speight (University of Cincinnati)

Title: A C^m Whitney Extension Theorem for Horizontal Curves in the Heisenberg Group

Abstract: We characterize those mappings from a compact subset of R into the Heisenberg group which can be extended to a C^m horizontal curve in the Heisenberg group. The characterization combines conditions from the classical Whitney extension theorem with an estimate comparing changes in the vertical coordinate with those predicted by the Taylor series of the horizontal coordinates.

Tuesday September 11, 12:30-1:30pm, room TBC: Panu Lahti (University of Jyvaskyla)

Title: Federer's characterization of sets of finite perimeter in metric spaces

Abstract: Federer's characterization of sets of finite perimeter states that a set is of finite perimeter if and only if the n-1-dimensional Hausdorff measure of the set's measure-theoretic boundary is finite. Federer's original proof of this result relies heavily on the structure of Euclidean space. I present a new proof of the characterization, relying instead on fine potential theory in the case p=1. An advantage of this proof is that it works in the general setting of a complete metric space equipped with a doubling measure and supporting a Poincaré inequality.

Thursday September 6, 4-5pm: 60W Charlton room 250: Guy C David (Ball State University) (colloquium)

Title: Lipschitz differentiability, embeddings, and rigidity for group actions

Abstract: We discuss a class of metric spaces that, despite being non-Euclidean, support a first-order calculus for Lipschitz functions due to Cheeger. After introducing these spaces, we will survey some of their embedding properties and explain a theorem of the speaker and Kyle Kinneberg concerning embeddings in Carnot groups. Then we will explain an application of this last result to a problem on group actions in hyperbolic geometry.

Thursday September 6, 12-1pm, French Hall seminar room (4206): Guy C David (Ball State University)

Title: Lipschitz and bi-Lipschitz maps between metric measure spaces

Abstract: An old rigidity question for Lipschitz mappings asks when a Lipschitz mapping between two metric spaces must have some "bi-Lipschitz behavior". We will survey some results and counterexamples related to this problem. We will then discuss new results of the speaker and Kyle Kinneberg concerning mappings between metric measure spaces and Carnot groups, addressing questions of Semmes.