2016-2017

Thursday April 20th, 2-3pm, seminar room: Gareth Speight (University of Cincinnati)

Maximal directional derivatives and universal differentiability sets in Carnot groups

Abstract: Rademacher's theorem asserts that Lipschitz functions from R^n to R^m are differentiable almost everywhere. Such a theorem may not be sharp: if n>1 then there exists a Lebesgue null set N in R^n containing a point of differentiability for every Lipschitz mapping f:R^n->R. Such sets are called universal differentiability sets and their construction relies on the fact that existence of an (almost) maximal directional derivative implies differentiability. We will see that maximality of directional derivatives implies differentiability in all Carnot groups where the Carnot-Caratheodory distance is suitably differentiable, which include all step 2 Carnot groups (in particular the Heisenberg group). Further, one may construct a measure zero universal differentiability set in any step 2 Carnot group. Finally, we will observe that in the Engel group, a Carnot group of step 3, things can go badly wrong... Based on joint work with Andrea Pinamonti and Enrico Le Donne.

Thursday April 13th, 2-3pm, seminar room: Tom Hill (University of Cincinnati)

A Presentation of Wiener's Lemma using Fourier Analysis

Thursday March 23rd, 2-3pm, seminar room: David Herron (University of Cincinnati)

Large Scale Hyperbolic Geometry (Part III)

Thursday March 9th, 2-3pm, seminar room: David Herron (University of Cincinnati)

Large Scale Hyperbolic Geometry (Part II)

Thursday March 2nd, 2-3pm, seminar room: David Herron (University of Cincinnati)

Large Scale Hyperbolic Geometry

Abstract: Each open connected proper subset Om=Omega of the Euclidean plane supports many non-Euclidean geometries. Of special importance are the hyperbolic and quasihyperbolic geometries obtained via hyperbolic distance h and quasihyperbolic distance k. We present an introduction to the large scale geometry of the metric spaces (Om,h) and (Om,k). These often quite different metric spaces have surprisingly similar geometries.

Thursday February 23rd, 2-3pm, seminar room: Panu Lahti (University of Cincinnati)

Lower semicontinuity of the total variation in quasiopen sets

Abstract: Consider a metric space equipped with a doubling measure and supporting a Poincaré inequality. It is well known that the total variation of a BV function is lower semicontinuous with respect to L^1-convergence in every open set. We show that lower semicontinuity holds also in every quasiopen set. To achieve this, we first prove a new characterization of the total variation in quasiopen sets. Then we discuss possible applications of the result to minimization problems.

Thursday February 16th, 2-3pm, seminar room: David Minda (University of Cincinnati)

Quotients of hyperbolic metrics near the boundary

Thursday February 9th, 2-3pm, seminar room: Michael Goldberg (University of Cincinnati)

On the differentiability of Fourier transforms

Abstract: In general, the Fourier transform of a function (on R^n) is differentiable if and only if the function itself decays rapidly at infinity. Furthermore, differentiability along a codimension-1 surface does not appear to imply good behavior in a neighborhood around that surface. Those appearances are slightly deceiving when the surface in question is curved. Even though regularity does not extend into the ambient space, one can still define a normal derivative along the surface itself. I will state a few claims of this nature, some of them proved and some still works in progress. This is joint work with Dmitriy Stolyarov.

Thursday February 2nd, 2-3pm, seminar room: Jim Gill (Saint Louis University)

Sobolev Spaces and Hyperbolic Fillings

Abstract: I will discuss a paper of the same title by M. Bonk and E. Saksman. In it they describe another construction of (Hajlasz)-type Sobolev spaces on an Ahlfors Q-regular metric space with a Poincare inequality.

Thursday December 1st, 2-3pm, French Hall Seminar Rm 4206. Wei Wang (Zhejiang University, China)

On the connections between different theories for nematic liquid crystals

Abstract: Nematic liquid crystals have properties between those of a conventional liquid and those of a solid crystal. There are three different types of theories to describe nematic liquid crystals: the molecular theory, the Q-tensor theory and the vector theory. The first is a microscopic theory based on statistical mechanics, and the later two are macroscopic theories based on continuum mechanics. In this talk, we will present some results on the inter-connection between these theories. In particular, by using the Hilbert expansion method, we will prove that the solution of the dynamic molecular model(Doi-Onsager model) converge to the solution of dynamic vector model(Ericksen-Leslie model) as the Deborah number goes to zero. Moreover, we will discuss the connection between Q-tensor theory and vector theories in different settings.

Tuesday November 29th, 12:30-1:30pm, French Hall Seminar Rm 4206. Nages Shanmugalingam (University of Cincinnati)

Dichotomy property for global capacity density associated with $p$-harmonic functions.

Abstract: Given an outer measure $\mu$ on a metric space $X$, and an unbounded set $U\subset X$, we say that $U$ has large scale density $1$ if

\[\liminf_{r\to\infty} \inf_{x\in X} \frac{\mu(B(x,r)\cap U)}{\mu(B(x,r))}=1.\]

We also say that $U$ has large scale density $0$ if

\[ \limsup_{r\to\infty} \inf_{x\in X} \frac{\mu(B(x,r)\cap U)}{\mu(B(x,r))}=0.\]

In the context of he complex plane, with $\mu$ the logarithmic capacity, it turns out that all analytic functions from the unit disc to a given planar domain belong to the class BMO if and only if the complement of the planar domain has large scale density $1$. This was discovered by Stegenga, who sought to extend a weaker result of Hayman and Pommerenke. Along the way Stegenga discovered a dichotomy property of logarithmic capacity; every unbounded subset of the plane must have either large scale density $1$ or else have large scale density $0$. This dichotomy property was extended to more general $p$-capacitary setting in Euclidean spaces by Aikawa and Itoh recently. In this talk we will describe the main ideas behind this property, and what extensions of this dichotomy result holds for metric measure spaces. This is joint work with H. Aikawa, A. Bjorn, and J. Bjorn.

Tuesday November 22nd, 12:30-1:30pm, French Hall Seminar Rm 4206. Abigail Richard (University of Cincinnati)

Quasisymmetric mappings and the quasihyperbolic metric

Abstract: Since the quasihyperbolic metric was introduced, it has proven to have many applications in quasiconformal and quasisymmetric mappings. Recently, Siaojun Huang and Jinsong Liu studied the relationship between the quasihyperbolic metric and quasisymmetric mappings. In one of the theorems that they proved, they were able to relate quasihyperbolic distance between points in quasiconvex spaces to their image under quasisymmetric maps. We will discuss this theorem and the ideas used by Huang and Liu in their proof.

Tuesday November 8th, 12:30-1:30pm, French Hall Seminar Room 4206. Lukas Maly (University of Cincinnati)

Trace and extension theorems for BV and Sobolev-type functions on domains in metric spaces

Abstract: In the general Dirichlet problem, one starts with a domain, prescribes boundary values, and looks at the set of functions on the interior of the domain whose trace on the boundary matches the prescribed boundary values. For domains in metric measure spaces, we investigate the class of functions on the boundary that can be extended to functions of some specified regularity on the interior. Under some rather mild requirements on regularity of the boundary, we find a linear extension operator from a certain Besov class on the boundary to the BV (or to the Newton–Sobolev class) on the interior of the domain. This operator can then be used to find BV or Newton–Sobolev extensions of L^p boundary data provided that the boundary is endowed with a codimension p regular Hausdorff measure.

We will also discuss the converse problem of establishing that generic BV and Sobolev-type functions defined inside the domain allow for a reasonable notion of a trace on the boundary. In particular, we will look into how smoothness (in terms of a Besov-type seminorm) and integrability of the trace depend on the codimension of the boundary and on the integrability of "gradients" of the Sobolev-type functions. Under somewhat stronger requirements on regularity of the domain and its boundary, we will see that the aforementioned extension results are sharp.

The talk is partly based on a joint work with N. Shanmugalingam and M. Snipes.

Thursday November 3rd, 2-3pm, French Hall Seminar Rm 4206. Zair Ibragimov (Cal State Fullerton)

Invariant Cassinian metrics

Abstract: The Cassinian metric of Euclidean domains were introduced in 2009. In this talk we discuss two recent modifications of the Cassinian metric, namely the scale-invariant and the M\"obis invariant Cassinian metrics. We will discuss basic properties of these metrics including their connections to other hyperbolic-type metrics.

Tuesday November 1st, 12:30-1:30pm, French Hall Seminar Rm 4206. Dr. Elizabeth Strouse (Univ Bordeaux). (Joint seminar with functional analysis and probability)

Different kinds of Hankel operators and different types of BMO

Abstract: Functions of Bounded Mean Oscillation on the circle are those whose values do not 'oscillate' too much on intervals. It is a remarkable and deep result that these functions can be characterized by their Fourier coefficents and are related to boundedness of certain 'Hankel operators'. I will give an elementary explanation of these results and their various generalizations to functions of several variables.

Tuesday October 18th, 12:30-1:30pm, French Hall Seminar Room 4206. Gareth Speight (University of Cincinnati).

Differentiability of Lipschitz maps in Euclidean spaces and Carnot groups

Abstract: Rademacher's theorem states that Lipschitz maps between Euclidean spaces are differentiable almost everywhere. Pansu's theorem asserts that the same holds for Lipschitz maps between Carnot groups. We discuss converses of these statements, where points of differentiability should be constructed inside measure zero sets. We then discuss applications of porous sets, which have relatively large holes on arbitrarily small scales, to differentiability of Lipschitz maps.

Thursday October 6th, 3:30-4:30pm, French Hall Seminar Room 4206. Guanying Peng (University of Cincinnati).

Regularity of the Eikonal equation with two vanishing entropies

Abstract: We study regularity of solutions to the Eikonal equation $|\nabla u|=1$ a.e. in a bounded simply-connected two dimensional domain. With the help of two vanishing entropies, we prove that solutions of the Eikonal equation are locally Lipschitz continuous, except at a locally finite set of points in the domain. The motivation of our problem comes from the zero energy state of the Aviles-Giga functional. Our results for the first time use only two entropies to characterize regularity properties in this direction. This is joint work with Andrew Lorent.

Tuesday October 4th. 12:30-1:30pm, French Hall Seminar Rm 4206. Panu Lahti (University of Cincinnati).

Strong approximation of sets of finite perimeter in metric spaces

Abstract: We show that in a metric space with a doubling measure and a Poincare inequality, a set of finite perimeter can be approximated in the BV norm by sets whose topological boundary almost coincides with the measure theoretic boundary.

Tuesday September 6th, 13th, 20th, 12:30-1:30pm, French Hall Seminar Rm 4206. Abigail Richard (University of Cincinnati).

Pointed Gromov-Hausdorff Convergence and the Embedding Theorem

Abstract: Just as we use distance and norms to measure how “close” points are to each other, we would like to have a method for describing how “close” metric spaces are to each other. One way of describing this is through the Gromov-Hausdorff distance between two spaces. In order to expand this idea of measuring the distance between metric spaces to pointed metric spaces, we will discuss what is known as pointed Gromov-Hausdorff distance. After discussing the notions of Gromov-Hausdorff distance and pointed Gromov-Hausdorff distance, we will present Dr. David Herron’s proof of an Embedding Theorem for the pointed Gromov-Hausdorff limit of a sequence of pointed proper metric spaces.