Metric Spaces: Analysis, Embeddings into Banach Spaces, Applications

Lecture Series

Gilles Lancien, Université de Franche-Comté

Approximation properties in Lipschitz free spaces

In their seminal work [GK1], G. Godefroy and N. Kalton revisited the notion of the Lipschitz-free space (also known as the Arens-Eells space) ℱ(M) over a metric space M, in order to use it as a fine analytic tool, perfectly suited to problems in nonlinear geometry. The free space ℱ(M) is a Banach space defined as a natural predual of the space of real valued Lipschitz functions on M, whose key property is that every Lipschitz map between metric spaces M and N is just the trace of a linear mapping between the free spaces ℱ(M) and ℱ(N). Using this concept, they proved a deep generalization of the Mazur-Ulam theorem: any isometric embedding of a separable Banach space into another Banach space is always accompanied by a linear isometric embedding. In the same paper, they showed that a Banach space X has the bounded approximation property (BAP) if and only if ℱ(X) has the BAP. One stricking consequence of this result is that the BAP is stable under Lipschitz isomorphisms between Banach spaces.

In various conferences and subsequent papers, G. Godefroy has promoted the natural program of studying the BAP (or its improved variants) for free spaces over metric and more specifically compact metric spaces. The aim of this short series of lecture is to describe some of the recent developments in this direction.

We will first recall briefly the basic definitions and the construction of Godefroy-Kalton's linear lifting isometry from a separable Banach space into its free space. Then, following [G1], we will explain the links between the BAP for a free space over a compact metric space and the existence of linear almost extension operators for Lipschitz functions. Next we will give a few counterexamples: there exist a compact convex metric space and a metric space homeomorphic to a Cantor space whose free spaces fail the approximation property ([GO] and [HLP]).

We hope to save most of our time to describe positive results about free spaces on doubling metric spaces [LP], over countable compact metric spaces [D] and compact convex subsets of ℝⁿ [PS]. If time allows we will also explain how one can build a Schauder basis of the free space over ℓ₁ or over an ultrametric space ([HP] and [CD]).

Bibliography:
[CD] M. Cúth and M. Doucha, Lipschitz-Free Spaces Over Ultrametric Spaces, Mediterr. J. Math., published electronically in 2015, DOI 10.1007/s00009-015-0566-7.
[D] A. Dalet, Free spaces over countable metric spaces, Proc. Amer. Math. Soc. 143 (2015), 3537--3546.
[HLP] P. Hájek, G. Lancien and E. Pernecká, Lipschitz-free spaces over metric spaces homeomorphic to the Cantor space, to appear in the Bull. of the Belgian Math. Soc.
[HP] P. Hájek and E. Pernecká, On Schauder bases in Lipschitz-free spaces, J. Math. Anal. Appl. 416 (2014), 629--646.
[G1] G. Godefroy, Extensions of Lipschitz functions and Grothendieck's bounded approximation property, North-Western European Journal of Mathematics Vol. 1 (2015), 1--6.
[GK1] G. Godefroy and N. Kalton, Lipschitz-free Banach spaces, Studia Math. 159 (2003), 121--141.
[GO] G. Godefroy and N. Ozawa, Free Banach spaces and the approximation properties, Proc. Amer. Math. Soc. 142 (5) (2014), 1681--1687.
[LP] G. Lancien and E. Pernecká, Approximation properties and Schauder decompositions in Lipschitz-free spaces, J. Funct. Anal. 264 (2013), 2323--2334.
[PS] E. Pernecká and R. J. Smith, The metric approximation property and Lipschitz-free spaces over subsets of ℝⁿ, J. Approx. Theory. 199 (2015), 29--44.

K. Makarychev (Microsoft Research) and Y. Makarychev (Toyota Technological Institute)

Metric spaces in theoretical computer science

In this lecture series, we will give an overview of metric techniques in computer science. We will discuss low distortion metric embeddings, ultrametrics, metric space decompositions, Lipschitz extensions, and their applications to graph partitioning problems, online algorithms, and vertex sparsification.

Jeremy Tyson, University of Illinois at Urbana-Champaign

Distortion of dimension by metric space-valued Sobolev mappings

Metric space-valued Sobolev mappings may be defined via an isometric embedding of the target into a Banach space. The theory of Sobolev mappings between metric spaces is motivated by geometric variational problems, quasiconformal mapping theory and geometric measure theory. In these lectures I will describe recent work on the dimension distortion behavior of Sobolev and quasiconformal mappings. The behavior of a Sobolev mapping on a generic member of a parameterized family of subsets may be significantly better than its worst-case behavior. For instance, quasiconformal mappings are absolutely continuous along almost every parallel line, but can distort the dimensions of individual lines rather badly. I will discuss the effect of quasiconformal and supercritical Sobolev mappings on the Hausdorff dimensions of generic elements in several parameterized families of sets: parallel affine Euclidean subspaces, Grassmannians, left and right cosets of homogeneous subgroups of the sub-Riemannian Heisenberg group, and fibers of a David-Semmes regular mapping between metric spaces of bounded geometry. The results which I will describe are new already for quasiconformal mappings of the plane. The supercritical integrability assumption is essential; by way of contrast, there exist critical Sobolev surjections from low-dimensional cubes onto a wide variety of metric spaces, including infinite-dimensional spaces.

These lectures are based on joint work with Zoltan Balogh, Pertti Mattila, Roberto Monti, Kevin Wildrick and Piotr Hajlasz.

Plenary Lectures

Mikael de la Salle, CNRS, ENS Lyon

A duality Banach spaces/Operators between L_p spaces

To a pair (X,T) of a Banach space and an operator between subspaces of L_p spaces, one associates a (possibly infinite) positive number, the norm of T between suitable subspaces of X-valued L_p spaces. I will motivate the study of this duality by examples from embeddings into Banach spaces, and I will explain a form of the bipolar theorem for this duality.

Stephen Dilworth, University of South Carolina

Metrical characterizations of operators between Banach spaces

This is a report on joint work with Ryan Causey. Non-superreflexivity of Banach spaces has been characterized by bilipschitz embeddability of various metric spaces or families of metric spaces: finite binary trees (Bourgain), diamond and Laakso graphs (Johnson and Schechtman), and the infinite binary tree (Baudier). Given a linear operator T:X→Y between Banach spaces, we consider bilipschitz embeddings into Y which factor through T. We obtain characterizations of non-super weakly compact operators which are the analogues for operators of the results for spaces mentioned above. We also consider the asymptotic version of superreflexivity known as property (β) and its isomorphic counterpart. We define property (β) for operators and characterize its isomorphic counterpart in terms of the nonexistence of bilipschitz embeddings which factor through T of infinitely branching trees, which is the analogue for operators of a theorem of Baudier, Kalton, and Lancien for spaces.

Denka Kutzarova, University of Illinois at Urbana-Champaign

Property (β) of Rolewicz and its applications to nonlinear analysis

S. Rolewicz introduced in 1987 a new geometric property (β) in connection to optimization problems. In 2012, Lima and Randrianarivony used it to solve an open problem in nonlinear analysis. We shall present isometric and isomorphic equivalent characterizations of property (β), which plays the role of an asymptotic version of uniform convexity. We shall discuss its connection to embeddings of different types of infinite graphs. The results are part of joint papers with various mathematicians.

Sean Li, University of Chicago

RNP Differentiability and Poincare-type inequalities in metric measure spaces

Over the last two decades, the Poincare inequality, in the sense of Heinonen and Koskela, has played a central role in the development of analysis on metric measure spaces. Cheeger and Kleiner showed that such spaces satisfy the differentiability theory of Cheeger for Lipschitz functions taking value in Banach spaces with the Radon-Nikodym property (RNP). We give a partial converse as well as a characterization of RNP Cheeger differentiability. Namely, we introduce the notion of an asymptotic nonhomogeneous Poincare inequality for metric measure spaces and show how it is equivalent to the space satisfying RNP Cheeger differentiability. (Joint work with David Bate.)

Masato Mimura, Tohoku University

Strong algebraization of fixed point properties

A group is said to have the fixed point property relative to a subgroup with respect to a class of Banach space if all affine isometric actions of the whole group on every Banach space in the class has a fixed point by the subgroup. If the subgroup equals the whole group, then this property is simply called the fixed point property. That represents rigidity: for instance, it has application to much more rigid expanders, from the view point of coarse geometry, than expanders with high girth.

One strategy for establishing the fixed point property directly for a discrete group is given by two breakthroughs of Y. Shalom (1999 in Publ. IHES and 2006 in ICM proceedings). This is called an "algebraization", and is a patch-working method of relative fixed point properties to the whole one. One bottle-neck WAS: this method always imposed so-called "Bounded Generation" assumptions, which is in general hard to expect.

I will talk on a resolution in the affirmative to the following problem since Shalom's first breakthrough (1999) : "Can we remove all forms of "Bounded Generation" assumptions from Shalom's algebraization strategy?" Our criterion is stated in terms of winning strategy for a certain "Game", and we call our new method a strong algebraization.

Damian Osajda , Uniwersytet Wroclawski

Groups containing expanders

I will present a construction of finitely generated groups containing isometrically embedded expanders. Such groups have many exotic properties. For instance, they do not embed coarsely into a Hilbert space, and the Baum-Connes conjecture with coefficients fails for them. The construction allows us to provide the first examples of groups that lack property A (i.e., they are not exact) but are still coarsely embeddable into a Hilbert space. Better still, these groups act properly on CAT(0) cubical complexes. To end with, I will also present some further applications of the main construction concerning aspherical manifolds and the asymptotic dimension.

Conrad Plaut , University of Tennessee

Essential Circles and their Applications

Essential Circles are special closed geodesics defined using discrete homotopies in compact geodesic spaces, whose length spectrum determines, among other things, a special set of covering spaces and corresponding subgroups of the fundamental group. Essential circles, and their discrete equivalents, essential triads, are particularly amenable to quantifying aspects related to the fundamental group. We will introduce the concepts and discuss applications in joint work with Jay Wilkins, including a quantitative version of a classical theorem of Gromov about fundamental groups of compact Riemannian manifolds and a resulting “curvature free” finiteness theorem generalizing theorems of Anderson and Shen-Wei. We also characterize the “closure” of the delta covers of Sormani-Wei, which we will discuss briefly for experts due to time constraints. The talk will conclude with some remaining mysteries concerning essential circles and their role in possible new topological invariants.

Beata Randrianantoanina , University of Miami, Ohio

On embeddings of series-parallel graphs in Banach spaces.

I will describe constructions of bilipschitz embeddings of some families of finite series-parallel graphs in Banach spaces. Joint work with Mikhail Ostrovskii.

Gideon Schechtman , Weizmann Institute

Pythagorean powers of hypercubes

For 1 ≤ p,q≤ 2, ℓ_p(ℓ_q) isomorphically embeds into L₁=L₁(0,1) if and only if p≤ q. The best proof of the non-embedability part of this statement, with the right estimates for the distance of ℓⁿ_p(ℓ^m_q) from a subspace of L₁, follows from an inequality of Kwapień and Schütt. We will be interested in lower bounding the distortion of Lipschitz embedding the ℓⁿ_p sum of the discrete cube 𝔽ⁿ₂ with the ℓ_q norm into L₁. Our main result is that
Theorem: For all 1≤ p < q, the distortion of embedding ℓⁿ_q(𝔽ⁿ₂,‖⋅‖_p) into L₁ is of order n^{¹⁄_p-¹⁄_q}.
The main part of the proof is establishing an inequality which is a non linear version of the inequality of Kwapień and Schütt. This inequality holds in L₁ and fails grossly in ℓⁿ_q(𝔽ⁿ₂,‖⋅‖_p). Joint work with Assaf Naor.

Viktor Schroeder , University of Zürich

Moebius structures on the boundary of hyperbolic spaces

A main feature of the classical hyperbolic space Hⁿ is the deep relation between the geometry of this space and the Moebius geometry of its boundary at infinity. For example the isometries of the hyperbolic space correspond to Moebius transformations of its boundary. Many of these relations can be generalized to Riemannian manifolds of negative curvature and even more general to so called CAT(-1) spaces or even to Gromov hyperbolic spaces. In the lecture we explain what a Moebius structure is and how one how can associate to generalized hyperbolic spaces a Moebius structure at infinity. Then we study the (much more complicated) inverse problem: to what extend can one reconstruct the hyperbolic space from the Moebius structure. This question is related to many classical problems: e.g. problems concerning the conjugacy problem of the geodesic flow on surfaces and questions the marked length spectrum. In general the question is very open and only partial answers are known.

Invited Lectures

Bruno de Mendonça Braga, University of Illinois at Chicago

On weak nonlinear embeddings between Banach spaces

We will define some weaker versions of coarse and uniform embeddability, and provide suggestive evidences that those weaker embeddings may be stronger than one would think. We do such by showing that many known results for coarse and uniform embeddings remain valid for those weaker notions of embeddings.

Marek Cuth , Charles University

Embedding of ℓ₁ into Lipschitz-free Banach spaces and ℓ_∞ into their duals

Given a metric space M, it is possible to construct a Banach space ℱ(M) in such a way that the Lipschitz structure of M corresponds to the linear structure of ℱ(M). This space ℱ(M) is sometimes called the "Lipschitz-free space over M". The study of Lipschitz-free Banach spaces became an active field of study (see e.g. the abstract of G. Lancien's lecture series above). I will present our recent result with M. Johanis that ℓ_∞ embeds isometrically into the dual of every infinite-dimensional Lipschitz-free Banach space and that it is often the case that a Lipschitz-free Banach space contains a 1-complemented subspace isometric to ℓ₁. We do not know whether the later is true for every infinite-dimensional Lipschitz-free Banach space.

Guy C. David, New York University

The Analyst's Traveling Salesman theorem in limits of metric graphs

We discuss a version of Peter Jones' "Analyst's Traveling Salesman" theorem in a class of highly non-Euclidean metric spaces introduced by Laakso and Cheeger-Kleiner. These spaces are constructed as inverse limits of metric graphs and have many interesting analytic properties. We show that a set in such a space is contained in a rectifiable curve if and only if it is quantitatively flat at most locations and scales, where flatness is measured with respect to so-called monotone geodesics. This provides a first examination of quantitative rectifiability within these spaces. This is joint work with Raanan Schul (Stony Brook).

Ana Khukhro, Université de Neuchâtel

Box spaces, diameter, and expanders

Box spaces are metric spaces which are built using finite quotients of residually finite groups. There are many intriguing connections between properties of groups, and coarse-geometric properties of their box spaces. These connections can be exploited to construct interesting examples and counterexamples, both in group theory and in metric geometry. In particular, group-theoretic information can be used to construct spaces with certain embedding properties with respect to classes of spaces such as Banach spaces. We will present recent work in this direction, as well as open questions which lie in the beautiful intersection of geometric group theory and coarse geometry.

Matthew D. Romney, University of Illinois at Urbana-Champaign

Bi-Lipschitz embeddings of Grushin spaces in Euclidean space

It is well-known that the Heisenberg group, equipped with its sub-Riemannian metric, cannot be embedded in any Euclidean space under a bi-Lipschitz mapping, even locally. In contrast, the Grushin plane is a sub-Riemannian manifold admitting such an embedding and so can be seen as an intermediate case in the theory. This talk will discuss a broad class of length spaces with geometry modeled on the Grushin plane, defined by a conformal deformation of the Euclidean metric outside of a prescribed singular set. We give conditions under which such spaces can similarly be bi-Lipschitz embedded in Euclidean space.

Hiroki Sako , Niigata University

Group approximation in Cayley topology and coarse geometry

The subject of my talk is correspondence between the following: (1) large scale structure of the disjoint union ⨆_m{Cay(G(m))} consisting of Cayley graphs of finite groups with k generators; (2) structure of groups which appear in the boundary of the set {G(m)}_m in the space of k-marked groups. In the first part, I prove that a coarse disjoint union has property A of Yu if and only if all Cayley limit groups are amenable. As an application, we construct a coarse disjoint union of finite special linear groups which has property A but is of very poor compression into uniformly convex Banach spaces. If possible, I will discuss the Haagerup property (coarse embeddability) and property (T) (geometric property (T)).

Joint work with M. Mimura, N. Ozawa, Y. Suzuki

Andrew Swift, Texas A&M University

A characterization of quasi-isometric embeddability into c₀(κ)

In 1994, Jan Pelant proved that a metric property related to the notion of paracompactness called the Uniform Stone Property characterizes a metric space's uniform embeddability into c₀(κ) for some cardinality κ. I will show how quasi-isometric embeddability of a metric space into c₀(κ) can be characterized in a similar manner. It follows that coarse, uniform, and Lipschitz embeddability into c₀(κ) are equivalent notions for normed linear spaces.