Fall School
''Metric Embeddings : Constructions and Obstructions''
3-7 November 2014, Paris, France


A. Dalet (Besançon)

D. Galicer (Buenos Aires)

A. Gournay (Dresden)

A. Kaïchouh (Lyon)

A. Khukhro (Neuchâtel)

M. Kraus (Prague)

S. Li (Chicago)

M. Mimura (Sendai)

O. Neiman (Beer-Sheva)

B. Olivier (Haifa)

T. Pillon (Neuchâtel)

S. Zhang (College Station)

Schedule of the Fall School.

Scientific Program

Below is a list of 12 topics, each of them will be presented by an active participant.

Topic 1: On stochastic decompositions of metric spaces.
Neiman's report.

The lecturer will describe the notion of stochastic padded decomposition and its utilization in the construction of embeddings. After defining stochastic padded decomposition bundles, the lecturer will explain why metric spaces with finite Assouad-Nagata dimension [2], admit such stochastic padded decompositions. The lecturer will then present in detail the construction of bi-Lipschitz embeddings of the snowflakings of (infinite) metric spaces admitting padded decompositions (Theorem 5.2 in [2]). The lecturer will also present, with as many details as time permits, the estimate on the lp-distortion of a finite metric space in terms of its modulus of padded decomposition (Theorem 1.4 in [1]).

Relevant references:

1. R. Krauthgamer, J. R. Lee, M. Mendel, and A. Naor, Measured descent: a new embedding method for finite metrics, Geom. Funct. Anal. 15 (2005), no. 4, 839–858.
2. A. Naor and L. Silberman, Poincaré inequalities, embeddings, and wild groups, Compos. Math. 147 (2011), no. 5, 1546–1572.
3. M. I. Ostrovskii, Metric Embeddings: Bilipschitz and coarse embeddings into Banach spaces, De Gruyter Studies in Mathematics, vol. 49, De Gruyter, Berlin, 2013.

Topic 2: On random walks and quantitative coarse embeddings.
Gournay's report.

The lecturer will present the utilization of random walks in order to estimate the compression exponent of groups. After introducing K. Ball's notion of Markov-type, the lecturer will present in detail the proof of Proposition 1.1 from [1], which gives an upper bound on the compression exponent of an amenable group in terms of the drift of the canonical simple random walk on the Cayley graph of G. If time permits, the lecturer will sketch the proof of Theorem 1.1 in [3], which takes care of the equivariant problem for, non necessarily amenable, groups.

Relevant references:

1. T. Austin, A. Naor, and Y. Peres, The wreath product of ℤ with ℤ has Hilbert compression exponent 2/3 , Proc. Amer. Math. Soc. 137 (2009), no. 1, 85–90.
2. A. Naor and Y. Peres, Lp compression, traveling salesmen, and stable walks, Duke Math. J. 157 (2011), no. 1, 53–108.
3. A. Naor and Y. Peres, Embeddings of discrete groups and the speed of random walks, Int. Math. Res. Not. IMRN (2008), Art. ID rnn 076, 34.

Topic 3: Local versus global embeddability of locally finite metric spaces.
Zhang's report.

The lecturer will present the techniques used by M. I. Ostrovskii to prove that the bi-Lipschitz (resp. coarse) embeddability (into an infinite dimensional Banach space) of a locally finite metric space is determined by its finite subsets (Theorem 1.2 (resp. Theorem 1.3) in [3]). One of the main technique is the gluing technique introduced in [1].

Relevant references:

1. F. Baudier, Metrical characterization of super-reflexivity and linear type of Banach spaces, Arch. Math. (Basel) 89 (2007), no. 5, 419–429.
2. F. Baudier and G. Lancien, Embeddings of locally finite metric spaces into Banach spaces, Proc. Amer. Math. Soc. 136 (2008), no. 3, 1029–1033 (electronic).
3. M. I. Ostrovskii, Embeddability of locally finite metric spaces into Banach spaces is finitely determined, Proc. Amer. Math. Soc. 140 (2012), no. 8, 2721–2730.

Topic 4: On Assouad's embedding technique.
Kraus' report.

The lecturer will present in detail the proof of Assouad's embedding of the snowflaking of a doubling metric space into a finite dimensional Euclidean space (Proposition 2.6 in [2]). If time permits, the lecturer could discuss recent developments and improvements of Assouad's embedding.

Relevant references:

1. I. Abraham, Y. Bartal, and O. Neiman, Advances in metric embedding theory, Adv. Math. 228 (2011), no. 6, 3026–3126.
2. P. Assouad, Plongements lipschitziens dans ℝn, Bull. Soc. Math. France 111 (1983), 429–448 (French, with English summary).
3. G. David and M. Snipes, A non-probabilistic proof of the Assouad embedding theorem with bounds on the dimension, Anal. Geom. Metr. Spaces 1 (2013), 36–41.
4. A. Naor and O. Neiman, Assouad’s theorem with dimension independent of the snowflaking, Rev. Mat. Iberoam. 28 (2012), no. 4, 1123–1142.

Topic 5: Obstruction to uniform or coarse embeddability into reflexive Banach spaces.
Dalet's report.

The lecturer will present a Ramsey type argument introduced by N. J. Kalton to rule out coarse or uniform embeddings into reflexive Banach spaces. The lecturer will present the material necessary to define Kalton's property Q and its applications in nonlinear geometry of Banach spaces. In particular, the lecturer will give a detailed proof of Theorem 4.2 and Corollary 4.3 in [1].

Relevant references:

1. N. J. Kalton, Coarse and uniform embeddings into reflexive spaces, Quart. J. Math. (Oxford) 58 (2007), 393–414.

Topic 6: Influence of asymptotic uniform smoothness on small compression coarse embeddability.
Olivier's report.

The goal is to present one occurrence of the influence of the asymptotic properties of a Banach space on its nonlinear geometry. The lecturer will present a Ramsey type argument introduced by Kalton and Randrianarivony to rule out coarse Lipschitz (a.k.a quasi-isometric) embeddings into reflexive Banach spaces which are asymptotically uniformly smooth of power type p>1. The lecturer will present a detailed proof of Theorem 4.2 in [2] and its applications. For instance, the optimal estimation of the lq-compression of lp for 1≤ p≤ q<∞ noticed in [1], could be presented.

Relevant references:

1. F. Baudier, Quantitative nonlinear embeddings into Lebesgue sequence spaces, J. Top. Anal. (to appear).
2. N. J. Kalton and N. L. Randrianarivony, The coarse Lipschitz structure of lp⊕ lq, Math. Ann. 341 (2008), 223–237.

Topic 7: Estimating the distortion using a metric invariant.
Li's report.

The lecturer will describe the utilization of metric invariants to estimate the distortion of a bi-Lipschitz embedding. The lecturer will briefly recall the notion of Enflo-type, and show how it was used to exhibit the first collection of metric spaces, namely the Hamming cubes, with unbounded Euclidean distortion [1]. The lecturer will then present the notion of Markov-convexity introduced and studied in [2] and [4]. The usefulness of this notion in estimating the lp-distortion of some families of metric spaces (for instance, the binary trees, the diamond graphs, the Laakso graphs...) will be discussed.

Relevant references:

1. P. Enflo, On the nonexistence of uniform homeomorphisms between Lp-spaces, Ark. Mat. 8 (1969), 103–105 (1969).
2. J. R. Lee, A. Naor, and Y. Peres, Trees and Markov convexity, Geom. Funct. Anal. 18 (2009), no. 5, 1609–1659.
3. J. Matoušek, Lectures on discrete geometry, Graduate Texts in Mathematics, 212. Springer-Verlag, New York, 2002. (Chapter 15)
4. M. Mendel and A. Naor, Markov convexity and local rigidity of distorted metrics, J. Eur. Math. Soc. (JEMS) 15 (2013), no. 1, 287–337.

Topic 8: On Poincaré type inequalities and non-embeddability.
Mimura's report.

The lecturer will discuss the transference of Poincaré type inequalities using Mazur maps or exponential maps, as well as the characterization of non-coarse embeddability in terms of Poincaré inequalties.

Relevant references:

1. M. Gromov, Spaces and questions, Geom. Funct. Anal. (2000), 118–161. GAFA 2000 (Tel Aviv, 1999).
2. G. Arzhantseva and R. Tessera, Relatively expanding box spaces with no expansion, arXiv:1402.1481.
3. M. I. Ostrovskii, Coarse embeddability into Banach spaces, Topology Proc. 33 (2009), 163–183.
4. N. Ozawa, A note on non-amenability of B(lp) for p = 1; 2, Internat. J. Math. 15 (2004), no. 6, 557–565.
5. R. Tessera, Coarse embeddings into a Hilbert space, Haagerup property and Poincaré inequalities, J. Topol. Anal. 1 (2009), no. 1, 87–100.
6. A. Wigderson, lecture notes on expanders: http://www.math.ias.edu/ boaz/ExpanderCourse/

Topic 9: On Følner type sets and coarse embeddings.
Pillon's report.

One of the most common and efficient way to produce coarse embeddings of spaces, and especially groups into a Hilbert space is Property A (introduced by G. Yu). It can be used for instance for constructing coarse embeddings of many classes of groups such as amenable or hyperbolic groups. A quantitative version of Property A can be used to produce embeddings with some lower bound on the compression modulus. For amenable groups, quantitative property A amounts to some quantitative estimate on the isoperimetric profile.

Relevant references:

1. P. Nowak and G. Yu, Large scale geometry, EMS Textbooks in Mathematics, European Mathematical Society (EMS), Zürich, 2012.
2. R. Tessera, Asymptotic isoperimetry on groups and uniform embeddings into Banach spaces, Comment. Math. Helv. 86 (2011), no. 3, 499–535.
3. R. Tessera, Quantitative property A, Poincaré inequalities, Lp-compression and Lp-distortion for metric measure spaces, Geom. Dedicata 136 (2008), 203–220.

Topic 10: On wall structures and coarse embeddings.
Khukhro's report.

Apart from Property A, there exists essentially one other technique, more combinatorial, which naturally yields embeddings (and most of the times proper actions by isometries) of groups into L1: these are spaces with walls. Here the emphasis will be put on a special but very interesting case, where such wall-structure is built on Cayley graphs of finite groups. The examples obtained by Arzhantseva, Guentner and Špakula are especially interesting as they provide the first instances of bounded geometry spaces which coarsely embed but do not have Property A.

Relevant references:

1. G. Arzhantseva, E. Guentner, and J. Špakula, Coarse non-amenability and coarse embeddings, Geom. Funct. Anal. 22 (2012), no. 1, 22–36.

Topic 11: On Ramsey techniques in quantitative metric geometry.
Galicer's report.

The lecturer will present the techniques used either in the proof of Theorem 1 in [2] or Theorem 1.9 in [1].

Relevant references:

1. Y. Bartal, N. Linial, M. Mendel, and A. Naor, On metric Ramsey-type phenomena, Ann. of Math. (2) 162 (2005), no. 2, 643–709.
2. J. Matoušek, On embedding trees into uniformly convex Banach spaces, Israel J. Math. 114 (1999), 221–237.

Topic 12: Non-embeddability of the Uryshon space; an algebraic obstruction.
Kaichouh's report.

The lecturer will present the algebraic techniques used in the proofs of Theorem 5.1 and Theorem 6.4 from [1].

Relevant references:

1. V. G. Pestov, A theorem of Hrushovski-Solecki-Vershik applied to uniform and coarse embeddings of the Urysohn metric space, Topology Appl. 155 (2008), no. 14, 1561–1575.

Scientific Committee

F. Baudier (IMJ-PRG and Texas A&M)

G. Godefroy (CNRS, IMJ-PRG)

P. Pansu (Paris-Sud 11, Orsay)

R. Tessera (CNRS, Paris-Sud 11, Orsay)

Organizing Committee

F. Baudier (IMJ-PRG and Texas A&M)

G. Godefroy (CNRS, IMJ-PRG)

P. Pansu (Paris-Sud 11, Orsay)

R. Tessera (CNRS, Paris-Sud 11, Orsay)