SPECIAL FEATURE: INTRODUCTION INTRODUCTION SPECIAL FEATURE: Quantitative geometry Assaf Naor1 on at least two generators (5), and Courant Institute, New York University, New York, NY 10012 pffiffiffiffiffiffiffiffiffiffi DGðnÞ ≍ log n if G is the Heisenberg group (8). Similar questions with Hilbert Rather than describing a traditional field of word metric in a bi-Lipschitz way. One is space replaced by L1 are of fundamental mathematics, “quantitative geometry” refers therefore led to consider groups as metric importance to theoretical computer science to a modern way of thinking about geometry spaces that are indistinguishable under bi- (9). Another way to quantify the extent and its applications that occur in a wide Lipschitz deformations. This viewpoint is of to which a group G that is generated by fi range of mathematical disciplines. The over- monumental importance to group theory a nite symmetric set S fails to embed bi- arching theme here is that certain geometric and geometry, Gromov’s characterization Lipschitzly into Hilbert space was introduced fi problems can only be formulated after the of groups of polynomial growth (1) being by Guentner and Kaminker (10): they de ne introduction of auxiliary quantitative param- a seminal work in this area (see also ref. 2). the Hilbert compression exponent of G, α eters and asking for their asymptotic or ap- Thethirdarticle(3)inthepresentvolume denoted ðGÞ, to be the supremum over α ∈ ; proximate behavior. This leads to meaningful (to be described below) answers a question of those ½0 1 for which there exists a : → ℓ questions that do not necessarily have coun- Gromov related to his polynomial growth Lipschitz mapping f G 2 satisfying − ≥ ; α ; ∈ terparts in classical “qualitative” geometric theorem, and the first article (4) in this vol- jjf ðxÞ f ðyÞjj2 cdSðx yÞ for every x y G. · ; · investigations, a viewpoint that allows one ume also resolves a basic question in geo- Here dSð Þ denotes the word metric in- ∈ ; ∞ to uncover deep and useful structural infor- metric group theory, exhibiting for the first duced by S and c ð0 Þ may depend on ; ; α ; mation that appears only in the quantitative time that two natural quantitative geometric G f but not on x y. Again, being in- fi fi regime. Quantitative reasoning allows one parameters of groups (homological and dependent of the speci c choice of a nite to tame complicated objects and phenom- homotopical Dehn functions) can have generating set, αðGÞ is an algebraic invariant ena that are unwieldy if one insists on exact different asymptotics for the same group. of G, yet there are only a few known meth- measurements. It is often the case that a rich Whether or not a given finitely generated ods to compute it, some of which involve yet structured picture emerges only if one group admits a geometrically faithful em- interesting links to harmonic analysis and asks the correct questions by allowing for bedding into a certain well-behaved space, probability theory. Note that both the controlled errors. say, into Hilbert space or into an L1ðμÞ space, asymptotics of DGðnÞ and the Hilbert The term quantitative geometry was coined is a basic question in geometric group theory, compression exponent αðGÞ are quantita- as an attempt to formulate the widespread with many important algebraic and topo- tive invariants that encode structural in- understanding of many researchers world- logical implications (see ref. 5 for a striking formation about the group G, yet they do wide that in recent years numerous natural example). Here the notion of a “geometrically not have qualitative counterparts. questions have emerged that mandate the faithful” embedding can take several mean- study of geometric problems from an ap- ings; there is, for example, great interest in bi- Geometric Topology. Topology is another proximate perspective. This trend is some- Lipschitz and coarse embeddings. Focusing source of a wealth of important quantitative times dictated by application areas in which on the bi-Lipschitz case, it is an intriguing geometric questions. Despite the numerous exact computation is infeasible yet approxi- and beautiful conjecture (6) that a finitely remarkable achievements of algebraic topol- fi mate measurements are of crucial impor- generated group G admits a bi-Lipschitz ogy, the dif culty of understanding classical tance, but it arises most often as a natural embedding into Hilbert space if and only if qualitative topological problems quantita- modern development of traditional mathe- G has an Abelian subgroup of finite index. tively has been emphasized by Gromov (11). matical disciplines, where quantitative refine- Therefore, to classify non-virtually-Abe- For example, suppose that X; Y are finite ments of classical investigations are necessary lian groups G in terms of their embeddability complexes and Y is simply connected. If to probe deeper and fully understand basic into Hilbert space, one must quantify the f ; g : X → Y are homotopic “well-behaved” mathematical objects. extent to which G does not admit a bi-Lip- maps, are they also homotopic through maps that are equally well behaved? There What Is Quantitative Geometry? schitz embedding into Hilbert space. There is more than one natural (and fruitful) way to are many ways to interpret what well be- Rather than attempting to give formal defi- do this. For every n ∈ N,onecanaskwhat haved means in this general context. Here nitions, we shall illustrate the above discus- is the largest Hilbertian bi-Lipschitz dis- we take this term to mean that the map- sion through examples from various areas tortion of an n-point subset of G,and pings in question are Lipschitz. Therefore, of mathematics. ; : → denoting this number by DGðnÞ,in-supposing that f g X Y are homotopic Geometric Group Theory. AgroupG that vestigate the rate at which DGðnÞ tends to L-Lipschitz functions, can they be homo- is generated by a finite symmetric set S infinity.Thisratedoesnotdependon toped through KL-Lipschitz functions, or becomes a geometric object if one endows aspecific finite generating set, so it is perhaps one can even find a homotopy it with the word metric that is induced by S. a genuine algebraic invariant of G. It turns that is itself KL-Lipschitz? Here K is If one nevertheless wants to study groups as out to be quite difficult to evaluate the as- Author contributions: A.N. wrote the paper. algebraic entities, the generating set itself is ymptotic behavior of DGðnÞ, and this has of minor or no importance, but replacing been computedpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi in only a few cases, e.g., The author declares no conflict of interest. 1 S by another generating set T changes the DGðnÞ ≍ log log n if G is a free group E-mail: [email protected].

www.pnas.org/cgi/doi/10.1073/pnas.1320388110 PNAS Early Edition | 1of4 Downloaded by guest on September 27, 2021 allowed to depend on X; Y but not on topological spaces, a lot of structure remains problems that are efficiently solvable, and f ; g; L. The seventh article (12) of the if one insists that homeomorphisms are the resulting “rounding problem” asks for present volume treats this question by quantitatively continuous. For example, an efficient way to generate a discrete solu- obtaining a necessary and sufficient con- since for distinct p; q ∈ ½1; ∞Þ the se- tion from a continuous solution while losing a dition for the existence of a KL-Lipschitz quence spaces ℓp and ℓq do not have the (hopefully small) definite multiplicative fac- homotopy but also shows that sometimes same isomorphic local linear structure, it tor. Conversely, through mechanisms such as one cannot find Lipschitz homotopies despite follows from Ribe’s theorem that ℓp and ℓq Khot’s Unique Games Conjecture (26), one the fact that homotopies through Lipschitz are not uniformly homeomorphic. can relate the computational complexity of functions are possible. We have indicated just Note a crucial feature of Ribe’s theorem: certain optimization problems and specifi- one example of quantitative geometric its conclusion involves an unspecified con- cally the “hardness threshold,” above which refinements of topological concepts and stant K, so it can be used only if one con- it becomes feasible to find an approximate questions, but the literature contains many siders properties of Banach spaces that are solution to continuous quantitative geo- investigations (and many open questions) of insensitive to deformations that result in metric questions; see ref. 22 for more on this this type, all of which belong under the a loss of such a constant factor. Moreover, topic. In graph theory, one sometimes quantitative geometry rubric. the conclusion of Ribe’s theorem involves relates geometric invariants to combinato- properties of finite dimensional subspaces rial graphs [e.g., the Lovász theta function Differential Geometry. Geometric flows (a.k.a. local properties), and since all of the (28)], leading naturally to interesting geo- and limiting constructions such as blow-ups norms on Rn are equivalent, finite di- metric problems of a quantitative/asymp- and tangent cones are commonly studied in mensional normed spaces become distin- totic nature (29). The fourth article (30) in differential geometry, and it is very natural guishable only if one studies them in the the present volume contains results along (and useful) to refine these investigations quantitative regime. This point of view these lines. As a last example, we consider quantitatively. The fifth article (13) of the gives rise to the local theory of Banach spaces: the area of incidence geometry. Classically, present volume describes interesting con- a deep and rich theory (18, 19) that over the one is interested here in understanding the tributions to this aspect of quantitative ge- last five decades has had a transformative constraints that are imposed on a finite set of ometry. It introduces new monotone quan- impact on functional analysis, in addition to points in Euclidean space, under the as- tities for the heat and Laplace equations many applications to a variety of mathe- sumption that the points obey certain alge- on manifolds and uses them to prove new matical disciplines. The sixth (20) and eighth braic restrictions of a qualitative nature, e.g., uniqueness results for tangent cones. While (21) articles of the present volume (to be the number of distinct pairwise distances is the existence of tangent cones is often described below) contain new applications small, or every line that passes through two proved through the use of monotone of the local theory of Banach spaces. of the points must also pass through a third quantities, proving the uniqueness of tan- Inspired by Ribe’s theorem and formulated point. The latter example is addressed by the gent cones is inherently quantitative, relying in print for the first time by Bourgain (22), classical Sylverster–Gallai theorem, which on showing that the monotone quantity theRibeprogramasksforanexplicitdictio- asserts that if a finite set S ⊆ Rk has the approaches its limit at a definite rate. The nary that translates local linear concepts and property that every line that passes through new quantities that are introduced in the phenomena from the linear setting of Banach two distinct points of S also passes through above-mentioned article are sufficiently well spaces to general metric spaces. Thus, besides a third point of S, then S consists of collinear behaved to yield the desired convergence being a very satisfactory rigidity statement in points. The second (31) article of the present rates and hence prove the uniqueness of itself, Ribe’s theorem is a beginning rather volume relaxes this rigid question by tangent cones at infinity for any Ricci-flat than an end: the source and motivation of a studying scenarios in which we only know manifold with Euclidean volume growth, quest to recast quantitative aspects of Banach that a constant fraction of the lines that provided that one such tangent cone has a space theory while using only the notion of pass through two distinct points of S also smooth cross section. distance, without reference to the linear passes through a third point of S. While structure whatsoever, and through this dic- this formulation leads to combinatorial Local Theory of Banach Spaces and the tionary proceeding to ask questions and ap- (counting) asymptotic questions, one can Ribe Program. Answering questions that ply insights that previously made sense only make it less qualitative by not considering were raised in the late 1920s and early 1930s for linear spaces to more general metric exact incidences but rather examining lines (14, 15), a deep theorem of Kadec (16) asserts spaces such as graphs, manifolds, and groups. that pass near points of S or equivalently that any two separable infinite dimensional We refer to the surveys (23, 24) for more on replacing lines by tubes. This is the topic of Banach spaces are homeomorphic. On the this topic. The ninth article (25) of the subsequent investigations (32) that will other hand, Ribe proved (17) that any two present volume (to be described below) appear elsewhere. Banach spaces X; Y that are uniformly ho- belongs to the Ribe program. meomorphic (i.e., the homeomorphism and Quantitative Geometry at Mathematical its inverse are both uniformly continuous) Aspects of Discrete Mathematics. Several Sciences Research Institute havethesameisomorphiclocallinearstruc- areas of discrete mathematics are well suited In the fall semester of 2011, the Mathe- ture, i.e., there exists K ∈ ð0; ∞Þ such that for for quantitative geometric investigations. matical Sciences Research Institute (MSRI) at any finite dimensional linear subspace E ⊆ X In theoretical computer science, the field Berkeley hosted a research program called there exists a linear subspace F ⊆ Y that is of approximation algorithms and the quantitative geometry (QG). This program linearly isomorphic to E via a linear mapping complementary viewpoint of hardness of was what MSRI calls a “jumbo program” in T : E → F satisfying jjTjj · jjT−1jj ≤ K.Thus, approximation are inherently related to geo- the sense that, while MSRI typically runs two while the rich world of separable infinite metric considerations of a quantitative nature. semester-long programs in parallel, the QG dimensional Banach spaces collapses to a NP-hard discrete optimization problems are program pooled the resources that are nor- single object when one considers them as often relaxed to continuous optimization mally intended for two separate programs

2of4 | www.pnas.org/cgi/doi/10.1073/pnas.1320388110 Naor Downloaded by guest on September 27, 2021 into one big program to which the MSRI showing that the (Gromov-Hausdorff) dis- pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi O log n log logn INTRODUCTION · ≤ ð Þ : SPECIAL FEATURE: activities of fall 2011 were devoted in their tance between the rescaled metric and Pansu’s jjBjj jjCjj e jjAjj entirety. The QG program featured extended limit metric is Oðk−2=ð3rÞÞ. À Á stays of more than 80 mathematicians, Thepffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi key point here is that the term O log n log logn ranging from senior researchers to post- Deterministic Volume Estimation. Sup- e is asymptotically smaller n doctoral fellows and graduate students. pose that a convex body K ⊆ R is given to than any power of n,butitremainsopenwhether Many additional mathematicians passed us in terms of a well-guaranteed membership or not it can be replaced by a universal constant. n through the QG program for shorter visits, oracle (32), i.e., for every point x ∈ R we can Itisshowninref.21howonecouldreplacethis some of which came to MSRI to participate asktheoraclewhetherornotx belongs to K, term by ðlog nÞOð1Þ if a sharp form of (the paving in one of the five focused workshops that and we also know that K is sandwiched be- formulation of) the Kadison–Singer problem tween two Euclidean balls whose ratio of took place as part of the program. The were true. Although the Kadison–Singer problem radii grows at most exponentially with n.The present special feature of PNAS contains was recently solved affirmatively by Marcus et al. goal is to find a fast algorithm that is guar- some highlights of new results that relate to (38), at present this solution does not seem to the QG program. Through the auspices of anteed to be within a hopefully small con- supply the desired quantitative estimates. MSRI, the QG program created an excep- stant factor of the volume of K, where fast is tionally fertile research environment that measured in terms of the number of times that the algorithm makes a membership Ultrametric Skeletons. The ninth article of led to the discovery of many additional the present volume (25) shows that for every results that appeared (or will appear) else- query to the oracle. The sixth article (20) of fi ɛ ∈ ð0; 1Þ there exists cɛ ∈ ½1; ∞Þ such that if where, so the articles that follow should be the present volume shows for the rst time ɛ ∈ ; ðX; dÞ is a compact metric space and μ is viewed as a sample of recent developments that for every ð0 1Þ there is a deter- ministic algorithm that computes a ð1 + ɛÞn a Borel probability measure on X, then there rather than a comprehensive account of the ⊆ ’ approximation to the volume of a centrally exists a compact subset S X, a Borel proba- program s outcome and impact. ν symmetric convex body that is given by bility measure that is supported on S,andan ρ : × → ; ∞ Quantitative Geometry Special Feature a well-guaranteed membership oracle in time ultrametric S S ½0 Þ such that of PNAS 1=ɛOðnÞ.Asshowninref.35,nodeterministic ∀x ; y ∈ S; dðx; yÞ ≤ ρðx; yÞ ≤ Oð1=ɛÞ · dðx; yÞ We shall now proceed to describe the content volume-computation algorithm can have a of some of the articles that appear herein. better performance [up to the implicit con- Our goal is to illustrate the flavor of the re- stants in the OðnÞ term]. The proof of this and search that is done in the context of quanti- fact relies on the design of the first de- tative geometry, and we do so by choosing terministic exponential time algorithm that ∀x ∈ X; ∀r ∈ ½0; ∞Þ; to discuss only a subset of the ensuing computes an M-ellipsoid of the given body, 1−ɛ νðBdðx; rÞÞ ≤ ðμðBdðx; cɛrÞÞÞ : works. We refer to the papers themselves which is an important type of ellipsoid whose for full details. existence was discovered by Milman (36). ; = ∈ : ; ≤ This is achieved through a clever iterative Here Bdðx rÞ fy X dðx yÞ rg is the ’ Rate of Convergence to Pansu s Limit. argument that uses important tools from the closed ball of radius r centered at x that is fi Let G be a nitely generated group, equip- local theory of Banach spaces—abeautiful induced by the metric d. Recall that the fact ped with a left-invariant word metric that is example of the interplay between geometry that ρ is an ultrametric means that, for every associated to a finite symmetric generating and algorithms. x; y; z ∈ S, it satisfies the improved triangle set. For n ∈ N let BðnÞ be the ball of radius n inequality ρðx; yÞ ≤ maxfρðx; zÞ; ρðy; zÞg. centered at the identity element. Gromov’s Quantitative Characterization of Com- The metric measure space ðS; ρ; νÞ is called famous polynomial growth theorem (1) mutators and the Kadison–Singer Prob- an “ultrametric skeleton” of the metric mea- asserts that there exists d ∈ N such the lem. A classical fact in linear algebra asserts d sure space ðX; d; μÞ for several reasons. One cardinality of BðnÞ satisfies jBðnÞj ≤ n for that if A ∈ M ðCÞ is an n × n matrix with n expects from the notion of a skeleton that it all n ∈ N if and only if G has a nilpo- complex entries whose trace vanishes then tent subgroup of finite index. Pansu ; ∈ C = − surrounds the macroscopic features of X to there exist B C Mnð Þ such that A BC μ proved (33) that if G is nilpotent then there CB. This identity clearly implies the operator which gives positive mass (by no means do exists c ∈ ð0; ∞Þ and d ∈ N such that norm bound jjAjj ≤ 2jjBjj · jjCjj. The eighth we expect a skeleton to be dense). This is jBðnÞj = cnd + oðndÞ. Answering a question article of the present volume (21) addresses indeed the case in the present setting, since if μ ; of Gromov (2), the third article (3) in the the question whether or not one can find assigns positive mass to two balls Bdðx rÞ fi and Bdðy; rÞ whose centers are sufficiently present volume obtains the rst known such matrices B; C ∈ MnðCÞ for which the quantitative estimate for the term oðndÞ:itis ratio jjBjj · jjCjj=jjAjj is small. It turns out distant with respect to the scale r, i.e., − = ; > + shown there that jBðnÞj = cnd + oðnd 3 ð3rÞÞ, that proving such a quantitative version of dðx yÞ ðcɛ 1Þr, then the probability mea- ν where r is the nilpotency class of G. In fact, a linear algebraic fact can be quite difficult, sure (which is supported on S)cannotassign Pansu proved that if we rescale the word and the argument here uses tools that were full mass to one of these balls. More impor- metric by 1=k, then the resulting metric developed in the context of the local theory tantly, the term skeleton appears here because spaces converge as k → ∞ in the Gromov- of Banach spaces [notably, the restricted of the way the above theorem is used in Hausdorff topology to a certain explicit left- invertibility principle of Bourgain and Tzafriri applications: one can prove theorems about invariant sub-Finsler metric on a nilpotent (37)]. The theorem proved in the eighth ar- general metric spaces by extracting a skeleton Lie group, and Gromov asked (2) for quan- ticle (21) of the present volume asserts and arguing that it can be used to imply titative estimates on the rate of convergence that given a matrix A ∈ MnðCÞ of trace a statement about the ambient metric space in Pansu’s theorem. The third paper (3) of the zero there exist B; C ∈ MnðCÞ with that does not involve the skeleton itself. In present volume solves Gromov’squestionby A = BC − CB and particular, the ultrametric skeleton theorem

Naor PNAS Early Edition | 3of4 Downloaded by guest on September 27, 2021 can be used to obtain a new proof of Tala- was inspired by the search for a metric counterpart, yet it (and its applications) was grand’s majorizing measure theorem (39), and space analog of a very important theorem unearthed because of the Ribe program’sca- it implies the only known proof that if ðX; dÞ of Dvoretzky (45) that originated in the pacity to generate meaningful and useful is a compact metric space of Hausdorff local theory of Banach spaces. Specifically, questions (through a dictionary that translates dimension bigger than n, then there exists it implies a sharp solution of the nonlinear Banach space phenomena into metric space a surjective Lipschitz function f : X → ½0; 1n Dvoretzky problem that was formulated by questions). In the present context of met- (40). This theorem also has algorithmic Bourgain et al. (46), and it also implies a nat- ric Dvoretzky-type problems, following the implications, including the best known ural Hausdorff-dimensional variant of the initial work of Bourgain et al. (46), this point lower bound for the randomized k-server nonlinear Dvoretzky problem (47) that was of view was later championed by Milman, problem on general metric spaces (41, 42) posed by Tao. This development is an example who proceeded to make similar (successful) and the only known construction of sharp of one of the most exciting features of the Ribe metric predictions inspired by his Quotient of approximate distance oracles with con- program and its impact on metric geometry: Subspace theorem (48–50). stant query time for general metric spaces analogy with Banach space theory initiates the (43, 44). search for a certain phenomenon, and it turns ACKNOWLEDGMENTS. This work was supported by National Science Foundation Grant CCF-0832795, United The ultrametric skeleton theorem is an out that this leads to a discovery of a phe- States–Israel Binational Science Foundation Grant 2010021, outgrowth of the Ribe program because it nomenon that is very different from its linear and the Packard Foundation.

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