DOCSLIB.ORG

An Invitation to Alexandrov Geometry: CAT(0) Spaces

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
An Invitation to Alexandrov Geometry: CAT(0) Spaces An invitation to Alexandrov geometry: CAT(0) spaces Stephanie Alexander, Vitali Kapovitch, Anton Petrunin Contents 1 Preliminaries5 1.1 Metric spaces........................5 1.2 Constructions........................6 1.3 Geodesics, triangles, and hinges..............7 1.4 Length spaces........................8 1.5 Model angles and triangles................. 11 1.6 Angles and the first variation............... 13 1.7 Space of directions and tangent space.......... 15 1.8 Hausdorff convergence................... 16 1.9 Gromov–Hausdorff convergence.............. 17 2 Gluing theorem and billiards 21 2.1 The 4-point condition................... 21 2.2 Thin triangles........................ 22 2.3 Reshetnyak’s gluing theorem............... 26 2.4 Reshetnyak puff pastry................... 28 2.5 Wide corners........................ 31 2.6 Billiards........................... 33 2.7 Comments.......................... 36 3 Globalization and asphericity 37 3.1 Locally CAT spaces.................... 37 3.2 Space of local geodesic paths............... 38 3.3 Globalization........................ 40 3.4 Polyhedral spaces...................... 44 3.5 Flag complexes....................... 46 3.6 Cubical complexes..................... 48 3.7 Exotic aspherical manifolds................ 49 3.8 Comments.......................... 52 iii iv CONTENTS 4 Subsets 55 4.1 Motivating examples.................... 55 4.2 Two-convexity........................ 56 4.3 Sets with smooth boundary................ 60 4.4 Open plane sets....................... 62 4.5 Shefel’s theorem...................... 63 4.6 Polyhedral case....................... 65 4.7 Two-convex hulls...................... 67 4.8 Proof of Shefel’s theorem................. 69 4.9 Comments.......................... 70 Semisolutions 73 Preface This short monograph arose as an offshoot of the book on Alexan- drov geometry we have been writing for a number of years. The notes were shaped in a number of lectures given by the third author to un- dergraduate students at different occasions at the MASS program at Penn State University and the Summer School “Algebra and Geome- try” in Yaroslavl. The idea is to demonstrate the beauty and power of Alexandrov geometry by reaching interesting applications and theorems with a minimum of preparation. Namely, we consider CAT(0) spaces — the metric spaces with nonpositive curvature in the sense of Alexandrov; these spaces can be loosely described as a non-linear generalization of a Hilbert space. In Chapter1, we discuss necessary preliminaries. In Chapter2, we discuss the Reshetnyak gluing theorem and apply it to a problem in billiards which was solved by Dmitri Burago, Serge Ferleger, and Alexey Kononenko. In Chapter3, we discuss the Hadamard–Cartan globalization the- orem, and apply it to the construction of exotic aspherical manifolds introduced by Michael Davis. In Chapter4, we discuss examples of Alexandrov spaces with cur- vature bounded above. This chapter is based largely on work of Samuel Shefel on nonsmooth saddle surfaces. Here is a list of some sources providing a good introduction to Alexandrov spaces with curvature bounded above, which we recom- mend for further information; we will not assume familiarity with any of these sources. The book by Martin Bridson and André Haefliger [18]; Lecture notes of Werner Ballmann [13]; Chapter 9 in the book [20] by Dmitri Burago, Yuri Burago and Sergei Ivanov. 1 2 CONTENTS Early history of Alexandov geometry The idea that the essence of curvature lies in a condition on quadruples of points apparently originated with Abraham Wald. It is found in his publication on “coordinate-free differential geometry” [68] written un- der the supervision of Karl Menger; the story of this discovery can be found in [45]. In 1941, similar definitions were rediscovered indepen- dently by Alexandr Danilovich Alexandrov, see [7]. In Alexandrov’s work the first fruitful applications of this approach were given. Mainly: Alexandrov’s embedding theorem — metrics of non-negative curvature on the sphere, and only they, are isometric to closed convex surfaces in Euclidean 3-space. Gluing theorem, which tells when the sphere obtained by gluing of two discs along their boundaries has non-negative curvature in the sense of Alexandrov. These two results together gave a very intuitive geometric tool for studying embeddings and bending of surfaces in Euclidean space, and changed this subject dramatically. They formed the foundation of the branch of geometry now called Alexandrov geometry. The study of spaces with curvature bounded above started later. The first paper on the subject was written by Alexandrov; it appeared in 1951, see [8]. It was based on work of Herbert Busemann, who studied spaces satisfying a weaker condition [25]. Yurii Grigorievich Reshetnyak proved fundamental results about general spaces with curvature bounded above, the most important of which is his gluing theorem. An equally important theorem is the Hadamard–Cartan theorem (globalization theorem). These theorems and their history are discussed in chapters2 and3. Surfaces with upper curvature bounds were studied extensively in the 50-s and 60-s, and are by now well understood; see the survey [59] and the references therein. Manifesto of Alexandrov geometry Alexandrov geometry can use “back to Euclid” as a slogan. Alexandrov spaces are defined via axioms similar to those given by Euclid, but certain equalities are changed to inequalities. Depending on the sign of the inequalities, we get Alexandrov spaces with curvature bounded above or curvature bounded below. The definitions of the two classes of spaces are similar, but their properties and known applications are quite different. Consider the space M4 of all isometry classes of 4-point metric spaces. Each element in M4 can be described by 6 numbers — the CONTENTS 3 distances between all 6 pairs of its points, say `i;j for 1 6 i < j 6 4 modulo permutations of the index set (1; 2; 3; 4). These 6 numbers are subject to 12 triangle inequalities; that is, `i;j + `j;k > `i;k holds for all i, j and k, where we assume that `j;i = `i;j and `i;i = 0. Consider the subset E4 ⊂ M4 of all isometry classes of 4-point metric spaces that admit isomet- ric embeddings into Euclidean N4 E4 P4 space. The complement M4nE4 has two connected components. 0.0.1. Exercise. Prove the lat- M4 ter statement. One of the components will be denoted by P4 and the other by N4. Here P and N stand for positive and negative curvature because spheres have no quadruples of type N4 and hyperbolic space has no quadruples of type P4. A metric space, with length metric, that has no quadruples of points of type P4 or N4 respectively is called an Alexandrov space with non-positive or non-negative curvature. Here is an exercise, solving which would force the reader to rebuild a considerable part of Alexandrov geometry. 0.0.2. Advanced exercise. Assume X is a complete metric space with length metric, containing only quadruples of type E4. Show that X is isometric to a convex set in a Hilbert space. In fact, it might be helpful to spend some time thinking about this exercise before proceeding. In the definition above, instead of Euclidean space one can take hy- perbolic space of curvature −1. In this case, one obtains the definition of spaces with curvature bounded above or below by −1. To define spaces with curvature bounded above or below by 1, one has to take the unit 3-sphere and specify that only the quadruples of points such that each of the four triangles has perimeter less than 2·π are checked. The latter condition could be considered as a part of the spherical triangle inequality. Acknowledgment We want to thank David Berg, Richard Bishop, Yurii Burago, Maxime Fortier Bourque, Sergei Ivanov, Michael Kapovich, Bernd Kirchheim, 4 CONTENTS Bruce Kleiner, Nikolai Kosovsky, Greg Kuperberg, Nina Lebedeva, John Lott, Alexander Lytchak, Dmitri Panov, Stephan Stadler, Wil- derich Tuschmann, and Sergio Zamora Barrera for a number of dis- cussions and suggestions. We thank the mathematical institutions where we worked on this material, including BIRS, MFO, Henri Poincaré Institute, University of Colone, Max Planck Institute for Mathematics. The first author was partially supported by the Simons Founda- tion grant #209053. The second author was partially supported by a Discovery grant from NSERC and by the Simons Foundation grant #390117. The third author was partially supported by the NSF grant DMS 1309340 and the Simons Foundation #584781. Chapter 1 Preliminaries In this chapter we fix some conventions and recall the main definitions. The chapter may be used as a quick reference when reading the book. To learn background in metric geometry, the reader may consult the book of Dmitri Burago, Yuri Burago, and Sergei Ivanov [20]. 1.1 Metric spaces The distance between two points x and y in a metric space X will be denoted by jx − yj or jx − yjX . The latter notation is used if we need to emphasize that the distance is taken in the space X . The function distx : y 7! jx − yj is called the distance function from x. The diameter of a metric space X is defined as diam X = supf jx − yjX : x; y 2 X g : Given R 2 [0; 1] and x 2 X , the sets B(x; R) = fy 2 X j jx − yj < Rg; B[x; R] = fy 2 X j jx − yj 6 Rg are called, respectively, the open and the closed balls of ra- dius R with center x. Again, if we need to emphasize that these balls are taken in the metric space X , we write B(x; R)X and B[x; R]X : 5 6 CHAPTER 1. PRELIMINARIES A metric space X is called proper if all closed bounded sets in X are compact. This condition is equivalent to each of the following statements: 1. For some (and therefore any) point p 2 X and any R < 1, the closed ball B[p; R] ⊂ X is compact. 2. The function distp : X! R is proper for some (and therefore any) point p 2 X ; that is, for any compact set K ⊂ R, its inverse image f x 2 X : jp − xjX 2 K g is compact.
Load more

Recommended publications
  • Describing the Universal Cover of a Compact Limit ∗
    Describing the Universal Cover of a Compact Limit ∗ John Ennis Guofang Wei Abstract If X is the Gromov-Hausdorff limit of a sequence of Riemannian manifolds n Mi with a uniform lower bound on Ricci curvature, Sormani and Wei have shown that the universal cover X˜ of X exists [13, 14]. For the case where X is compact, we provide a description of X˜ in terms of the universal covers M˜ i of the manifolds. More specifically we show that if X¯ is the pointed Gromov- Hausdorff limit of the universal covers M˜ i then there is a subgroup H of Iso(X¯) such that X˜ = X/H.¯ 1 Introduction In 1981 Gromov proved that any finitely generated group has polynomial growth if and only if it is almost nilpotent [7]. In his proof, Gromov introduced the Gromov- Hausdorff distance between metric spaces [7, 8, 9]. This distance has proven to be especially useful in the study of n-dimensional manifolds with Ricci curvature uniformly bounded below since any sequence of such manifolds has a convergent subsequence [10]. Hence we can follow an approach familiar to analysts, and consider the closure of the class of all such manifolds. The limit spaces of this class have path metrics, and one can study these limit spaces from a geometric or topological perspective. Much is known about the limit spaces of n-dimensional Riemannian manifolds with a uniform lower bound on sectional curvature. These limit spaces are Alexandrov spaces with the same curvature bound [1], and at all points have metric tangent cones which are metric cones.
    [Show full text]
  • The Materiality & Ontology of Digital Subjectivity
    THE MATERIALITY & ONTOLOGY OF DIGITAL SUBJECTIVITY: GRIGORI “GRISHA” PERELMAN AS A CASE STUDY IN DIGITAL SUBJECTIVITY A Thesis Submitted to the Committee on Graduate Studies in Partial Fulfillment of the Requirements for the Degree of Master of Arts in the Faculty of Arts and Science TRENT UNIVERSITY Peterborough, Ontario, Canada Copyright Gary Larsen 2015 Theory, Culture, and Politics M.A. Graduate Program September 2015 Abstract THE MATERIALITY & ONTOLOGY OF DIGITAL SUBJECTIVITY: GRIGORI “GRISHA” PERELMAN AS A CASE STUDY IN DIGITAL SUBJECTIVITY Gary Larsen New conditions of materiality are emerging from fundamental changes in our ontological order. Digital subjectivity represents an emergent mode of subjectivity that is the effect of a more profound ontological drift that has taken place, and this bears significant repercussions for the practice and understanding of the political. This thesis pivots around mathematician Grigori ‘Grisha’ Perelman, most famous for his refusal to accept numerous prestigious prizes resulting from his proof of the Poincaré conjecture. The thesis shows the Perelman affair to be a fascinating instance of the rise of digital subjectivity as it strives to actualize a new hegemonic order. By tracing first the production of aesthetic works that represent Grigori Perelman in legacy media, the thesis demonstrates that there is a cultural imperative to represent Perelman as an abject figure. Additionally, his peculiar abjection is seen to arise from a challenge to the order of materiality defended by those with a vested interest in maintaining the stability of a hegemony identified with the normative regulatory power of the heteronormative matrix sustaining social relations in late capitalism.
    [Show full text]
  • On the Sormani-Wenger Intrinsic Flat Convergence of Alexandrov Spaces 3
    ON THE SORMANI-WENGER INTRINSIC FLAT CONVERGENCE OF ALEXANDROV SPACES NAN LI AND RAQUEL PERALES Abstract. We study sequences of integral current spaces (X j, d j, T j) such that the integral current structure T j has weight 1 and no boundary and, all (X j, d j) are closed Alexandrov spaces with curvature uniformly bounded from below and diameter uniformly bounded from above. We prove that for such sequences either their limits collapse or the Gromov- Hausdorff and Sormani-Wenger Intrinsic Flat limits agree. The latter is done showing that the lower n dimensional density of the mass measure at any regular point of the Gromov- Hausdorff limit space is positive by passing to a filling volume estimate. In an appendix we show that the filling volume of the standard n dimensional integral current space coming from an n dimensional sphere of radius r > 0 in Euclidean space equals rn times the filling volume of the n dimensional integral current space coming from the n dimensional sphere of radius 1. Introduction Burago, Gromov and Perelman proved that sequences of Alexandrov spaces with cur- vature uniformly bounded from below, diameter and dimension uniformly bounded from above, have subsequences which converge in the Gromov-Hausdorff sense to an Alexan- drov space with the same curvature and diameter bounds. The properties of Alexandrov spaces and their Gromov-Hausdorff limit spaces have been amply studied by Alexander- Bishop [1], Alexander-Kapovitch-Petrunin [2], Burago-Gromov-Perelman [5], Burago- Burago-Ivanov [4], Li-Rong [15], Otsu-Shioya [20], and many others. The Gromov Hausdorff distance between two metric spaces, Xi, is defined as, Z (0.1) dGH (X , X ) = inf d (ϕ (X ) , ϕ (X )) , 1 2 { H 1 1 2 2 } Z ff where dH denotes the Hausdor distance and the infimum is taken over all complete metric spaces Z and all distance preserving maps ϕi : Xi Z, [7].
    [Show full text]
  • Large Scale Geometry by Nowak and Yu
    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 52, Number 1, January 2015, Pages 141–149 S 0273-0979(2014)01460-0 Article electronically published on August 20, 2014 Large scale geometry, by P. Nowak and G. Yu, EMS Textbooks in Mathematics, European Mathematical Society, Zu¨rich, 2012, xiv+189 pp., ISBN 978-3-03719- 112-5, AC38.00, $41.80 Everyone knows that many phenomena are best studied at one scale or another, and that deep problems often involve the interactions of several scales. From the broad conception of our understanding of physics, based on the atomic hypothesis to the germ theory of disease to the success of calculus with its move to the infin- itesimal, the value of the study of phenomena at small scales and the techniques of integrating such information to the macroscopic level are now the most basic intuition in the scientific world view—with attendant backlashes against “blind reductionism”. In the subject of global geometry, the important general direction of “under- standing manifolds under some condition of curvature” is of this sort. Curvature is the infinitesimal measure of how the space di®ers from Euclidean space, and one tries to obtain global conclusions from uniform assumptions on this type of local hypothesis. There is also another extreme: going from the cosmic, the large scale, back to the local. In retrospect, at least, one can see this trend in, say, Liouville’s theorem that bounded analytic functions are constant: boundedness is a large scale condition; surely, if one looks from a very large scale, a bounded function should not be viewed as di®erent than a constant, and Liouville’s theorem tells us that under a condition of analyticity, the large scale information tells all.
    [Show full text]
  • An Intrinsic Parallel Transport in Wasserstein Space
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 145, Number 12, December 2017, Pages 5329–5340 http://dx.doi.org/10.1090/proc/13655 Article electronically published on July 10, 2017 AN INTRINSIC PARALLEL TRANSPORT IN WASSERSTEIN SPACE JOHN LOTT (Communicated by Guofang Wei) Abstract. If M is a smooth compact connected Riemannian manifold, let P (M) denote the Wasserstein space of probability measures on M. We describe a geometric construction of parallel transport of some tangent cones along geodesics in P (M). We show that when everything is smooth, the geometric parallel transport agrees with earlier formal calculations. 1. Introduction Let M be a smooth compact connected Riemannian manifold without boundary. The space P (M) of probability measures of M carries a natural metric, the Wasser- stein metric, and acquires the structure of a length space. There is a close relation between minimizing geodesics in P (M) and optimal transport between measures. For more information on this relation, we refer to Villani’s book [13]. Otto discovered a formal Riemannian structure on P (M), underlying the Wasser- stein metric [10]. One can do formal geometric calculations for this Riemannian structure [6]. It is an interesting problem to make these formal considerations into rigorous results in metric geometry. If M has nonnegative sectional curvature, then P (M) is a compact length space with nonnegative curvature in the sense of Alexandrov [8, Theorem A.8], [12, Propo- sition 2.10]. Hence one can define the tangent cone TμP (M)ofP (M)atameasure μ ∈ P (M). If μ is absolutely continuous with respect to the volume form dvolM , then TμP (M) is a Hilbert space [8, Proposition A.33].
    [Show full text]
  • ON GLUING ALEXANDROV SPACES with LOWER RICCI CURVATURE BOUNDS 3 and the Boundary (Defined Either Way) Is a Closed Subset in the Ambient Space
    ON GLUING ALEXANDROV SPACES WITH LOWER RICCI CURVATURE BOUNDS VITALI KAPOVITCH, CHRISTIAN KETTERER, AND KARL-THEODOR STURM Abstract. In this paper we prove that in the class of metric measure space with Alexandrov curvature bounded from below the Riemannian curvature- dimension condition RCD(K,N) with K ∈ R & N ∈ [1, ∞) is preserved under doubling and gluing constructions. Contents 1. Introduction and Statement of Main Results 1 1.1. Application to heat flow with Dirichlet boundary condition 4 2. Preliminaries 5 2.1. Curvature-dimension condition 5 2.2. Alexandrov spaces 7 2.3. Gluing 8 2.4. Semi-concave functions 10 2.5. 1D localisation of generalized Ricci curvature bounds. 11 2.6. Characterization of curvature bounds via 1D localisation 13 3. Applying 1D localisation 14 3.1. First application 14 3.2. Second application 15 4. Semiconcave functions on glued spaces 18 5. Proof of Theorem 1.1 21 References 23 1. Introduction and Statement of Main Results arXiv:2003.06242v1 [math.DG] 13 Mar 2020 A way to construct Alexandrov spaces is by gluing together two or more given Alexandrov spaces along isometric connected components of their intrinsic bound- aries. The isometry between the boundaries is understood w.r.t. induced length metric. A special case of this construction is the double space where one glues together two copies of the same Alexandrov space with nonempty boundary. It was shown by Perelman that the double of an Alexandrov space of curvature k is again Alexandrov of curvature k. Petrunin later showed [Pet97] that the≥ lower ≥ CK is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Projektnummer 396662902.
    [Show full text]
  • Mathematical Congress of the Americas
    656 Mathematical Congress of the Americas First Mathematical Congress of the Americas August 5–9, 2013 Guanajuato, México José A. de la Peña J. Alfredo López-Mimbela Miguel Nakamura Jimmy Petean Editors American Mathematical Society Mathematical Congress of the Americas First Mathematical Congress of the Americas August 5–9, 2013 Guanajuato, México José A. de la Peña J. Alfredo López-Mimbela Miguel Nakamura Jimmy Petean Editors 656 Mathematical Congress of the Americas First Mathematical Congress of the Americas August 5–9, 2013 Guanajuato, México José A. de la Peña J. Alfredo López-Mimbela Miguel Nakamura Jimmy Petean Editors American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor Michael Loss Kailash Misra Martin J. Strauss 2000 Mathematics Subject Classification. Primary 00-02, 00A05, 00A99, 00B20, 00B25. Library of Congress Cataloging-in-Publication Data Library of Congress Cataloging-in-Publication (CIP) Data has been requested for this volume. Contemporary Mathematics ISSN: 0271-4132 (print); ISSN: 1098-3627 (online) DOI: http://dx.doi.org/10.1090/conm/656 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service.
    [Show full text]
  • Calculus, Heat Flow and Curvature-Dimension Bounds in Metric Measure Spaces
    P. I. C. M. – 2018 Rio de Janeiro, Vol. 1 (301–340) CALCULUS, HEAT FLOW AND CURVATURE-DIMENSION BOUNDS IN METRIC MEASURE SPACES L A Abstract The theory of curvature-dimension bounds for nonsmooth spaces has several motivations: the study of functional and geometric inequalities in structures which are very far from being Euclidean, therefore with new non-Riemannian tools, the description of the “closure” of classes of Riemannian manifolds under suitable geo- metric constraints, the stability of analytic and geometric properties of spaces (e.g. to prove rigidity results). Even though these goals may occasionally be in conflict, in the last few years we have seen spectacular developments in all these directions, and my text is meant both as a survey and as an introduction to this quickly devel- oping research field. 1 Introduction I will mostly focus on metric measure spaces (m.m.s. in brief), namely triples (X; d; m), where (X; d) is a complete and separable metric space and m is a non-negative Borel measure, finite on bounded sets, typically with supp m = X. The model case that should always be kept in mind is a weighted Riemannian manifold (M; g; m), with m given by V (1-1) m := e volg for a suitable weight function V : M R. It can be viewed as a metric measure space ! by taking as d = dg the Riemannian distance induced by g. In order to achieve the goals I mentioned before, it is often necessary to extend many basic calculus tools from smooth to nonsmooth structures.
    [Show full text]
  • Grigori Perelman
    Grigori Perelman Biography Grigori Perelman was born in Leningrad, Soviet Union (now Saint Petersburg, Russia) on 13 June 1966, to Jewish parents, Yakov (who now lives in Israel) and Lubov. Grigori's mother Lubov gave up graduate work in mathematics to raise him. Grigori's mathematical talent became apparent at the age of ten, and his mother enrolled him in Sergei Rukshin's after-school math training program. His mathematical education continued at the Leningrad Secondary School №239, a specialized school with advanced mathematics and physics programs. Grigori excelled in all subjects except physical education. In 1982, as a member of the Soviet Union team competing in the International Mathematical Olympiad, an international competition for high school students, he won a gold medal, achieving a perfect score. In the late 1980s, Perelman went on to earn a Candidate of Sciences degree (the Soviet equivalent to the Ph.D.) at the School of Mathematics and Mechanics of the Leningrad State University, one of the leading universities in the former Soviet Union. His dissertation was titled "Saddle surfaces in Euclidean spaces." After graduation, Perelman began work at the renowned Leningrad Department of Steklov Institute of Mathematics of the USSR Academy of Sciences, where his advisors were Aleksandr Aleksandrov and Yuri Burago. In the late 1980s and early 1990s, Perelman held research positions at several universities in the United States. In 1991 Perelman won the Young Mathematician Prize of the St. Petersburg Mathematical Society for his work on Aleksandrov's spaces of curvature bounded from below . In 1992, he was invited to spend a semester each at the Courant Institute in New York University and State University of New York at Stony Brook where he began work on manifolds with lower bounds on Ricci curvature.
    [Show full text]
  • Optimal Transport Maps on Alexandrov Spaces Revisited 3
    OPTIMAL TRANSPORT MAPS ON ALEXANDROV SPACES REVISITED TAPIO RAJALA AND TIMO SCHULTZ Abstract. We give an alternative proof for the fact that in n-dimensional Alexandrov spaces with curvature bounded below there exists a unique optimal transport plan from any purely (n − 1)-unrectifiable starting measure, and that this plan is induced by an optimal map. 1. Introduction The problem of optimal mass transportation has a long history, starting from the work of Monge [27] in the late 18th century. In the original formulation of the problem, nowadays called the Monge-formulation, the problem is to find the transport map T minimizing the transportation cost c(x, T (x)) dµ0(x), (1.1) ZRn n n among all Borel maps T : R → R transporting a given probability measure µ0 to another given probability measure µ1, that is, T♯µ0 = µ1. In the original problem of Monge, the cost function c(x, y) was the Euclidean distance. Later, other cost functions have been considered, in particular much of the study has involved the distance squared cost, c(x, y)= |x − y|2, which is the cost studied also in this paper. In the Monge-formulation (1.1) of the optimal mass transportation problem the class of admissible maps T that send µ0 to µ1 is in most cases not closed in any suitable topology. To overcome this problem, Kantorovich [20, 19] considered a larger class of optimal transports, n n namely, measures π on R ×R such that the first marginal of π is µ0 and the second is µ1. Such measures π are called transport plans.
    [Show full text]
  • Grigory Perelman Biography
    Grigory Perelman Life & Education Soul conjecture Poincare conjecture Topology Mathematics Genealogy Park Hong-gil Early life and education Born in June 13, 1966 (Leningrad, Soviet Union) lived in Saint Petersburg, Russia Blood Russian-Jewish parents Education Leningrad Secondary School #239 Leningrad State University Doctoral advisor Aleksandr Aleksandrov Yuri Burago Prize International Mathematical Olympiad Gold Medal After his PhD in 1990 : work at the Leningrad Department of Steklov Institute of Mathematics of USSR Academy of Sciences The late 1980s and early 1990s : research positions at several universities in the United States 1991 : the Young Mathematician Prize of the St. Petersburg Mathematical Society 1992 : invited to spend a semester each at the Courant Institute in New York University and Stony Brook University 1993 : accepted a two-year Miller Research Fellowship at the University of California, Berkeley . 1994 : proved the soul conjecture The summer of 1995 : returned to the Steklov Institute in Saint Petersburg for a research-only position. 2003 : proved Thurston's geometrization conjecture. 2006 : confirmed the above August, 2006 : offered the Fields Medal but he declined the award December 22, 2006 : the scientific journal Science recognized Perelman's proof of the Poincaré conjecture March 18, 2010 : the first Clay Millennium Prize July 1, 2010 : rejected the prize of one million dollars Soul conjecture Cheeger and Gromoll's soul conjecture states: Suppose (M, g) is complete, connected and non-compact with sectional curvature K ≥ 0, and there exists a point in M where the sectional curvature (in all sectional directions) is strictly positive. Then the soul of M is a point; equivalently M is diffeomorphic to Rn.
    [Show full text]
  • January 2014 Prizes and Awards
    January 2014 Prizes and Awards 4:25 P.M., Thursday, January 16, 2014 PROGRAM SUMMARY OF AWARDS OPENING REMARKS FOR AMS Bob Devaney, President AWARD FOR DISTINGUISHED PUBLIC SERVICE: PHILIP KUTZKO Mathematical Association of America BÔCHER MEMORIAL PRIZE: SIMON BRENDLE AWARD FOR DISTINGUISHED PUBLIC SERVICE LEVI L. CONANT PRIZE: ALEX KONTOROVICH American Mathematical Society JOSEPH L. DOOB PRIZE: CÉDRIC VILLANI BÔCHER MEMORIAL PRIZE FRANK NELSON COLE PRIZE IN NUMBER THEORY: YITANG ZHANG, AND DANIEL GOLDSTON, JÁNOS American Mathematical Society PINTZ, AND CEM Y. YILDIRIM EONARD ISENBUD RIZE FOR ATHEMATICS AND HYSICS REGORY OORE FRANK NELSON COLE PRIZE IN NUMBER THEORY L E P M P : G W. M American Mathematical Society LEROY P. STEELE PRIZE FOR LIFETIME ACHIEVEMENT: PHILLIP A. GRIFFITHS LEROY P. STEELE PRIZE FOR MATHEMATICAL EXPOSITION: DMITRI Y. BURAGO, YURI D. BURAGO, AND LEVI L. CONANT PRIZE SERGEI V. IVANOV American Mathematical Society LEROY P. STEELE PRIZE FOR SEMINAL CONTRIBUTION TO RESEARCH: LUIS A. CAFFARELLI, ROBERT KOHN, LEONARD EISENBUD PRIZE FOR MATHEMATICS AND PHYSICS AND LOUIS NIRENBERG American Mathematical Society FOR AMS-MAA-SIAM DEBORAH AND FRANKLIN TEPPER HAIMO AWARDS FOR DISTINGUISHED COLLEGE OR UNIVERSITY TEACHING OF MATHEMATICS FRANK AND BRENNIE MORGAN PRIZE FOR OUTSTANDING RESEARCH IN MATHEMATICS BY Mathematical Association of America AN UNDERGRADUATE STUDENT: ERIC LARSON EULER BOOK PRIZE FOR AWM Mathematical Association of America LOUISE HAY AWARD FOR CONTRIBUTIONS TO MATHEMATICS EDUCATION: SYBILLA BECKMANN CHAUVENET PRIZE M. GWENETH HUMPHREYS AWARD FOR MENTORSHIP OF UNDERGRADUATE WOMEN IN MATHEMATICS: Mathematical Association of America WILLIAM YSLAS VÉLEZ ALICE T. SCHAFER PRIZE FOR EXCELLENCE IN MATHEMATICS BY AN UNDERGRADUATE WOMAN ALICE T.
    [Show full text]