DOCSLIB.ORG

The Title of the Dissertation

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Title of the Dissertation UNIVERSITY OF CALIFORNIA, SAN DIEGO Some Structural Results for Measured Equivalence Relations and Their Associated von Neumann Algebras A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Daniel J. Hoff Committee in charge: Professor Adrian Ioana, Chair Professor Benjamin Grinstein Professor Bill Helton Professor John McGreevy Professor Alireza Salehi Golsefidy Professor Hans Wenzl 2016 Copyright Daniel J. Hoff, 2016 All rights reserved. The dissertation of Daniel J. Hoff is approved, and it is acceptable in quality and form for publication on microfilm and electronically: Chair University of California, San Diego 2016 iii DEDICATION For Burcu, Keeper of my spirit, Home to my joy iv TABLE OF CONTENTS Signature Page.................................. iii Dedication..................................... iv Table of Contents.................................v Acknowledgements................................ viii Vita........................................x Abstract of the Dissertation........................... xi Chapter 1 Introduction............................1 1.1 Background.........................1 1.1.1 Organization.....................2 1.2 Preliminaries........................2 1.2.1 Von Neumann algebras...............2 1.2.2 Tracial von Neumann algebras...........3 1.2.3 Amenable von Neumann algebras.........4 1.2.4 Probability measure preserving equivalence relations.......................4 1.2.5 Von Neumann algebras of equivalence relations..5 1.2.6 Amenable equivalence relations..........6 1.2.7 Orbit equivalence relations.............6 1.2.8 Amplifications....................7 Chapter 2 Von Neumann Algebras of Equivalence Relations with Non- trivial One-Cohomology.....................8 2.1 Introduction.........................8 2.1.1 Background and statement of results........8 2.1.2 Organization and strategy............. 12 2.2 Preliminaries........................ 12 2.2.1 Representations of equivalence relations...... 12 2.2.2 Orbit equivalence relations............. 17 2.2.3 Relative mixingness and weak containment of bimodules...................... 18 2.2.4 Relative amenability................ 20 2.2.5 Amplifications.................... 20 2.2.6 Popa's intertwining by bimodules......... 21 2.3 Deducing Primeness from an s-Malleable Deformation......................... 21 v 2.4 Gaussian Extension of R and s-Malleable Deformation of L(R)............................. 25 2.4.1 Gaussian extension of R .............. 25 2.4.2 s-Malleable deformation of L(R).......... 27 2.5 Primeness of L(R)..................... 28 2.5.1 L(R)-L(R) bimodules arising from representations of R ......................... 29 2.5.2 Proof of Theorem 2A................ 32 2.5.3 Remark....................... 33 2.6 Unique Prime Factorization................ 33 2.6.1 An obstruction to unique factorization...... 33 2.6.2 Unique prime factorization via s-malleable deformation..................... 38 2.6.3 Unique prime factorization for equivalence relations....................... 42 2.7 Application to Measure Equivalent Groups........ 45 Chapter 3 Von Neumann's Problem and Extensions of Non-amenable Equiv- alence Relations.......................... 48 3.1 Introduction and statement of main results........ 48 3.1.1 Background..................... 48 3.1.2 Von Neumann's problem for Bernoulli extensions 49 3.1.3 Uncountably many non-isomorphic extensions.. 51 3.1.4 Actions of locally compact groups......... 53 3.1.5 Outline of the proof of Theorem 3A........ 54 3.1.6 Organization..................... 55 3.2 Preliminaries........................ 55 3.2.1 Amenable equivalence relations.......... 55 3.2.2 Extensions and expansions of equivalence relations 56 3.2.3 Graphed equivalence relations and isoperimetric constants....................... 59 3.2.4 Cost of equivalence relations............ 60 3.3 Bernoulli extensions of equivalence relations....... 61 3.3.1 Bernoulli extensions and ergodicity........ 61 3.3.2 Isomorphisms of Bernoulli extensions....... 62 3.3.3 Bernoulli extensions restricted to subequivalence relations....................... 63 3.3.4 Compressions of Bernoulli extensions....... 64 3.4 Bernoulli percolation on graphed equivalence relations.................... 68 3.4.1 Bernoulli percolation on graphs.......... 69 3.4.2 Infinitely many infinite clusters.......... 70 3.4.3 Proof of Theorem 3.4.3............... 72 vi 3.5 Ergodicity of the cluster equivalence relation....... 74 3.6 Proofs of Theorem 3A and Corollary 3B......... 78 3.6.1 A generalization of Theorem 3A.......... 78 3.6.2 Proof of Theorem 3.6.1............... 78 3.6.3 Proof of Corollary 3B................ 83 3.7 Uncountably many ergodic extensions of nonamenable R ....................... 84 3.7.1 Co-induced equivalence relation.......... 84 3.7.2 A separability argument.............. 88 3.7.3 Proof of Theorem 3C................ 90 3.8 Actions of locally compact groups............. 91 3.8.1 Proof of Corollary 3D................ 94 3.8.2 Deducing [GM15, Theorem B] from Theorem 3A. 95 References..................................... 97 vii ACKNOWLEDGEMENTS There are many people without whom this dissertation would not exist. First and foremost I would like to thank Adrian Ioana for proposing the topics of study herein and for the countless hours he spent sharing both his knowledge and his wisdom as my advisor. I cannot overstate how much I appreciated our many stimulating discussions when I had progress to report, and perhaps most importantly, his encouragement, patience, and unfailing kindness when I struggled. I am also grateful for the time and efforts of the other committee members, and for all of the faculty whose stimulating courses I had the opportunity to enjoy. I would also like to thank R´emiBoutonnet for organizing many engaging learning seminars and for patiently answering my many questions regardless of their quality. I am so grateful to have been able to show him the full extent of my ignorance, and for his selfless efforts to whittle away at it. I would like to thank Daniel Drimbe for his always-encouraging belief in me. I am also very indebted to my fellow graduate students, especially to Jay, Rob, and Tait, for celebrating in the good times, commiserating in the bad, and providing all kinds of rejuvenating diversions in between. I owe much to all those I studied with, especially the algebros, Grimm, Rob, and Robert. A special thanks is due to Rob, without whose efforts our class would not have enjoyed such synergy. Away from math, thank you to all members of the UN of Windansea, especially Darlene, David, Giulio, Michael, Moses, and Paul, who provided so many of my best memories in San Diego. A great thanks is due to my perpetually ahead brother and sister who gave me the stubborn desire to understand that which I do not. I thank Eric Lauer- Hunt for sparking my interest in math, and my many excellent math teachers for building it, especially Mrs. Eldredge, who sent me to college with a love of math. Once there I owe so much to Peter Olver, who patiently guided my transition to research, and provided the passion for discovery that has driven me ever since. It is only the very fortunate whose lives allow for the pursuit of these studies. For this opportunity I thank my parents, who have given me so much, and on whose unconditional love and support I rely in all things. viii Chapter 2 is, in part, a reprint of the material as it appears in [Ho15] Daniel J. Hoff, Von Neumann algebras of equivalence relations with nontrivial one-cohomology, J. Funct. Anal. 270 (2016), no. 4, 1501{1536. MR 3447718. of which the dissertation author was the primary investigator and author. Chapter 3 is, in part, material submitted for publication as it appears in [BHI15] Lewis Bowen, Daniel J. Hoff, and Adrian Ioana, Von Neumann's problem and extensions of non-amenable equivalence relations, preprint arXiv:1509.01723, 2015. of which the dissertation author was one of the primary investigators and authors. This material is based upon work supported by the National Science Foun- dation Graduate Research Fellowship Program under Grant No. DGE-1144086. Any opinions, findings, and conclusions or recommendations expressed in this ma- terial are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. ix VITA 2011 B.S. in Mathematics University of Minnesota, Twin Cities 2011-2013 Graduate Teaching Assistant University of California, San Diego 2013 M.A. in Mathematics University of California, San Diego 2013-2016 NSF Graduate Research Fellow University of California, San Diego 2014 C.Phil. in Mathematics University of California, San Diego 2016 Ph.D. in Mathematics University of California, San Diego x ABSTRACT OF THE DISSERTATION Some Structural Results for Measured Equivalence Relations and Their Associated von Neumann Algebras by Daniel J. Hoff Doctor of Philosophy in Mathematics University of California San Diego, 2016 Professor Adrian Ioana, Chair Using Popa's deformation/rigidity theory, we investigate prime decomposi- tions of von Neumann algebras of the form L(R) for countable probability measure preserving (pmp) equivalence relations R. We show that L(R) is prime whenever R is non-amenable, ergodic, and admits an unbounded 1-cocycle into a mixing orthogonal representation weakly contained in the regular representation. This is accomplished by constructing the Gaussian extension R~ of R and subsequently an s-malleable deformation of the inclusion L(R) ⊂ L(R~). We go on to note a general
Load more

Recommended publications
  • Abstract We Survey the Published Work of Harry Kesten in Probability Theory, with Emphasis on His Contributions to Random Walks
    Abstract We survey the published work of Harry Kesten in probability theory, with emphasis on his contributions to random walks, branching processes, perco- lation, and related topics. Keywords Probability, random walk, branching process, random matrix, diffusion limited aggregation, percolation. Mathematics Subject Classification (2010) 60-03, 60G50, 60J80, 60B20, 60K35, 82B20. Noname manuscript No. 2(will be inserted by the editor) Geoffrey R. Grimmett Harry Kesten's work in probability theory Geoffrey R. Grimmett In memory of Harry Kesten, inspiring colleague, valued friend April 8, 2020 1 Overview Harry Kesten was a prominent mathematician and personality in a golden period of probability theory from 1956 to 2018. At the time of Harry's move from the Netherlands to the USA in 1956, as a graduate student aged 24, much of the foundational infrastructure of probability was in place. The central characters of probability had long been identified (including random walk, Brownian motion, the branching process, and the Poisson process), and connections had been made and developed between `pure theory' and cognate areas ranging from physics to finance. In the half-century or so since 1956, a coordinated and refined theory has been developed, and probability has been recognised as a crossroads discipline in mathematical science. Few mathematicians have contributed as much during this period as Harry Kesten. Following a turbulent childhood (see [59]), Harry studied mathematics with David van Dantzig and Jan Hemelrijk in Amsterdam, where in 1955 he attended a lecture by Mark Kac entitled \Some probabilistic aspects of potential theory". This encounter appears to have had a decisive effect, in that Harry moved in 1956 to Cornell University to work with Kac.
    [Show full text]
  • I. Overview of Activities, April, 2005-March, 2006 …
    MATHEMATICAL SCIENCES RESEARCH INSTITUTE ANNUAL REPORT FOR 2005-2006 I. Overview of Activities, April, 2005-March, 2006 …......……………………. 2 Innovations ………………………………………………………..... 2 Scientific Highlights …..…………………………………………… 4 MSRI Experiences ….……………………………………………… 6 II. Programs …………………………………………………………………….. 13 III. Workshops ……………………………………………………………………. 17 IV. Postdoctoral Fellows …………………………………………………………. 19 Papers by Postdoctoral Fellows …………………………………… 21 V. Mathematics Education and Awareness …...………………………………. 23 VI. Industrial Participation ...…………………………………………………… 26 VII. Future Programs …………………………………………………………….. 28 VIII. Collaborations ………………………………………………………………… 30 IX. Papers Reported by Members ………………………………………………. 35 X. Appendix - Final Reports ……………………………………………………. 45 Programs Workshops Summer Graduate Workshops MSRI Network Conferences MATHEMATICAL SCIENCES RESEARCH INSTITUTE ANNUAL REPORT FOR 2005-2006 I. Overview of Activities, April, 2005-March, 2006 This annual report covers MSRI projects and activities that have been concluded since the submission of the last report in May, 2005. This includes the Spring, 2005 semester programs, the 2005 summer graduate workshops, the Fall, 2005 programs and the January and February workshops of Spring, 2006. This report does not contain fiscal or demographic data. Those data will be submitted in the Fall, 2006 final report covering the completed fiscal 2006 year, based on audited financial reports. This report begins with a discussion of MSRI innovations undertaken this year, followed by highlights
    [Show full text]
  • Divergence of Shape Fluctuation for General Distributions in First-Passage Percolation
    DIVERGENCE OF SHAPE FLUCTUATION FOR GENERAL DISTRIBUTIONS IN FIRST-PASSAGE PERCOLATION SHUTA NAKAJIMA d Abstract. We study the shape fluctuation in the first-passage percolation on Z . It is known that it diverges when the distribution obeys Bernoulli in [Yu Zhang. The divergence of fluctuations for shape in first passage percolation. Probab. Theory. Related. Fields. 136(2) 298–320, 2006]. In this paper, we extend the result to general distributions. 1. Introduction First-passage percolation is a random growth model, which was first introduced by Hammersley and Welsh in 1965. The model is defined as follows. The vertices are the d d elements of Z . Let us denote the set edges by E : d d E = ffv; wgj v; w 2 Z ; jv − wj1 = 1g; Pd where we set jv − wj1 = i=1 jvi − wij for v = (v1; ··· ; vd), w = (w1; ··· ; wd). Note that we consider non-oriented edges in this paper, i.e., fv; wg = fw; vg and we sometimes d regard fv; wg as a subset of Z with a slight abuse of notation. We assign a non-negative d random variable τe on each edge e 2 E , called the passage time of the edge e. The collec- tion τ = fτege2Ed is assumed to be independent and identically distributed with common distribution F . d A path γ is a finite sequence of vertices (x1; ··· ; xl) ⊂ Z such that for any i 2 d f1; ··· ; l − 1g, fxi; xi+1g 2 E . It is customary to regard a path as a subset of edges as follows: given an edge e 2 Ed, we write e 2 γ if there exists i 2 f1 ··· ; l − 1g such that e = fxi; xi+1g.
    [Show full text]
  • Ericsson's Conjecture on the Rate of Escape Of
    TRANSACTIONS of the AMERICAN MATHEMATICAL SOCIETY Volume 240, lune 1978 ERICSSON'SCONJECTURE ON THE RATEOF ESCAPEOF ¿-DIMENSIONALRANDOM WALK BY HARRYKESTEN Abstract. We prove a strengthened form of a conjecture of Erickson to the effect that any genuinely ¿-dimensional random walk S„, d > 3, goes to infinity at least as fast as a simple random walk or Brownian motion in dimension d. More precisely, if 5* is a simple random walk and B, a Brownian motion in dimension d, and \j/i [l,oo)-»(0,oo) a function for which /"'/^(OiO, then \¡/(n)~'\S*\-» oo w.p.l, or equivalent^, ifr(t)~'\Bt\ -»oo w.p.l, iff ffif/(tY~2t~d/2 < oo; if this is the case, then also $(ri)~'\S„\ -» oo w.p.l for any random walk Sn of dimension d. 1. Introduction. Let XX,X2,... be independent identically distributed ran- dom ¿-dimensional vectors and S,, = 2"= XX¡.Assume throughout that d > 3 and that the distribution function F of Xx satisfies supp(P) is not contained in any hyperplane. (1.1) The celebrated Chung-Fuchs recurrence criterion [1, Theorem 6] implies that1 |5n|->oo w.p.l irrespective of F. In [4] Erickson made the much stronger conjecture that there should even exist a uniform escape rate for S„, viz. that «""iSj -* oo w.p.l for all a < \ and all F satisfying (1.1). Erickson proved his conjecture in special cases and proved in all cases that «-a|5„| -» oo w.p.l for a < 1/2 — l/d. Our principal result is the following theorem which contains Erickson's conjecture and which makes precise the intuitive idea that a simple random walk S* (which corresponds to the distribution F* which puts mass (2d)~x at each of the points (0, 0,..., ± 1, 0,..., 0)) goes to infinity slower than any other random walk.
    [Show full text]
  • An Interview with Martin Davis
    Notices of the American Mathematical Society ISSN 0002-9920 ABCD springer.com New and Noteworthy from Springer Geometry Ramanujan‘s Lost Notebook An Introduction to Mathematical of the American Mathematical Society Selected Topics in Plane and Solid Part II Cryptography May 2008 Volume 55, Number 5 Geometry G. E. Andrews, Penn State University, University J. Hoffstein, J. Pipher, J. Silverman, Brown J. Aarts, Delft University of Technology, Park, PA, USA; B. C. Berndt, University of Illinois University, Providence, RI, USA Mediamatics, The Netherlands at Urbana, IL, USA This self-contained introduction to modern This is a book on Euclidean geometry that covers The “lost notebook” contains considerable cryptography emphasizes the mathematics the standard material in a completely new way, material on mock theta functions—undoubtedly behind the theory of public key cryptosystems while also introducing a number of new topics emanating from the last year of Ramanujan’s life. and digital signature schemes. The book focuses Interview with Martin Davis that would be suitable as a junior-senior level It should be emphasized that the material on on these key topics while developing the undergraduate textbook. The author does not mock theta functions is perhaps Ramanujan’s mathematical tools needed for the construction page 560 begin in the traditional manner with abstract deepest work more than half of the material in and security analysis of diverse cryptosystems. geometric axioms. Instead, he assumes the real the book is on q- series, including mock theta Only basic linear algebra is required of the numbers, and begins his treatment by functions; the remaining part deals with theta reader; techniques from algebra, number theory, introducing such modern concepts as a metric function identities, modular equations, and probability are introduced and developed as space, vector space notation, and groups, and incomplete elliptic integrals of the first kind and required.
    [Show full text]
  • Harry Kesten's Work in Probability Theory
    Abstract We survey the published work of Harry Kesten in probability theory, with emphasis on his contributions to random walks, branching processes, perco- lation, and related topics. A complete bibliography is included of his publications. Keywords Probability, random walk, branching process, random matrix, diffusion limited aggregation, percolation. Mathematics Subject Classification (2010) 60-03, 60G50, 60J80, 60B20, 60K35, 82B20. arXiv:2004.03861v2 [math.PR] 13 Nov 2020 Noname manuscript No. 2(will be inserted by the editor) Geoffrey R. Grimmett Harry Kesten's work in probability theory Geoffrey R. Grimmett In memory of Harry Kesten, inspiring colleague, valued friend Submitted 20 April 2020, revised 26 October 2020 Contents 1 Overview . .2 2 Highlights of Harry Kesten's research . .4 3 Random walk . .8 4 Products of random matrices . 17 5 Self-avoiding walks . 17 6 Branching processes . 19 7 Percolation theory . 20 8 Further work . 31 Acknowledgements . 33 References . 34 Publications of Harry Kesten . 40 1 Overview Harry Kesten was a prominent mathematician and personality in a golden period of probability theory from 1956 to 2018. At the time of Harry's move from the Netherlands to the USA in 1956, as a graduate student aged 24, much of the foundational infrastructure of probability was in place. The central characters of probability had long been identified (including random walk, Brownian motion, the branching process, and the Poisson process), and connections had been made and developed between `pure theory' and cognate areas ranging from physics to finance. In the half-century or so since 1956, a coordinated and refined theory has been developed, and probability has been recognised as a crossroads discipline in mathematical science.
    [Show full text]
  • Jennifer Tour Chayes March 2016 CURRICULUM VITAE Office
    Jennifer Tour Chayes March 2016 CURRICULUM VITAE Office Address: Microsoft Research New England 1 Memorial Drive Cambridge, MA 02142 e-mail: [email protected] Personal: Born September 20, 1956, New York, New York Education: 1979 B.A., Physics and Biology, Wesleyan University 1983 Ph.D., Mathematical Physics, Princeton University Positions: 1983{1985 Postdoctoral Fellow in Mathematical Physics, Departments of Mathematics and Physics, Harvard University 1985{1987 Postdoctoral Fellow, Laboratory of Atomic and Solid State Physics and Mathematical Sciences Institute, Cornell University 1987{1990 Associate Professor, Department of Mathematics, UCLA 1990{2001 Professor, Department of Mathematics, UCLA 1997{2005 Senior Researcher and Head, Theory Group, Microsoft Research 1997{2008 Affiliate Professor, Dept. of Physics, U. Washington 1999{2008 Affiliate Professor, Dept. of Mathematics, U. Washington 2005{2008 Principal Researcher and Research Area Manager for Mathematics, Theoretical Computer Science and Cryptography, Microsoft Research 2008{ Managing Director, Microsoft Research New England 2010{ Distinguished Scientist, Microsoft Corporation 2012{ Managing Director, Microsoft Research New York City Long-Term Visiting Positions: 1994-95, 1997 Member, Institute for Advanced Study, Princeton 1995, May{July ETH, Z¨urich 1996, Sept.{Dec. AT&T Research, New Jersey Awards and Honors: 1977 Johnston Prize in Physics, Wesleyan University 1979 Graham Prize in Natural Sciences & Mathematics, Wesleyan University 1979 Graduated 1st in Class, Summa Cum Laude,
    [Show full text]
  • Annual Report 1998–99
    Department of Mathematics Cornell University Annual Report 1998Ð99 The image on the cover is taken from a catalogue of several hundred “symmetric tensegrities” developed by Professor Robert Connelly and Allen Back, director of our Instructional Computer Lab. (The whole catalogue can be viewed at http://mathlab.cit.cornell.edu/visualization/tenseg/tenseg.html.) The 12 balls represent points in space placed at the midpoints of the edges of a cube. The thin dark edges represent “cables,” each connecting a pair of vertices. If two vertices are connected by a cable, then they are not permitted to get further apart. The thick lighter shaded diagonals represent “struts.” If two vertices are connected by a strut, then they are not permitted to get closer together. The examples in the catalogue — including the one represented on the cover — are constructed so that they are “super stable.” This implies that any pair of congurations of the vertices satisfying the cable and strut constraints are congruent. So if one builds this structure with sticks for the struts and string for the cables, then it will be rigid and hold its shape. The congurations in the catalogue are constructed so that they are “highly” symmetric. By this we mean that there is a congruence of space that permutes all the vertices, and any vertex can be superimposed onto any other by one of those congruences. In the case of the image at hand, all the congruences can be taken to be rotations. Department of Mathematics Annual Report 1998Ð99 Year in Review: Mathematics Instruction and Research Cornell University Þrst among private institutions in undergraduates who later earn Ph.D.s.
    [Show full text]
  • Random Graph Dynamics
    1 Random Graph Dynamics Rick Durrett Copyright 2007, All rights reserved. Published by Cambridge University Press 2 Preface Chapter 1 will explain what this book is about. Here I will explain why I chose to write the book, how it is written, where and when the work was done, and who helped. Why. It would make a good story if I was inspired to write this book by an image of Paul Erd¨osmagically appearing on a cheese quesadilla, which I later sold for thousands on dollars on eBay. However, that is not true. The three main events that led to this book were (i) the use of random graphs in the solution of a problem that was part of Nathanael Berestycki’s thesis, (ii) a talk that I heard Steve Strogatz give on the CHKNS model, which inspired me to prove some rigorous results about their model, and (iii) a book review I wrote on the books by Watts and Barab´asifor the Notices of the American Math Society. The subject of this book was attractive for me, since many of the papers were outside the mathematics literature, so the rigorous proofs of the results were, in some cases, interesting mathematical problems. In addition, since I had worked for a number of years on the proper- ties of stochastic spatial models on regular lattices, there was the natural question of how did the behavior of these systems change when one introduced long range connections between individuals or considered power law degree distributions. Both of these modifications are reasonable if one considers the spread of influenza in a town where children bring the disease home from school, or the spread of sexually transmitted diseases through a population of individuals that have a widely varying number of contacts.
    [Show full text]
  • A Model for Random Chain Complexes
    A MODEL FOR RANDOM CHAIN COMPLEXES MICHAEL J. CATANZARO AND MATTHEW J. ZABKA Abstract. We introduce a model for random chain complexes over a finite field. The randomness in our complex comes from choosing the entries in the matrices that represent the boundary maps uniformly over Fq, conditioned on ensuring that the composition of consecutive boundary maps is the zero map. We then investigate the combinatorial and homological properties of this random chain complex. 1. Introduction There have been a variety of attempts to randomize topological constructions. Most famously, Erd¨os and R´enyi introduced a model for random graphs [6]. This work spawned an entire industry of probabilistic models and tools used for under- standing other random topological and algebraic phenomenon. These include vari- ous models for random simplicial complexes, random networks, and many more [7, 13]. Further, this has led to beautiful connections with statistical physics, for exam- ple through percolation theory [1, 3, 12]. Our ultimate goal is to understand higher dimensional topological constructions arising in algebraic topology from a random perspective. In this manuscript, we begin to address this goal with the much sim- pler objective of understanding an algebraic construction commonly associated with topological spaces, known as a chain complex. Chain complexes are used to measure a variety of different algebraic, geoemtric, and topological properties Their usefulness lies in providing a pathway for homolog- ical algebra computations. They arise in a variety of contexts, including commuta- tive algebra, algebraic geometry, group cohomology, Hoschild homology, de Rham arXiv:1901.00964v1 [math.CO] 4 Jan 2019 cohomology, and of course algebraic topology [2, 4, 9, 10, 11].
    [Show full text]
  • Harry Kesten
    Harry Kesten November 19, 1931 – March 29, 2019 Harry Kesten, Ph.D. ’58, the Goldwin Smith Professor Emeritus of Mathematics, died March 29, 2019 in Ithaca at the age of 87. His wife, Doraline, passed three years earlier. He is survived by his son Michael Kesten ’90. Through his pioneering work on key models of Statistical Mechanics including Percolation Theory, and his novel solutions to highly significant problems, Harry Kesten shaped and transformed entire areas of Mathematics. He is recognized worldwide as one of the greatest contributors to Probability Theory during the second half of the twentieth century. Harry’s work was mostly theoretical in nature but he always maintained a keen interest for the applications of probability theory to the real world and studied problems arising from other sciences including physics, population growth and biology. Harry was born in Duisburg, Germany, in 1931. He moved to Holland with his parents in 1933 and, somehow, survived the Second World War. In 1956, he was studying at the University of Amsterdam and working half-time as a research assistant for Professor D. Van Dantzig. As he was finishing his undergraduate degree, he wrote to the world famous mathematician Mark Kac to enquire if he could possibly receive a graduate fellowship to study probability theory at Cornell, perhaps for a year. At the time, Harry was considered a Polish subject (even though he had never been in Poland) and carried a passport of the international refugee organization. A fellowship was arranged and Harry joined the mathematics graduate program at Cornell that summer.
    [Show full text]
  • Frank L. Spitzer
    NATIONAL ACADEMY OF SCIENCES F R A N K L UDVI G Sp ITZER 1926—1992 A Biographical Memoir by H A R R Y KESTEN Any opinions expressed in this memoir are those of the author(s) and do not necessarily reflect the views of the National Academy of Sciences. Biographical Memoir COPYRIGHT 1996 NATIONAL ACADEMIES PRESS WASHINGTON D.C. Courtesy of Geoffrey Grimmett FRANK LUDVIG SPITZER July 24, 1926–February 1, 1992 BY HARRY KESTEN RANK SPITZER WAS A highly original probabilist and a hu- Fmorous, charismatic person, who had warm relations with students and colleagues. Much of his earlier work dealt with the topics of random walk and Brownian motion, which are quite familiar to probabilists. Spitzer invented or devel- oped quite new aspects of these, such as fluctuation theory and potential theory of random walk (more about these later); however, his most influential work is undoubtedly the creation of a good part of the theory of interacting particle systems. Through the many elegant models that Frank constructed and intriguing phenomena he demon- strated, a whole new set of questions was raised. These have attracted and stimulated a large number of young probabi- lists and have made interacting particle systems one of the most exciting and active subfields of probability today. Frank Spitzer was born in Vienna, Austria, on July 24, 1926, into a Jewish family. His father was a lawyer. When Frank was about twelve years old his parents sent him to a summer camp for Jewish children in Sweden. Quite possi- bly the intention of this camp was to bring out Jewish chil- dren from Nazi-held or Nazi-threatened territory.
    [Show full text]