Journal of the American Mathematical Society This Journal Is Devoted to Research Articles of the Highest Quality in All Areas of Pure and Applied Mathematics
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[The PROOF of FERMAT's LAST THEOREM] and [OTHER MATHEMATICAL MYSTERIES] the World's Most Famous Math Problem the World's Most Famous Math Problem
0Eft- [The PROOF of FERMAT'S LAST THEOREM] and [OTHER MATHEMATICAL MYSTERIES] The World's Most Famous Math Problem The World's Most Famous Math Problem [ THE PROOF OF FERMAT'S LAST THEOREM AND OTHER MATHEMATICAL MYSTERIES I Marilyn vos Savant ST. MARTIN'S PRESS NEW YORK For permission to reprint copyrighted material, grateful acknowledgement is made to the following sources: The American Association for the Advancement of Science: Excerpts from Science, Volume 261, July 2, 1993, C 1993 by the AAAS. Reprinted by permission. Birkhauser Boston: Excerpts from The Mathematical Experience by Philip J. Davis and Reuben Hersh © 1981 Birkhauser Boston. Reprinted by permission of Birkhau- ser Boston and the authors. The Chronicleof Higher Education: Excerpts from The Chronicle of Higher Education, July 7, 1993, C) 1993 Chronicle of HigherEducation. Reprinted by permission. The New York Times: Excerpts from The New York Times, June 24, 1993, X) 1993 The New York Times. Reprinted by permission. Excerpts from The New York Times, June 29, 1993, © 1993 The New York Times. Reprinted by permission. Cody Pfanstieh/ The poem on the subject of Fermat's last theorem is reprinted cour- tesy of Cody Pfanstiehl. Karl Rubin, Ph.D.: The sketch of Dr. Wiles's proof of Fermat's Last Theorem in- cluded in the Appendix is reprinted courtesy of Karl Rubin, Ph.D. Wesley Salmon, Ph.D.: Excerpts from Zeno's Paradoxes by Wesley Salmon, editor © 1970. Reprinted by permission of the editor. Scientific American: Excerpts from "Turing Machines," by John E. Hopcroft, Scientific American, May 1984, (D 1984 Scientific American, Inc. -
On Selmer Groups of Geometric Galois Representations Tom Weston
On Selmer Groups of Geometric Galois Representations Tom Weston Department of Mathematics, Harvard University, Cambridge, Mass- chusetts 02140 E-mail address: [email protected] iii Dedicated to the memory of Annalee Henderson and to Arnold Ross Contents Introduction ix Acknowledgements xii Notation and terminology xv Fields xv Characters xv Galois modules xv Schemes xv Sheaves xvi Cohomology xvi K-theory xvi Part 1. Selmer groups and deformation theory 1 Chapter 1. Local cohomology groups 3 1. Local finite/singular structures 3 2. Functorialities 4 3. Local exact sequences 5 4. Examples of local structures 6 5. Ordinary representations 7 6. Cartier dual structures 8 7. Local structures for archimedean fields 9 Chapter 2. Global cohomology groups 11 1. Selmer groups 11 2. Functorialities 13 3. The global exact sequence 13 4. A finiteness theorem for Selmer groups 14 5. The Kolyvagin pairing 16 6. Shafarevich-Tate groups 18 7. The Bockstein pairing 20 Chapter 3. Annihilation theorems for Selmer groups 21 1. Partial geometric Euler systems 21 2. The key lemmas 22 3. The annihilation theorem 25 4. Right non-degeneracy of the Bockstein pairing 27 5. A δ-vanishing result 28 Chapter 4. Flach systems 31 1. Minimally ramified deformations 31 v vi CONTENTS 2. Tangent spaces and Selmer groups 34 3. Good primes 36 4. Flach systems 38 5. Cohesive Flach systems 39 6. Cohesive Flach systems of Eichler-Shimura type 40 Chapter 5. Flach systems of Eichler-Shimura type 43 1. The map on differentials 43 2. The Tate pairing 45 3. A special case 47 4. -
FOCS 2005 Program SUNDAY October 23, 2005
FOCS 2005 Program SUNDAY October 23, 2005 Talks in Grand Ballroom, 17th floor Session 1: 8:50am – 10:10am Chair: Eva´ Tardos 8:50 Agnostically Learning Halfspaces Adam Kalai, Adam Klivans, Yishay Mansour and Rocco Servedio 9:10 Noise stability of functions with low influences: invari- ance and optimality The 46th Annual IEEE Symposium on Elchanan Mossel, Ryan O’Donnell and Krzysztof Foundations of Computer Science Oleszkiewicz October 22-25, 2005 Omni William Penn Hotel, 9:30 Every decision tree has an influential variable Pittsburgh, PA Ryan O’Donnell, Michael Saks, Oded Schramm and Rocco Servedio Sponsored by the IEEE Computer Society Technical Committee on Mathematical Foundations of Computing 9:50 Lower Bounds for the Noisy Broadcast Problem In cooperation with ACM SIGACT Navin Goyal, Guy Kindler and Michael Saks Break 10:10am – 10:30am FOCS ’05 gratefully acknowledges financial support from Microsoft Research, Yahoo! Research, and the CMU Aladdin center Session 2: 10:30am – 12:10pm Chair: Satish Rao SATURDAY October 22, 2005 10:30 The Unique Games Conjecture, Integrality Gap for Cut Problems and Embeddability of Negative Type Metrics Tutorials held at CMU University Center into `1 [Best paper award] Reception at Omni William Penn Hotel, Monongahela Room, Subhash Khot and Nisheeth Vishnoi 17th floor 10:50 The Closest Substring problem with small distances Tutorial 1: 1:30pm – 3:30pm Daniel Marx (McConomy Auditorium) Chair: Irit Dinur 11:10 Fitting tree metrics: Hierarchical clustering and Phy- logeny Subhash Khot Nir Ailon and Moses Charikar On the Unique Games Conjecture 11:30 Metric Embeddings with Relaxed Guarantees Break 3:30pm – 4:00pm Ittai Abraham, Yair Bartal, T-H. -
Property Testing on Boolean Functions Thesis Proposal
Property testing on boolean functions Thesis proposal Eric Blais Computer Science Department Carnegie Mellon University Pittsburgh, PA [email protected] May 20, 2010 Abstract Property testing deals with the following general question: given query access to some combinatorial object, what properties of the ob- ject can one test with only a small number of queries? In this thesis, we will study property testing on boolean functions. Specifically, we will focus on two basic questions in the field: Can we characterize the set of properties on boolean functions that can be tested with a constant number of queries? And can we determine the exact number of queries needed to test some fundamental properties of boolean functions? 1 Contents 1 Introduction 3 2 Definitions and notation 4 2.1 Boolean functions . 4 2.2 Property testing . 5 3 Characterization of testable properties 5 3.1 Related work . 6 3.2 Testing function isomorphism . 7 3.3 Completed work on testing function isomorphism . 7 3.4 Proposed research . 8 4 Exact query complexity bounds 10 4.1 Related work . 10 4.2 Completed work on testing juntas . 11 4.3 Proposed research . 11 5 Other proposed research 12 5.1 Alternative models of property testing . 13 5.2 Application of property testing ideas to other domains . 14 6 Suggested timeline 14 2 1 Introduction This thesis is primarily concerned with the study of boolean functions. Boolean functions play a central role in many areas of computer science: complexity theory [44], machine learning [19], coding theory [43], cryptog- raphy [18], circuit design [34], data structures [30], and combinatorics [16]. -
2021 Leroy P. Steele Prizes
FROM THE AMS SECRETARY 2021 Leroy P. Steele Prizes The 2021 Leroy P. Steele Prizes were presented at the Annual Meeting of the AMS, held virtually January 6–9, 2021. Noga Alon and Joel Spencer received the Steele Prize for Mathematical Exposition. Murray Gerstenhaber was awarded the Prize for Seminal Contribution to Research. Spencer Bloch was honored with the Prize for Lifetime Achievement. Citation for Mathematical Biographical Sketch: Noga Alon Exposition: Noga Alon Noga Alon is a Professor of Mathematics at Princeton and Joel Spencer University and a Professor Emeritus of Mathematics and The 2021 Steele Prize for Math- Computer Science at Tel Aviv University, Israel. He received ematical Exposition is awarded his PhD in Mathematics at the Hebrew University of Jeru- to Noga Alon and Joel Spencer salem in 1983 and had visiting and part-time positions in for the book The Probabilistic various research institutes, including the Massachusetts Method, published by Wiley Institute of Technology, Harvard University, the Institute and Sons, Inc., in 1992. for Advanced Study in Princeton, IBM Almaden Research Now in its fourth edition, Center, Bell Laboratories, Bellcore, and Microsoft Research The Probabilistic Method is an (Redmond and Israel). He joined Tel Aviv University in invaluable toolbox for both 1985, served as the head of the School of Mathematical Noga Alon the beginner and the experi- Sciences in 1999–2001, and moved to Princeton in 2018. enced researcher in discrete He supervised more than twenty PhD students. He serves probability. It brings together on the editorial boards of more than a dozen international through one unifying perspec- technical journals and has given invited lectures in numer- tive a head-spinning variety of ous conferences, including plenary addresses in the 1996 results and methods, linked to European Congress of Mathematics and in the 2002 Inter- applications in graph theory, national Congress of Mathematicians. -
CURRICULUM VITAE Rafail Ostrovsky
last updated: December 26, 2020 CURRICULUM VITAE Rafail Ostrovsky Distinguished Professor of Computer Science and Mathematics, UCLA http://www.cs.ucla.edu/∼rafail/ mailing address: Contact information: UCLA Computer Science Department Phone: (310) 206-5283 475 ENGINEERING VI, E-mail: [email protected] Los Angeles, CA, 90095-1596 Research • Cryptography and Computer Security; Interests • Streaming Algorithms; Routing and Network Algorithms; • Search and Classification Problems on High-Dimensional Data. Education NSF Mathematical Sciences Postdoctoral Research Fellow Conducted at U.C. Berkeley 1992-95. Host: Prof. Manuel Blum. Ph.D. in Computer Science, Massachusetts Institute of Technology, 1989-92. • Thesis titled: \Software Protection and Simulation on Oblivious RAMs", Ph.D. advisor: Prof. Silvio Micali. Final version appeared in Journal of ACM, 1996. Practical applications of thesis work appeared in U.S. Patent No.5,123,045. • Minor: \Management and Technology", M.I.T. Sloan School of Management. M.S. in Computer Science, Boston University, 1985-87. B.A. Magna Cum Laude in Mathematics, State University of New York at Buffalo, 1980-84. Department of Mathematics Graduation Honors: With highest distinction. Personal • U.S. citizen, naturalized in Boston, MA, 1986. Data Appointments UCLA Computer Science Department (2003 { present): Distinguished Professor of Computer Science. Recruited in 2003 as a Full Professor with Tenure. UCLA School of Engineering (2003 { present): Director, Center for Information and Computation Security. (See http://www.cs.ucla.edu/security/.) UCLA Department of Mathematics (2006 { present): Distinguished Professor of Mathematics (by courtesy). 1 Appointments Bell Communications Research (Bellcore) (cont.) (1999 { 2003): Senior Research Scientist; (1995 { 1999): Research Scientist, Mathematics and Cryptography Research Group, Applied Research. -
Derandomized Graph Products
Derandomized Graph Products Noga Alon ∗ Uriel Feigey Avi Wigderson z David Zuckermanx February 22, 2002 Abstract Berman and Schnitger [10] gave a randomized reduction from approximating MAX- SNP problems [24] within constant factors arbitrarily close to 1 to approximating clique within a factor of n (for some ). This reduction was further studied by Blum [11], who gave it the name randomized graph products. We show that this reduction can be made deterministic (derandomized), using random walks on expander graphs [1]. The main technical contribution of this paper is in lower bounding the probability that all steps of a random walk stay within a specified set of vertices of a graph. (Previous work was mainly concerned with upper bounding this probability.) This lower bound extends also to the case that different sets of vertices are specified for different time steps of the walk. 1 Introduction We present lower bounds on the probability that all steps of a random walk stay within a specified set of vertices of a graph. We then apply these lower bounds to amplify unapprox- imability results about certain NP-hard optimization problems. Our work was motivated by the problem of approximating the size of the maximum clique in graphs. This motivat- ing problem, which serves also as an example of how our lower bounds can be applied, is described in section 1.1. ∗Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv Uni- versity, Tel Aviv, Israel and AT & T Bell Labs, Murray Hill, NJ, 07974, USA. Email: [email protected]. -
Algebra & Number Theory
Algebra & Number Theory Volume 4 2010 No. 2 mathematical sciences publishers Algebra & Number Theory www.jant.org EDITORS MANAGING EDITOR EDITORIAL BOARD CHAIR Bjorn Poonen David Eisenbud Massachusetts Institute of Technology University of California Cambridge, USA Berkeley, USA BOARD OF EDITORS Georgia Benkart University of Wisconsin, Madison, USA Susan Montgomery University of Southern California, USA Dave Benson University of Aberdeen, Scotland Shigefumi Mori RIMS, Kyoto University, Japan Richard E. Borcherds University of California, Berkeley, USA Andrei Okounkov Princeton University, USA John H. Coates University of Cambridge, UK Raman Parimala Emory University, USA J-L. Colliot-Thel´ ene` CNRS, Universite´ Paris-Sud, France Victor Reiner University of Minnesota, USA Brian D. Conrad University of Michigan, USA Karl Rubin University of California, Irvine, USA Hel´ ene` Esnault Universitat¨ Duisburg-Essen, Germany Peter Sarnak Princeton University, USA Hubert Flenner Ruhr-Universitat,¨ Germany Michael Singer North Carolina State University, USA Edward Frenkel University of California, Berkeley, USA Ronald Solomon Ohio State University, USA Andrew Granville Universite´ de Montreal,´ Canada Vasudevan Srinivas Tata Inst. of Fund. Research, India Joseph Gubeladze San Francisco State University, USA J. Toby Stafford University of Michigan, USA Ehud Hrushovski Hebrew University, Israel Bernd Sturmfels University of California, Berkeley, USA Craig Huneke University of Kansas, USA Richard Taylor Harvard University, USA Mikhail Kapranov Yale -
Discrete Mathematics: Methods and Challenges 121 Ous Interesting Applications
ICM 2002 Vol. I 119–135 · · Discrete Mathematics: Methods and Challenges Noga Alon∗ Abstract Combinatorics is a fundamental mathematical discipline as well as an es- sential component of many mathematical areas, and its study has experienced an impressive growth in recent years. One of the main reasons for this growth is the tight connection between Discrete Mathematics and Theoretical Com- puter Science, and the rapid development of the latter. While in the past many of the basic combinatorial results were obtained mainly by ingenuity and detailed reasoning, the modern theory has grown out of this early stage, and often relies on deep, well developed tools. This is a survey of two of the main general techniques that played a crucial role in the development of mod- ern combinatorics; algebraic methods and probabilistic methods. Both will be illustrated by examples, focusing on the basic ideas and the connection to other areas. 2000 Mathematics Subject Classification: 05-02. Keywords and Phrases: Combinatorial nullstellensatz, Shannon capacity, Additive number theory, List coloring, The probabilistic method, Ramsey theory, Extremal graph theory. 1. Introduction The originators of the basic concepts of Discrete Mathematics, the mathemat- ics of finite structures, were the Hindus, who knew the formulas for the number of arXiv:math/0212390v1 [math.CO] 1 Dec 2002 permutations of a set of n elements, and for the number of subsets of cardinality k in a set of n elements, already in the sixth century. The beginning of Combinatorics as we know it today started with the work of Pascal and De Moivre in the 17th century, and continued in the 18th century with the seminal ideas of Euler in Graph Theory, with his work on partitions and their enumeration, and with his interest in latin squares. -
CV and Bibliography Karl Rubin Education 1981 Ph.D., Mathematics, Harvard University 1977 M.A., Mathematics, Harvard University 1976 A.B
Karl Rubin phone: 949-824-1645 Department of Mathematics fax: 508-374-0599 UC Irvine [email protected] Irvine, CA 92697-3875 http://www.math.uci.edu/~krubin CV and Bibliography Karl Rubin Education 1981 Ph.D., Mathematics, Harvard University 1977 M.A., Mathematics, Harvard University 1976 A.B. summa cum laude, Mathematics, Princeton University Employment 2019{ Distinguished Professor Emeritus, University of California Irvine 2004{2020 Thorp Professor of Mathematics, University of California Irvine 2013{2016 Chair, Department of Mathematics, UC Irvine 1997{2006 Professor, Stanford University 1996{1999 Distinguished University Professor, Ohio State University 1987{1996 Professor, Ohio State University 1988{1989 Professor, Columbia University 1984{1987 Assistant Professor, Ohio State University 1982{1983 Instructor, Princeton University Selected visiting positions Universit¨atErlangen-N¨urnberg Harvard University Institute for Advanced Study (Princeton) Institut des Hautes Etudes Scientifiques (Paris) Mathematical Sciences Research Institute (Berkeley) Max-Planck-Institut f¨urMathematik (Bonn) Selected honors and awards 2012 Fellow of the American Mathematical Society 1999 Humboldt-Forschungspreis (Humboldt Foundation Research Award) 1994 Guggenheim Fellowship 1992 AMS Cole Prize in Number Theory 1988 NSF Presidential Young Investigator Award 1987 Ohio State University Distinguished Scholar Award 1985 Sloan Fellowship 1981 NSF Postdoctoral Fellowship 1979 Harvard University Graduate School of Arts and Sciences Fellow 1976 NSF Graduate Fellowship -
Self-Sorting SSD: Producing Sorted Data Inside Active Ssds
Foreword This volume contains the papers presentedat the Third Israel Symposium on the Theory of Computing and Systems (ISTCS), held in Tel Aviv, Israel, on January 4-6, 1995. Fifty five papers were submitted in response to the Call for Papers, and twenty seven of them were selected for presentation. The selection was based on originality, quality and relevance to the field. The program committee is pleased with the overall quality of the acceptedpapers and, furthermore, feels that many of the papers not used were also of fine quality. The papers included in this proceedings are preliminary reports on recent research and it is expected that most of these papers will appear in a more complete and polished form in scientific journals. The proceedings also contains one invited paper by Pave1Pevzner and Michael Waterman. The program committee thanks our invited speakers,Robert J. Aumann, Wolfgang Paul, Abe Peled, and Avi Wigderson, for contributing their time and knowledge. We also wish to thank all who submitted papers for consideration, as well as the many colleagues who contributed to the evaluation of the submitted papers. The latter include: Noga Alon Alon Itai Eric Schenk Hagit Attiya Roni Kay Baruch Schieber Yossi Azar Evsey Kosman Assaf Schuster Ayal Bar-David Ami Litman Nir Shavit Reuven Bar-Yehuda Johan Makowsky Richard Statman Shai Ben-David Yishay Mansour Ray Strong Allan Borodin Alain Mayer Eli Upfal Dorit Dor Mike Molloy Moshe Vardi Alon Efrat Yoram Moses Orli Waarts Jack Feldman Dalit Naor Ed Wimmers Nissim Francez Seffl Naor Shmuel Zaks Nita Goyal Noam Nisan Uri Zwick Vassos Hadzilacos Yuri Rabinovich Johan Hastad Giinter Rote The work of the program committee has been facilitated thanks to software developed by Rob Schapire and FOCS/STOC program chairs. -
Algebraic Tori in Cryptography
Fields Institute Communications Volume 00, 0000 Algebraic tori in cryptography Karl Rubin Department of Mathematics, Stanford University, Stanford CA 94305, USA [email protected] Alice Silverberg Department of Mathematics, Ohio State University, Columbus OH 43210, USA [email protected] Abstract. We give a mathematical interpretation in terms of algebraic tori of the LUC and XTR cryptosystems and a conjectured generaliza- tion. 1 Introduction In a series of papers culminating in [7], Edouard Lucas introduced and explored the properties of certain recurrent sequences that became known as Lucas functions. Since then, generalizations and applications of Lucas functions have been studied (see [17, 18, 20]), and public key discrete log based cryptosystems (such as LUC) have been based on them (see [8, 12, 13, 19, 1]). The Lucas functions arise when studying quadratic field extensions. Cryptographic applications of generalizations to cubic and sextic field extensions are given in [4] and [3, 6], respectively. The cryptosystem in [6] is called XTR. An approach for constructing a generalization of these cryptosystems to the case of degree 30 extensions is suggested in [3, 2]. The × idea of these cryptosystems is to represent certain elements of Fqn (for n = 2, 6, and 30, respectively) using only ϕ(n) elements of Fq, and do a variant of the Diffie- Hellman key exchange protocol. For XTR and LUC, traces are used to represent the elements. In [2], symmetric functions are proposed in place of the trace (which is the first symmetric function). In [11] we use algebraic tori to construct public key cryptosystems. These × systems are based on the discrete log problem in a subgroup of Fqn in which the elements can be represented by only ϕ(n) elements of Fq.