2021 Leroy P. Steele Prizes
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Motives, Volume 55.2
http://dx.doi.org/10.1090/pspum/055.2 Recent Titles in This Series 55 Uwe Jannsen, Steven Kleiman, and Jean-Pierre Serre, editors, Motives (University of Washington, Seattle, July/August 1991) 54 Robert Greene and S. T. Yau, editors, Differential geometry (University of California, Los Angeles, July 1990) 53 James A. Carlson, C. Herbert Clemens, and David R. Morrison, editors, Complex geometry and Lie theory (Sundance, Utah, May 1989) 52 Eric Bedford, John P. D'Angelo, Robert £. Greene, and Steven G. Krantz, editors, Several complex variables and complex geometry (University of California, Santa Cruz, July 1989) 51 William B. Arveson and Ronald G. Douglas, editors, Operator theory/operator algebras and applications (University of New Hampshire, July 1988) 50 James Glimm, John Impagliazzo, and Isadore Singer, editors, The legacy of John von Neumann (Hofstra University, Hempstead, New York, May/June 1988) 49 Robert C. Gunning and Leon Ehrenpreis, editors, Theta functions - Bowdoin 1987 (Bowdoin College, Brunswick, Maine, July 1987) 48 R. O. Wells, Jr., editor, The mathematical heritage of Hermann Weyl (Duke University, Durham, May 1987) 47 Paul Fong, editor, The Areata conference on representations of finite groups (Humboldt State University, Areata, California, July 1986) 46 Spencer J. Bloch, editor, Algebraic geometry - Bowdoin 1985 (Bowdoin College, Brunswick, Maine, July 1985) 45 Felix E. Browder, editor, Nonlinear functional analysis and its applications (University of California, Berkeley, July 1983) 44 William K. Allard and Frederick J. Almgren, Jr., editors, Geometric measure theory and the calculus of variations (Humboldt State University, Areata, California, July/August 1984) 43 Francois Treves, editor, Pseudodifferential operators and applications (University of Notre Dame, Notre Dame, Indiana, April 1984) 42 Anil Nerode and Richard A. -
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]. -
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]. -
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. -
Summer School Special Values of L-Functions
Summer school Special values of L-functions Contents 1. Aim of the summer school 1 2. Abstract of courses 1 A. L-functions and Galois cohomology 1 B. Regulators and motives 2 C. Conjectures on special values 2 D. Modular aspects 2 References 3 1. Aim of the summer school The summer school will be aimed at Ph. D. students and young researchers. The aim is to give an overview of Beilinson's conjectures and its refinement by Bloch and Kato on special values of L functions, relying on the abundant literature on the subject. More particularly, we will focus on the classical computations which justify the conjectures as well as on the motivic theory which constitutes the theoretical basis. 2. Abstract of courses A. L-functions and Galois cohomology. (2 talks, J. Johnson-Leung) The aim of this course is to introduce the material on L-functions and Galois co- homology which is needed to formulate the Bloch-Kato conjecture. More precisely, it will cover the following topics : • Definition of L-functions associated to (´etalerealizations of) motives. Pos- sibly, it could also cover equivariant L-functions (i.e. associated to motives with commutative coefficients). • Definition of Fontaine's period rings. • Cohomology of local Galois representations : definition of H1 and its sub- 1 group Hf . • Definition of local Tamagawa numbers. References. [Ser94], [BK90, x3-4], [FPR94, Chap. I, x3-4]. Pre-requisite. Etale´ cohomology. 1 2 B. Regulators and motives. (3 talks, J. Wildeshaus) The first aim of this course is to present Beilinson's regulator, from rational motivic cohomology to Deligne cohomology. -
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. -
Problems and Results in Extremal Combinatorics, Part I
Problems and results in Extremal Combinatorics, Part I Noga Alon ∗ Abstract Extremal Combinatorics is an area in Discrete Mathematics that has developed spectacularly during the last decades. This paper contains a collection of problems and results in the area, including solutions or partial solutions to open problems suggested by various researchers in Extremal Graph Theory, Extremal Finite Set Theory and Combinatorial Geometry. This is not meant to be a comprehensive survey of the area, it is merely a collection of various extremal problems, which are hopefully interesting. The choice of the problems is inevitably somewhat biased, and as the title of the paper suggests I hope to write a related paper in the future. Each section of this paper is essentially self contained, and can be read separately. 1 Introduction Extremal Combinatorics deals with the problem of determining or estimating the maximum or min- imum possible cardinality of a collection of finite objects that satisfies certain requirements. Such problems are often related to other areas including Computer Science, Information Theory, Number Theory and Geometry. This branch of Combinatorics has developed spectacularly over the last few decades, see, e.g., [10], [29], and their many references. This paper contains a collection of problems and results in the area, including solutions or partial solutions to open problems suggested by various researchers in Extremal Graph Theory, Extremal Finite Set Theory and Combinatorial Geometry. This is not meant to be a comprehensive survey of the area, but rather a collection of several extremal problems, which are hopefully interesting. The techniques used include combinatorial, probabilistic and algebraic tools. -
Motives: an Introductory Survey for Physicists
MOTIVES: AN INTRODUCTORY SURVEY FOR PHYSICISTS ABHIJNAN REJ (WITH AN APPENDIX BY MATILDE MARCOLLI) Abstract. We survey certain accessible aspects of Grothendieck's theory of motives in arith- metic algebraic geometry for mathematical physicists, focussing on areas that have recently found applications in quantum field theory. An appendix (by Matilde Marcolli) sketches further connections between motivic theory and theoretical physics. post hoc, ergo ante hoc { Umberto Eco, Interpretation and overinterpretation Contents 1. Introduction 1 2. The Grothendieck ring 2 3. The Tannakian formalism 6 4. Weil cohomology 12 5. Classical motives 15 6. Mixed motives 18 7. Motivic measures and zeta functions 22 8. Appendix: Motivic ideas in physics (by M.Marcolli) 24 References 29 1. Introduction This survey paper is based on lectures given by the author at Boston University, the Max Planck Institute in Bonn, at Durham university and at the Indian Statistical Institute and the SN Bose National Center for the Basic Sciences in Kolkata and the Indian Institute of Technology in Mumbai. The purpose of these introductory notes are to familiarize an audience of physicists with some of the algebraic and algebro-geometric background upon which Grothendieck's theory of motives of algebraic varieties relies. There have been many recent developments in the interactions between high energy physics and motives, mostly within the framework of perturbative quantum field theory and the evaluation of Feynman diagrams as periods of algebraic varieties, though motives are beginning to play an important role in other branched of theoretical physics, such as string theory, especially through the recent interactions with the Langlands program, and through the theory of BPS states. -
Arxiv:2012.06421V2 [Cs.LG] 21 Jul 2021
When is Memorization of Irrelevant Training Data Necessary for High-Accuracy Learning? Gavin Brown∗ Mark Bun∗ Vitaly Feldman† Adam Smith∗ Kunal Talwar† July 23, 2021 Abstract Modern machine learning models are complex and frequently encode surprising amounts of information about individual inputs. In extreme cases, complex models appear to memorize entire input examples, including seemingly irrelevant information (social secu- rity numbers from text, for example). In this paper, we aim to understand whether this sort of memorization is necessary for accurate learning. We describe natural prediction problems in which every sufficiently accurate training algorithm must encode, in the pre- diction model, essentially all the information about a large subset of its training examples. This remains true even when the examples are high-dimensional and have entropy much higher than the sample size, and even when most of that information is ultimately irrel- evant to the task at hand. Further, our results do not depend on the training algorithm or the class of models used for learning. Our problems are simple and fairly natural variants of the next-symbol prediction and the cluster labeling tasks. These tasks can be seen as abstractions of text- and image- related prediction problems. To establish our results, we reduce from a family of one-way communication problems for which we prove new information complexity lower bounds. Additionally, we present synthetic-data experiments demonstrating successful attacks on logistic regression and neural network classifiers. arXiv:2012.06421v2 [cs.LG] 21 Jul 2021 ∗Computer Science Department, Boston University. {grbrown,mbun,ads22}@bu.edu. GB and AS are supported in part by NSF award CCF-1763786 as well as a Sloan Foundation research award. -
The Shannon Capacity of a Union
The Shannon Capacity of a union Noga Alon ∗ To the memory of Paul Erd}os Abstract For an undirected graph G = (V; E), let Gn denote the graph whose vertex set is V n in which two distinct vertices (u1; u2; : : : ; un) and (v1; v2; : : : ; vn) are adjacent iff for all i between 1 and n 1=n n either ui = vi or uivi E. The Shannon capacity c(G) of G is the limit limn (α(G )) , 2 !1 where α(Gn) is the maximum size of an independent set of vertices in Gn. We show that there are graphs G and H such that the Shannon capacity of their disjoint union is (much) bigger than the sum of their capacities. This disproves a conjecture of Shannon raised in 1956. 1 Introduction For an undirected graph G = (V; E), let Gn denote the graph whose vertex set is V n in which two distinct vertices (u1; u2; : : : ; un) and (v1; v2; : : : ; vn) are adjacent iff for all i between 1 and n either n 1=n n ui = vi or uivi E. The Shannon capacity c(G) of G is the limit limn (α(G )) , where α(G ) is 2 !1 the maximum size of an independent set of vertices in Gn. This limit exists, by super-multiplicativity, and it is always at least α(G). (It is worth noting that it is sometimes customary to call log c(G) the Shannon capacity of G, but we prefer to use here the above definition, following Lov´asz[12].) The study of this parameter was introduced by Shannon in [14], motivated by a question in Information Theory.