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Kloosterman Sheaves for Reductive Groups
Annals of Mathematics 177 (2013), 241{310 http://dx.doi.org/10.4007/annals.2013.177.1.5 Kloosterman sheaves for reductive groups By Jochen Heinloth, Bao-Chau^ Ngo,^ and Zhiwei Yun Abstract 1 Deligne constructed a remarkable local system on P − f0; 1g attached to a family of Kloosterman sums. Katz calculated its monodromy and asked whether there are Kloosterman sheaves for general reductive groups and which automorphic forms should be attached to these local systems under the Langlands correspondence. Motivated by work of Gross and Frenkel-Gross we find an explicit family of such automorphic forms and even a simple family of automorphic sheaves in the framework of the geometric Langlands program. We use these auto- morphic sheaves to construct `-adic Kloosterman sheaves for any reductive group in a uniform way, and describe the local and global monodromy of these Kloosterman sheaves. In particular, they give motivic Galois rep- resentations with exceptional monodromy groups G2;F4;E7 and E8. This also gives an example of the geometric Langlands correspondence with wild ramification for any reductive group. Contents Introduction 242 0.1. Review of classical Kloosterman sums and Kloosterman sheaves 242 0.2. Motivation and goal of the paper 244 0.3. Method of construction 246 0.4. Properties of Kloosterman sheaves 247 0.5. Open problems 248 0.6. Organization of the paper 248 1. Structural groups 249 P1 1.1. Quasi-split group schemes over nf0;1g 249 1.2. Level structures at 0 and 1 250 1.3. Affine generic characters 252 1.4. Principal bundles 252 2. -
Hodge Theory in Combinatorics
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 55, Number 1, January 2018, Pages 57–80 http://dx.doi.org/10.1090/bull/1599 Article electronically published on September 11, 2017 HODGE THEORY IN COMBINATORICS MATTHEW BAKER Abstract. If G is a finite graph, a proper coloring of G is a way to color the vertices of the graph using n colors so that no two vertices connected by an edge have the same color. (The celebrated four-color theorem asserts that if G is planar, then there is at least one proper coloring of G with four colors.) By a classical result of Birkhoff, the number of proper colorings of G with n colors is a polynomial in n, called the chromatic polynomial of G.Read conjectured in 1968 that for any graph G, the sequence of absolute values of coefficients of the chromatic polynomial is unimodal: it goes up, hits a peak, and then goes down. Read’s conjecture was proved by June Huh in a 2012 paper making heavy use of methods from algebraic geometry. Huh’s result was subsequently refined and generalized by Huh and Katz (also in 2012), again using substantial doses of algebraic geometry. Both papers in fact establish log-concavity of the coefficients, which is stronger than unimodality. The breakthroughs of the Huh and Huh–Katz papers left open the more general Rota–Welsh conjecture, where graphs are generalized to (not neces- sarily representable) matroids, and the chromatic polynomial of a graph is replaced by the characteristic polynomial of a matroid. The Huh and Huh– Katz techniques are not applicable in this level of generality, since there is no underlying algebraic geometry to which to relate the problem. -
Notices of the American Mathematical Society Karim Adiprasito Karim Adiprasito January 2017 Karim Adiprasito FEATURES
THE AUTHORS THE AUTHORS THE AUTHORS Notices of the American Mathematical Society Karim Adiprasito Karim Adiprasito January 2017 Karim Adiprasito FEATURES June Huh June Huh 8 32 2626 June Huh JMM 2017 Lecture Sampler The Graduate Student Hodge Theory of Barry Simon, Alice Silverberg, Lisa Section Matroids Jeffrey, Gigliola Staffilani, Anna Interview with Arthur Benjamin Karim Adiprasito, June Huh, and Wienhard, Donald St. P. Richards, by Alexander Diaz-Lopez Eric Katz Tobias Holck Colding, Wilfrid Gangbo, and Ingrid Daubechies WHAT IS?...a Complex Symmetric Operator by Stephan Ramon Garcia Graduate Student Blog Enjoy the Joint Mathematics Meetings from near or afar through our lecture sampler and our feature on Hodge Theory, the subject of a JMM Current Events Bulletin Lecture. This month's Graduate Student Section includes an interview Eric Katz Eric Katz with Arthur Benjamin, "WHAT IS...a Complex Symmetric Operator?" and the latest "Lego Graduate Student" craze. Eric Katz The BackPage announces the October caption contest winner and presents a new caption contest for January. As always, comments are welcome on our webpage. And if you're not already a member of the AMS, I hope you'll consider joining, not just because you will receive each issue of these Notices but because you believe in our goal of supporting and opening up mathematics. —Frank Morgan, Editor-in-Chief ALSO IN THIS ISSUE 40 My Year as an AMS Congressional Fellow Anthony J. Macula 44 AMS Trjitzinsky Awards Program Celebrates Its Twenty-Fifth Year 47 Origins of Mathematical Words A Review by Andrew I. Dale 54 Aleksander (Olek) Pelczynski, 1932–2012 Edited by Joe Diestel 73 2017 Annual Meeting of the American Association for the Advancement of Science Matroids (top) and Dispersion in a Prism (bottom), from this month's contributed features; see pages 8 and 26. -
Mathematics People
NEWS Mathematics People faculty at the University of Washington, where she will Toro Named Director retain tenure during her term at MSRI. Her honors and awards include the Landolt Distin- of MSRI guished Graduate Mentor Award of the University of Wash- Tatiana Toro of the University of ington and the 2020 Blackwell–Tapia Prize. She has been Washington has been appointed as the recipient of an NSF Postdoctoral Research Fellowship, director of the Mathematical Sci- a Sloan Research Fellowship, a Guggenheim Fellowship, ences Research Institute (MSRI). She and two Simons Foundation Fellowships. She was elected a will serve a five-year term beginning Fellow of the AMS in 2016. Her service to AMS includes as August 1, 2022. an editor of Proceedings of the American Mathematical Society Toro’s primary research interests (2009–2015) and of Transactions and Memoirs of the Ameri- lie in the interface of partial differ- can Mathematical Society (2015–2019); as an elected mem- ential equations, harmonic analysis, ber of the AMS Editorial Boards Committee (2016–2019) and of the AMS Nominating Committee; and as a member Tatiana Toro calculus of variations, and geometric measure theory. According to the of the AMS Colloquium Lecture Committee. She previously MSRI announcement, “the main premise of Toro’s work served on the board of directors of the Pacific Institute is that under the right lens, objects that at first glance for the Mathematical Sciences at the University of British might appear to be very irregular do exhibit quantifiable Columbia. She is currently a trustee of the Institute of Pure regular characteristics. -
C Copyright 2007 Atri Rudra
c Copyright 2007 Atri Rudra List Decoding and Property Testing of Error Correcting Codes Atri Rudra A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Washington 2007 Program Authorized to Offer Degree: Computer Science and Engineering University of Washington Graduate School This is to certify that I have examined this copy of a doctoral dissertation by Atri Rudra and have found that it is complete and satisfactory in all respects, and that any and all revisions required by the final examining committee have been made. Chair of the Supervisory Committee: Venkatesan Guruswami Reading Committee: Paul Beame Venkatesan Guruswami Dan Suciu Date: In presenting this dissertation in partial fulfillment of the requirements for the doctoral degree at the University of Washington, I agree that the Library shall make its copies freely available for inspection. I further agree that extensive copying of this dissertation is allowable only for scholarly purposes, consistent with “fair use” as prescribed in the U.S. Copyright Law. Requests for copying or reproduction of this dissertation may be referred to Proquest Information and Learning, 300 North Zeeb Road, Ann Arbor, MI 48106-1346, 1-800-521-0600, to whom the author has granted “the right to reproduce and sell (a) copies of the manuscript in microform and/or (b) printed copies of the manuscript made from microform.” Signature Date University of Washington Abstract List Decoding and Property Testing of Error Correcting Codes Atri Rudra Chair of the Supervisory Committee: Associate Professor Venkatesan Guruswami Department of Computer Science and Engineering Error correcting codes systematically introduce redundancy into data so that the original in- formation can be recovered when parts of the redundant data are corrupted. -
Focm 2017 Foundations of Computational Mathematics Barcelona, July 10Th-19Th, 2017 Organized in Partnership With
FoCM 2017 Foundations of Computational Mathematics Barcelona, July 10th-19th, 2017 http://www.ub.edu/focm2017 Organized in partnership with Workshops Approximation Theory Computational Algebraic Geometry Computational Dynamics Computational Harmonic Analysis and Compressive Sensing Computational Mathematical Biology with emphasis on the Genome Computational Number Theory Computational Geometry and Topology Continuous Optimization Foundations of Numerical PDEs Geometric Integration and Computational Mechanics Graph Theory and Combinatorics Information-Based Complexity Learning Theory Plenary Speakers Mathematical Foundations of Data Assimilation and Inverse Problems Multiresolution and Adaptivity in Numerical PDEs Numerical Linear Algebra Karim Adiprasito Random Matrices Jean-David Benamou Real-Number Complexity Alexei Borodin Special Functions and Orthogonal Polynomials Mireille Bousquet-Mélou Stochastic Computation Symbolic Analysis Mark Braverman Claudio Canuto Martin Hairer Pierre Lairez Monique Laurent Melvin Leok Lek-Heng Lim Gábor Lugosi Bruno Salvy Sylvia Serfaty Steve Smale Andrew Stuart Joel Tropp Sponsors Shmuel Weinberger 2 FoCM 2017 Foundations of Computational Mathematics Barcelona, July 10th{19th, 2017 Books of abstracts 4 FoCM 2017 Contents Presentation . .7 Governance of FoCM . .9 Local Organizing Committee . .9 Administrative and logistic support . .9 Technical support . 10 Volunteers . 10 Workshops Committee . 10 Plenary Speakers Committee . 10 Smale Prize Committee . 11 Funding Committee . 11 Plenary talks . 13 Workshops . 21 A1 { Approximation Theory Organizers: Albert Cohen { Ron Devore { Peter Binev . 21 A2 { Computational Algebraic Geometry Organizers: Marta Casanellas { Agnes Szanto { Thorsten Theobald . 36 A3 { Computational Number Theory Organizers: Christophe Ritzenhaler { Enric Nart { Tanja Lange . 50 A4 { Computational Geometry and Topology Organizers: Joel Hass { Herbert Edelsbrunner { Gunnar Carlsson . 56 A5 { Geometric Integration and Computational Mechanics Organizers: Fernando Casas { Elena Celledoni { David Martin de Diego . -
Regularity of the Singular Set in a Two-Phase Problem for Harmonic Measure with Holder¨ Data
REGULARITY OF THE SINGULAR SET IN A TWO-PHASE PROBLEM FOR HARMONIC MEASURE WITH HOLDER¨ DATA MATTHEW BADGER, MAX ENGELSTEIN, AND TATIANA TORO Abstract. In non-variational two-phase free boundary problems for harmonic measure, we examine how the relationship between the interior and exterior harmonic measures of a domain Ω ⊂ Rn influences the geometry of its boundary. This type of free boundary problem was initially studied by Kenig and Toro in 2006 and was further examined in a series of separate and joint investigations by several authors. The focus of the present paper is on the singular set in the free boundary, where the boundary looks infinitesimally like zero sets of homogeneous harmonic polynomials of degree at least 2. We prove that if the Radon-Nikodym derivative of the exterior harmonic measure with respect to the interior harmonic measure has a H¨oldercontinuous logarithm, then the free boundary admits unique geometric blowups at every singular point and the singular set can be covered by countably many C1,β submanifolds of dimension at most n − 3. This result is partly obtained by adapting tools such as Garofalo and Petrosyan's Weiss type monotonicity formula and an epiperimetric inequality for harmonic functions from the variational to the non-variational setting. Contents 1. Introduction 2 2. Blowups of harmonic measure on NTA domains 8 3. Start of the proof of the main theorems 14 4. Almost monotonicity via Almgren's frequency functional 16 5. Bounding the growth of the Weiss type functional 20 6. C1,β convergence to the blowup at singular points 23 7. -
Ashwin Nayak Project Suggestions
C&O 739-2 Information Theory and Applications University of Waterloo, Winter 2020 Instructor: Ashwin Nayak Project Suggestions The following is a partial list of research articles related to the theme of the course. For related papers and other ideas, please talk to me. Please confirm the topic and the scope of the project with me. Applications of entropic quantities. • Sorting a partially ordered sequence: Jeff Kahn and Jeong Han Kim. Entropy and sorting. Journal of Computer and System Sciences, 51(3):390{399, December 1995. • Monotone functions: Jeff Kahn. Entropy, independent sets and antichains: a new approach to Dedekind's problem. Proceedings of the American Mathematical Society, 130(2):371{378, 2001. • Predecessor-search: Pranab Sen and Srinivasan Venkatesh. Lower bounds for predecessor searching in the cell probe model. Journal of Computer and System Sciences, 74(3):364{385, 2008. • More data-structures: Mihai Patrascu and Erik Demaine. Logarithmic Lower Bounds in the Cell- Probe Model. SIAM Journal on Computing, 35(4):932{963, 2006. • Set Disjointness and extended formulations: Mark Braverman and Ankur Moitra. An information complexity approach to extended formulations. In Proceedings of the forty-fifth Annual ACM Sym- posium on Theory of Computing, pp. 161{170, 2013. • Explicit polytopes with high extension complexity: Mika G¨o¨os,Rahul Jain, and Thomas Watson. Extension Complexity of Independent Set Polytopes. SIAM Journal on Computing, 47(1):241{269, 2018. Compression of interactive protocols. • Compression, direct sum: Boaz Barak, Mark Braverman, Xi Chen, Anup Rao. How to Compress Interactive Communication. SIAM Journal on Computing, 42(3):1327{1363, 2013. -
Issue 118 ISSN 1027-488X
NEWSLETTER OF THE EUROPEAN MATHEMATICAL SOCIETY S E European M M Mathematical E S Society December 2020 Issue 118 ISSN 1027-488X Obituary Sir Vaughan Jones Interviews Hillel Furstenberg Gregory Margulis Discussion Women in Editorial Boards Books published by the Individual members of the EMS, member S societies or societies with a reciprocity agree- E European ment (such as the American, Australian and M M Mathematical Canadian Mathematical Societies) are entitled to a discount of 20% on any book purchases, if E S Society ordered directly at the EMS Publishing House. Recent books in the EMS Monographs in Mathematics series Massimiliano Berti (SISSA, Trieste, Italy) and Philippe Bolle (Avignon Université, France) Quasi-Periodic Solutions of Nonlinear Wave Equations on the d-Dimensional Torus 978-3-03719-211-5. 2020. 374 pages. Hardcover. 16.5 x 23.5 cm. 69.00 Euro Many partial differential equations (PDEs) arising in physics, such as the nonlinear wave equation and the Schrödinger equation, can be viewed as infinite-dimensional Hamiltonian systems. In the last thirty years, several existence results of time quasi-periodic solutions have been proved adopting a “dynamical systems” point of view. Most of them deal with equations in one space dimension, whereas for multidimensional PDEs a satisfactory picture is still under construction. An updated introduction to the now rich subject of KAM theory for PDEs is provided in the first part of this research monograph. We then focus on the nonlinear wave equation, endowed with periodic boundary conditions. The main result of the monograph proves the bifurcation of small amplitude finite-dimensional invariant tori for this equation, in any space dimension. -
The Rota-Welsh Conjecture for Representable Matroids
THE ROTA-WELSH CONJECTURE FOR REPRESENTABLE MATROIDS ETHAN WILLIAMS Abstract. The Rota-Welsh Conjecture claims that the coefficients of specific polynomials associated to certain mathematical objects called matroids form a log concave sequence. This paper will define matroids and the associated polynomials referenced in the conjecture. It will then prove the conjecture for the special class of matroids representable over the complex field and explain how the strategies for this special case can be used to tackle parts of the general case of the conjecture. Contents 1. Introduction 1 2. Unimodality and Log Concavity 2 3. Algebraic Varieties 2 4. Matroids 3 5. Hyperplane Arrangements and Complements 7 6. Representable Matroids 9 7. Chow Rings 10 8. Proving the Rota-Welsh Conjecture for Representable Matroids 13 Acknowledgments 18 References 18 1. Introduction A matroid is an object that abstractly captures combinatorial information related to independence. Matroids can be constructed from sets of vectors, graphs, and many other situations or exist abstractly on their own unrelated to any underlying structure. Every matroid has an associated characteristic polynomial associated with it, the coefficients of which were conjectured to form a log concave sequence in the Rota-Welsh Conjecture. Surprisingly, for a special class of matroids the Rota-Welsh Conjecture can be reformulated into a problem in intersection theory, a field that had already been studied for some time before the Rota-Welsh Conjecture and has many classical results that can be utilized to prove the conjecture. More recently, Adiprasito, Huh, and Katz were able to prove that the main result gained from the special case's connection to intersection theory still hold when we are not in the special case. -
Arxiv:1711.02045V3 [Math.AG] 11 Sep 2018
CONVEXITY OF COMPLEMENTS OF TROPICAL VARIETIES, AND APPROXIMATIONS OF CURRENTS KARIM ADIPRASITO AND FARHAD BABAEE ABSTRACT. The goal of this note is to affirm a local version of conjecture of Nisse–Sottile [NS16] on higher convexity of complements of tropical varieties, while providing a family of counter- examples for the global Nisse–Sottle conjecture in any codimension and dimension higher than 1. Moreover, it is shown that, surprisingly, this family also provides a family of counter-examples for the generalized Hodge conjecture for positive currents in these dimensions, and gives rise to further approximability obstruction. 1. INTRODUCTION Following a classical theorem of Aumann [Aum36], compact convex sets can be characterized using a homological condition on section by linear subspaces of a fixed dimension. Gromov [Gro91] intuited from this and subsequent results that such homological conditions for non-compact sets should be thought of as lying on the same spectrum as the Lefschetz sec- tion property for smooth compact varieties. Hence, a natural notion of global k-convexity, for short k-convexity, is that any nontrivial homology (k − 1)-dimensional cycle in the section of the given set with an affine k-plane extends to a nontrivial global homology cycle. He related this to curvature properties of submanifolds, in particular providing a necessary and sufficient condition for a smooth submanifold of Euclidean n-dimensional space of dimension k to be (n − k)-convex. The advent of the study of algebraic aspects of tropical and log-geometry brought with it a desire to sink also Lefschetz’ topological harpoon into tropical varieties and amoebas and their complements. -
Curriculum Vitae Wei Zhang 09/2016
Curriculum Vitae Wei Zhang 09/2016 Contact Department of Mathematics Phone: +1-212-854 9919 Information Columbia University Fax: +1-212-854-8962 MC 4423, 2990 Broadway URL: www.math.columbia.edu/∼wzhang New York, NY, 10027 E-mail: [email protected] Birth 1981 year/place Sichuan, China. Citizenship P.R. China, and US permanent resident. Research Number theory, automorphic forms and related area in algebraic geometry. Interests Education 09/2000-06/2004 B.S., Mathematics, Peking University, China. 08/2004-06/2009 Ph.D, Mathematics, Columbia University. Advisor: Shouwu Zhang Employment 07/2015{Now, Professor, Columbia University. 01/2014-06/2015 Associate professor, Columbia University. 07/2011-12/2013 Assistant professor, Columbia University. 07/2010-06/2011 Benjamin Peirce Fellow, Harvard University. 07/2009-06/2010 Postdoctoral fellow, Harvard University. Award,Honor The 2010 SASTRA Ramanujan Prize. 2013 Alfred P. Sloan Research Fellowship. 2016 Morningside Gold Medal of Mathematics of ICCM. Grants 07/2010 - 06/2013: PI, NSF Grant DMS 1001631, 1204365. 07/2013 - 06/2016: PI, NSF Grant DMS 1301848. 07/2016 - 06/2019: PI, NSF Grant DMS 1601144. 1 of 4 Selected recent invited lectures Automorphic Forms, Galois Representations and L-functions, Rio de Janeiro, July 23-27, 2018 The 30th Journ´eesArithm´etiques,July 3-7, 2017, Caen, France. Workshop, June 11-17, 2017 Weizmann, Israel. Arithmetic geometry, June 5-9, 2017, BICMR, Beijing Co-organizer (with Z. Yun) Arbeitsgemeinschaft: Higher Gross Zagier Formulas, 2 Apr - 8 Apr 2017, Oberwolfach AIM Dec. 2016, SQUARE, March 2017 Plenary speaker, ICCM Beijing Aug. 2016 Number theory conference, MCM, Beijing, July 31-Aug 4, 2016 Luminy May 23-27 2016 AMS Sectional Meeting Invited Addresses, Fall Eastern Sectional Meeting, Nov.