Orbit Parametrizations for K3 Surfaces
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Program of the Sessions San Diego, California, January 9–12, 2013
Program of the Sessions San Diego, California, January 9–12, 2013 AMS Short Course on Random Matrices, Part Monday, January 7 I MAA Short Course on Conceptual Climate Models, Part I 9:00 AM –3:45PM Room 4, Upper Level, San Diego Convention Center 8:30 AM –5:30PM Room 5B, Upper Level, San Diego Convention Center Organizer: Van Vu,YaleUniversity Organizers: Esther Widiasih,University of Arizona 8:00AM Registration outside Room 5A, SDCC Mary Lou Zeeman,Bowdoin upper level. College 9:00AM Random Matrices: The Universality James Walsh, Oberlin (5) phenomenon for Wigner ensemble. College Preliminary report. 7:30AM Registration outside Room 5A, SDCC Terence Tao, University of California Los upper level. Angles 8:30AM Zero-dimensional energy balance models. 10:45AM Universality of random matrices and (1) Hans Kaper, Georgetown University (6) Dyson Brownian Motion. Preliminary 10:30AM Hands-on Session: Dynamics of energy report. (2) balance models, I. Laszlo Erdos, LMU, Munich Anna Barry*, Institute for Math and Its Applications, and Samantha 2:30PM Free probability and Random matrices. Oestreicher*, University of Minnesota (7) Preliminary report. Alice Guionnet, Massachusetts Institute 2:00PM One-dimensional energy balance models. of Technology (3) Hans Kaper, Georgetown University 4:00PM Hands-on Session: Dynamics of energy NSF-EHR Grant Proposal Writing Workshop (4) balance models, II. Anna Barry*, Institute for Math and Its Applications, and Samantha 3:00 PM –6:00PM Marina Ballroom Oestreicher*, University of Minnesota F, 3rd Floor, Marriott The time limit for each AMS contributed paper in the sessions meeting will be found in Volume 34, Issue 1 of Abstracts is ten minutes. -
Proceedings of the Conference on Promoting Undergraduate Research in Mathematics Proceedings of the Conference on Promoting Undergraduate Research in Mathematics
In the span of twenty years, research in Proceedings mathematics by undergraduates has gone from rare to commonplace. To examine Promoting Undergraduate Research in Mathematics Gallian, Editor of the and promote this trend, the AMS and Proceedings NSA sponsored a three-day conference Conference in September 2006 to bring together of the mathematicians who were involving on Promoting undergraduates across the country in Conference on mathematics research to exchange ideas, Undergraduate discuss issues of common concern, establish contacts, and gather information Research in that would be of value to others interested Promoting in engaging undergraduates in mathematics Mathematics research. The conference featured plenary talks, panel discussions, and small group sessions. The topics included summer Undergraduate research programs, academic year research Joseph A. Gallian opportunities, diversity issues, assessment Editor methods, and perspectives from alumni Research in of research programs. This volume is the proceedings of that conference. Mathematics Joseph A. Gallian Editor AMS on the Web PURM/GALLIAN2 www.ams.org AMS 472 pages • Spine Width: 7/8 inch • Two Color Cover: This plate PMS 4515 (Beige) This plate PMS 323 (Teal) Proceedings of the Conference on Promoting Undergraduate Research in Mathematics Proceedings of the Conference on Promoting Undergraduate Research in Mathematics Joseph A. Gallian Editor American Mathematical Society Providence, Rhode Island Cover photography courtesy of Cindy Wyels, CSU Channel Islands, Camarillo, CA, and Darren Narayan, Rochester Institute of Technology, Rochester, NY. This material is based upon work supported by National Security Agency under grant H98230-06-1-0095. ISBN-13: 978-0-8218-4321-5 ISBN-10: 0-8218-4321-4 Copyright c 2007 American Mathematical Society Printed in the United States of America. -
Coregular Spaces and Genus One Curves
Cambridge Journal of Mathematics Volume 4, Number 1, 1–119, 2016 Coregular spaces and genus one curves Manjul Bhargava∗ and Wei Ho† A coregular space is a representation of an algebraic group for which the ring of polynomial invariants is freely generated. We show that the orbits of many coregular irreducible representations of algebraic groups where the number of generating invariants is at least two, over a (not necessarily algebraically closed) field k, correspond to genus one curves over k together with line bundles, vector bundles, and/or points on their Jacobian curves. In particu- lar, we give explicit descriptions of certain moduli spaces of genus one curves with extra structure as quotients by algebraic groups of open subsets of affine spaces. In most cases, we also describe how the generators of the in- variant rings are geometrically manifested, often as coefficients for the Jacobian families of elliptic curves for the genus one curves. We also show how many of the correspondences, including their proofs, are special cases of two general constructions related to Hermitian matrices and cubic Jordan algebras. In forthcoming work, we use these orbit parametrizations to determine the average sizes of Selmer groups for various families of elliptic curves. 1 Introduction 3 2 Main results I: Genus one curves and 2×2×2×2 hypercubes 9 2.1 Preliminaries on 2 × 2 × 2 cubes 10 2.2 On 2 × 2 × 2 × 2 hypercubes 11 2.3 Genus one curves from hypercubes 13 2.4 Orbit classification of 2 × 2 × 2 × 2 hypercubes 17 2.5 Symmetrization 18 2.6 Triple symmetrization 19 ∗Supported by a Packard Foundation Fellowship and NSF grant DMS-1001828. -
Alladi's Article on Bhargava's Fields Medal
MANJUL BHARGAVA'S FIELDS MEDAL AND BEYOND Krishnaswami Alladi University of Florida The award of the 2014 Fields Medal to Professor Manjul Bhargava of Princeton Uni- versity is a crowning recognition for his path-breaking contributions to some of the most famous and difficult problems in Number Theory. The Fields Medal, known as the \Nobel Prize of Mathematics", has an age limit of 40, and therefore recognizes outstanding con- tributions by young mathematicians who are expected to influence the development of the subject in the years ahead. Indeed when the Fields Medals were instituted in 1936, it was stressed that not only were these to recognise young mathematicians for what they had achieved, but also to encourage them to continue to influence the progress of the discipline in the future, and to inspire researchers around the world to collaborate with them or work on the ideas laid out by them to shape the development of mathematics. Manjul Bhar- gava was awarded the Fields Medal for his pioneering work on several long standing and important problems in number theory, and for the methods he has introduced to achieve this phenomenal progress - methods that will influence research for years to come. Here I will focus on his recent work and its future influence, but to provide a background, I will first briefly describe work stemming from his PhD thesis. Bhargava's PhD thesis concerns a problem going back to Carl Friedrich Gauss, one of the greatest mathematicians in history. In the nineteenth century, Gauss had discov- ered a fundamental composition law for binary quadratic forms which are homogeneous polynomial functions of degree two in two variables. -
Current Events Bulletin
A MERICAN M ATHEMATICAL S OCIETY 2013 CURRENT CURRENT EVENTS BULLETIN EVENTS BULLETIN Friday, January 11, 2013, 1:00 PM to 5:00 PM Committee Room 6F, Upper Level, San Diego Convention Center Joint Mathematics Meetings, San Diego, CA Hélène Barcelo, Mathematical Sciences Research Institute Jeffrey Brock, Brown University David Eisenbud, University of California, Berkeley, Chair 1:00 PM Wei Ho Dan Freed, University of Texas, Austin How many rational points does a random curve have? Eric Friedlander, University of Southern California Susan Friedlander, University of Southern California Bhargava and his school have turned classical questions about cubic polynomial equations in two variables over the integers in a whole new direction. Andrew Granville, Université de Montréal Richard Karp, University of California, Berkeley Isabella Laba, University of British Columbia David Nadler, University of California, Berkeley Thomas Scanlon, University of California, Berkeley 2:00 PM Sam Payne Gigliola Staffilani, Massachusetts Institute of Technology Ulrike Tillmann, Oxford University Topology of nonarchimedean analytic spaces Luca Trevisan, Stanford University "Tropical" algebra and geometry is a burgeoning field, suggesting interesting Yuri Tschinkel, New York University paths into geometry over fields like the p-adic numbers. Akshay Venkatesh, Stanford University Andrei Zelevinsky, Northeastern University 3:00 PM Mladen Bestvina Geometric group theory and 3-manifolds hand in hand: the fulfillment of Thurston's vision for three-manifolds Another vindication of Thurston's fabulous insights into three-dimensional geometry comes with a wonderful group-theoretic construction. 4:00 PM Lauren Williams Cluster algebras An exotic but simple structure that depends on a finite directed graph was brought to light in representation theory by Fomin and Zelevinsky. -
2021 2008 • Teichmüller Theory and Low-Dimensional Topology, June
Summer Conferences, 2008 – 2021 2008 • Teichmüller Theory and Low-Dimensional Topology, June 14–20, 2008. Organizers: Francis Bonahon, University of Southern California; Howard Masur, University of Illinois at Chicago; Abigail Thompson, University of California at Davis; and Genevieve Walsh, Tufts University. • Scientific Computing and Advanced Computation, June 21–27, 2008. Organizers: John Bell, Lawrence Berkeley National Laboratory; Randall LeVeque, University of Washington; Juan Meza, Lawrence Berkeley National Laboratory. • Computational Algebra and Convexity, June 21–27, 2008. Organizers: Henry Schenck, University of Illinois at Urbana-Champaign; Michael Stillman, Cornell University; Jan Verschelde, University of Illinois at Chicago. 2009 • Mathematical Challenges of Relativity, June 13-19, 2009. Organizers: Mihalis Dafermos, University of Cambridge; Alexandru Ionescu, University of Wisconsin, Madison; Sergiu Klainerman, Princeton University; Richard Schoen, Stanford University. • Inverse Problems, June 20-26, 2009. Organizers: Guillaume Bal, Columbia University; Allan Greenleaf, University of Rochester; Todd Quinto, Tufts University; Gunther Uhlmann, University of Washington • Modern Markov Chains and their Statistical Applications, June 27-July 3, 2009. Organizers: Persi Diaconis, Stanford University; Jim Hobert, University of Florida; Susan Holmes, Stanford University • Harmonic Analysis, June 27- July 3, 2009. Organizers: Ciprian Demeter, Indiana University; Michael Lacey, Georgia Institute of Technology; Christoph Thiele, University -
How Many Rational Points Does a Random Curve Have?
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY S 0273-0979(2013)01433-2 Article electronically published on September 18, 2013 HOW MANY RATIONAL POINTS DOES A RANDOM CURVE HAVE? WEI HO Abstract. A large part of modern arithmetic geometry is dedicated to or motivated by the study of rational points on varieties. For an elliptic curve over Q, the set of rational points forms a finitely generated abelian group. The ranks of these groups, when ranging over all elliptic curves, are conjectured to be evenly distributed between rank 0 and rank 1, with higher ranks being negligible. We will describe these conjectures and discuss some results on bounds for average rank, highlighting recent work of Bhargava and Shankar. 1. Introduction The past decade has seen a resurgence of classical techniques such as invariant theory and Minkowski’s geometry of numbers applied to problems in number theory. In particular, the newly coined phrase arithmetic invariant theory loosely refers to studying the orbits of representations of groups over rings or nonalgebraically closed fields, especially the integers Z or the rational numbers Q. Combining such ideas with improved geometry-of-numbers methods has proved fruitful for some questions in arithmetic statistics (another neologism!), the subject of which con- cerns the distribution of arithmetic quantities in families. Perhaps the earliest example of this phenomenon is well known to many number theorists: using Gauss’s study of the relationship between equivalence classes of binary quadratic forms and ideal classes for quadratic rings [Gau01], Mertens and Siegel determined the asymptotic behavior of ideal class groups for quadratic fields [Mer74, Sie44]. -
Compositio Mathematica
COMPOSITIO MATHEMATICA VOLUME 149 NUMBER 12 DECEMBER 2013 ISSN 0010-437X VOLUME 149 NUMBER 12 DECEMBER 2013 Kaoru Hiraga and Tamotsu Ikeda On the Kohnen plus space for Hilbert modular forms of half-integral weight I 1963–2010 David Grant and Su-Ion Ih Integral division points on curves 2011–2035 Bhargav Bhatt, Wei Ho, Zsolt Patakfalvi and Christian Schnell Moduli of products of stable varieties 2036–2070 Daniel Murfet Residues and duality for singularity categories of isolated Gorenstein singularities 2071–2100 Valentino Tosatti and Ben Weinkove The Chern–Ricci flow on complex surfaces 2101–2138 VOLUME 149 NUMBER 12 PP 1963–2183 Noriyuki Abe On a classification of irreducible admissible modulo p representations of a p-adic split reductive group 2139–2168 Ngo Dac Tuan Comptage des G-chtoucas : la partie elliptique 2169–2183 DECEMBER 2013 Downloaded from https://www.cambridge.org/core. IP address: 170.106.202.226, on 02 Oct 2021 at 03:03:20, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1112/S0010437X12001145 Managing Editors: B. J. J. Moonen E. M. Opdam B. J. Totaro Radboud University Nijmegen Korteweg–de Vries Institute DPMMS IMAPP, Faculty of Science University of Amsterdam University of Cambridge P. O. Box 9010 P. O. Box 94248 Wilberforce Road NL-6500 GL Nijmegen NL-1090 GE Amsterdam Cambridge CB3 0WB Ownership and Publication of Compositio Mathematica. The journal was founded in 1934 and has been owned The Netherlands The Netherlands United Kingdom since 1950 by the Foundation Compositio Mathematica. -
From the Chair Abel Prize for John Nash *50 Fields Medal for Manjul
Spring 2015 Issue 4 Department of Mathematics Princeton University Abel Prize for John Nash *50 From the Chair John Nash *50 received the 2015 Abel Prize This has been quite a year for from the Norwegian Academy of Science and the Mathematics Department; Letters for his work on partial differential however, space allows for mentioning but equations. Nash shares the $800,000 prize a few of the highlights. To start with, we with Louis Nirenberg, a professor emeritus are overjoyed to have Maria Chudnovsky, at NYU's Courant Institute of Mathemati- Fernando Marques and Assaf Naor as new cal Sciences. The prize recognized Nash and members of the senior faculty. Their pres- Nirenberg for “striking and seminal contri- ence has already made a major impact on butions to the theory of nonlinear partial the Department. differential equations and its applications to Manjul Bhargava *01 won the Fields medal geometric analysis.” Nash’s name is attached at last summer’s International Congress of to a range of influential work in mathematics, Mathematicians in Seoul, Korea. Manjul is including the Nash-Moser inverse function not only an extraordinary mathematician theorem, the Nash-De Giorgi theorem (which stemmed from a problem Nash undertook but also an unusually gifted teacher and at the suggestion of Nirenberg), and the Nash embedding theorems, which the academy tabla player. described as “among the most original results in geometric analysis of the twentieth century.” According to David Gabai, the Nash embedding/immersion theorems, which John Nash *50 and Louis Nirenberg from required unusual insight as well as tremendous technical expertise, played an important NYU are this year’s Able Prize winners. -
SMS 2014: Director's Report
SMS 2014: Director’s report. The 53rd S´eminaire de Math´ematiques Sup´erieures took place in Montr´eal in the period June 23- July 4, 2014. It was the largest summer school in recent years, with 112 participants. Focused on the celebrated work of Manjul Bhargava, who was himself one of the featured lecturers, this school was an exceptional event for number theory and beyond, with a great impact on making Bhargava’s breakthroughs accessible to a larger audience. Remarkably this year, many of the speakers, Bhargava in particular, were present the full two-week period. The organizers, Henri Darmon, Andrew Granville and Benedict Gross have done a tremendous job by putting together a fantastic group of speakers and motivated students and in coordinating tightly the lectures in such a way as to insure the highest training impact. Moreover, they have already started to prepare a volume tied to the school activity that promises to play an important role in the field. I thank all three of them for their hard work as well as Ms. Sakina Benhima from the CRM who assisted them and me with the administrative matters required in running this activity. As in past years, this edition of the SMS was only possible with the co-operation of our main partners the CRM, Fields Institute, PIMS and MSRI as well as with support from the ISM, the University of Montreal and with support from the Canadian Mathematical Society. I thank all these institutions for their contributions and I also thank the board of directors of the SMS for their work and support. -
2018 2008 • Teichmüller Theory And
The American Mathematical Society Mathematics Research Communities, 2008 – 2018 2008 • Teichmüller Theory and Low-Dimensional Topology , June 14–20, 2008. Organizers: Francis Bonahon , University of Southern California; Howard Masur , University of Illinois at Chicago; Abigail Thompson , University of California at Davis; and Genevieve Walsh , Tufts University. • Scientific Computing and Advanced Computation , June 21–27, 2008. Organizers: John Bell , Lawrence Berkeley National Laboratory; Randall LeVeque , University of Washington; Juan Meza , Lawrence Berkeley National Laboratory. • Computational Algebra and Convexity , June 21–27, 2008 . Organizers: Henry Schenck , University of Illinois at Urbana-Champaign; Michael Stillman , Cornell University; Jan Verschelde , University of Illinois at Chicago. 2009 • Mathematical Challenges of Relativity , June 13-19, 2009. Organizers: Mihalis Dafermos , University of Cambridge; Alexandru Ionescu , University of Wisconsin, Madison; Sergiu Klainerman , Princeton University; Richard Schoen , Stanford University. • Inverse Problems , June 20-26, 2009. Organizers: Guillaume Bal , Columbia University; Allan Greenleaf, University of Rochester; Todd Quinto , Tufts University; Gunther Uhlmann , University of Washington • Modern Markov Chains and their Statistical Applications , June 27-July 3, 2009. Organizers: Persi Diaconis , Stanford University; Jim Hobert, University of Florida; Susan Holmes , Stanford University • Harmonic Analysis , June 27- July 3, 2009. Organizers: Ciprian Demeter , Indiana University; Michael Lacey , Georgia Institute of Technology; Christoph Thiele , University of California, Los Angeles. 2010 • Birational Geometry and Moduli Spaces , June 12-18, 2010. Organizers: Dan Abramovich , Brown University; Izzet Coskun , University of Illinois at Chicago; Tommàso de Fernex , University of Utah; Angela Gibney , University of Georgia; James McKernan , Massachusetts Institute of Technology, assisted by Emanuele Macri , University of Utah • Model Theory of Fields , June 19–25, 2010.