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Geometric Measure Theory and the Calculus of Variations PROCEEDINGS of SYMPOSIA in PURE MATHEMATICS Volume 44
http://dx.doi.org/10.1090/pspum/044 Geometric Measure Theory and the Calculus of Variations PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS Volume 44 Geometric Measure Theory and the Calculus of Variations William K. Allard and Frederick J. Almgren, Jr., Editors AMERICAN MATHEMATICAL SOCIETY PROVIDENCE, RHODE ISLAND PROCEEDINGS OF THE SUMMER INSTITUTE ON GEOMETRIC MEASURE THEORY AND THE CALCULUS OF VARIATIONS HELD AT HUMBOLDT STATE UNIVERSITY ARCATA, CALIFORNIA JULY 16-AUGUST 3, 1984 with support from the National Science Foundation, Grant DMS-8317890 1980 Mathematics Subject Classification. Primary 49F22, 49F20, 53A10, 53C42; Secondary 49F10, 53C20, 35J65, 58-XX. Library of Congress Cataloging-in-Publication Data Main entry under title: Geometric measure theory and the calculus of variations. (Proceedings of symposia in pure mathematics; v. 44) Includes bibliographies. 1. Geometric measure theory-Addresses, essays, lectures. I. Allard, William K., 1941- II. Almgren, Frederick J. III. Series. QA312.G394 1985 516.3'6 85-18641 ISBN 0-8218-1470-2 (alk. paper) COPYING AND REPRINTING. Individual leaders of this publication, and nonprofit libraries acting for them are permitted to make fair use of the material, such as to copy an article for use in teaching or research. Permission is granted to quote brief passages from this pubUcation in reviews provided the customary acknowledgement of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Executive Director, American Mathemat ical Society, P.O. Box 6248, Providence, Rhode Island 02940. -
The Geometry of Asymptotically Hyperbolic Manifolds a Dissertation Submitted to the Department of Mathematics and the Committee
THE GEOMETRY OF ASYMPTOTICALLY HYPERBOLIC MANIFOLDS A DISSERTATION SUBMITTED TO THE DEPARTMENT OF MATHEMATICS AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Otis Chodosh June 2015 © 2015 by Otis Avram Chodosh. All Rights Reserved. Re-distributed by Stanford University under license with the author. This work is licensed under a Creative Commons Attribution- Noncommercial 3.0 United States License. http://creativecommons.org/licenses/by-nc/3.0/us/ This dissertation is online at: http://purl.stanford.edu/mp634xn8004 ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Simon Brendle, Primary Adviser I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Michael Eichmair, Co-Adviser I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Leon Simon I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Brian White Approved for the Stanford University Committee on Graduate Studies. Patricia J. Gumport, Vice Provost for Graduate Education This signature page was generated electronically upon submission of this dissertation in electronic format. -
David Donoho COMMENTARY 52 Cliff Ord J
ISSN 0002-9920 (print) ISSN 1088-9477 (online) of the American Mathematical Society January 2018 Volume 65, Number 1 JMM 2018 Lecture Sampler page 6 Taking Mathematics to Heart y e n r a page 19 h C th T Ru a Columbus Meeting l i t h i page 90 a W il lia m s r e lk a W ca G Eri u n n a r C a rl ss on l l a d n a R na Da J i ll C . P ip her s e v e N ré F And e d e r i c o A rd ila s n e k c i M . E d al Ron Notices of the American Mathematical Society January 2018 FEATURED 6684 19 26 29 JMM 2018 Lecture Taking Mathematics to Graduate Student Section Sampler Heart Interview with Sharon Arroyo Conducted by Melinda Lanius Talithia Williams, Gunnar Carlsson, Alfi o Quarteroni Jill C. Pipher, Federico Ardila, Ruth WHAT IS...an Acylindrical Group Action? Charney, Erica Walker, Dana Randall, by omas Koberda André Neves, and Ronald E. Mickens AMS Graduate Student Blog All of us, wherever we are, can celebrate together here in this issue of Notices the San Diego Joint Mathematics Meetings. Our lecture sampler includes for the first time the AMS-MAA-SIAM Hrabowski-Gates-Tapia-McBay Lecture, this year by Talithia Williams on the new PBS series NOVA Wonders. After the sampler, other articles describe modeling the heart, Dürer's unfolding problem (which remains open), gerrymandering after the fall Supreme Court decision, a story for Congress about how geometry has advanced MRI, “My Father André Weil” (2018 is the 20th anniversary of his death), and a profile on Donald Knuth and native script by former Notices Senior Writer and Deputy Editor Allyn Jackson. -
METRICS of CONSTANT SCALAR CURVATURE CONFORMAL to RIEMANNIAN PRODUCTS 1. Introduction Any Closed Manifold Admits Metrics of Cons
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 138, Number 8, August 2010, Pages 2897–2905 S 0002-9939(10)10293-7 Article electronically published on March 29, 2010 METRICS OF CONSTANT SCALAR CURVATURE CONFORMAL TO RIEMANNIAN PRODUCTS JIMMY PETEAN (Communicated by Richard A. Wentworth) Abstract. We consider the conformal class of the Riemannian product g0 +g, m where g0 is the constant curvature metric on S and g is a metric of constant scalar curvature on some closed manifold. We show that the number of metrics of constant scalar curvature in the conformal class grows at least linearly with respect to the square root of the scalar curvature of g. This is obtained by studying radial solutions of the equation Δu − λu + λup =0onSm and the number of solutions in terms of λ. 1. Introduction Any closed manifold admits metrics of constant scalar curvature. Given any Riemannian metric g on M n we consider its conformal class [g] and define the Yamabe constant of [g] as the minimum of the (normalized) total scalar curvature functional restricted to [g]: M sh dvolh Y (M,[g]) = inf − , ∈ n 2 h [g] Vol(M,h) n where sh and dvolh are the scalar curvature and volume element of h. It is elementary that the functional restricted to [g] is bounded below, and the fact that the infimum is actually achieved is a fundamental result, obtained in a series of steps by Hidehiko Yamabe [17], Thierry Aubin [2], Neil Trudinger [16] and Richard Schoen [14]. Since the critical points of the functional (restricted to [g]) are the metrics of constant scalar curvature in [g], it follows that minimizers are metrics of constant scalar curvature. -
The Existence of Embedded Minimal Hypersurfaces 3
THE EXISTENCE OF EMBEDDED MINIMAL HYPERSURFACES CAMILLO DE LELLIS AND DOMINIK TASNADY Abstract. We give a shorter proof of the existence of nontrivial closed minimal hypersur- faces in closed smooth (n + 1)–dimensional Riemannian manifolds, a theorem proved first by Pitts for 2 ≤ n ≤ 5 and extended later by Schoen and Simon to any n. 0. Introduction In this paper we give a proof of the following theorem, a natural generalization of the classical existence of nontrivial simple closed geodesics in closed 2–d Riemannian manifolds. Theorem 0.1. Let M be an (n+1)-dimensional smooth closed Riemannian manifold. Then there is a nontrivial embedded minimal hypersurface Σ ⊂ M without boundary with a singular set Sing Σ of Hausdorff dimension at most n − 7. More precisely, Σ is a closed set of finite Hn–measure and Sing Σ ⊂ Σ is the smallest closed set S such that M \ S is a smooth embedded hypersurface (Σ \ Sing Σ is in fact analytic if M is analytic). In this paper smooth will always mean C∞. In fact, the result remains true for any C4 Riemannian manifold M, Σ then will be of class C2 (see [19]). Moreover Σ\Sing Σ ω = 0 for any exact n–form on M. The case 2 ≤ n ≤ 5 was proved by Pitts inR his groundbreaking monograph [16], an outstanding contribution which triggered all the subsequent research in the topic. The general case was proved by Schoen and Simon in [19], building heavily upon the work of Pitts. The monograph [16] can be ideally split into two parts. -
Geometric Relativity
GRADUATE STUDIES IN MATHEMATICS 201 Geometric Relativity Dan A. Lee 10.1090/gsm/201 Geometric Relativity GRADUATE STUDIES IN MATHEMATICS 201 Geometric Relativity Dan A. Lee EDITORIAL COMMITTEE Daniel S. Freed (Chair) Bjorn Poonen Gigliola Staffilani Jeff A. Viaclovsky 2010 Mathematics Subject Classification. Primary 53-01, 53C20, 53C21, 53C24, 53C27, 53C44, 53C50, 53C80, 83C05, 83C57. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-201 Library of Congress Cataloging-in-Publication Data Names: Lee, Dan A., 1978- author. Title: Geometric relativity / Dan A. Lee. Description: Providence, Rhode Island : American Mathematical Society, [2019] | Series: Gradu- ate studies in mathematics ; volume 201 | Includes bibliographical references and index. Identifiers: LCCN 2019019111 | ISBN 9781470450816 (alk. paper) Subjects: LCSH: General relativity (Physics)–Mathematics. | Geometry, Riemannian. | Differ- ential equations, Partial. | AMS: Differential geometry – Instructional exposition (textbooks, tutorial papers, etc.). msc | Differential geometry – Global differential geometry – Global Riemannian geometry, including pinching. msc | Differential geometry – Global differential geometry – Methods of Riemannian geometry, including PDE methods; curvature restrictions. msc | Differential geometry – Global differential geometry – Rigidity results. msc — Differential geometry – Global differential geometry – Spin and Spin. msc | Differential geometry – Global differential geometry – Geometric evolution equations (mean curvature flow, -
The Shape of Inner Space Provides a Vibrant Tour Through the Strange and Wondrous Possibility SPACE INNER
SCIENCE/MATHEMATICS SHING-TUNG $30.00 US / $36.00 CAN Praise for YAU & and the STEVE NADIS STRING THEORY THE SHAPE OF tring theory—meant to reconcile the INNER SPACE incompatibility of our two most successful GEOMETRY of the UNIVERSE’S theories of physics, general relativity and “The Shape of Inner Space provides a vibrant tour through the strange and wondrous possibility INNER SPACE THE quantum mechanics—holds that the particles that the three spatial dimensions we see may not be the only ones that exist. Told by one of the Sand forces of nature are the result of the vibrations of tiny masters of the subject, the book gives an in-depth account of one of the most exciting HIDDEN DIMENSIONS “strings,” and that we live in a universe of ten dimensions, and controversial developments in modern theoretical physics.” —BRIAN GREENE, Professor of © Susan Towne Gilbert © Susan Towne four of which we can experience, and six that are curled up Mathematics & Physics, Columbia University, SHAPE in elaborate, twisted shapes called Calabi-Yau manifolds. Shing-Tung Yau author of The Fabric of the Cosmos and The Elegant Universe has been a professor of mathematics at Harvard since These spaces are so minuscule we’ll probably never see 1987 and is the current department chair. Yau is the winner “Einstein’s vision of physical laws emerging from the shape of space has been expanded by the higher them directly; nevertheless, the geometry of this secret dimensions of string theory. This vision has transformed not only modern physics, but also modern of the Fields Medal, the National Medal of Science, the realm may hold the key to the most important physical mathematics. -
Arxiv:1904.02289V1 [Math.DG]
Stationary one-sided area-minimizing hypersurfaces with isolated singularities Zhenhua Liu Dedicated to Xunjing Wei Abstract. We extend the results of Hardt and Simon in [8] to prove that isolated singularities of stationary one-sided area-minimizing hypersurfaces can be locally perturbed away on the side that they are minimizing. 1. Introduction Area-minimizing hypersurfaces in n + 1-dimensional manifolds M n+1 are known to be smooth outside a set of Hausdorff dimension at most n − 7 ([7]). However, little is known about the geometry of the singular sets. In this regard, the best results to date are the structure theory of area-minimizing hypersurfaces with isolated singularities developed by Robert Hardt and Leon Simon in [16] and [8]. Roughly speaking, they have proved that (i) On any side of an area-minimizing hypercone, all area-minimizing bound- aries confined in that side are smooth and unique up to scalings. (ii) (Based on (i)) On any side of an area-minimizing hypersurface, if the n − 1 boundary of a Plateau problem is close enough to a smooth n − 1 submanifold of the cone, then the solution to the Plateau problem is smooth. arXiv:1904.02289v1 [math.DG] 4 Apr 2019 We will explain the terminologies contained in those statements later. These state- ments basically say that isolated singularities of area-minimizing hypersurfaces be- have as well as one can imagine, in that one can always perturb them away locally in a suitable sense to get smooth objects. In this paper, we extend these results to hypersurfaces that are only area-minimizing on one side. -
Regularity of Capillary Surfaces Over Domains with Corners
Pacific Journal of Mathematics REGULARITY OF CAPILLARY SURFACES OVER DOMAINS WITH CORNERS LEON M. SIMON Vol. 88, No. 2 April 1980 PACIFIC JOURNAL OF MATHEMATICS Vol. 88, No. 2, 1980 REGULARITY OF CAPILLARY SURFACES OVER DOMAINS WITH CORNERS LEON SIMON Using the usual mathematical model (capillary surface equation with contact angle boundary condition) we discuss regularity of the equilibrium free surface of a fluid in a cylin- drical container in case the container cross-section has corners. It is shown that good regularity holds at a corner if the "corner angle" θ satisfies O<0<τr and θ + 2β>ττ, where 0</3< π/2 is the contact angle between the fluid surface and the container wall. It is known that no regularity holds in case θ + 2β<π, hence only the borderline case θ + 2β = π remains open. We here want to examine the regularity of solutions of capillary surface type equations (subject to contact angle boundary conditions) on domain Ω ciί2 in a neighbourhood of a point of dΩ where there is a corner. To be specific let Ω (as depicted in the diagram) be a region 2 contained in DR = {x e R : | x | < R) (R > 0 given) such that dΩ consists of a circular segment of dDR together with two compact Jordan arcs 7i, 72 such that 7i Π 72 = {0}. 7i, 72 are supposed to be Qua fQΐ some o < a < 1, and to meet at 0 with angle (measured in Ω) θ, 0 < θ < π. We also suppose (without loss of generality, since we can always take a smaller R) that 7* intersects dD0 in a single point for each i = 1, 2, 0 < p < R. -
Arxiv:1102.2999V2 [Math.DG] 30 Jan 2013 N Ftesfgat2-77348-12
LARGE ISOPERIMETRIC SURFACES IN INITIAL DATA SETS MICHAEL EICHMAIR AND JAN METZGER Abstract. We study the isoperimetric structure of asymptotically flat Riemannian 3–manifolds (M,g) that are C0-asymptotic to Schwarzschild of mass m > 0. Refining an argument due to H. Bray we obtain an ef- fective volume comparison theorem in Schwarzschild. We use it to show that isoperimetric regions exist in (M,g) for all sufficiently large vol- umes, and that they are close to centered coordinate spheres. This im- plies that the volume-preserving stable constant mean curvature spheres constructed by G. Huisken and S.-T. Yau as well as R. Ye as pertur- bations of large centered coordinate spheres minimize area among all competing surfaces that enclose the same volume. This confirms a con- jecture of H. Bray. Our results are consistent with the uniqueness results for volume-preserving stable constant mean curvature surfaces in initial data sets obtained by G. Huisken and S.-T. Yau and strengthened by J. Qing and G. Tian. The additional hypotheses that the surfaces be spherical and far out in the asymptotic region in their results are not necessary in our work. 1. Introduction In this paper we describe completely the large isoperimetric surfaces of asymptotically flat Riemannian 3-manifolds (M, g) that are 0-asymptotic to the Schwarzschild metric of mass m > 0. Such RiemannianC manifolds arise naturally as initial data for the time-symmetric Cauchy problem for arXiv:1102.2999v2 [math.DG] 30 Jan 2013 the Einstein equations in general relativity. For brevity we will refer to such (M, g) as initial data sets in the introduction. -
Mathematics People
Mathematics People Packard Fellowships Awarded Royal Society of London The David and Lucile Packard Foundation has awarded Elections twenty Fellowships for Science and Engineering for the year Six mathematical scientists are among those elected as new 2002. Among the new fellows are two with links to the fellows of the Royal Society of London for 2003. They are ONALD EDKIW mathematical sciences. R P. F of Stanford Uni- ELEANOR DODSON, University of York; ROGER FLETCHER, versity and RAJESH P. RAO of the University of Washington, University of Dundee; PETER GREEN, University of Bristol; both of whom work in computer science, will each receive JOHN PAPALOIZOU, University of London; and LEON SIMON, a fellowship of $625,000 over five years. Stanford University. DONALD KNUTH, Stanford University, was The fellowships are awarded to researchers in mathe- elected as a foreign member. matics, natural sciences, computer science, and engineering who are in the first three years of a faculty appointment. —From a Royal Society announcement —From a Packard Foundation announcement AMS Menger Prizes at the 2003 ISEF Werschulz Receives Prize for The 2003 Intel-International Science and Engineering Fair Achievement in Information- (ISEF) was held May 11–17 in Cleveland, Ohio. This year marked the fifty-fourth anniversary of the ISEF. More than Based Complexity 1,200 ninth- through twelfth-graders competed in the fair. The participants had qualified by winning competitions in ARTHUR G. WERSCHULZ of Fordham University is the fifth local, regional, and state fairs in the United States or winner of the Prize for Information-Based Complexity. national science fairs abroad. -
Notices of the American Mathematical Society ISSN 0002-9920
Notices of the American Mathematical Society ISSN 0002-9920 springer.com New and Noteworthy from Springer Lie Sphere How Does One Cut a Triangle? Random Curves nd of the American Mathematical Society Geometry 2 A. Soifer, University of Colorado, Colorado Journeys of a Mathematician EDITION November 2007 Volume 54, Number 10 With Applications Springs, CO, USA N. I. Koblitz , University of Washington, Seattle, to Submanifolds This title demonstrates how diff erent areas WA, USA T. E. Cecil , College of the Holy Cross, of mathematics can be juxtaposed in the These autobiographical memoirs of Neal Worcester, MA, USA solution of any given problem by presenting Koblitz, coinventor of one of the two most an interesting problem, “How does one cut a This book begins with Lie’s construction of the popular forms of encryption and digital triangle?”. By providing analytical proofs and space of spheres, including the fundamental signature, cover many topics besides his own counterexamples, he proves that research is notions of oriented contact, parabolic pencils personal career in mathematics and a collection of mathematical ideas developed of spheres and Lie sphere transformation. The cryptography - travels to the Soviet Union, throughout the course of history. Latin America, Vietnam and elsewhere, link with Euclidean submanifold theory is established via the Legendre map, providing a I very warmly recommend the book and political activism, and academic controversies framework for the study of submanifolds, hope the readers will have pleasure in thinking relating to math education, the C. P. Snow especially those characterized by restrictions about the unsolved problems and will nd new two-culture problem, and mistreatment of on their curvature spheres.