DOCSLIB.ORG

Analysis & PDE Vol. 5 (2012)

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Analysis & PDE Vol. 5 (2012) ANALYSIS & PDE Volume 5 No. 1 2012 mathematical sciences publishers Analysis & PDE msp.berkeley.edu/apde EDITORS EDITOR-IN-CHIEF Maciej Zworski University of California Berkeley, USA BOARD OF EDITORS Michael Aizenman Princeton University, USA Nicolas Burq Université Paris-Sud 11, France [email protected] [email protected] Luis A. Caffarelli University of Texas, USA Sun-Yung Alice Chang Princeton University, USA [email protected] [email protected] Michael Christ University of California, Berkeley, USA Charles Fefferman Princeton University, USA [email protected] [email protected] Ursula Hamenstaedt Universität Bonn, Germany Nigel Higson Pennsylvania State Univesity, USA [email protected] [email protected] Vaughan Jones University of California, Berkeley, USA Herbert Koch Universität Bonn, Germany [email protected] [email protected] Izabella Laba University of British Columbia, Canada Gilles Lebeau Université de Nice Sophia Antipolis, France [email protected] [email protected] László Lempert Purdue University, USA Richard B. Melrose Massachussets Institute of Technology, USA [email protected] [email protected] Frank Merle Université de Cergy-Pontoise, France William Minicozzi II Johns Hopkins University, USA [email protected] [email protected] Werner Müller Universität Bonn, Germany Yuval Peres University of California, Berkeley, USA [email protected] [email protected] Gilles Pisier Texas A&M University, and Paris 6 Tristan Rivière ETH, Switzerland [email protected] [email protected] Igor Rodnianski Princeton University, USA Wilhelm Schlag University of Chicago, USA [email protected] [email protected] Sylvia Serfaty New York University, USA Yum-Tong Siu Harvard University, USA [email protected] [email protected] Terence Tao University of California, Los Angeles, USA Michael E. Taylor Univ. of North Carolina, Chapel Hill, USA [email protected] [email protected] Gunther Uhlmann University of Washington, USA András Vasy Stanford University, USA [email protected] [email protected] Dan Virgil Voiculescu University of California, Berkeley, USA Steven Zelditch Northwestern University, USA [email protected] [email protected] PRODUCTION [email protected] Silvio Levy, Scientific Editor Sheila Newbery, Senior Production Editor See inside back cover or msp.berkeley.edu/apde for submission instructions. The subscription price for 2012 is US $140/year for the electronic version, and $240/year for print and electronic. Subscriptions, requests for back issues from the last three years and changes of subscribers address should be sent to Mathematical Sciences Publishers, Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA. Analysis & PDE, at Mathematical Sciences Publishers, Department of Mathematics, University of California, Berkeley, CA 94720-3840 is published continuously online. Periodical rate postage paid at Berkeley, CA 94704, and additional mailing offices. APDE peer review and production are managed by EditFLOW™ from Mathematical Sciences Publishers. PUBLISHED BY mathematical sciences publishers http://msp.org/ A NON-PROFIT CORPORATION Typeset in LATEX Copyright ©2012 by Mathematical Sciences Publishers ANALYSIS AND PDE Vol. 5, No. 1, 2012 dx.doi.org/10.2140/apde.2012.5.1 msp A CHARACTERIZATION OF TWO WEIGHT NORM INEQUALITIES FOR MAXIMAL SINGULAR INTEGRALS WITH ONE DOUBLING MEASURE MICHAEL LACEY,ERIC T. SAWYER AND IGNACIO URIARTE-TUERO Let σ and ! be positive Borel measures on R with σ doubling. Suppose first that 1 < p ≤ 2. We p characterize boundedness of certain maximal truncations of the Hilbert transform T\ from L (σ / to p L .!/ in terms of the strengthened A p condition Z 1=p Z 1=p0 p p0 sQ.x/ d!.x/ sQ.x/ dσ .x/ ≤ CjQj; R R where sQ.x/ D jQj=.jQj C jx − xQj/, and two testing conditions. The first applies to a restricted class of functions and is a strong-type testing condition, Z Z p T\(χE σ /.x/ d!.x/ ≤ C1 dσ .x/ for all E ⊂ Q; Q Q and the second is a weak-type or dual interval testing condition, Z Z 1=p Z 1=p0 p T\(χQ f σ /.x/ d!.x/ ≤ C2 j f .x/j dσ .x/ d!.x/ Q Q Q for all intervals Q in R and all functions f 2 L p(σ /. In the case p > 2 the same result holds if we include an additional necessary condition, the Poisson condition 1 1 1 Z −` p X p0−1 X 2 X p0 jI j jI j jχ .`/ .y/ d!.y/ ≤ C jI j jI j ; r σ r .`/ .Ir / r σ r j.Ir / R rD1 `D0 rD1 S1 for all pairwise disjoint decompositions Q D rD1 Ir of the dyadic interval Q into dyadic intervals Ir . We prove that analogues of these conditions are sufficient for boundedness of certain maximal singular integrals in Rn when σ is doubling and 1 < p < 1. Finally, we characterize the weak-type two weight n inequality for certain maximal singular integrals T\ in R when 1 < p < 1, without the doubling assumption on σ, in terms of analogues of the second testing condition and the A p condition. 1. Introduction Sawyer [1984; 1982; 1988] characterized two weight inequalities for maximal functions and other pos- itive operators, in terms of the obviously necessary conditions that the operators be uniformly bounded on a restricted class of functions, namely indicators of intervals and cubes. Thus, these characterizations have a form reminiscent of the T 1 theorem of David and Journé. Lacey is supported in part by the NSF, through grant DMS-0456538. Sawyer is supported in part by NSERC. Uriarte-Tuero is supported in part by the NSF, through grant DMS-0901524. MSC2000: 42B20. Keywords: two weight, singular integral, maximal function, maximal truncation. 1 2 MICHAELLACEY,ERICT.SAWYER AND IGNACIO URIARTE-TUERO Corresponding results for even the Hilbert transform have only recently been obtained [Nazarov et al. 2010; Lacey et al. 2011] and even then only for p D 2; evidently these are much harder to obtain. We comment in more detail on prior results below, including the innovative work of Nazarov, Treil and Volberg [1999; 2008; 2010; 2003]. Our focus is on providing characterizations of the boundedness of certain maximal truncations of a fixed operator of singular integral type. The singular integrals will be of the usual type, for example the Hilbert transform or paraproducts. Only size and smoothness conditions on the kernel are assumed; see (1-9). The characterizations are in terms of certain obviously necessary conditions, in which the class of functions being tested is simplified. For such examples, we prove unconditional characterizations of both strong-type and weak-type two weight inequalities for certain maximal truncations of the Hilbert transform, but with the additional assumption that σ is doubling for the strong-type inequality. A major point of our characterizations is that they hold for all 1 < p < 1. The methods in [Lacey et al. 2011] and those of Nazarov, Treil and Volberg apply only to the case p D 2, where the orthogonality of measure- adapted Haar bases prove critical. The doubling hypothesis on σ may not be needed in our theorems, but is required by the use of Calderón–Zygmund decompositions in our method. As the precise statements of our general results are somewhat complicated, we illustrate them with an important case here. Let Z 1 T f .x/ D lim f .x − y/ dy "!0 Rn.−";"/ y denote the Hilbert transform, let Z D 1 − T[ f .x/ sup f .x y/ dy 0<"<1 Rn.−";"/ y denote the usual maximal singular integral associated with T , and finally let Z 1 T f .x/ D sup f .x − y/ dy \ y 0<"1;"2<1 Rn.−"1;"2/ 1=4<"2="1<4 denote the new strongly (or noncentered) maximal singular integral associated with T that is defined more precisely below. Suppose σ and ! are two locally finite positive Borel measures on R that have no point masses in common. Then we have the following weak and strong-type characterizations, which we emphasize hold for all 1 < p < 1. • The operator T[ is weak type .p; p/ with respect to (σ; !/, that is, kT[. f σ /kL p;1.!/ ≤ Ck f kL p(σ / (1-1) for all f bounded with compact support if and only if the two weight A p condition 1 Z 1 Z p−1 d! dσ ≤ C; jQj Q jQj Q TWOWEIGHTNORMINEQUALITIESFORMAXIMALSINGULARINTEGRALS 3 holds for all intervals Q and the dual T[ interval testing condition Z Z 1=p Z 1=p0 p T[(χQ f σ / d! ≤ C j f j dσ d! ; (1-2) Q Q Q p holds for all intervals Q and f 2 L Q(σ / (part 4 of Theorem 1.8). The same is true for T\. It is easy to see that (1-2) is equivalent to the more familiar dual interval testing condition Z Z ∗ p0 jL (χQ!/j dσ ≤ C d!; (1-3) Q Q for all intervals Q and linearizations L of the maximal singular integral T[ (see (2-10)). • Suppose in addition that σ is doubling and 1 < p < 1. Then the operator T\ is strong-type .p; p/ with respect to (σ; !/, that is, kT\. f σ /kL p.!/ ≤ Ck f kL p(σ / for all f bounded with compact support if and only if these four conditions hold: (1) the strengthened A p condition Z 1=p Z 1=p0 p p0 sQ.x/ d!.x/ sQ.x/ dσ .x/ ≤ CjQj; Q I jQj where sQ.x/ D , holds for all intervals Q; (2) the dual T interval testing condition jQjCjx−xQ j \ Z Z 1=p Z 1=p0 p T\(χQ f σ / d! ≤ C j f j dσ d! ; Q Q Q p holds for all intervals Q and f 2 L Q(σ /; (3) the forward T\ testing condition Z Z p T\(χE σ / d! ≤ C dσ; (1-4) Q Q holds for all intervals Q and all compact subsets E of Q; and (4) the Poisson condition 1 1 1 Z −` p X p0−1 X 2 X p0 jI j jI j χ .`/ .y/ d!.y/ ≤ C jI j jI j ; r σ r .`/ .Ir / r σ r j.Ir / j R rD1 `D0 rD1 S1 for all pairwise disjoint decompositions Q D rD1 Ir of the dyadic interval Q into dyadic inter- vals Ir for any fixed dyadic grid.
Load more

Recommended publications
  • Rapport Annuel 2014-2015
    RAPPORT ANNUEL 2014-2015 Présentation du rapport annuel 1 Programme thématique 2 Autres activités 12 Grandes Conférences et colloques 16 Les laboratoires du CRM 20 Les prix du CRM 30 Le CRM et la formation 34 Les partenariats du CRM 38 Les publications du CRM 40 Comités à la tête du CRM 41 Le CRM en chiffres 42 Luc Vinet Présentation En 2014-2015, contrairement à ce qui était le cas dans (en physique mathématique) à Charles Gale de l’Université les années récentes, le programme thématique du CRM a McGill et le prix CRM-SSC (en statistique) à Matías été consacré à un seul thème (très vaste !) : la théorie des Salibián-Barrera de l’Université de Colombie-Britannique. nombres. L’année thématique, intitulée « La théorie des Les Grandes conférences du CRM permirent au grand public nombres : de la statistique Arithmétique aux éléments Zêta », de s’initier à des sujets variés, présentés par des mathémati- a été organisée par les membres du CICMA, un laboratoire ciens chevronnés : Euler et les jets d’eau de Sans-Souci du CRM à la fine pointe de la recherche mondiale, auxquels il (par Yann Brenier), la mesure des émotions en temps réel faut ajouter Louigi Addario-Berry (du Groupe de probabilités (par Chris Danforth), le mécanisme d’Anticythère (par de Montréal). Je tiens à remercier les quatre organisateurs de James Evans) et l’optique et les solitons (par John Dudley). cette brillante année thématique : Henri Darmon de l’Univer- L’année 2014-2015 fut également importante du point de sité McGill, Chantal David de l’Université Concordia, Andrew vue de l’organisation et du financement du CRM.
    [Show full text]
  • Department of Mathematics
    Department of Mathematics The Department of Mathematics seeks to maintain its top ranking in mathematics research and education in the United States. The department is a key part of MIT’s educational mission at both the undergraduate and graduate levels and produces the most sought-after young researchers. Key to the department’s success is recruitment of the best faculty members, postdoctoral associates, and graduate students in an ever more competitive environment. The department strives to be diverse at all levels in terms of race, gender, and ethnicity. It continues to serve the varied needs of its graduate students, undergraduate students majoring in mathematics, and the broader MIT community. Awards and Honors The faculty received numerous distinctions this year. Professor Victor Kac received the Leroy P. Steele Prize for Lifetime Achievement, for “groundbreaking contributions to Lie Theory and its applications to Mathematics and Mathematical Physics.” Larry Guth (together with Netz Katz of the California Institute of Technology) won the Clay Mathematics Institute Research Award. Tomasz Mrowka was elected as a member of the National Academy of Sciences and William Minicozzi was elected as a fellow of the American Academy of Arts and Sciences. Alexei Borodin received the 2015 Loève Prize in Probability, given by the University of California, Berkeley, in recognition of outstanding research in mathematical probability by a young researcher. Alan Edelman received the 2015 Babbage Award, given at the IEEE International Parallel and Distributed Processing Symposium, for exceptional contributions to the field of parallel processing. Bonnie Berger was elected vice president of the International Society for Computational Biology.
    [Show full text]
  • Department of Mathematics, Report to the President 2015-2016
    Department of Mathematics The Department of Mathematics continues to be the top-ranked mathematics department in the United States. The department is a key part of MIT’s educational mission at both the undergraduate and graduate levels and produces top sought- after young researchers. Key to the department’s success is recruitment of the very best faculty, postdoctoral fellows, and graduate students in an ever-more competitive environment. The department aims to be diverse at all levels in terms of race, gender, and ethnicity. It continues to serve the varied needs of the department’s graduate students, mathematics majors, and the broader MIT community. Awards and Honors The faculty received numerous distinctions this year. Professor Emeritus Michael Artin was awarded the National Medal of Science. In 2016, President Barack Obama presented this honor to Artin for his outstanding contributions to mathematics. Two other emeritus professors also received distinctions: Professor Bertram Kostant was selected to receive the 2016 Wigner Medal, in recognition of “outstanding contributions to the understanding of physics through Group Theory.” Professor Alar Toomre was elected member of the American Philosophical Society. Among active faculty, Professor Larry Guth was awarded the New Horizons in Mathematics Prize for “ingenious and surprising solutions to long standing open problems in symplectic geometry, Riemannian geometry, harmonic analysis, and combinatorial geometry.” He also received a 2015 Teaching Prize for Graduate Education from the School of Science. Professor Alexei Borodin received the 2015 Henri Poincaré Prize, awarded every three years at the International Mathematical Physics Congress to recognize outstanding contributions in mathematical physics. He also received a 2016 Simons Fellowship in Mathematics.
    [Show full text]
  • Mathematician Unleashes 'A Wave of New Results' in Geometric Analysis 4 June 2014, by Helen Knight
    Mathematician unleashes 'a wave of new results' in geometric analysis 4 June 2014, by Helen Knight would form with that shape as its boundary. Although intuition would tell you that it should do this, there is no way to physically test the infinite number of possible variations that could be made to the shape of the wire in order to provide mathematical proof, Minicozzi says. A top geometric analyst Answering Plateau's question—and addressing subsequent conjectures on the properties of complex minimal surfaces—has kept mathematicians busy ever since. The most notable of these researchers in recent years have been Minicozzi and his colleague Tobias Colding, the MIT mathematics professor William Minicozzi in his Cecil and Ida Green Distinguished Professor of office at Building E17. Minicozzi studies the theory of Mathematics at MIT. Together, Minicozzi and surface tension in solutions. Credit: Dominick Reuter Colding are widely considered to be the world's leading geometric analysts of their generation. In 2004 the duo jointly published a series of papers It's something children do every day when blowing in the Annals of Mathematics that resolved a bubbles: Stick a circular wire in a pot of soapy number of longstanding conjectures in the field; this water, pull it out, and behold the film forming earned them the prestigious Oswald Veblen Prize across it. in Geometry. But it's not only children who are amused by this Of particular interest to Minicozzi and Colding was phenomenon—which has also kept mathematicians whether it is possible to describe what all minimal occupied since the 18th century, says William surfaces look like.
    [Show full text]
  • Arxiv:2012.02249V3 [Math.DG] 24 May 2021
    BUILDING MANIFOLDS FROM QUANTUM CODES MICHAEL FREEDMAN AND MATTHEW HASTINGS ABSTRACT. We give a procedurefor “reverseengineering”a closed, simply connected, Riemannian manifold with boundedlocal geometry from a sparse chain complex over Z. Applying this procedure to chain complexes obtained by “lifting” recently developed quantum codes, which correspond to chain complexes over Z2, we construct the first examples of power law Z2 systolic freedom. As a result that may be of independent interest in graph theory, we give an efficient randomized algorithm to construct a weakly fundamental cycle basis for a graph, such that each edge appears only polylogarithmically times in the basis. We use this result to trivialize the fundamental group of the manifold we construct. 1. BUILDING MANIFOLDS FROM QUANTUM CODES 1.1. Introduction. The last fifty years has seen a glorious exploration of the possible “shapes” of Riemannian manifolds. Often the new ideas contained a combinatorial element reminiscent of geometric topology and rather far from the study of homogenous spaces in which Riemann- ian geometry had incubated during the previous fifty years. This paper is in the modern spirit of building and studying Riemannian manifolds from a combinatorial perspective. In this case we use recently discovered quantum codes to build closed Riemannian manifolds with a surprising prop- erty: power law Z2-systolic freedom. This discovery in effect reverses the historical flow of infor- mation. Twenty years ago one of us helped to create quantum codes exhibiting a weaker poly(log) Z2-systolic freedom [FML02,Fre99], starting with a certain family of Riemannian 3-manifolds. In retrospect it is not surprising that the most efficient constructions will necessarily have a combina- torial complexity beyond a geometer’s intuition, perhaps even requiring computer search, and are better discovered within coding theory and then translated into geometry.
    [Show full text]
  • Joint Journals Catalogue EMS / MSP 2020
    Joint Journals Cataloguemsp EMS / MSP 1 2020 Super package deal inside! msp 1 EEuropean Mathematical Society Mathematical Science Publishers msp 1 The EMS Publishing House is a not-for-profit Mathematical Sciences Publishers is a California organization dedicated to the publication of high- nonprofit corporation based in Berkeley. MSP quality peer-reviewed journals and high-quality honors the best traditions of quality publishing books, on all academic levels and in all fields of while moving with the cutting edge of information pure and applied mathematics. The proceeds from technology. We publish more than 16,000 pages the sale of our publications will be used to keep per year, produce and distribute scientific and the Publishing House on a sound financial footing; research literature of the highest caliber at the any excess funds will be spent in compliance lowest sustainable prices, and provide the top with the purposes of the European Mathematical quality of mathematically literate copyediting and Society. The prices of our products will be set as typesetting in the industry. low as is practicable in the light of our mission and We believe scientific publishing should be an market conditions. industry that helps rather than hinders scholarly activity. High-quality research demands high- Contact addresses quality communication – widely, rapidly and easily European Mathematical Society Publishing House accessible to all – and MSP works to facilitate it. Technische Universität Berlin, Mathematikgebäude Straße des 17. Juni 136, 10623 Berlin, Germany Contact addresses Email: [email protected] Mathematical Sciences Publishers Web: www.ems-ph.org 798 Evans Hall #3840 c/o University of California Managing Director: Berkeley, CA 94720-3840, USA Dr.
    [Show full text]
  • THE UNIVERSITY of CALIFORNIA, IRVINE on Singularities and Weak
    THE UNIVERSITY OF CALIFORNIA, IRVINE On Singularities and Weak Solutions of Mean Curvature Flow DISSERTATION Submitted in partial satisfaction of the requirements for the degree of DOCTOR OF PHILOSOPHY in mathematics by Alexander Everest Mramor Dissertation Committee: Professor Richard Schoen, Chair Associate Professor Li-Sheng Tseng Assistant Professor Xiangwen Zhang 2019 • Portions of chapter 4 of this thesis were originally published in Entropy and the generic mean curvature flow in curved ambient spaces, Proc. Amer. Math. Soc. 146, 2663-2677, American Mathematical Society, Providence, RI, 2018. • Portions of chapter 6 of this thesis were originally published in A finiteness the- orem via the mean curvature flow with surgery, Journal of geometric analysis, December 2018, Volume 28, Issue 4, pp 3348{3372, Springer Nature Switzer- land AG. • Portions of chapter 7 of this thesis are to appear in Regularity and stability results for the level set flow via the mean curvature flow with surgery in Com- munications in Analysis and Geometry, International Press of Boston, Boston, Mass • Portions of chapter 8 of this thesis are to appear in On the topological rigidity of self shrinkers in R3 in Int. Math. Res. Not., Oxford univerisity press, Oxford, UK, 2019. All other materials c 2019 Alexander Mramor ii Table of contents List of Figures . vi Acknowledgements . vii Curriculum Vitae . ix Abstract of the dissertation . .x Preface 1 1 Introduction to the mean curvature flow 4 1.1 Classical formulation of the mean curvature flow . .5 1.2 Mean curvature flow with surgery for compact 2-convex hypersurfaces 11 1.2.1 Mean curvature flow with surgery according to Huisken and Sinestrari .
    [Show full text]
  • Princeton University Press Spring 2019 Catalog
    Mathematics 2019 press.princeton.edu NEW & FORTHCOMING “We o en claim that education should not just teach facts; it should help us learn how to think clearly. [ is] is a book that takes that goal seriously. It is brilliantly constructed, clearly written, and fun.” —William C. Powers Jr., former president of the University of Texas, Austin Making Up Your Own Mind We solve countless problems—big and small—every day. With so much practice, why do we o en have trouble making simple decisions—much less arriving at optimal solutions to important questions? Is there a practical way to learn to think more e ectively and creatively? Edward Burger shows how we can become far better at solving real-world problems by learning creative puzzle-solving skills using simple, e ective thinking techniques. EDWARD B. BURGER is the president of Southwestern University, a mathematics professor, and a leading teacher on thinking, innovation, and creativity. He 2018. 136 pages. 35 b/w illus. 4 ½ x 7 ½ . has written more than seventy research articles, video Hardback 9780691182780 $19.95 | £14.99 series, and books, including e 5 Elements of E ective E-book 9780691188881 Audiobook 9780691193014 inking (with Michael Starbird) (Princeton). “Much of today’s college talk revolves around getting in—but this book meaningfully shi s the focus to how to be successful once getting to college. Johnson provides expert advice to make this book an important and eye-opening read.” —Sarah Graham, director of college counseling, Princeton Day School Will This Be on the Test? is is the essential survival guide for high-school students making the transition to college academics.
    [Show full text]
  • NOTICES of the AMS VOLUME 42, NUMBER 8 People.Qxp 5/8/98 3:47 PM Page 887
    people.qxp 5/8/98 3:47 PM Page 886 Mathematics People anybody in need; where others might slough it off, he is Franklin Institute Awards right there.” In May, the Franklin Institute presented medals to seven Chen Ning Yang of the State University of New York at individuals whose pioneering work in science, engineering, Stony Brook received the Bower Award and Prize for and education made significant contributions to under- Achievement in Science. This is the first year that this standing of the world. In addition to an awards dinner, a $250,000 award was given. The citation for the award to series of lectures and symposia highlighted the work of the Yang reads: “For the formulation of a general field theory honorees. Among those receiving awards were three who which synthesizes the physical laws of nature and provides have made important mathematical contributions. us with an understanding of the fundamental forces of the Joel L. Lebowitz of Rutgers University received the universe. As one of the conceptual masterpieces of the twen- Delmer S. Fahrney Medal “for his dynamic leadership which tieth century explaining the interaction of subatomic par- has advanced the field of statistical mechanics, and for his ticles, his theory has profoundly reshaped the development inspiring humanitarian efforts on behalf of oppressed sci- of physics and modern geometry during the last forty entists throughout the world.” Lebowitz, the George William years. This theoretical model, already ranked alongside the Hill Professor of Mathematics and Physics at Rutgers, is not works of Newton, Maxwell, and Einstein, will surely have only a renowned mathematician and physicist but also a a comparable influence on future generations.
    [Show full text]
  • Singular Behavior of Minimal Surfaces and Mean Curvature Flow
    Singular Behavior of Minimal Surfaces and Mean Curvature Flow by Stephen James Kleene A dissertation submitted to The Johns Hopkins University in conformity with the requirements for the degree of Doctor of Philosophy. Baltimore, Maryland April, 2010 c Stephen James Kleene 2010 All rights reserved Abstract This document records three distinct theorems that the author proved, along with his collaborators, while a graduate student at The Johns Hopkins University. The author, with M. Calle and J. Kramer, generalized a sharp estimate of Tobias Colding and William Minicozzi for the extinction time of of convex hyper-surfaces in euclidean space moving my their mean curvature vector to a much broader class of evolutions studied by Ben Andrews in (1). Also, the author gave an alternate proof, first given by D. Hoffman and B. White in (17), of very poor limiting behavior for sequences of minimal surfaces in euclidean three space. Finally, the author, together with N. Moller in (23) constructed a new family o asymptotically f conical ends that satisfy the mean curvature self shrinking equation in euclidean three space of all dimensions. ii Acknowledgements Thanks Bill Minicozzi for his continuing advice and support during my time as his student. Thanks to Bill, Chika Mese, and Joel Spruck for the many excellent classes on Geometry and PDE’s over the years; they have benefited me immeasurably. Thanks to all my wonderful friends here in Baltimore, who provided vital advice and support in all things non-mathematical. Thanks to Morgan and Nevena, who always gave me a place to stay and food to eat (and cash) when I needed it.
    [Show full text]
  • Integral 2017
    Spring 2017 Integral Volume 10 NEWS FROM THE MATHEMATICS DEPARTMENT AT MIT Dear Friends, t’s been a while since our last issue of IIntegral. The past two years have been busy, including our move back into the Simons Building in January 2016. There are a lot of new faces in the department. We are also happy that several department members won major honors for their work and service. This issue will catch you up on happenings in the department, up to spring 2017. Move to the Simons Building Our move to the Simons Building went smoothly, and we are thrilled with the results of the renovation. If you haven’t seen the department, come by and take a look. While still familiar, there are many changes. The first floor now houses the newly renovated first-year graduate studentGwen McKinley Simons Lectures Math Majors’ Lounge and the Samberg and Simons Professor , who Bonnie Berger The annual Simons Lecture series this spring Administrative Suite next door, which includes shared their thoughts on how the new spaces featured talks by from Microsoft Headquarters and Math Academic Services. The are changing our lives in the department. Yuval Peres Research and from the Calderón Lecture Hall now has tiered seating Martin Hairer Thanks also go out to the MIT administration University of Warwick, and in 2016, with state-of-the-art audio/visual, including Michael for their support on this huge project, and from Harvard University. lecture capture. Three grand views highlight the Brenner to Michael Sipser, our former department spacious Norbert Wiener Common Room.
    [Show full text]
  • Mean Curvature Flow
    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 52, Number 2, April 2015, Pages 297–333 S 0273-0979(2015)01468-0 Article electronically published on January 13, 2015 MEAN CURVATURE FLOW TOBIAS HOLCK COLDING, WILLIAM P. MINICOZZI II, AND ERIK KJÆR PEDERSEN Abstract. Mean curvature flow is the negative gradient flow of volume, so any hypersurface flows through hypersurfaces in the direction of steepest descent for volume and eventually becomes extinct in finite time. Before it becomes extinct, topological changes can occur as it goes through singularities. If the hypersurface is in general or generic position, then we explain what singulari- ties can occur under the flow, what the flow looks like near these singularities, and what this implies for the structure of the singular set. At the end, we will briefly discuss how one may be able to use the flow in low-dimensional topology. 0. Introduction Imagine that a closed surface in R3 flows in time to decrease its area as rapidly as possible. Convex points will move inward, while concave points move outward, the speed is slower where the surface is flatter. Independently of whether points move inward or outward, the total area will decrease along the flow and eventually go to zero in finite time. In particular, any closed surface becomes extinct in finite time and, thus, the flow can only be continued smoothly for some finite amount of time before singularities occur. Mean curvature flow (MCF) is the negative gradient flow for area. This is a nonlinear partial differential equation for the evolving hypersurface that is formally similar to the ordinary heat equation, with some important differences.
    [Show full text]