CMS 75+1 Anniversary 75+1 Anniversaire
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The Weil Proof and the Geometry of the Adèles Class Space
The Weil Proof and the Geometry of the Adèles Class Space Alain Connes,1 Caterina Consani,2 and Matilde Marcolli3 1 Collège de France, 3, rue d’Ulm, Paris, F-75005 France [email protected] 2 Mathematics Department, Johns Hopkins University, Baltimore, MD 21218 USA [email protected] 3 Department of Mathematics, California Institute of Technology 1200 E California Blvd, Pasadena, CA 91101, USA [email protected] Dedicated to Yuri Manin on the occasion of his 70th birthday O simili o dissimili che sieno questi mondi non con minor raggione sarebe bene a l’uno l’essere che a l’altro Giordano Bruno – De l’infinito, universo e mondi Summary. This paper explores analogies between the Weil proof of the Riemann hypothesis for function fields and the geometry of the adèles class space, which is the noncommutative space underlying Connes’ spectral realization of the zeros of the Riemann zeta function. We consider the cyclic homology of the cokernel (in the abelian category of cyclic modules) of the “restriction map” defined by the inclu- sion of the idèles class group of a global field in the noncommutative adèles class space. Weil’s explicit formula can then be formulated as a Lefschetz trace formula for the induced action of the idèles class group on this cohomology. In this formu- lation the Riemann hypothesis becomes equivalent to the positivity of the relevant trace pairing. This result suggests a possible dictionary between the steps in the Weil proof and corresponding notions involving the noncommutative geometry of the adèles class space, with good working notions of correspondences, degree, and codegree etc. -
I. Overview of Activities, April, 2005-March, 2006 …
MATHEMATICAL SCIENCES RESEARCH INSTITUTE ANNUAL REPORT FOR 2005-2006 I. Overview of Activities, April, 2005-March, 2006 …......……………………. 2 Innovations ………………………………………………………..... 2 Scientific Highlights …..…………………………………………… 4 MSRI Experiences ….……………………………………………… 6 II. Programs …………………………………………………………………….. 13 III. Workshops ……………………………………………………………………. 17 IV. Postdoctoral Fellows …………………………………………………………. 19 Papers by Postdoctoral Fellows …………………………………… 21 V. Mathematics Education and Awareness …...………………………………. 23 VI. Industrial Participation ...…………………………………………………… 26 VII. Future Programs …………………………………………………………….. 28 VIII. Collaborations ………………………………………………………………… 30 IX. Papers Reported by Members ………………………………………………. 35 X. Appendix - Final Reports ……………………………………………………. 45 Programs Workshops Summer Graduate Workshops MSRI Network Conferences MATHEMATICAL SCIENCES RESEARCH INSTITUTE ANNUAL REPORT FOR 2005-2006 I. Overview of Activities, April, 2005-March, 2006 This annual report covers MSRI projects and activities that have been concluded since the submission of the last report in May, 2005. This includes the Spring, 2005 semester programs, the 2005 summer graduate workshops, the Fall, 2005 programs and the January and February workshops of Spring, 2006. This report does not contain fiscal or demographic data. Those data will be submitted in the Fall, 2006 final report covering the completed fiscal 2006 year, based on audited financial reports. This report begins with a discussion of MSRI innovations undertaken this year, followed by highlights -
Noncommutative Geometry Models for Particle Physics and Cosmology, Lecture I
Noncommutative Geometry models for Particle Physics and Cosmology, Lecture I Matilde Marcolli Villa de Leyva school, July 2011 Matilde Marcolli NCG models for particles and cosmology, I Plan of lectures 1 Noncommutative geometry approach to elementary particle physics; Noncommutative Riemannian geometry; finite noncommutative geometries; moduli spaces; the finite geometry of the Standard Model 2 The product geometry; the spectral action functional and its asymptotic expansion; bosons and fermions; the Standard Model Lagrangian; renormalization group flow, geometric constraints and low energy limits 3 Parameters: relations at unification and running; running of the gravitational terms; the RGE flow with right handed neutrinos; cosmological timeline and the inflation epoch; effective gravitational and cosmological constants and models of inflation 4 The spectral action and the problem of cosmic topology; cosmic topology and the CMB; slow-roll inflation; Poisson summation formula and the nonperturbative spectral action; spherical and flat space forms Matilde Marcolli NCG models for particles and cosmology, I Geometrization of physics Kaluza-Klein theory: electromagnetism described by circle bundle over spacetime manifold, connection = EM potential; Yang{Mills gauge theories: bundle geometry over spacetime, connections = gauge potentials, sections = fermions; String theory: 6 extra dimensions (Calabi-Yau) over spacetime, strings vibrations = types of particles NCG models: extra dimensions are NC spaces, pure gravity on product space becomes gravity -
Richard Schoen – Mathematics
Rolf Schock Prizes 2017 Photo: Private Photo: Richard Schoen Richard Schoen – Mathematics The Rolf Schock Prize in Mathematics 2017 is awarded to Richard Schoen, University of California, Irvine and Stanford University, USA, “for groundbreaking work in differential geometry and geometric analysis including the proof of the Yamabe conjecture, the positive mass conjecture, and the differentiable sphere theorem”. Richard Schoen holds professorships at University of California, Irvine and Stanford University, and is one of three vice-presidents of the American Mathematical Society. Schoen works in the field of geometric analysis. He is in fact together with Shing-Tung Yau one of the founders of the subject. Geometric analysis can be described as the study of geometry using non-linear partial differential equations. The developments in and around this field has transformed large parts of mathematics in striking ways. Examples include, gauge theory in 4-manifold topology, Floer homology and Gromov-Witten theory, and Ricci-and mean curvature flows. From the very beginning Schoen has produced very strong results in the area. His work is characterized by powerful technical strength and a clear vision of geometric relevance, as demonstrated by him being involved in the early stages of areas that later witnessed breakthroughs. Examples are his work with Uhlenbeck related to gauge theory and his work with Simon and Yau, and with Yau on estimates for minimal surfaces. Schoen has also established a number of well-known and classical results including the following: • The positive mass conjecture in general relativity: the ADM mass, which measures the deviation of the metric tensor from the imposed flat metric at infinity is non-negative. -
Lectures on Factorization Homology, ∞-Categories, and Topological
SPRINGER BRIEFS IN MATHEMATICAL PHYSICS 39 Hiro Lee Tanaka Lectures on Factorization Homology, ∞-Categories, and Topological Field Theories Contributed by Araminta Amabel · Artem Kalmykov · Lukas Müller SpringerBriefs in Mathematical Physics Volume 39 Series Editors Nathanaël Berestycki, University of Vienna, Vienna, Austria Mihalis Dafermos, Mathematics Department, Princeton University, Princeton, NJ, USA Atsuo Kuniba, Institute of Physics, The University of Tokyo, Tokyo, Japan Matilde Marcolli, Department of Mathematics, University of Toronto, Toronto, Canada Bruno Nachtergaele, Department of Mathematics, Davis, CA, USA Hirosi Ooguri, Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA, USA SpringerBriefs are characterized in general by their size (50–125 pages) and fast production time (2–3 months compared to 6 months for a monograph). Briefs are available in print but are intended as a primarily electronic publication to be included in Springer’s e-book package. Typical works might include: • An extended survey of a field • A link between new research papers published in journal articles • A presentation of core concepts that doctoral students must understand in order to make independent contributions • Lecture notes making a specialist topic accessible for non-specialist readers. SpringerBriefs in Mathematical Physics showcase, in a compact format, topics of current relevance in the field of mathematical physics. Published titles will encompass all areas of theoretical and mathematical physics. -
President's Report
AWM ASSOCIATION FOR WOMEN IN MATHE MATICS Volume 36, Number l NEWSLETTER March-April 2006 President's Report Hidden Help TheAWM election results are in, and it is a pleasure to welcome Cathy Kessel, who became President-Elect on February 1, and Dawn Lott, Alice Silverberg, Abigail Thompson, and Betsy Yanik, the new Members-at-Large of the Executive Committee. Also elected for a second term as Clerk is Maura Mast.AWM is also pleased to announce that appointed members BettyeAnne Case (Meetings Coordi nator), Holly Gaff (Web Editor) andAnne Leggett (Newsletter Editor) have agreed to be re-appointed, while Fern Hunt and Helen Moore have accepted an extension of their terms as Member-at-Large, to join continuing members Krystyna Kuperberg andAnn Tr enk in completing the enlarged Executive Committee. I look IN THIS ISSUE forward to working with this wonderful group of people during the coming year. 5 AWM ar the San Antonio In SanAntonio in January 2006, theAssociation for Women in Mathematics Joint Mathematics Meetings was, as usual, very much in evidence at the Joint Mathematics Meetings: from 22 Girls Just Want to Have Sums the outstanding mathematical presentations by women senior and junior, in the Noerher Lecture and the Workshop; through the Special Session on Learning Theory 24 Education Column thatAWM co-sponsored withAMS and MAA in conjunction with the Noether Lecture; to the two panel discussions thatAWM sponsored/co-sponsored.AWM 26 Book Review also ran two social events that were open to the whole community: a reception following the Gibbs lecture, with refreshments and music that was just right for 28 In Memoriam a networking event, and a lunch for Noether lecturer Ingrid Daubechies. -
LECTURE Series
University LECTURE Series Volume 52 Koszul Cohomology and Algebraic Geometry Marian Aprodu Jan Nagel American Mathematical Society http://dx.doi.org/10.1090/ulect/052 Koszul Cohomology and Algebraic Geometry University LECTURE Series Volume 52 Koszul Cohomology and Algebraic Geometry Marian Aprodu Jan Nagel M THE ATI A CA M L ΤΡΗΤΟΣ ΜΗ N ΕΙΣΙΤΩ S A O C C I I R E E T ΑΓΕΩΜΕ Y M A F O 8 U 88 NDED 1 American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Jerry L. Bona NigelD.Higson Eric M. Friedlander (Chair) J. T. Stafford 2000 Mathematics Subject Classification. Primary 14H51, 14C20, 14H60, 14J28, 13D02, 16E05. For additional information and updates on this book, visit www.ams.org/bookpages/ulect-52 Library of Congress Cataloging-in-Publication Data Aprodu, Marian. Koszul cohomology and algebraic geometry / Marian Aprodu, Jan Nagel. p. cm. — (University lecture series ; v. 52) Includes bibliographical references and index. ISBN 978-0-8218-4964-4 (alk. paper) 1. Koszul algebras. 2. Geometry, Algebraic. 3. Homology theory. I. Nagel, Jan. II. Title. QA564.A63 2010 512.46—dc22 2009042378 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. -
Curriculum Vitae.Pdf
Lan-Hsuan Huang Department of Mathematics Phone: (860) 486-8390 University of Connecticut Fax: (860) 486-4238 Storrs, CT 06269 Email: [email protected] USA http://lhhuang.math.uconn.edu Research Geometric Analysis and General Relativity Employment University of Connecticut Professor 2020-present Associate Professor 2016-2020 Assistant Professor 2012-2016 Institute for Advanced Study Member (with the title of von Neumann fellow) 2018-2019 Columbia University Ritt Assistant Professor 2009-2012 Education Ph.D. Mathematics, Stanford University 2009 Advisor: Professor Richard Schoen B.S. Mathematics, National Taiwan University 2004 Grants • NSF DMS-2005588 (PI, $250,336) 2020-2023 & Honors • von Neumann Fellow, Institute for Advanced Study 2018-2019 • Simons Fellow in Mathematics, Simons Foundation ($122,378) 2018-2019 • NSF CAREER Award (PI, $400,648) 2015-2021 • NSF Grant DMS-1308837 (PI, $282,249) 2013-2016 • NSF Grant DMS-1005560 and DMS-1301645 (PI, $125,645) 2010-2013 Visiting • Erwin Schr¨odingerInternational Institute July 2017 Positions • National Taiwan University Summer 2016 • MSRI Research Member Fall 2013 • Max-Planck Institute for Gravitational Physics, Germany Fall 2010 • Institut Mittag-Leffler, Sweden Fall 2008 1 Journal 1. Equality in the spacetime positive mass theorem (with D. Lee), Commu- Publications nications in Mathematical Physics 376 (2020), no. 3, 2379{2407. 2. Mass rigidity for hyperbolic manifolds (with H. C. Jang and D. Martin), Communications in Mathematical Physics 376 (2020), no. 3, 2329- 2349. 3. Localized deformation for initial data sets with the dominant energy condi- tion (with J. Corvino), Calculus Variations and Partial Differential Equations (2020), no. 1, No. 42. 4. Existence of harmonic maps into CAT(1) spaces (with C. -
Research Notices
AMERICAN MATHEMATICAL SOCIETY Research in Collegiate Mathematics Education. V Annie Selden, Tennessee Technological University, Cookeville, Ed Dubinsky, Kent State University, OH, Guershon Hare I, University of California San Diego, La jolla, and Fernando Hitt, C/NVESTAV, Mexico, Editors This volume presents state-of-the-art research on understanding, teaching, and learning mathematics at the post-secondary level. The articles are peer-reviewed for two major features: (I) advancing our understanding of collegiate mathematics education, and (2) readability by a wide audience of practicing mathematicians interested in issues affecting their students. This is not a collection of scholarly arcana, but a compilation of useful and informative research regarding how students think about and learn mathematics. This series is published in cooperation with the Mathematical Association of America. CBMS Issues in Mathematics Education, Volume 12; 2003; 206 pages; Softcover; ISBN 0-8218-3302-2; List $49;AII individuals $39; Order code CBMATH/12N044 MATHEMATICS EDUCATION Also of interest .. RESEARCH: AGul<lelbrthe Mathematics Education Research: Hothomatldan- A Guide for the Research Mathematician --lllll'tj.M...,.a.,-- Curtis McKnight, Andy Magid, and -- Teri J. Murphy, University of Oklahoma, Norman, and Michelynn McKnight, Norman, OK 2000; I 06 pages; Softcover; ISBN 0-8218-20 16-8; List $20;AII AMS members $16; Order code MERN044 Teaching Mathematics in Colleges and Universities: Case Studies for Today's Classroom Graduate Student Edition Faculty -
RM Calendar 2017
Rudi Mathematici x3 – 6’135x2 + 12’545’291 x – 8’550’637’845 = 0 www.rudimathematici.com 1 S (1803) Guglielmo Libri Carucci dalla Sommaja RM132 (1878) Agner Krarup Erlang Rudi Mathematici (1894) Satyendranath Bose RM168 (1912) Boris Gnedenko 1 2 M (1822) Rudolf Julius Emmanuel Clausius (1905) Lev Genrichovich Shnirelman (1938) Anatoly Samoilenko 3 T (1917) Yuri Alexeievich Mitropolsky January 4 W (1643) Isaac Newton RM071 5 T (1723) Nicole-Reine Etable de Labrière Lepaute (1838) Marie Ennemond Camille Jordan Putnam 2002, A1 (1871) Federigo Enriques RM084 Let k be a fixed positive integer. The n-th derivative of (1871) Gino Fano k k n+1 1/( x −1) has the form P n(x)/(x −1) where P n(x) is a 6 F (1807) Jozeph Mitza Petzval polynomial. Find P n(1). (1841) Rudolf Sturm 7 S (1871) Felix Edouard Justin Emile Borel A college football coach walked into the locker room (1907) Raymond Edward Alan Christopher Paley before a big game, looked at his star quarterback, and 8 S (1888) Richard Courant RM156 said, “You’re academically ineligible because you failed (1924) Paul Moritz Cohn your math mid-term. But we really need you today. I (1942) Stephen William Hawking talked to your math professor, and he said that if you 2 9 M (1864) Vladimir Adreievich Steklov can answer just one question correctly, then you can (1915) Mollie Orshansky play today. So, pay attention. I really need you to 10 T (1875) Issai Schur concentrate on the question I’m about to ask you.” (1905) Ruth Moufang “Okay, coach,” the player agreed. -
Homogenization 2001, Proceedings of the First HMS2000 International School and Conference on Ho- Mogenization
in \Homogenization 2001, Proceedings of the First HMS2000 International School and Conference on Ho- mogenization. Naples, Complesso Monte S. Angelo, June 18-22 and 23-27, 2001, Ed. L. Carbone and R. De Arcangelis, 191{211, Gakkotosho, Tokyo, 2003". On Homogenization and Γ-convergence Luc TARTAR1 In memory of Ennio DE GIORGI When in the Fall of 1976 I had chosen \Homog´en´eisationdans les ´equationsaux d´eriv´eespartielles" (Homogenization in partial differential equations) for the title of my Peccot lectures, which I gave in the beginning of 1977 at Coll`egede France in Paris, I did not know of the term Γ-convergence, which I first heard almost a year after, in a talk that Ennio DE GIORGI gave in the seminar that Jacques-Louis LIONS was organizing at Coll`egede France on Friday afternoons. I had not found the definition of Γ-convergence really new, as it was quite similar to questions that were already discussed in control theory under the name of relaxation (which was a more general question than what most people mean by that term now), and it was the convergence in the sense of Umberto MOSCO [Mos] but without the restriction to convex functionals, and it was the natural nonlinear analog of a result concerning G-convergence that Ennio DE GIORGI had obtained with Sergio SPAGNOLO [DG&Spa]; however, Ennio DE GIORGI's talk contained a quite interesting example, for which he referred to Luciano MODICA (and Stefano MORTOLA) [Mod&Mor], where functionals involving surface integrals appeared as Γ-limits of functionals involving volume integrals, and I thought that it was the interesting part of the concept, so I had found it similar to previous questions but I had felt that the point of view was slightly different. -
Stable Minimal Hypersurfaces in Four-Dimensions
STABLE MINIMAL HYPERSURFACES IN R4 OTIS CHODOSH AND CHAO LI Abstract. We prove that a complete, two-sided, stable minimal immersed hyper- surface in R4 is flat. 1. Introduction A complete, two-sided, immersed minimal hypersurface M n → Rn+1 is stable if 2 2 2 |AM | f ≤ |∇f| (1) ZM ZM ∞ for any f ∈ C0 (M). We prove here the following result. Theorem 1. A complete, connected, two-sided, stable minimal immersion M 3 → R4 is a flat R3 ⊂ R4. This resolves a well-known conjecture of Schoen (cf. [14, Conjecture 2.12]). The corresponding result for M 2 → R3 was proven by Fischer-Colbrie–Schoen, do Carmo– Peng, and Pogorelov [21, 18, 36] in 1979. Theorem 1 (and higher dimensional analogues) has been established under natural cubic volume growth assumptions by Schoen– Simon–Yau [37] (see also [45, 40]). Furthermore, in the special case that M n ⊂ Rn+1 is a minimal graph (implying (1) and volume growth bounds) flatness of M is known as the Bernstein problem, see [22, 17, 3, 45, 6]. Several authors have studied Theorem 1 under some extra hypothesis, see e.g., [41, 8, 5, 44, 11, 32, 30, 35, 48]. We also note here some recent papers [7, 19] concerning stability in related contexts. It is well-known (cf. [50, Lecture 3]) that a result along the lines of Theorem 1 yields curvature estimates for minimal hypersurfaces in R4. Theorem 2. There exists C < ∞ such that if M 3 → R4 is a two-sided, stable minimal arXiv:2108.11462v2 [math.DG] 2 Sep 2021 immersion, then |AM (p)|dM (p,∂M) ≤ C.