DOCSLIB.ORG

Mirror Symmetry for Honeycombs

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Mirror Symmetry for Honeycombs TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 373, Number 1, January 2020, Pages 71–107 https://doi.org/10.1090/tran/7909 Article electronically published on September 10, 2019 MIRROR SYMMETRY FOR HONEYCOMBS BENJAMIN GAMMAGE AND DAVID NADLER Abstract. We prove a homological mirror symmetry equivalence between the A-brane category of the pair of pants, computed as a wrapped microlocal sheaf category, and the B-brane category of its mirror LG model, understood as a category of matrix factorizations. The equivalence improves upon prior results in two ways: it intertwines evident affine Weyl group symmetries on both sides, and it exhibits the relation of wrapped microlocal sheaves along different types of Lagrangian skeleta for the same hypersurface. The equivalence proceeds through the construction of a combinatorial realization of the A-model via arboreal singularities. The constructions here represent the start of a program to generalize to higher dimensions many of the structures which have appeared in topological approaches to Fukaya categories of surfaces. Contents 1. Introduction 71 2. Combinatorial A-model 78 3. Mirror symmetry 95 4. Symplectic geometry 100 Acknowledgments 106 References 106 1. Introduction This paper fits into the framework of homological mirror symmetry, as introduced in [23] and expanded in [19, 20, 22]. The formulation of interest to us relates the A-model of a hypersurface X in a toric variety to the mirror Landau-Ginzburg B- model of a toric variety X∨ equipped with superpotential W ∨ ∈O(X∨). Following Mikhalkin [27], a distinguished “atomic” case is when the hypersurface is the pair of pants ∗ n ∼ ∗ 1 n Pn−1 = {z1 + ···+ zn +1=0}⊂(C ) = T (S ) n+1 with mirror Landau-Ginzburg model (A ,z1 ···zn+1). In this paper, we will also be interested in the universal abelian cover Pn−1 of the pair of pants, which fits in Received by the editors December 7, 2018, and, in revised form, April 12, 2019. 2010 Mathematics Subject Classification. Primary 14J33, 53D37. Key words and phrases. Mirror symmetry, microlocal sheaves. The first author is grateful to the NSF for the support of a Graduate Research Fellowship. The second author is grateful for the support of grant DMS-1502178. c 2019 American Mathematical Society 71 Licensed to Univ of Calif, Berkeley. Prepared on Fri May 1 11:16:12 EDT 2020 for download from IP 128.32.10.230. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 72 BENJAMIN GAMMAGE AND DAVID NADLER the Cartesian diagram / ∗ n Pn−1 T R / ∗ 1 n Pn−1 T (S ) ∗ n ∗ 1 n as the pullback of Pn−1 along the universal covering map T R → T (S ) ; it has mirror a torus-equivariant version of the Landau-Ginzburg model (An+1, z1 ···zn+1). This paper expands upon prior mirror symmetry equivalences for pairs of pants found in [1, 36, 39]; however, it differs from those in its understanding of the A- model. The traditional mathematical realization of the A-model is the Fukaya category, with objects decorated Lagrangian submanifolds, morphisms their deco- rated intersections, and structure constants defined by integrals over moduli spaces of pseudoholomorphic polygons. There is increasing evidence (for example, [3, 9, 14–16, 24, 30, 33, 38, 41]) that the Fukaya category of a Weinstein manifold is equivalent to microlocal sheaves (as developed by Kashiwara-Schapira [21]) along a Lagrangian skeleton. In this paper, we follow [32] and study the A-model of the pair of pants in its guise as wrapped microlocal sheaves. A calculation of microlocal sheaves on a skeleton for the pair of pants was per- formed already in [32]; our calculation here involves a different skeleton, which is of independent interest. The skeleton we study is more symmetrical, having an action of the symmetric group Σn+1 rather than just Σn, but more importantly, the skeleton here is of a different “flavor” compared to the one constructed there. The calculations from [32] are well-adapted to considerations of mirror symmetry which relate a hypersurface in (C×)n to a toric degeneration and were used in [17] for this purpose. The skeleton we study in this paper is more adapted to mirror symmetry equivalences which relate a hypersurface in (C×)n to a Landau-Ginzburg model. The first sort of skeleton can be considered as a “degeneration” of the sec- ond; indeed, the relation between these two flavors of skeleton is very interesting and will be studied further in future work. The skeleton from this paper is very well-suited to a combinatorial perspective, since singularities are all arboreal in the sense of [28]. The form of our calculations should be understood as a paradigm for extending the substantial literature devoted to understanding Fukaya categories of Riemann surfaces through topological skeleta and ribbon graphs (for example, [2, 6, 18, 35, 40]) to higher-dimensional examples. Moreover, the type of skeleton described here has close relations to the dimer models which have appeared in earlier mirror symmetry contexts (e.g., [10, 11]). In future work, we hope to explore further the relation between skeleta and dimer models, along with generalizations to higher dimensions. This correspondence was noticed (in a slightly different form) in [11], and in Section 4.2 we make use of the Lefschetz fibrations described in that paper. 1.1. Symplectic geometry. 1.1.1. Cotangent bundles. Fix a characteristic zero coefficient field k, and let L ⊂ T ∗X be a closed conic Lagrangian submanifold of a cotangent bundle. There are ♦ ∗ conic sheaves of dg categories μShL and μShL on T X, localized along L, which ⊂ ∗ ♦ to a conic open set Ω T X assign, respectively, the dg category μShL(Ω) of Licensed to Univ of Calif, Berkeley. Prepared on Fri May 1 11:16:12 EDT 2020 for download from IP 128.32.10.230. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use MIRROR SYMMETRY FOR HONEYCOMBS 73 unbounded-rank microlocal sheaves and the dg category μShL(Ω) of traditional microlocal sheaves along L ∩ Ω. wr Definition 1.1.2. The category μShL (Ω) of wrapped microlocal sheaves along ∩ ♦ c ♦ L Ω is the category μShL(Ω) of compact objects inside μShL(Ω). → wr Proposition 1.1.3 ([32, Proposition 3.16]). The assignment Ω μShL (Ω) forms wr ∗ a cosheaf μShL of dg categories on T X, localized on L. If x is a smooth point of L and Ω is a contractible conic neighborhood of x,then ♦ the stalk of the sheaf μShL is equivalent to the dg category Modk of (unbounded- w rank) k-modules, while the stalk of μShL and the costalk of μShL at x are both equivalent to the dg category Perfk of perfect k-modules. If x is a singular point, the local calculation is more complicated, but this cal- culation has already been performed in [28] for a certain class of Legendrian sin- gularities termed arboreal. In this paper we will only be concerned with the An L arboreal singularity An , a certain singular Legendrian in the projectivized cotan- ∞ n n gent bundle T (R ) which is homeomorphic to Cone(skn−2Δ ), the cone on the (n − 2)-skeleton of an n-simplex. Proposition 1.1.4 ([28]). Let L be a conic Lagrangian in T ∗Rn which is locally ∈ L equivalent, near a point x L, to the cone on An . Then to a neighborhood of x, wr the sheaf μShL and the cosheaf μShL each assign the category An -Perfk of perfect modules over the An quiver. 1.1.5. Weinstein manifolds. Let W be a Weinstein manifold. The Weinstein struc- ture of W endows it with a Lagrangian skeleton Λ, onto which W deformation retracts. Since our definition of wrapped microlocal sheaf categories applies only in the setting of cotangent bundles, in order to apply it here we have to relate the geometry of our Weinstein manifold to the geometry of a cotangent bundle. Let Λ be the skeleton of a Weinstein manifold W and let U be an open neighborhood of Λ which is conic for the flow of the Liouville vector field of W , and suppose that there exist a manifold X and a closed conic Lagrangian L ⊂ T ∗X such that U is exact symplectomorphic to a neighborhood Ω of L by a symplectomorphism taking Λ to L. w Definition 1.1.6. In the situation described above, the category μShL (Ω) is the microlocal A-model category associated to the Weinstein manifold W .(Tomake explicit the dependence on Weinstein structure, we will sometimes call this category the wrapped microlocal A-model category of W associated to Λ.) In practice, the symplectomorphism relating Λ to a conic Lagrangian in a cotan- gent bundle might not exist. However, such symplectomorphisms always exist lo- cally, so that we can obtain a cosheaf of categories on the skeleton by defining these categories locally, gluing them together, and checking that the resulting cosheaf didn’t depend on choices. A more detailed explanation of our expectations can be found in Conjecture 4.1.8. One skeleton for the pair of pants Pn−1 was described in [32], where it was used to prove a mirror symmetry equivalence. In this paper, we study a more symmetric skeleton of the pair of pants, which we can describe using the geometry of the permutohedron. Licensed to Univ of Calif, Berkeley. Prepared on Fri May 1 11:16:12 EDT 2020 for download from IP 128.32.10.230. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 74 BENJAMIN GAMMAGE AND DAVID NADLER n+1 Let Vn be the quotient of R by the span of the vector λ1 + ···+ λn+1, where n+1 {λi} is the standard coordinate basis of R .
Load more

Recommended publications
  • TWAS Fellowships Worldwide
    CDC Round Table, ICTP April 2016 With science and engineering, countries can address challenges in agriculture, climate, health TWAS’s and energy. guiding principles 2 Food security Challenges Water quality for a Energy security new era Biodiversity loss Infectious diseases Climate change 3 A Globally, 81 nations fall troubling into the category of S&T- gap lagging countries. 48 are classified as Least Developed Countries. 4 The role of TWAS The day-to-day work of TWAS is focused in two critical areas: •Improving research infrastructure •Building a corps of PhD scholars 5 TWAS Research Grants 2,202 grants awarded to individuals and research groups (1986-2015) 6 TWAS’ AIM: to train 1000 PhD students by 2017 Training PhD-level scientists: •Researchers and university-level educators •Future leaders for science policy, business and international cooperation Rapidly growing opportunities P BRAZIL A K I N D I CA I RI A S AF TH T SOU A N M KENYA EX ICO C H I MALAYSIA N A IRAN THAILAND TWAS Fellowships Worldwide NRF, South Africa - newly on board 650+ fellowships per year PhD fellowships +460 Postdoctoral fellowships +150 Visiting researchers/professors + 45 17 Programme Partners BRAZIL: CNPq - National Council MALAYSIA: UPM – Universiti for Scientific and Technological Putra Malaysia WorldwideDevelopment CHINA: CAS - Chinese Academy of KENYA: icipe – International Sciences Centre for Insect Physiology and Ecology INDIA: CSIR - Council of Scientific MEXICO: CONACYT– National & Industrial Research Council on Science and Technology PAKISTAN: CEMB – National INDIA: DBT - Department of Centre of Excellence in Molecular Biotechnology Biology PAKISTAN: ICCBS – International Centre for Chemical and INDIA: IACS - Indian Association Biological Sciences for the Cultivation of Science PAKISTAN: CIIT – COMSATS Institute of Information INDIA: S.N.
    [Show full text]
  • Relative Quantum Field Theory 3
    RELATIVE QUANTUM FIELD THEORY DANIEL S. FREED AND CONSTANTIN TELEMAN Abstract. We highlight the general notion of a relative quantum field theory, which occurs in several contexts. One is in gauge theory based on a compact Lie algebra, rather than a compact Lie group. This is relevant to the maximal superconformal theory in six dimensions. 1. Introduction The (0, 2)-superconformal field theory in six dimensions, which we term Theory X for brevity, was discovered as a limit of superstring theories [W1, S]. It is thought not to have a lagrangian description, so is difficult to access directly, yet some expectations can be deduced from the string theory description [W2, GMN]. Two features are particularly relevant: (i) it is not an ordinary quantum field theory, and (ii) the theory depends on a Lie algebra, not on a Lie group. A puzzle, emphasized by Greg Moore, is that the dimensional reduction of Theory X to five dimensions is usually understood to be an ordinary quantum field theory—contrary to (i)—and it is a supersym- metric gauge theory so depends on a particular choice of Lie group—contrary to (ii). In this paper we spell out the modified notion indicated in (i), which we call a relative quantum field theory, and use it to resolve this puzzle about Theory X by pointing out that the dimensional reduction is also a relative theory. Relative gauge theories are not particular to dimension five. In fact, the possibility of studying four-dimensional gauge theory as a relative theory was exploited in [VW] and [W3].
    [Show full text]
  • Breakthrough Prize in Mathematics Awarded to Terry Tao
    156 Breakthrough Prize in Mathematics awarded to Terry Tao Our congratulations to Australian Mathematician Terry Tao, one of five inaugu- ral winners of the Breakthrough Prizes in Mathematics, announced on 23 June. Terry is based at the University of California, Los Angeles, and has been awarded the prize for numerous breakthrough contributions to harmonic analysis, combi- natorics, partial differential equations and analytic number theory. The Breakthrough Prize in Mathematics was launched by Mark Zuckerberg and Yuri Milner at the Breakthrough Prize ceremony in December 2013. The other winners are Simon Donaldson, Maxim Kontsevich, Jacob Lurie and Richard Tay- lor. All five recipients of the Prize have agreed to serve on the Selection Committee, responsible for choosing subsequent winners from a pool of candidates nominated in an online process which is open to the public. From 2015 onwards, one Break- through Prize in Mathematics will be awarded every year. The Breakthrough Prizes honor important, primarily recent, achievements in the categories of Fundamental Physics, Life Sciences and Mathematics. The prizes were founded by Sergey Brin and Anne Wojcicki, Mark Zuckerberg and Priscilla Chan, and Yuri and Julia Milner, and aim to celebrate scientists and generate excitement about the pursuit of science as a career. Laureates will receive their trophies and $3 million each in prize money at a televised award ceremony in November, designed to celebrate their achievements and inspire the next genera- tion of scientists. As part of the ceremony schedule, they also engage in a program of lectures and discussions. See https://breakthroughprize.org and http://www.nytimes.com/2014/06/23/us/ the-multimillion-dollar-minds-of-5-mathematical-masters.html? r=1 for further in- formation..
    [Show full text]
  • NEWSLETTER No
    NEWSLETTER No. 458 May 2016 NEXT DIRECTOR OF THE ISAAC NEWTON INSTITUTE In October 2016 David Abrahams will succeed John Toland as Director of the Isaac Newton Institute for Mathematical Sciences and NM Rothschild and Sons Professor of Mathematics in Cambridge. David, who is a Royal Society Wolfson Research Merit Award holder, has been Beyer Professor of Applied Mathematics at the University of Man- chester since 1998. From 2014-16 he was Scientific Director of the International Centre for Math- ematical Sciences in Edinburgh and was President of the Institute of Mathematics and its Applica- tions from 2007-2009. David’s research has been in the broad area of applied mathematics, mainly focused on the theoretical understanding of wave processes including scattering, diffraction, localisation and homogenisation. In recent years his research has broadened somewhat, to now cover topics as diverse as mathematical finance, nonlinear vis- He has also been involved in a range of public coelasticity and glaciology. He has close links with engagement activities over the years. He a number of industrial partners. regularly offers mathematics talks of interest David plays an active role within the internation- to school students and the general public, al mathematics community, having served on over and ran the annual Meet the Mathematicians 30 national and international working parties, outreach events for sixth form students with panels and committees over the past decade. Chris Howls (Southampton). With Chris Budd This has included as a Member of the Applied (Bath) he has organised a training conference Mathematics sub-panel for the 2008 Research As- in 2010 on How to Talk Maths in Public, and in sessment Exercise and Deputy Chair for the Math- 2014 co-chaired the inaugural Festival of Math- ematics sub-panel in the 2014 Research Excellence ematics and its Applications.
    [Show full text]
  • On The\'Etale Homotopy Type of Higher Stacks
    ON THE ETALE´ HOMOTOPY TYPE OF HIGHER STACKS DAVID CARCHEDI Abstract. A new approach to ´etale homotopy theory is presented which applies to a much broader class of objects than previously existing approaches, namely it applies not only to all schemes (without any local Noetherian hypothesis), but also to arbitrary higher stacks on the ´etale site of such schemes, and in particular to all algebraic stacks. This approach also produces a more refined invariant, namely a pro-object in the infinity category of spaces, rather than in the homotopy category. We prove a profinite comparison theorem at this level of generality, which states that if X is an arbitrary higher stack on the ´etale site of affine schemes of finite type over C, then the ´etale homotopy type of X agrees with the homotopy type of the underlying stack Xtop on the topological site, after profinite completion. In particular, if X is an Artin stack locally of finite type over C, our definition of the ´etale homotopy type of X agrees up to profinite completion with the homotopy type of the underlying topological stack Xtop of X in the sense of Noohi [30]. In order to prove our comparison theorem, we provide a modern reformulation of the theory of local systems and their cohomology using the language of -categories which we believe to be of independent interest. ∞ 1. Introduction Given a complex variety V, one can associate topological invariants to this variety by computing invariants of its underlying topological space Van, equipped with the complex analytic topology. However, for a variety over an arbitrary base ring, there is no good notion of underlying topological space which plays the same role.
    [Show full text]
  • Clark Barwick
    Curriculum Vitæ — Clark Barwick Massachusetts Institute of Technology Department of Mathematics, E17-332 77 Massachusetts Avenue Cambridge, MA 02139-4307 Web http://www.math.mit.edu/∼clarkbar/ Email [email protected] or [email protected] Citizenship United States Employment History 2015– Massachusetts Institute of Technology (Cambridge, MA, USA). Cecil and Ida Green Career Development Associate Professor of Mathematics. 2013–15 Massachusetts Institute of Technology (Cambridge, MA, USA). Cecil and Ida Green Career Development Assistant Professor of Mathematics. 2010–13 Massachusetts Institute of Technology (Cambridge, MA, USA). Assistant Professor. 2008–10 Harvard University (Cambridge, MA, USA). Benjamin Peirce Lecturer. 2007–08 Institute for Advanced Study (Princeton, NJ, USA). Visitor, Term I; Member, Term II. New connections of representation theory to algebraic geometry and physics. Project leader: R. Bezrukavnikov. 2006–07 Matematisk Institutt, Universitetet i Oslo (Oslo, Norway). YFF Postdoctoral fellow. Geometry and arithmetic of structured ring spectra. Project leader: J. Rognes. 2005–06 Mathematisches Institut Göttingen (Göttingen, Germany). DFG Postdoctoral fellow. Homotopical algebraic geometry. Project leader: Yu. Tschinkel. 1 Education 2005 University of Pennsylvania (Philadelphia, PA, USA). Ph.D, Mathematics. Thesis advisor: Tony Pantev. 2001 University of North Carolina at Chapel Hill (Chapel Hill, NC, USA). B.S., Mathematics. Papers in progress 23. Higher Mackey functors and Grothendieck–Verdier duality. In progress. 22. Algebraic K-theory of Thom spectra (with J. Shah). In progress. 21. Equivariant higher categories and equivariant higher algebra (with E. Dotto, S. Glasman, D. Nardin, and J. Shah). In progress. 20. Higher plethories and chromatic redshift. In progress. 19. On the algebraic K-theory of algebraic K-theory.
    [Show full text]
  • 2017 Breakthrough Prizes in Mathematics and Fundamental
    COMMUNICATION 2017 Breakthrough Prizes Awarded Jean Bourgain Joseph Polchinski Andrew Strominger Cumrun Vafa Breakthrough Prize in Mathematics be obtained from the results of Weil and Deligne on the Jean Bourgain of the Institute for Advanced Study, Riemann hypothesis over finite fields), as well as the de- Princeton, New Jersey, has been selected as the recipient velopment (with Gamburd and Sarnak) of the affine sieve of the 2017 Breakthrough Prize in Mathematics by the that has proven to be a powerful tool for analyzing thin Breakthrough Prize Foundation. Bourgain was honored for groups. Most recently, with Demeter and Guth, Bourgain “major contributions across an incredibly diverse range of has established several important decoupling theorems areas, including harmonic analysis, functional analysis, er- in Fourier analysis that have had applications to partial godic theory, partial differential equations, mathematical differential equations, combinatorial incidence geometry, physics, combinatorics, and theoretical computer science.” and analytic number theory, in particular solving the Main The prize carries a cash award of US$3 million. Conjecture of Vinogradov, as well as obtaining new bounds The Notices asked Terence Tao of the University of Cal- on the Riemann zeta function.” ifornia Los Angeles to comment on the work of Bourgain (Tao was on the Breakthrough Prize committee and was Biographical Sketch: Jean Bourgain also one of the nominators of Bourgain). Tao responded: Jean Bourgain was born in 1954 in Oostende, Belgium. He “Jean Bourgain is an unparalleled problem-solver in anal- received his PhD (1977) and his habilitation (1979) from ysis who has revolutionized many areas of the subject by the Free University of Brussels.
    [Show full text]
  • Quantum Field Theory and the Volume Conjecture
    Contemporary Mathematics Volume 541, 2011 Quantum Field Theory and the Volume Conjecture Thdor Dimofte and Sergei Gukov ABSTRACT. The volume conjecture states that for a hyperbolic knot K in the three-sphere S 3 the asymptotic growth of the colored Jones polynomial of K is governed by the hyperbolic volume of the knot complement S3 \K. The conjecture relates two topological invariants, one combinatorial and one geometric, in a very nonobvious, nontrivial manner. The goal of the present lectures is to review the original statement of the volume conjecture and its recent extensions and generalizations, and to show how, in the most general context, the conjecture can be understood in terms of topological quantum field theory. In particular, we consider: a) generalization of the volume conjecture to families of incomplete hyperbolic metrics; b) generalization that involves not only the leading (volume) term, but the entire asymptotic expansion in 1/N; c) generalization to quantum group invariants for groups of higher rank; and d) generalization to arbitrary links in arbitrary three-manifolds. 1. Preliminaries 3 Let K be an oriented knot (or link) in the three-sphere S . The original volume conjecture [21, 26] relates theN-colored Jones polynomial of K to the hyperbolic volume of the knot complement S3\K: N-colored Jones poly of K hyperbolic volume of S3 \K (1.1) (combinatorial, rep. theory) (geometric) . We begin by reviewing some of the definitions and ingredients that enter on the two sides here in order to make this statement more precise, and to serve as a precursor for its subsequent generalization.
    [Show full text]
  • Prizes and Awards
    SAN DIEGO • JAN 10–13, 2018 January 2018 SAN DIEGO • JAN 10–13, 2018 Prizes and Awards 4:25 p.m., Thursday, January 11, 2018 66 PAGES | SPINE: 1/8" PROGRAM OPENING REMARKS Deanna Haunsperger, Mathematical Association of America GEORGE DAVID BIRKHOFF PRIZE IN APPLIED MATHEMATICS American Mathematical Society Society for Industrial and Applied Mathematics BERTRAND RUSSELL PRIZE OF THE AMS American Mathematical Society ULF GRENANDER PRIZE IN STOCHASTIC THEORY AND MODELING American Mathematical Society CHEVALLEY PRIZE IN LIE THEORY American Mathematical Society ALBERT LEON WHITEMAN MEMORIAL PRIZE American Mathematical Society FRANK NELSON COLE PRIZE IN ALGEBRA American Mathematical Society LEVI L. CONANT PRIZE American Mathematical Society AWARD FOR DISTINGUISHED PUBLIC SERVICE American Mathematical Society LEROY P. STEELE PRIZE FOR SEMINAL CONTRIBUTION TO RESEARCH American Mathematical Society LEROY P. STEELE PRIZE FOR MATHEMATICAL EXPOSITION American Mathematical Society LEROY P. STEELE PRIZE FOR LIFETIME ACHIEVEMENT American Mathematical Society SADOSKY RESEARCH PRIZE IN ANALYSIS Association for Women in Mathematics LOUISE HAY AWARD FOR CONTRIBUTION TO MATHEMATICS EDUCATION Association for Women in Mathematics M. GWENETH HUMPHREYS AWARD FOR MENTORSHIP OF UNDERGRADUATE WOMEN IN MATHEMATICS Association for Women in Mathematics MICROSOFT RESEARCH PRIZE IN ALGEBRA AND NUMBER THEORY Association for Women in Mathematics COMMUNICATIONS AWARD Joint Policy Board for Mathematics FRANK AND BRENNIE MORGAN PRIZE FOR OUTSTANDING RESEARCH IN MATHEMATICS BY AN UNDERGRADUATE STUDENT American Mathematical Society Mathematical Association of America Society for Industrial and Applied Mathematics BECKENBACH BOOK PRIZE Mathematical Association of America CHAUVENET PRIZE Mathematical Association of America EULER BOOK PRIZE Mathematical Association of America THE DEBORAH AND FRANKLIN TEPPER HAIMO AWARDS FOR DISTINGUISHED COLLEGE OR UNIVERSITY TEACHING OF MATHEMATICS Mathematical Association of America YUEH-GIN GUNG AND DR.CHARLES Y.
    [Show full text]
  • Annual Report 2018 Edition TABLE of CONTENTS 2018
    Annual Report 2018 Edition TABLE OF CONTENTS 2018 GREETINGS 3 Letter From the President 4 Letter From the Chair FLATIRON INSTITUTE 7 Developing the Common Language of Computational Science 9 Kavli Summer Program in Astrophysics 12 Toward a Grand Unified Theory of Spindles 14 Building a Network That Learns Like We Do 16 A Many-Method Attack on the Many Electron Problem MATHEMATICS AND PHYSICAL SCIENCES 21 Arithmetic Geometry, Number Theory and Computation 24 Origins of the Universe 26 Cracking the Glass Problem LIFE SCIENCES 31 Computational Biogeochemical Modeling of Marine Ecosystems (CBIOMES) 34 Simons Collaborative Marine Atlas Project 36 A Global Approach to Neuroscience AUTISM RESEARCH INITIATIVE (SFARI) 41 SFARI’s Data Infrastructure for Autism Discovery 44 SFARI Research Roundup 46 The SPARK Gambit OUTREACH AND EDUCATION 51 Science Sandbox: “The Most Unknown” 54 Math for America: The Muller Award SIMONS FOUNDATION 56 Financials 58 Flatiron Institute Scientists 60 Mathematics and Physical Sciences Investigators 62 Mathematics and Physical Sciences Fellows 63 Life Sciences Investigators 65 Life Sciences Fellows 66 SFARI Investigators 68 Outreach and Education 69 Simons Society of Fellows 70 Supported Institutions 71 Advisory Boards 73 Board of Directors 74 Simons Foundation Staff 3 LETTER FROM THE PRESIDENT As one year ends and a new one begins, it is always a In the pages that follow, you will also read about the great pleasure to look back over the preceding 12 months foundation’s grant-making in Mathematics and Physical and reflect on all the fascinating and innovative ideas Sciences, Life Sciences, autism science (SFARI), Outreach conceived, supported, researched and deliberated at the and Education, and our Simons Collaborations.
    [Show full text]
  • Mathematisches Forschungsinstitut Oberwolfach Non-Archimedean
    Mathematisches Forschungsinstitut Oberwolfach Report No. 57/2015 DOI: 10.4171/OWR/2015/57 Non-Archimedean Geometry and Applications Organised by Vladimir Berkovich, Rehovot Walter Gubler, Regensburg Peter Schneider, M¨unster Annette Werner, Frankfurt 13 December – 19 December 2015 Abstract. The workshop focused on recent developments in non-Archi- medean analytic geometry with various applications to other fields, in partic- ular to number theory and algebraic geometry. These applications included Mirror Symmetry, the Langlands program, p-adic Hodge theory, tropical ge- ometry, resolution of singularities and the geometry of moduli spaces. Much emphasis was put on making the list of talks to reflect this diversity, thereby fostering the mutual inspiration which comes from such interactions. Mathematics Subject Classification (2010): 03 C 98, 11 G 20, 11 G 25, 14 F 20, 14 G 20, 14 G 22, 32 P 05. Introduction by the Organisers The workshop on Non-Archimedean Analytic Geometry and Applications, or- ganized by Vladimir Berkovich (Rehovot), Walter Gubler (Regensburg), Peter Schneider (M¨unster) and Annette Werner (Frankfurt) had 53 participants. Non- Archimedean analytic geometry is a central area of arithmetic geometry. The first analytic spaces over fields with a non-Archimedean absolute value were introduced by John Tate and explored by many other mathematicians. They have found nu- merous applications to problems in number theory and algebraic geometry. In the 1990s, Vladimir Berkovich initiated a different approach to non-Archimedean an- alytic geometry, providing spaces with good topological properties which behave similarly as complex analytic spaces. Independently, Roland Huber developed a similar theory of adic spaces. Recently, Peter Scholze has introduced perfectoid 3272 Oberwolfach Report 57/2015 spaces as a ground breaking new tool to attack deep problems in p-adic Hodge theory and representation theory.
    [Show full text]
  • 2007 Integral
    Autumn 2007 Volume 2 Massachusetts Institute of Technology 1ntegral news from the mathematics department at mit Inside • New faculty • Leighton and Akamai • Awards and achievements • Campaign for Math update • The UMO • Faculty chairs 7 • Project Laboratory • News bits • Putnam Competition • Simons Lectures Dear Friends, Welcome to the second edition of Integral, Speaking of undergraduate education, a careers. That conference is planned for our department’s annual newsletter. major MIT task force has just completed April 12-13, 2008, and will follow a A year has flown by since we published a comprehensive review of the General meeting of the department’s Visiting our inaugural edition in September 2006. Institute Requirements (GIRs) that shape Committee. Much has happened and much more is the undergraduate experience of all MIT planned. students and it has made recommendations Thanks to the overwhelming support for various changes in these requirements. of our alumni and other friends, our First of all, we’ve had an ambitious recruit- MIT faculty and administrators are cur- $15 million fundraising “Campaign for ment effort and are delighted to be rently reviewing these proposed changes, Mathematics” has been going exceedingly welcoming six new faculty to the depart- some of which will undoubtedly be imple- well. At the present time, we are 90 percent ment this year: Professors Paul Seidel mented. Our department is considering the of the way to our goal, so that means you and James McKernan; tenured Associate impact of these changes on our programs still have time to participate and we hope Professors Ju-Lee Kim and Jacob Lurie; and whether we ought to review our own you do! Please look inside for additional and Assistant Professors Jon Kelner and offerings and requirements.
    [Show full text]