Mathematical Sciences 2010
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CMI Summer Schools
CMI Summer Schools 2007 Homogeneous flows, moduli spaces, and arithmetic De Giorgi Center, Pisa 2006 Arithmetic Geometry The Clay Mathematics Institute has Mathematisches Institut, conducted a program of research summer schools Georg-August-Universität, Göttingen since 2000. Designed for graduate students and PhDs within five years of their degree, the aim of 2005 Ricci Flow, 3-manifolds, and Geometry the summer schools is to furnish a new generation MSRI, Berkeley of mathematicians with the knowledge and tools needed to work successfully in an active research CMI summer schools 2004 area. Three introductory courses, each three weeks Floer Homology, Gauge Theory, and in duration, make up the core of a typical summer Low-dimensional Topology school. These are followed by one week of more Rényi Institute, Budapest advanced minicourses and individual talks. Size is limited to roughly 100 participants in order to 2003 Harmonic Analysis, Trace Formula, promote interaction and contact. Venues change and Shimura Varieties from year to year, and have ranged from Cambridge, Fields Institute, Toronto Massachusetts to Pisa, Italy. The lectures from each school are published in the CMI–AMS proceedings 2002 Geometry and String Theory series, usually within two years’ time. Newton Institute, Cambridge UK Summer Schools 2001 www.claymath.org/programs/summer_school Minimal surfaces MSRI, Berkeley Summer School Proceedings www.claymath.org/publications 2000 Mirror Symmetry Pine Manor College, Boston 2006 Arithmetic Geometry Summer School in Göttingen techniques are drawn from the theory of elliptic The 2006 summer school program will introduce curves, including modular curves and their para- participants to modern techniques and outstanding metrizations, Heegner points, and heights. -
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. -
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). -
Convergence of Complete Ricci-Flat Manifolds Jiewon Park
Convergence of Complete Ricci-flat Manifolds by Jiewon Park Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY May 2020 © Massachusetts Institute of Technology 2020. All rights reserved. Author . Department of Mathematics April 17, 2020 Certified by. Tobias Holck Colding Cecil and Ida Green Distinguished Professor Thesis Supervisor Accepted by . Wei Zhang Chairman, Department Committee on Graduate Theses 2 Convergence of Complete Ricci-flat Manifolds by Jiewon Park Submitted to the Department of Mathematics on April 17, 2020, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics Abstract This thesis is focused on the convergence at infinity of complete Ricci flat manifolds. In the first part of this thesis, we will give a natural way to identify between two scales, potentially arbitrarily far apart, in the case when a tangent cone at infinity has smooth cross section. The identification map is given as the gradient flow of a solution to an elliptic equation. We use an estimate of Colding-Minicozzi of a functional that measures the distance to the tangent cone. In the second part of this thesis, we prove a matrix Harnack inequality for the Laplace equation on manifolds with suitable curvature and volume growth assumptions, which is a pointwise estimate for the integrand of the aforementioned functional. This result provides an elliptic analogue of matrix Harnack inequalities for the heat equation or geometric flows. Thesis Supervisor: Tobias Holck Colding Title: Cecil and Ida Green Distinguished Professor 3 4 Acknowledgments First and foremost I would like to thank my advisor Tobias Colding for his continuous guidance and encouragement, and for suggesting problems to work on. -
Notices of the AMS 595 Mathematics People NEWS
NEWS Mathematics People contrast electrical impedance Takeda Awarded 2017–2018 tomography, as well as model Centennial Fellowship reduction techniques for para- bolic and hyperbolic partial The AMS has awarded its Cen- differential equations.” tennial Fellowship for 2017– Borcea received her PhD 2018 to Shuichiro Takeda. from Stanford University and Takeda’s research focuses on has since spent time at the Cal- automorphic forms and rep- ifornia Institute of Technology, resentations of p-adic groups, Rice University, the Mathemati- especially from the point of Liliana Borcea cal Sciences Research Institute, view of the Langlands program. Stanford University, and the He will use the Centennial Fel- École Normale Supérieure, Paris. Currently Peter Field lowship to visit the National Collegiate Professor of Mathematics at Michigan, she is Shuichiro Takeda University of Singapore and deeply involved in service to the applied and computa- work with Wee Teck Gan dur- tional mathematics community, in particular on editorial ing the academic year 2017–2018. boards and as an elected member of the SIAM Council. Takeda obtained a bachelor's degree in mechanical The Sonia Kovalevsky Lectureship honors significant engineering from Tokyo University of Science, master's de- contributions by women to applied or computational grees in philosophy and mathematics from San Francisco mathematics. State University, and a PhD in 2006 from the University —From an AWM announcement of Pennsylvania. After postdoctoral positions at the Uni- versity of California at San Diego, Ben-Gurion University in Israel, and Purdue University, since 2011 he has been Pardon Receives Waterman assistant and now associate professor at the University of Missouri at Columbia. -
2.1 Harmonic Map Heat Flow from a Circle
Self-shrinkers of Mean Curvature Flow and Harmonic Map Heat Flow with Rough Boundary Data by 0 TECHF 2L ' Lu Wang SEP 0 2 20 Bachelor of Science, Peking University, July 2006 LIRARIS Submitted to the Department of Mathematics ARCHNES in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2011 @ Lu Wang, MMXI. All rights reserved. The author hereby grants to MIT permission to reproduce and distribute publicly paper and electronic copies of this thesis document in whole or in part. A uthor .......... ........................................... Department of Mathematics Lr'\ '~I April 19, 2011 Certified by -- ------ - ........................ Tobias H. Colding Levinson Professor of Mathematics Thesis Supervisor Accepted by 37 Bjorn Poonen Chairman, Department Committee on Graduate Students 2 Self-shrinkers of Mean Curvature Flow and Harmonic Map Heat Flow with Rough Boundary Data by Lu Wang Submitted to the Department of Mathematics on April 19, 2011, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract In this thesis, first, joint with Longzhi Lin, we establish estimates for the harmonic map heat flow from the unit circle into a closed manifold, and use it to construct sweepouts with the following good property: each curve in the tightened sweepout, whose energy is close to the maximal energy of curves in the sweepout, is itself close to a closed geodesic. Second, we prove the uniqueness for energy decreasing weak solutions of the har- monic map heat flow from the unit open disk into a closed manifold, given any H' initial data and boundary data, which is the restriction of the initial data on the boundary of the disk. -
Completions of Period Mappings: Progress Report
COMPLETIONS OF PERIOD MAPPINGS: PROGRESS REPORT MARK GREEN, PHILLIP GRIFFITHS, AND COLLEEN ROBLES Abstract. We give an informal, expository account of a project to construct completions of period maps. 1. Introduction The purpose of this this paper is to give an expository overview, with examples to illus- trate some of the main points, of recent work [GGR21b] to construct \maximal" completions of period mappings. This work is part of an ongoing project, including [GGLR20, GGR21a], to study the global properties of period mappings at infinity. 1.1. Completions of period mappings. We consider triples (B; Z; Φ) consisting of a smooth projective variety B and a reduced normal crossing divisor Z whose complement B = BnZ has a variation of (pure) polarized Hodge structure p ~ F ⊂ V B ×π (B) V (1.1a) 1 B inducing a period map (1.1b) Φ : B ! ΓnD: arXiv:2106.04691v1 [math.AG] 8 Jun 2021 Here D is a period domain parameterizing pure, weight n, Q{polarized Hodge structures on the vector space V , and π1(B) Γ ⊂ Aut(V; Q) is the monodromy representation. Without loss of generality, Φ : B ! ΓnD is proper [Gri70a]. Let } = Φ(B) Date: June 10, 2021. 2010 Mathematics Subject Classification. 14D07, 32G20, 32S35, 58A14. Key words and phrases. period map, variation of (mixed) Hodge structure. Robles is partially supported by NSF DMS 1611939, 1906352. 1 2 GREEN, GRIFFITHS, AND ROBLES denote the image. The goal is to construct both a projective completion } of } and a surjective extension Φe : B ! } of the period map. We propose two such completions T B Φ }T (1.2) ΦS }S : The completion ΦT : B ! }T is maximal, in the sense that it encodes all the Hodge- theoretic information associated with the triple (B; Z; Φ). -
Hodge Theory and Geometry
HODGE THEORY AND GEOMETRY PHILLIP GRIFFITHS This expository paper is an expanded version of a talk given at the joint meeting of the Edinburgh and London Mathematical Societies in Edinburgh to celebrate the centenary of the birth of Sir William Hodge. In the talk the emphasis was on the relationship between Hodge theory and geometry, especially the study of algebraic cycles that was of such interest to Hodge. Special attention will be placed on the construction of geometric objects with Hodge-theoretic assumptions. An objective in the talk was to make the following points: • Formal Hodge theory (to be described below) has seen signifi- cant progress and may be said to be harmonious and, with one exception, may be argued to be essentially complete; • The use of Hodge theory in algebro-geometric questions has become pronounced in recent years. Variational methods have proved particularly effective; • The construction of geometric objects, such as algebraic cycles or rational equivalences between cycles, has (to the best of my knowledge) not seen significant progress beyond the Lefschetz (1; 1) theorem first proved over eighty years ago; • Aside from the (generalized) Hodge conjecture, the deepest is- sues in Hodge theory seem to be of an arithmetic-geometric character; here, especially noteworthy are the conjectures of Grothendieck and of Bloch-Beilinson. Moreover, even if one is interested only in the complex geometry of algebraic cycles, in higher codimensions arithmetic aspects necessarily enter. These are reflected geometrically in the infinitesimal structure of the spaces of algebraic cycles, and in the fact that the convergence of formal, iterative constructions seems to involve arithmetic 1 2 PHILLIP GRIFFITHS as well as Hodge-theoretic considerations. -
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]. -
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. -
AMS Bookstore
35 Monticello Place, Pawtucket, RI 02861 USA Society Distribution Center American Mathematical AMERICAN MATHEMATICAL SOCIETY Featured below are some of the major books released Mathematical in 2013. The selected titles are representative of the many Surveys and Monographs Volume 191 diverse book series published by the AMS. An Introduction to Central Simple Algebras and Their Applications to Wireless An Introduction to Central Simple Experiencing Mathematics Communication Grégory Berhuy Algebras and Their Applications to What do we do, when we do mathematics? Frédérique Oggier Wireless Communication Reuben Hersh Grégory Berhuy and Frédérique Oggier American Mathematical Society This book of selected articles and essays provides an An introduction to the theory of central simple alge- honest, coherent, and clearly understandable account Combinatorial bras intertwined with its applications to coding theory. of mathematicians’ proof as it really is, and of the exis- Game Theory Mathematical Surveys and Monographs, Volume 191; 2013; tence and reality of mathematical entities. Aaron N. Siegel 276 pages; Hardcover; ISBN: 978-0-8218-4937-8; List US$98; 2014; approximately 259 pages; Softcover; ISBN: 978-0-8218- AMS members US$78.40; Order code SURV/191 Conference Board of the Mathematical Sciences 9420-0; List US$39; AMS members US$31.20; Order code Graduate Studies in Mathematics CBMS Volume 146 MBK/83 Regional Conference Series in Mathematics Number 118 Combinatorial Game Theory American Mathematical Society Hodge Theory, The Endoscopic Classification of Complex Geometry, and Aaron N. Siegel Representation Theory Representations A comprehensive and up-to-date introduction to the Mark Green Phillip Griffiths subject of combinatorial game theory, tracing its devel- Orthogonal and Symplectic Groups Matt Kerr opment from first principles and examples through American Mathematical Society with support from the James Arthur National Science Foundation many of its most recent advances. -
Mathematics Opportunities
Mathematics Opportunities will be published by the American Mathematical Society, NSF Integrative Graduate by the Society for Industrial and Applied Mathematics, or jointly by the American Statistical Association and the Education and Research Institute of Mathematical Statistics. Training Support is provided for about thirty participants at each conference, and the conference organizer invites The Integrative Graduate Education and Research Training both established researchers and interested newcomers, (IGERT) program was initiated by the National Science including postdoctoral fellows and graduate students, to Foundation (NSF) to meet the challenges of educating Ph.D. attend. scientists and engineers with the interdisciplinary back- The proposal due date is April 8, 2005. For further grounds and the technical, professional, and personal information on submitting a proposal, consult the CBMS skills needed for the career demands of the future. The website, http://www.cbms.org, or contact: Conference program is intended to catalyze a cultural change in grad- Board of the Mathematical Sciences, 1529 Eighteenth Street, uate education for students, faculty, and universities by NW, Washington, DC 20036; telephone: 202-293-1170; establishing innovative models for graduate education in fax: 202-293-3412; email: [email protected] a fertile environment for collaborative research that tran- or [email protected]. scends traditional disciplinary boundaries. It is also intended to facilitate greater diversity in student participation and —From a CBMS announcement to contribute to the development of a diverse, globally aware science and engineering workforce. Supported pro- jects must be based on a multidisciplinary research theme National Academies Research and administered by a diverse group of investigators from U.S.