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A Tour Through Mirzakhani's Work on Moduli Spaces of Riemann Surfaces
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 57, Number 3, July 2020, Pages 359–408 https://doi.org/10.1090/bull/1687 Article electronically published on February 3, 2020 A TOUR THROUGH MIRZAKHANI’S WORK ON MODULI SPACES OF RIEMANN SURFACES ALEX WRIGHT Abstract. We survey Mirzakhani’s work relating to Riemann surfaces, which spans about 20 papers. We target the discussion at a broad audience of non- experts. Contents 1. Introduction 359 2. Preliminaries on Teichm¨uller theory 361 3. The volume of M1,1 366 4. Integrating geometric functions over moduli space 367 5. Generalizing McShane’s identity 369 6. Computation of volumes using McShane identities 370 7. Computation of volumes using symplectic reduction 371 8. Witten’s conjecture 374 9. Counting simple closed geodesics 376 10. Random surfaces of large genus 379 11. Preliminaries on dynamics on moduli spaces 382 12. Earthquake flow 386 13. Horocyclic measures 389 14. Counting with respect to the Teichm¨uller metric 391 15. From orbits of curves to orbits in Teichm¨uller space 393 16. SL(2, R)-invariant measures and orbit closures 395 17. Classification of SL(2, R)-orbit closures 398 18. Effective counting of simple closed curves 400 19. Random walks on the mapping class group 401 Acknowledgments 402 About the author 402 References 403 1. Introduction This survey aims to be a tour through Maryam Mirzakhani’s remarkable work on Riemann surfaces, dynamics, and geometry. The star characters, all across Received by the editors May 12, 2019. 2010 Mathematics Subject Classification. Primary 32G15. c 2020 American Mathematical Society 359 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 360 ALEX WRIGHT 2 3117 4 5 12 14 16 18 19 9106 13 15 17 8 Figure 1.1. -
2010 Table of Contents Newsletter Sponsors
OKLAHOMA/ARKANSAS SECTION Volume 31, February 2010 Table of Contents Newsletter Sponsors................................................................................ 1 Section Governance ................................................................................ 6 Distinguished College/University Teacher of 2009! .............................. 7 Campus News and Notes ........................................................................ 8 Northeastern State University ............................................................. 8 Oklahoma State University ................................................................. 9 Southern Nazarene University ............................................................ 9 The University of Tulsa .................................................................... 10 Southwestern Oklahoma State University ........................................ 10 Cameron University .......................................................................... 10 Henderson State University .............................................................. 11 University of Arkansas at Monticello ............................................... 13 University of Central Oklahoma ....................................................... 14 Minutes for the 2009 Business Meeting ............................................... 15 Preliminary Announcement .................................................................. 18 The Oklahoma-Arkansas Section NExT ............................................... 21 The 2nd Annual -
President's Report
Newsletter Volume 43, No. 3 • mAY–JuNe 2013 PRESIDENT’S REPORT Greetings, once again, from 35,000 feet, returning home from a major AWM conference in Santa Clara, California. Many of you will recall the AWM 40th Anniversary conference held in 2011 at Brown University. The enthusiasm generat- The purpose of the Association ed by that conference gave rise to a plan to hold a series of biennial AWM Research for Women in Mathematics is Symposia around the country. The first of these, the AWM Research Symposium 2013, took place this weekend on the beautiful Santa Clara University campus. • to encourage women and girls to study and to have active careers The symposium attracted close to 150 participants. The program included 3 plenary in the mathematical sciences, and talks, 10 special sessions on a wide variety of topics, a contributed paper session, • to promote equal opportunity and poster sessions, a panel, and a banquet. The Santa Clara campus was in full bloom the equal treatment of women and and the weather was spectacular. Thankfully, the poster sessions and coffee breaks girls in the mathematical sciences. were held outside in a courtyard or those of us from more frigid climates might have been tempted to play hooky! The event opened with a plenary talk by Maryam Mirzakhani. Mirzakhani is a professor at Stanford and the recipient of multiple awards including the 2013 Ruth Lyttle Satter Prize. Her talk was entitled “On Random Hyperbolic Manifolds of Large Genus.” She began by describing how to associate a hyperbolic surface to a graph, then proceeded with a fascinating discussion of the metric properties of surfaces associated to random graphs. -
Computations in Algebraic Geometry with Macaulay 2
Computations in algebraic geometry with Macaulay 2 Editors: D. Eisenbud, D. Grayson, M. Stillman, and B. Sturmfels Preface Systems of polynomial equations arise throughout mathematics, science, and engineering. Algebraic geometry provides powerful theoretical techniques for studying the qualitative and quantitative features of their solution sets. Re- cently developed algorithms have made theoretical aspects of the subject accessible to a broad range of mathematicians and scientists. The algorith- mic approach to the subject has two principal aims: developing new tools for research within mathematics, and providing new tools for modeling and solv- ing problems that arise in the sciences and engineering. A healthy synergy emerges, as new theorems yield new algorithms and emerging applications lead to new theoretical questions. This book presents algorithmic tools for algebraic geometry and experi- mental applications of them. It also introduces a software system in which the tools have been implemented and with which the experiments can be carried out. Macaulay 2 is a computer algebra system devoted to supporting research in algebraic geometry, commutative algebra, and their applications. The reader of this book will encounter Macaulay 2 in the context of concrete applications and practical computations in algebraic geometry. The expositions of the algorithmic tools presented here are designed to serve as a useful guide for those wishing to bring such tools to bear on their own problems. A wide range of mathematical scientists should find these expositions valuable. This includes both the users of other programs similar to Macaulay 2 (for example, Singular and CoCoA) and those who are not interested in explicit machine computations at all. -
Prizes and Awards Session
PRIZES AND AWARDS SESSION Wednesday, July 12, 2021 9:00 AM EDT 2021 SIAM Annual Meeting July 19 – 23, 2021 Held in Virtual Format 1 Table of Contents AWM-SIAM Sonia Kovalevsky Lecture ................................................................................................... 3 George B. Dantzig Prize ............................................................................................................................. 5 George Pólya Prize for Mathematical Exposition .................................................................................... 7 George Pólya Prize in Applied Combinatorics ......................................................................................... 8 I.E. Block Community Lecture .................................................................................................................. 9 John von Neumann Prize ......................................................................................................................... 11 Lagrange Prize in Continuous Optimization .......................................................................................... 13 Ralph E. Kleinman Prize .......................................................................................................................... 15 SIAM Prize for Distinguished Service to the Profession ....................................................................... 17 SIAM Student Paper Prizes .................................................................................................................... -
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Arxiv:2010.06953V2 [Math.RA] 23 Apr 2021
IDENTITIES AND BASES IN THE HYPOPLACTIC MONOID ALAN J. CAIN, ANTÓNIO MALHEIRO, AND DUARTE RIBEIRO Abstract. This paper presents new results on the identities satisfied by the hypoplactic monoid. We show how to embed the hypoplactic monoid of any rank strictly greater than 2 (including infinite rank) into a direct product of copies of the hypoplactic monoid of rank 2. This confirms that all hypoplactic monoids of rank greater than or equal to 2 satisfy exactly the same identities. We then give a complete characterization of those identities, and prove that the variety generated by the hypoplactic monoid has finite axiomatic rank, by giving a finite basis for it. 1. Introduction A (non-trivial) identity is a formal equality u ≈ v, where u and v are words over some alphabet of variables, which is not of the form u ≈ u. If a monoid is known to satisfy an identity, an important question is whether the set of identities it satisfies is finitely based, that is, if all these identities are consequences of those in some finite subset (see [37, 42]). The plactic monoid plac, also known as the monoid of Young tableaux, is an important algebraic structure, first studied by Schensted [38] and Knuth [23], and later studied in depth by Lascoux and Schützenberger [27]. It is connected to many different areas of Mathematics, such as algebraic combinatorics, symmetric functions [29], crystal bases [4] and representation theory [14]. In particular, the question of identities in the plactic monoid has received a lot of attention recently [26, 20]. Finitely-generated polynomial-growth groups are virtually nilpotent and so sat- isfy identities [16]. -
Lms Elections to Council and Nominating Committee 2017: Candidate Biographies
LMS ELECTIONS TO COUNCIL AND NOMINATING COMMITTEE 2017: CANDIDATE BIOGRAPHIES Candidate for election as President (1 vacancy) Caroline Series Candidates for election as Vice-President (2 vacancies) John Greenlees Catherine Hobbs Candidate for election as Treasurer (1 vacancy) Robert Curtis Candidate for election as General Secretary (1 vacancy) Stephen Huggett Candidate for election as Publications Secretary (1 vacancy) John Hunton Candidate for election as Programme Secretary (1 vacancy) Iain A Stewart Candidates for election as Education Secretary (1 vacancy) Tony Gardiner Kevin Houston Candidate for election as Librarian (Member-at-Large) (1 vacancy) June Barrow-Green Candidates for election as Member-at-Large of Council (6 x 2-year terms vacant) Mark AJ Chaplain Stephen J. Cowley Andrew Dancer Tony Gardiner Evgenios Kakariadis Katrin Leschke Brita Nucinkis Ronald Reid-Edwards Gwyneth Stallard Alina Vdovina Candidates for election to Nominating Committee (2 vacancies) H. Dugald Macpherson Martin Mathieu Andrew Treglown 1 CANDIDATE FOR ELECTION AS PRESIDENT (1 VACANCY) Caroline Series FRS, Professor of Mathematics (Emeritus), University of Warwick Email address: [email protected] Home page: http://www.maths.warwick.ac.uk/~cms/ PhD: Harvard University 1976 Previous appointments: Warwick University (Lecturer/Reader/Professor)1978-2014; EPSRC Senior Research Fellow 1999- 2004; Research Fellow, Newnham College, Cambridge 1977-8; Lecturer, Berkeley 1976-77. Research interests: Hyperbolic Geometry, Kleinian Groups, Dynamical Systems, Ergodic Theory. LMS service: Council 1989-91; Nominations Committee 1999- 2001, 2007-9, Chair 2009-12; LMS Student Texts Chief Editor 1990-2002; LMS representative to various other bodies. LMS Popular Lecturer 1999; Mary Cartwright Lecture 2000; Forder Lecturer 2003. -
Vita BERND STURMFELS Department of Mathematics, University of California, Berkeley, CA 94720 Phone
Vita BERND STURMFELS Department of Mathematics, University of California, Berkeley, CA 94720 Phone: (510) 642 4687, Fax: (510) 642 8204, [email protected] M.A. [Diplom] TH Darmstadt, Germany, Mathematics and Computer Science, 1985 Ph.D. [Dr. rer. nat.] TH Darmstadt, Germany, Mathematics, 1987 Ph.D. University of Washington, Seattle, Mathematics, 1987 Professional Experience: 1987–1988 Postdoctoral Fellow, I.M.A., University of Minnesota, Minneapolis 1988–1989 Assistant Professor, Research Institute for Symbolic Computation, (RISC-Linz), Linz, Austria 1989–1991 Assistant Professor, Department of Mathematics, Cornell University 1992–1996 Associate Professor, Department of Mathematics, Cornell University 1994–2001 Professor, Department of Mathematics, University of California, Berkeley 2001– Professor, Department of Mathematics and Computer Science, UC Berkeley Academic Honors: 1986 - 1987 Alfred P. Sloan Doctoral Dissertation Fellowship 1991 - 1993 Alfred P. Sloan Research Fellow 1992 - 1997 National Young Investigator (NSF) 1992 - 1997 David and Lucile Packard Fellowship 1999 Lester R. Ford Prize for Expository Writing (MAA) 2000-2001 Miller Research Professorship, UC Berkeley Spring 2003 John von Neumann Professor, Technical University M¨unchen 2003-2004 Hewlett-Packard Research Professor at MSRI Berkeley July 2004 Clay Mathematics Institute Senior Scholar Research Interests: Computational Algebra, Combinatorics, Algebraic Geometry Selected Professional Activities: Visiting Positions: D´epartment de Math´ematiques, Universit´ede Nice, France, Spring 1989 Mathematical Sciences Research Institute, Berkeley, Fall 1992 Courant Institute, New York University, 1994–95 RIMS, Kyoto University, Japan, 1997–98 Current Editorial Board Membership: Journal of the American Mathematical Society, Duke Mathematical Journal, Collecteana Mathematica, Beitr¨agezur Geometrie und Algebra, Order, Discrete and Computational Geometry, Applicable Algebra (AAECC) Journal of Combinatorial Theory (Ser. -
April 2017 Table of Contents
ISSN 0002-9920 (print) ISSN 1088-9477 (online) of the American Mathematical Society April 2017 Volume 64, Number 4 AMS Prize Announcements page 311 Spring Sectional Sampler page 333 AWM Research Symposium 2017 Lecture Sampler page 341 Mathematics and Statistics Awareness Month page 362 About the Cover: How Minimal Surfaces Converge to a Foliation (see page 307) MATHEMATICAL CONGRESS OF THE AMERICAS MCA 2017 JULY 2428, 2017 | MONTREAL CANADA MCA2017 will take place in the beautiful city of Montreal on July 24–28, 2017. The many exciting activities planned include 25 invited lectures by very distinguished mathematicians from across the Americas, 72 special sessions covering a broad spectrum of mathematics, public lectures by Étienne Ghys and Erik Demaine, a concert by the Cecilia String Quartet, presentation of the MCA Prizes and much more. SPONSORS AND PARTNERS INCLUDE Canadian Mathematical Society American Mathematical Society Pacifi c Institute for the Mathematical Sciences Society for Industrial and Applied Mathematics The Fields Institute for Research in Mathematical Sciences National Science Foundation Centre de Recherches Mathématiques Conacyt, Mexico Atlantic Association for Research in Mathematical Sciences Instituto de Matemática Pura e Aplicada Tourisme Montréal Sociedade Brasileira de Matemática FRQNT Quebec Unión Matemática Argentina Centro de Modelamiento Matemático For detailed information please see the web site at www.mca2017.org. AMERICAN MATHEMATICAL SOCIETY PUSHING LIMITS From West Point to Berkeley & Beyond PUSHING LIMITS FROM WEST POINT TO BERKELEY & BEYOND Ted Hill, Georgia Tech, Atlanta, GA, and Cal Poly, San Luis Obispo, CA Recounting the unique odyssey of a noted mathematician who overcame military hurdles at West Point, Army Ranger School, and the Vietnam War, this is the tale of an academic career as noteworthy for its o beat adventures as for its teaching and research accomplishments. -
Curriculum Vitae
Paul Breiding | Curriculum Vitae Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22, 04103 Leipzig, Germany Q [email protected] • paulbreiding.org • PBrdng u Paul Breiding • 7 @_pbrdng • born 12th of May 1988, german citizenship Max-Plack-Institute for Mathematics in the Sciences Leipzig Head of Emmy Noether Research Group: Numerical and Probabilistic Nonlinear Algebra Since 04/2021 University of Kassel Substitute Professor for Computeralgebra 11/2020 – 03/2021 Akademie der Wissenschaften und der Literatur Mainz Member of the Junge Akademie 04/2020 – 03/2024 Parental leave 7 months in total 10/2019 – 11/2019 and 04/2020 – 10/2020 Technische Universität Berlin Postdoctoral researcher in the algorithmic algebra research group 04/2019 – 10/2020 Max-Plack-Institute for Mathematics in the Sciences Leipzig Postdoctoral researcher in the nonlinear algebra research group 10/2017 – 03/2019 Technische Universität Berlin PhD student with Prof. Dr. Bürgisser 12/2013 – 09/2017 Date of thesis defense: July 25, 2017. Evaluation ’summa cum laude’. Simons Institute for the Theory of Computing Visiting graduate student 08/2014 – 10/2014 Algorithms and Complexity in Algebraic Geometry Education Georg-August-Universität Göttingen Master of Science 10/2011 – 11/2013 Evaluation: excellent. Universidad de Sevilla Undergraduate studies, part of the Erasmus exchange program 02/2011 – 09/2011 Georg-August Universität Göttingen Bachelor of Science 10/2008 – 09/2011 Languages........................................................................................................................ -
Lectures on Algebraic Statistics
Lectures on Algebraic Statistics Mathias Drton, Bernd Sturmfels, Seth Sullivant September 27, 2008 2 Contents 1 Markov Bases 7 1.1 Hypothesis Tests for Contingency Tables . 7 1.2 Markov Bases of Hierarchical Models . 17 1.3 The Many Bases of an Integer Lattice . 26 2 Likelihood Inference 35 2.1 Discrete and Gaussian Models . 35 2.2 Likelihood Equations for Implicit Models . 46 2.3 Likelihood Ratio Tests . 54 3 Conditional Independence 67 3.1 Conditional Independence Models . 67 3.2 Graphical Models . 75 3.3 Parametrizations of Graphical Models . 85 4 Hidden Variables 95 4.1 Secant Varieties in Statistics . 95 4.2 Factor Analysis . 105 5 Bayesian Integrals 113 5.1 Information Criteria and Asymptotics . 113 5.2 Exact Integration for Discrete Models . 122 6 Exercises 131 6.1 Markov Bases Fixing Subtable Sums . 131 6.2 Quasi-symmetry and Cycles . 136 6.3 A Colored Gaussian Graphical Model . 139 6.4 Instrumental Variables and Tangent Cones . 143 6.5 Fisher Information for Multivariate Normals . 150 6.6 The Intersection Axiom and Its Failure . 152 6.7 Primary Decomposition for CI Inference . 155 6.8 An Independence Model and Its Mixture . 158 7 Open Problems 165 4 Contents Preface Algebraic statistics is concerned with the development of techniques in algebraic geometry, commutative algebra, and combinatorics, to address problems in statis- tics and its applications. On the one hand, algebra provides a powerful tool set for addressing statistical problems. On the other hand, it is rarely the case that algebraic techniques are ready-made to address statistical challenges, and usually new algebraic results need to be developed. -
Coding of Geodesics and Lorenz-Like Templates for Some Geodesic Flows
CODING OF GEODESICS AND LORENZ-LIKE TEMPLATES FOR SOME GEODESIC FLOWS PIERRE DEHORNOY AND TALI PINSKY Abstract. We construct a template with two ribbons that describes the topology of all periodic orbits of the geodesic flow on the unit tangent bundle to any sphere with three cone points with hyperbolic metric. The construction relies on the existence of a particular coding with two letters for the geodesics on these orbifolds. 1. Introduction 1 1 1 For p; q; r three positive integers|r being possibly infinite—satisfying p + q + r < 1, we consider the associated hyperbolic triangle and the associated orientation preserving 2 Fuchsian group Gp;q;r. The quotient H =Gp;q;r is a sphere with three cone points of an- 2π 2π 2π 1 2 gles p ; q ; r obtained by gluing two triangles. The unit tangent bundle T H =Gp;q;r is a 3-manifold that is a Seifert fibered space. It naturally supports a flow whose orbits are 2 1 2 lifts of geodesics on H =Gp;q;r. It is called the geodesic flow on T H =Gp;q;r and is denoted by 'p;q;r. These flows are of Anosov type [Ano67] and, as such, are important for at least two reasons: they are among the simpliest chaotic systems [Had1898] and they are funda- mental objects in 3-dimensional topology [Thu88]. Each of these flows has infinitely many periodic orbits, which are all pairwise non-isotopic. The study of the topology of these periodic orbits began with David Fried who showed that many collections of such periodic orbits form fibered links [Fri83].