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Pila, O-Minimality and Diophantine Geometry
O-minimality and Diophantine geometry Jonathan Pila Abstract. This lecture is concerned with some recent applications of mathematical logic to Diophantine geometry. More precisely it concerns applications of o-minimality, a branch of model theory which treats tame structures in real geometry, to certain finiteness problems descending from the classical conjecture of Mordell. Mathematics Subject Classification (2010). Primary 03C64, 11G18. Keywords. O-minimal structure, Andr´e-Oortconjecture, Zilber-Pink conjecture. 1. Introduction This is a somewhat expanded version of my lecture at ICM 2014 in Seoul. It surveys some recent interactions between model theory and Diophantine geometry. The Diophantine problems to be considered are of a type descending from the classical Mordell conjecture (theorem of Faltings). I will describe the passage from Mordell's conjecture to the far-reaching Zilber-Pink conjecture, which is very much open and the subject of lively study by a variety of methods on several fronts. The model theory is \o-minimality", which studies tame structures in real geometry, and offers powerful tools applicable to certain “definable” sets. In combination with an elementary analytic method for \counting rational points" it leads to a general result about the height distribution of rational points on definable sets. This result can be successfully applied to Zilber-Pink problems in the presence of certain functional transcendence and arithmetic ingredients which are known in many cases but seemingly quite difficult in general. Both the methods and problems have connections with transcendental number theory. My further objective is to explain these connections and to bring out the pervasive presence of Schanuel's conjecture. -
Algebra & Number Theory Vol. 7 (2013)
Algebra & Number Theory Volume 7 2013 No. 3 msp Algebra & Number Theory msp.org/ant EDITORS MANAGING EDITOR EDITORIAL BOARD CHAIR Bjorn Poonen David Eisenbud Massachusetts Institute of Technology University of California Cambridge, USA Berkeley, USA BOARD OF EDITORS Georgia Benkart University of Wisconsin, Madison, USA Susan Montgomery University of Southern California, USA Dave Benson University of Aberdeen, Scotland Shigefumi Mori RIMS, Kyoto University, Japan Richard E. Borcherds University of California, Berkeley, USA Raman Parimala Emory University, USA John H. Coates University of Cambridge, UK Jonathan Pila University of Oxford, UK J-L. Colliot-Thélène CNRS, Université Paris-Sud, France Victor Reiner University of Minnesota, USA Brian D. Conrad University of Michigan, USA Karl Rubin University of California, Irvine, USA Hélène Esnault Freie Universität Berlin, Germany Peter Sarnak Princeton University, USA Hubert Flenner Ruhr-Universität, Germany Joseph H. Silverman Brown University, USA Edward Frenkel University of California, Berkeley, USA Michael Singer North Carolina State University, USA Andrew Granville Université de Montréal, Canada Vasudevan Srinivas Tata Inst. of Fund. Research, India Joseph Gubeladze San Francisco State University, USA J. Toby Stafford University of Michigan, USA Ehud Hrushovski Hebrew University, Israel Bernd Sturmfels University of California, Berkeley, USA Craig Huneke University of Virginia, USA Richard Taylor Harvard University, USA Mikhail Kapranov Yale University, USA Ravi Vakil Stanford University, -
All Souls College
All Souls College Annual Report and Financial Statements for the year ended 31 July 2014 Registered as a Charity in England and Wales, no: 1138057. Registered address: High Street, Oxford OX1 4AL ALL SOULS COLLEGE Year ended 31 July 2014 Table of Contents Pages Report of the Governing Body 2 - 23 Reference and Administrative Information 24 - 27 Auditor’s Report 28 - 29 Principal Accounting Policies 30 - 33 Consolidated Statement of Financial Activities 34 Consolidated and College Balance Sheets 35 Consolidated Cashflow Statement 36 Notes to the Financial Statements 37 - 57 1 ALL SOULS COLLEGE Report of the Governing Body Year ended 31 July 2014 REPORT OF THE GOVERNING BODY The Warden and Fellows of All Souls College present their Annual Report for the year ended 31 July 2014 under the Charities Act 2011 and the Charities SORP 2005 together with the audited financial statements for the year. INTRODUCTION The College of All Souls of the Faithful Departed, of Oxford – known as All Souls College – was founded by Henry VI and Henry Chichele (Archbishop of Canterbury) in 1438. Today the College is primarily an academic research institution with particular strengths in the Humanities, Mathematics, Social and Natural Sciences and an outstanding library. It also has strong ties to public life. Although the Warden and Fellows of the College are involved in teaching and supervision of research in the University, there are no undergraduate members. On 31 July 2014 there were seventy-seven Fellows of All Souls, twenty-nine Emeritus (i.e. retired academic) and seven Honorary Fellows, many of whose continuing research the College was actively supporting. -
Ax-Schanuel and Strong Minimality for the $ J $-Function
AX-SCHANUEL AND STRONG MINIMALITY FOR THE j-FUNCTION VAHAGN ASLANYAN Abstract. Let K := (K;+, ·, D, 0, 1) be a differentially closed field of character- istic 0 with field of constants C. In the first part of the paper we explore the connection between Ax-Schanuel type theorems (predimension inequalities) for a differential equation E(x, y) and the geometry of the fibres Us := {y : E(s,y) ∧ y∈ / C} where s is a non-constant element. We show that certain types of predimension inequalities imply strong minimality and geometric triviality of Us. Moreover, the induced structure on the Cartesian powers of Us is given by special subvarieties. In particular, since the j-function satisfies an Ax-Schanuel inequality of the required form (due to Pila and Tsimerman), applying our results to the j-function we recover a theorem of Freitag and Scanlon stating that the differential equation of j defines a strongly minimal set with trivial geometry. In the second part of the paper we study strongly minimal sets in the j-reducts of differentially closed fields. Let Ej (x, y) be the (two-variable) differential equation of the j-function. We prove a Zilber style classification result for strongly minimal sets in the reduct K := (K;+, ·, Ej ). More precisely, we show that in K all strongly minimal sets are geometrically trivial or non-orthogonal to C. Our proof is based on the Ax-Schanuel theorem and a matching Existential Closedness statement which asserts that systems of equations in terms of Ej have solutions in K unless having a solution contradicts Ax-Schanuel. -
PGR Faculty List 2021 ALL Departments 24August2021 Draft
Faculty Lists fall 2021 Email: [email protected] for corrections. Current update: 8/24/2021 #=75 or older in 2021 (* was over 70 in 2017 list) Part-time faculty are half-time, unless otherwise noted. UNITED STATES (the top 50 will be ranked) FACULTY # Arizona Faculty: Sara Aronowitz, Thomas Christiano, Stewart Cohen, Juan Comesaña, Reza Hadisi, RiChard Healey, Laura, Howard, J. Christopher Maloney, MiChael McKenna, Bill OberdiCK, Guido PinCione, Marga Reimer, Daniel Russell, Carolina Sartorio, David SChmidtz, Houston Smit, MarK Timmons, Joseph Tolliver, Jason Turner, Steven Wall, Jonathan Weinberg. Part-time faculty: *Allen Buchanan (.25 time) Cognate faculty and philosophers in other units: Martin FriCKe, Massimo Piattelli-Palmarini, Christopher Robertson, Simone Sepe. FACULTY # Arizona State Faculty: RiChard Amesbury, Brad Armendt, Thomas BlaCKson, Cheshire Calhoun, Peter de Marneffe, Typer DesRoChes, MarCello Di Bello, Peter Kung, Joan MCGregor, Shyam Nair, Ben Phillips, Nestor Ángel Pinillos, Douglas W. Portmore, Maura Priest, Steven Reynolds. Part-time faculty: Cognate faculty and philosophers in other units: RiChard Creath, Tyler DesRoChes, ZaChary Horne, Ted Humphrey, PatriCia J. Huntingon, Manfred LaubiChler, Jane MainesChein, Martin BeCK MatuštíK, Ben A. Minteer, *Jeffrie G. Murphy, BeCKett Sterner, Jason Robert, Hava Tirosh-Samuelson, Norbert Samuelson, BeCKett Sterner. FACULTY # BerKeley Faculty: Olivia Bailey, John Campbell, Timothy ClarKe, Shamik Dasgupta, Johann FriCK, Hannah Ginsborg, Florian Grosser, Wesley H. Holliday, NiKo Kolodny, Geoffrey Lee, John MaCFarlane, Paolo ManCosu, Alva Noë, Andreja NovaKoviC, Kristin Primus, R. Jay WallaCe, Daniel Warren, Seth Yalcin, Xueyin (Snow) Zhang. Part-time Faculty: Joshua Cohen (.25 time), MiChael (M.G.F.) Martin, Veronique Munoz Darde, Kwong-Loi Shun. Cognate Faculty and Philosophers in Other Units: Asad Q. -
Staff Magazine for the University of Oxford | October 2014
blueprint Staff magazine for the University of Oxford | October 2014 Chu’s views | Vegetables propagate smiles | Women on the wall News in brief u The Dickson Poon University of Oxford u Make sure you’re effectively engaged in China Centre Building was formally opened social media by taking advantage of this by the Duke of Cambridge on 8 September. term’s Engage (#oxengage) programme The centre, which is in the grounds of run by IT Services in partnership with the St Hugh’s College, brings academics with Bodleian Libraries. Now in its third year, an interest in China under the same roof. the programme comprises a series of talks, The five-floor building houses a dedicated seminars and workshops to explore social library and reading room, which will provide media strategies and digital tools, and is a permanent home for 60,000 volumes and designed to inspire academics, researchers UniversityOxford/Jonathanof Hordle a significant part of the Bodleian Libraries’ and graduate students to consider using Chinese book collection. It also features a social media and digital technology to lecture theatre, language laboratory, study develop their online presence for outreach areas and a dining room. The £21m cost was and public engagement. Sessions range from largely met through benefactions, including academic blogging and tweeting to sharing £10m from Hong Kong philanthropist research findings via infographics. View the Mr Dickson Poon CBE. programme at blogs.it.ox.ac.uk/engage/social- media-michaelmas/programme-2014. Robotics Alcock / Ed Aldebaran u 15 September saw the inaugural Oxford India Lecture take place in New Delhi. -
O-Minimality and the André-Oort Conjecture for Cn
Annals of Mathematics 173 (2011), 1779{1840 doi: 10.4007/annals.2011.173.3.11 O-minimality and the Andr´e-Oortconjecture for Cn By Jonathan Pila Abstract We give an unconditional proof of the Andr´e-Oortconjecture for arbi- trary products of modular curves. We establish two generalizations. The first includes the Manin-Mumford conjecture for arbitrary products of el- liptic curves defined over Q as well as Lang's conjecture for torsion points in powers of the multiplicative group. The second includes the Manin- Mumford conjecture for abelian varieties defined over Q. Our approach uses the theory of o-minimal structures, a part of Model Theory, and follows a strategy proposed by Zannier and implemented in three recent papers: a new proof of the Manin-Mumford conjecture by Pila-Zannier; a proof of a special (but new) case of Pink's relative Manin-Mumford conjecture by Masser-Zannier; and new proofs of certain known results of Andr´e-Oort- Manin-Mumford type by Pila. 1. Introduction In this paper we give an unconditional proof of the Andr´e-Oortconjecture for arbitrary products of modular curves. Under the Generalized Riemann Hypothesis for imaginary quadratic fields this result is due to Edixhoven [32], [34]; for n = 2 it is an unconditional result of Andr´e[3]. Our approach uses the theory of o-minimal structures, a part of Model Theory. It leads naturally to a more general result that is an \Andr´e-Oort-Manin-Mumford-Lang"statement for varieties of the form ` X = Y1 × · · · × Yn × E1 × · · · × Em × G ; where n; m; ` are nonnegative integers, Y1 = Γ1nH;:::;Yn = ΓnnH are modular curves corresponding to the quotient of the upper half-plane H by congruence subgroups Γi of SL2(Z), E1;:::;Em are elliptic curves defined over Q, and G = Gm(C) is the multiplicative group of nonzero complex numbers. -
PGR Faculty Lists Non-US Departments 19August2021
Faculty Lists fall 2021 Email: [email protected] for corrections. Current update: 8/17/2021 #=75 or older in 2021 (* was over 70 in 2017 list) Part-time faculty are half-time, unless otherwise noted. NON-US SCHOOLS FACULTIES IN THE UNITED KINGDOM (the top 15 will be ranked) FACULTY # Birkbeck Faculty: Kristoffer Ahlstrom-Vij, Sophia Connell, Cristian Constantinescu, Stacie Friend, Michael Garnett, Alex Grzankowski, Andrew Huddleston, Susan James, Hallvard Lillehammer, Robert Northcott, Sarah Patterson, Florian Steinberger. Part-time faculty: Ken Gemes, Jennifer Hornsby, Keith Hossack. FACULTY # Birmingham Faculty: Lisa Bortolotti, Darragh Byrne, Nikk Effingham, Salvatore Florio, Iain Law, Herjeet Marway, Yujin Nagasawa, Wouter Peeters, Martin Pickup, Merten Reglitz, Michael Rush, Alex Silk, Maja Spener, Scott Sturgeon, Jussi Suikkanen, Ema Sullivan-Bissett, Henry Taylor, Heather Widdows, Jeremy Williams, Alastair Wilson, Mohammad Saleh Zarepour. Part-time faculty: *Kit Fine (.20 time), *Alison Jaggar (.20 time). Cognate Faculty and Philosophers in Other Units: Kick Adams, Michael Hand, Richard Kaye, Ben Kotzee, Kristján Kristjánsson. FACULTY # Bristol Faculty: Joanna Burch-Brown, Catrin Campbell-Moore, Havi Carel, Anthony Everett, Jonathan Grose, Max Jones, Jason Konek, James Ladyman, Seiriol Morgan, Samir Okasha, Giles Pearson, Richard Pettigrew, Martin Sticker, Tuomas Tahko, Karim Thebault, Tuomas Tahko, Tzuchien Tho, Alan Wilson, Lena Zuchowski. Cognate Faculty and Philosophers in Other Units: *Ken Binmore, David Leech, Philip Welch. FACULTY # Cambridge (Philosophy and HPS) Faculty: Arif Ahmed, Anna Alexandrova, Alexander Bird, Julia Borcherding, Mary Augusta Brazelton, Angela Breitenbach, Clare Chambers, Hasok Chang, Helen Anne Curry, Sarah Fine, Neil Dewar, Marta Halina, Richard Holton, Nick Hopwood, Stephen John, Rae Langton, Lauren Kassell, Tim Lewens, John Marenbon, Dániel Margócsy, Staffan Müller-Wille, Natasha McCarthy, Jessie Munton, Alex Oliver, Michael Potter, Simon Schaffer, Jim Secord, Paulina Sliwa, Richard Staley, Jacob Stegenga, Liba Taub. -
Logic, Diophantine Geometry, and Transcendence Theory
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 49, Number 1, January 2012, Pages 51–71 S 0273-0979(2011)01354-4 Article electronically published on October 24, 2011 COUNTING SPECIAL POINTS: LOGIC, DIOPHANTINE GEOMETRY, AND TRANSCENDENCE THEORY THOMAS SCANLON Abstract. We expose a theorem of Pila and Wilkie on counting rational points in sets definable in o-minimal structures and some applications of this theorem to problems in diophantine geometry due to Masser, Peterzil, Pila, Starchenko, and Zannier. 1. Introduction Over the past decade and a half, starting with Hrushovski’s proof of the func- tion field Mordell-Lang conjecture [12], some of the more refined theorems from model theory in the sense of mathematical logic have been applied to problems in diophantine geometry. In most of these cases, the technical results underlying the applications concern the model theory of fields considered with some additional distinguished structure, and the model theoretic ideas fuse algebraic model theory (the study of algebraic structures with a special emphasis on questions of defin- ability) and stability theory (the development of abstract notions of dimension, dependence, classification, etc.) for the purpose of analyzing the class of models of a theory. Over this period, there has been a parallel development of the model theory of theories more suited for real analysis carried out under the rubric of o-minimality, but this theory did not appear to have much to say about number theory. Some spectacular recent theorems demonstrate the error of this impression. In the paper [30], Pila presents an unconditional proof of a version of the so- called Andr´e-Oort conjecture about algebraic relations amongst the j-invariants of elliptic curves with complex multiplication using a novel technique that comes from model theory. -
Report for the Academic Year 2017–2018
Institute for Advanced Study Re port for 2 0 1 7–2 0 INSTITUTE FOR ADVANCED STUDY 1 8 EINSTEIN DRIVE PRINCETON, NEW JERSEY 08540 Report for the Academic Year (609) 734-8000 www.ias.edu 2017–2018 Cover: SHATEMA THREADCRAFT, Ralph E. and Doris M. Hansmann Member in the School of Social Science (right), gives a talk moderated by DIDIER FASSIN (left), James D. Wolfensohn Professor, on spectacular black death at Ideas 2017–18. Opposite: Fuld Hall COVER PHOTO: DAN KOMODA Table of Contents DAN KOMODA DAN Reports of the Chair and the Director 4 The Institute for Advanced Study 6 School of Historical Studies 8 School of Mathematics 20 School of Natural Sciences 30 School of Social Science 40 Special Programs and Outreach 48 Record of Events 57 80 Acknowledgments 88 Founders, Trustees, and Officers of the Board and of the Corporation 89 Administration 90 Present and Past Directors and Faculty 91 Independent Auditors’ Report THOMAS CLARKE REPORT OF THE CHAIR The Institute for Advanced Study’s independence and excellence led by Sanjeev Arora, Visiting Professor in the School require the dedication of many benefactors, and in 2017–18, of Mathematics. we celebrated the retirements of our venerable Vice Chairs The Board was delighted to welcome new Trustees Mark Shelby White and Jim Simons, whose extraordinary service has Heising, Founder and Managing Director of the San Francisco enhanced the Institute beyond measure. I am immensely grateful investment firm Medley Partners, and Dutch astronomer and and feel exceptionally privileged to have worked with both chemist Ewine Fleur van Dishoeck, Professor of Molecular Shelby and Jim in shaping and guiding the Institute into the Astrophysics at the University of Leiden. -
Notices of the American Mathematical Society ABCD Springer.Com
ISSN 0002-9920 Notices of the American Mathematical Society ABCD springer.com More Math Number Theory NEW Into LaTeX An Intro duc tion to NEW G. Grätzer , Mathematics University of W. A. Coppel , Australia of the American Mathematical Society Numerical Manitoba, National University, Canberra, Australia Models for Winnipeg, MB, Number Theory is more than a May 2009 Volume 56, Number 5 Diff erential Canada comprehensive treatment of the Problems For close to two subject. It is an introduction to topics in higher level mathematics, and unique A. M. Quarte roni , Politecnico di Milano, decades, Math into Latex, has been the in its scope; topics from analysis, Italia standard introduction and complete modern algebra, and discrete reference for writing articles and books In this text, we introduce the basic containing mathematical formulas. In mathematics are all included. concepts for the numerical modelling of this fourth edition, the reader is A modern introduction to number partial diff erential equations. We provided with important updates on theory, emphasizing its connections consider the classical elliptic, parabolic articles and books. An important new with other branches of mathematics, Climate Change and and hyperbolic linear equations, but topic is discussed: transparencies including algebra, analysis, and discrete also the diff usion, transport, and Navier- the Mathematics of (computer projections). math Suitable for fi rst-year under- Stokes equations, as well as equations graduates through more advanced math Transport in Sea Ice representing conservation laws, saddle- 2007. XXXIV, 619 p. 44 illus. Softcover students; prerequisites are elements of point problems and optimal control ISBN 978-0-387-32289-6 $49.95 linear algebra only A self-contained page 562 problems. -
2014 Annual Report
CLAY MATHEMATICS INSTITUTE www.claymath.org ANNUAL REPORT 2014 1 2 CMI ANNUAL REPORT 2014 CLAY MATHEMATICS INSTITUTE LETTER FROM THE PRESIDENT Nicholas Woodhouse, President 2 contents ANNUAL MEETING Clay Research Conference 3 The Schanuel Paradigm 3 Chinese Dragons and Mating Trees 4 Steenrod Squares and Symplectic Fixed Points 4 Higher Order Fourier Analysis and Applications 5 Clay Research Conference Workshops 6 Advances in Probability: Integrability, Universality and Beyond 6 Analytic Number Theory 7 Functional Transcendence around Ax–Schanuel 8 Symplectic Topology 9 RECOGNIZING ACHIEVEMENT Clay Research Award 10 Highlights of Peter Scholze’s contributions by Michael Rapoport 11 PROFILE Interview with Ivan Corwin, Clay Research Fellow 14 PROGRAM OVERVIEW Summary of 2014 Research Activities 16 Clay Research Fellows 17 CMI Workshops 18 Geometry and Fluids 18 Extremal and Probabilistic Combinatorics 19 CMI Summer School 20 Periods and Motives: Feynman Amplitudes in the 21st Century 20 LMS/CMI Research Schools 23 Automorphic Forms and Related Topics 23 An Invitation to Geometry and Topology via G2 24 Algebraic Lie Theory and Representation Theory 24 Bounded Gaps between Primes 25 Enhancement and Partnership 26 PUBLICATIONS Selected Articles by Research Fellows 29 Books 30 Digital Library 35 NOMINATIONS, PROPOSALS AND APPLICATIONS 36 ACTIVITIES 2015 Institute Calendar 38 1 ach year, the CMI appoints two or three Clay Research Fellows. All are recent PhDs, and Emost are selected as they complete their theses. Their fellowships provide a gener- ous stipend, research funds, and the freedom to carry on research for up to five years anywhere in the world and without the distraction of teaching and administrative duties.