MATTHEW JAMES EMERTON Curriculum Vitae
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Arxiv:1401.5507V2
FAMILIES OF L-FUNCTIONS AND THEIR SYMMETRY PETER SARNAK, SUG WOO SHIN, AND NICOLAS TEMPLIER Abstract. In [100] the first-named author gave a working definition of a family of automorphic L-functions. Since then there have been a number of works [32], [118], [73] [49], [72] and especially [108] by the second and third-named authors which make it possible to give a conjectural answer for the symmetry type of a family and in particular the universality class predicted in [68] for 1 the distribution of the zeros near s = 2 . In this note we carry this out after introducing some basic invariants associated to a family. 1. Definition of families and Conjectures The zoo of automorphic cusp forms π on G = GLn over Q correspond bijectively to their standard completed L-functions Λ(s, π) and they constitute a countable set containing species of different types. For example there are self-dual forms, ones corresponding to finite Galois representations, to Hasse–Weil zeta functions of varieties defined over Q, to Maass forms, etc. From a number of points of view (including the nontrivial problem of isolating special forms) one is led to study such Λ(s, π)’s in families in which the π’s have similar characteristics. Some applications demand the understanding of the behavior of the L-functions as π varies over a family. Other applications involve questions about an individual L-function. In practice a family is investigated as it arises. For example the density theorems of Bombieri [13] and Vinogradov [113] are concerned with showing that in a suitable sense most Dirichlet L-functions have few violations of the Riemann hypothesis, and as such it is a powerful substitute for the latter. -
Counting Points on Igusa Varieties of Hodge Type by Thomas a Mack
Counting Points on Igusa Varieties of Hodge Type by Thomas A Mack-Crane A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Sug Woo Shin, Chair Professor Xinyi Yuan Professor Richard Bamler Professor Ori Ganor Spring 2021 Abstract Counting Points on Igusa Varieties of Hodge Type by Thomas A Mack-Crane Doctor of Philosophy in Mathematics University of California, Berkeley Professor Sug Woo Shin, Chair The Langlands-Kottwitz method seeks to understand Shimura varieties in terms of automorphic forms by deriving a trace formula for the cohomology of Shimura varieties which can be compared to the automorphic trace formula. This method was pioneered by Langlands [Lan77, Lan79], and developed further by Kottwitz in [Kot90, Kot92] in the case of Shimura varieties of PEL type with good reduction. Igusa varieties were introduced in their modern form by Harris-Taylor [HT01] in the course of studying the bad reduction of certain simple Shimura varieties. The relation between Igusa varieties and Shimura varieties was expanded to PEL type by Mantovan [Man04, Man05], and their study of the cohomology of Igusa varieties was expanded to PEL type and streamlined in the Langlands-Kottwitz style by Shin [Shi09, Shi10]. Following the generalization of Mantovan’s work to Hodge type by Hamacher and Hamacher-Kim [Ham19, HK19], we carry out the Langlands-Kottwitz method for Igusa varieties of Hodge type, generalizing the work of [Shi09]. That is, we derive a trace formula for the cohomology of Igusa varieties suitable for eventual comparison with the automorphic trace formula. -
Construction of Automorphic Galois Representations: the Self-Dual Case
CONSTRUCTION OF AUTOMORPHIC GALOIS REPRESENTATIONS: THE SELF-DUAL CASE SUG WOO SHIN This survey paper grew out of the author's lecture notes for the Clay Summer School in 2009. Proofs are often omitted or sketched, but a number of references are given to help the reader find more details if he or she wishes. The article aims to be a guide to the outline of argument for constructing Galois representations from the so-called regular algebraic conju- gate self-dual automorphic representations, implementing recent improvements in the trace formula and endoscopy. Warning. Although it is important to keep track of various constants (e.g. sign matters!), twists by characters, etc in proving representation-theoretic identities, I often ignore them in favor of simpler notation and formulas. Thus many identities and statements in this article should be taken with a grain of salt and not cited in academic work. Correct and precise statements can be found in original papers. I should also point out that significant progress has been made, especially in the case of non-self-dual automorphic representations ([HLTT16], [Sch15]) since the original version of these notes was written. I have kept updates minimal (mostly taking place in introduction, adding references to some recent developments) and do not discuss these exciting new results here. The interested readers are referred to survey papers such as [Wei16, Mor16, Scha, Sch16, Car]. Acknowledgments. I heartily thank the Clay Mathematics Institute for its generous support toward the 2009 summer school and choosing an exciting location as Hawaii. I am grateful to the participants for their interest and wonderful questions. -
![Arxiv:1009.0785V3 [Math.NT] 7 Sep 2015 Ersnaininstead](https://docslib.b-cdn.net/cover/9245/arxiv-1009-0785v3-math-nt-7-sep-2015-ersnaininstead-419245.webp)
Arxiv:1009.0785V3 [Math.NT] 7 Sep 2015 Ersnaininstead
THE CONJECTURAL CONNECTIONS BETWEEN AUTOMORPHIC REPRESENTATIONS AND GALOIS REPRESENTATIONS KEVIN BUZZARD AND TOBY GEE Abstract. We state conjectures on the relationships between automorphic representations and Galois representations, and give evidence for them. Contents 1. Introduction. 1 2. L-groups and local definitions. 4 3. Global definitions, and the first conjectures. 16 4. The case of tori. 22 5. Twisting and Gross’ η. 27 6. Functoriality. 34 7. Reality checks. 35 8. Relationship with theorems/conjectures in the literature. 37 References 41 1. Introduction. 1.1. Given an algebraic Hecke character for a number field F , a classical con- struction of Weil produces a compatible system of 1-dimensional ℓ-adic representa- tions of Gal(F /F ). In the late 1950s, Taniyama’s work [Tan57] on L-functions of abelian varieties with complex multiplications led him to consider certain higher- dimensional compatible systems of Galois representations, and by the 1960s it was realised by Serre and others that Weil’s construction might well be the tip of a arXiv:1009.0785v3 [math.NT] 7 Sep 2015 very large iceberg. Serre conjectured the existence of 2-dimensional ℓ-adic repre- sentations of Gal(Q/Q) attached to classical modular eigenforms for the group GL2 over Q, and their existence was established by Deligne not long afterwards. More- over, Langlands observed that one way to attack Artin’s conjecture on the analytic continuation of Artin L-functions might be via first proving that any non-trivial n-dimensional irreducible complex representation of the absolute Galois group of a number field F came (in some precise sense) from an automorphic representation for GLn /F , and then analytically continuing the L-function of this automorphic representation instead. -
Chapter 1 Introduction 1.1 Algebraic Setting
Towards a Functor Between Affine and Finite Hecke Categories in Type A by Kostiantyn Tolmachov Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2018 Massachusetts Institute of Technology 2018. All rights reserved. -Signature redacted A uthor .. ....................... Department of Mathematics May 1, 2018 Certified by. Signature redacted.................... Roman Bezrukavnikov Professor of Mathematics Thesis Supervisor Accepted by. Signature redacted ................ -.. - -William Minicozzi G duate Co-Chair Department of Mathematics MASSACHUlS INSITUTE OF TECHNOLOGY MAY 3 0 2018 LIBRARIES fRCHIVES Towards a Functor Between Affine and Finite Hecke Categories in Type A by Kostiantyn Tolmachov Submitted to the Department of Mathematics on May 1, 2018, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract In this thesis we construct a functor from the perfect subcategory of the coherent version of the affine Hecke category in type A to the finite constructible Hecke ca- tegory, partly categorifying a certain natural homomorphism of the corresponding Hecke algebras. This homomorphism sends generators of the Bernstein's commutative subalgebra inside the affine Hecke algebra to Jucys-Murphy elements in the finite Hecke algebra. Construction employs the general strategy devised by Bezrukavnikov to prove the equivalence of coherent and constructible variants of the affine Hecke category. Namely, we identify an action of the category Rep(GL,) on the finite Hecke category, and lift this action to a functor from the perfect derived category of the Steinberg variety, by equipping it with various additional data. Thesis Supervisor: Roman Bezrukavnikov Title: Professor of Mathematics 3 4 Acknowledgments I would like to thank my advisor, Roman Bezrukavnikov, for his support and guidance over these years, as well as for many helpful discussions and sharing his expertise. -
Report 2018/2019 ETH Institute for Theoretical Studies
Report 2018/2019 ETH Institute for Theoretical Studies ETH-ITS Table of Contents Foreword 5 The ETH Institute for Theoretical Studies 6 History and aims 6 Fellows at the ITS 6 Collaborations 7 Activities 8 Meetings, talks, minicourses 8 The ITS Science Colloquium 10 Programme 2018–2019 11 Fellows’ seminar 12 Programme 2018–2019 12 Awards 13 Fellows’ report 14 Outlook 22 People at the ETH-ITS 24 Director 24 Coordinator 24 Board of Patrons 24 Advisory Committee 24 2013–2017 2017–2019 Fellows 2014–2019 25 Senior Fellows Junior Fellows (with current affiliation of former Junior Fellows) Contact 27 Clausiusstrasse 47, the address of the ETH Institute for Theoretical Studies. 4 Foreword The academic year 2018/2019 was in several ways special for the Institute for Theoretical Studies. Firstly, it experimen- ted a new way of functioning, dedicating a semester to the interdisciplinary subject of «modular forms, periods and scattering amplitudes», hosting over forty physicists and mathematicians, with a school, a workshop and other events involving several young scientists and putting into contact participants with scientists of ETH. Secondly, the Institute was evaluated by an international committee, who was very positively impressed by the achievements of the Institute and gave interesting suggestions for improvement. Thirdly, on a more personal note, this was my last year as the director of the ETH-ITS; the last six years were an intense period in which I had the unique opportunity to meet scientists in a variety of subjects who are shaping their respective disciplines. I thank the donors for making this possible, the Fellows of the Institute for their excellent scientific contributions, the school board of ETH Zurich for their continuous support, the Advisory Committee for its commitment and advice, and Christina Buchmann, who is retiring this year as coordinator, for her invaluable help and competence in running the Institute. -
November 2013
LONDONLONDON MATHEMATICALMATHEMATICAL SOCIETYSOCIETY NEWSLETTER No. 430 November 2013 Society MeetingsSociety ONE THOUSAND to publicise LMS activities and Meetings AND COUNTING mathematics more generally, and Events and leave with a range of pub- and Events Three hundred people visited De lications including the Annual 2013 Morgan House on Sunday 22 Sep- Review and information about tember 2013 as part of the an- membership, grants and Women Friday 15 November nual London Open House event. in Mathematics. LMS Graduate Student Since first participating in Open The feedback from visitors Meeting, London House three years ago over 1,000 was again very positive. The LMS page 4 people have visited De Morgan will continue to develop its pres- Friday 15 November House, learning about the Society ence at the event and is already LMS AGM, London and mathematics more generally. discussing a more comprehen- page 5 sive programme for next year. 1 Monday 16 December SW & South Wales 2013 ELECTIONS Regional Meeting, TO COUNCIL AND Swansea page 13 NOMINATING 18-20 December COMMITTEE LMS Prospects in Members should now have re- Mathematics, Durham ceived a communication from the page 14 Electoral Reform Society (ERS) for both e-voting and paper ballot. 2014 For online voting, members may cast a vote by going to www.vote- Friday 28 February byinternet.com/LMS2013 and us- Mary Cartwright ing the two part security code on Lecture, York the email sent by the ERS and also Monday 31 March on their ballot paper. Northern Regional All members are asked to look Meeting, Durham out for communication from page 19 At this year’s event visitors the ERS. -
PETER SCHOLZE to RECEIVE 2013 SASTRA RAMANUJAN PRIZE the 2013 SASTRA Ramanujan Prize Will Be Awarded to Professor Peter Scholze
PETER SCHOLZE TO RECEIVE 2013 SASTRA RAMANUJAN PRIZE The 2013 SASTRA Ramanujan Prize will be awarded to Professor Peter Scholze of the University of Bonn, Germany. The SASTRA Ramanujan Prize was established in 2005 and is awarded annually for outstanding contributions by young mathematicians to areas influenced by the genius Srinivasa Ramanujan. The age limit for the prize has been set at 32 because Ramanujan achieved so much in his brief life of 32 years. The prize will be awarded in late December at the International Conference on Number Theory and Galois Representations at SASTRA University in Kumbakonam (Ramanujan's hometown) where the prize has been given annually. Professor Scholze has made revolutionary contributions to several areas at the interface of arithmetic algebraic geometry and the theory of automorphic forms. Already in his master's thesis at the University of Bonn, Scholze gave a new proof of the Local Langlands Conjecture for general linear groups. There were two previous approaches to this problem, one by Langlands{Kottwitz, and another by Harris and Taylor. Scholze's new approach was striking for its efficiency and simplicity. Scholze's proof is based on a novel approach to calculate the zeta function of certain Shimura varieties. This work completed in 2010, appeared in two papers in Inventiones Mathematicae in 2013. Scholze has generalized his methods partly in collaboration with Sug Woo Shin to determine the `-adic Galois representations defined by a class of Shimura varieties. These results are contained in two papers published in 2013 in the Journal of the American Mathematical Society. While this work for his master's was groundbreaking, his PhD thesis written under the direction of Professor Michael Rapoport at the University of Bonn was a more mar- velous breakthrough and a step up in terms of originality and insight. -
Junehyuk Jung –
Junehyuk Jung B [email protected] • Í sites.google.com/brown.edu/junehyuk/ Positions Brown University, Department of Mathematics Providence, RI Assistant Professor 7/2020– Texas A&M University, Department of Mathematics College Station, TX Assistant Professor 8/2017–6/2020 KAIST, Department of Mathematical Science Daejeon, South Korea Researcher 6/2013–7/2016 Visiting Positions Rice University, Department of Mathematics Houston, TX Adjunct Assistant Professor 7/2019–3/2020 Yale University, Department of Mathematics New Haven, CT Visitor 4/2017–6/2017 University of California Berkeley, Department of Mathematics Berkeley, CA Visitor 2/2017–3/2017 Northwestern University, Mathematics Department Evanston, IL Visitor 10/2016–11/2016 Institute for Advanced Study, School of Mathematics Princeton, NJ Member 9/2014–12/2015 Northwestern University, Mathematics Department Evanston, IL Visiting Postdoctoral Fellow 9/2013–12/2013 Education Princeton University Princeton, NJ Doctor of Philosophy 9/2008–6/2013 Mathematics Advisor: Peter C. Sarnak Thesis Title: “On the zeros of automorphic forms” University of Chicago Chicago, IL Master of Science 9/2006–8/2008 Mathematics University of Chicago Chicago, IL Bachelor of Arts 9/2005–8/2008 Mathematics with Honors Honors and Awards National Science Foundation grant Awarded, DMS-1900993 7/2019–6/2022 Sloan Research Fellowship Awarded, $ 35,000 per year 7/2019–6/2021 Posco TJ Park Science Fellowship Awarded, $ 35,000 per year 3/2014–2/2016 Samsung Scholarship for Graduate Studies Awarded, $ 50,000 per -
The Work of Robert Macpherson 1
The work of Robert MacPherson 1 along with a huge supporting cast of coauthors: Paul Baum, Sasha Beilinson, Walter Borho, Tom Braden, JeanPaul Brasselet, JeanLuc Brylinski, Jeff Cheeger, Robert Coveyou, Corrado DeConcini, Pierre Deligne, Bill Fulton, Israel Gelfand, Sergei Gelfand, Mark Goresky, Dick Hain, Masaki Hanamura, Gunter Harder, Lizhen Ji, Steve Kleiman, Bob Kot twitz, George Lusztig, Mark MacConnell, Arvind Nair, Claudio Procesi, Vadim Schechtman, Vera Serganova, Frank Sottile, Berndt Sturmfels, and Kari Vilonen 2 and students: Paul Dippolito (Brown 1974), Mark Goresky (B 1976), Thomas DeLio (B 1978—in music!), David Damiano (B 1980), Kari Vilonen (B 1983), Kevin Ryan (B 1984), Allen Shepard (B 1985), Mark McConnell (B 1987), Masaki Hanamura (B 1988), Wolfram Gerdes (MIT 1989), David Yavin (M 1991), Yi Hu (M 1991), Richard Scott (M 1993), Eric Babson (M 1994), Paul Gunnells (M 1994), Laura Anderson (M 1994), Tom Braden (M 1995), Mikhail Grinberg (Harvard 1997), Francis Fung (Princeton 1997), Jared Anderson (P 2000), David Nadler (P 2001), Julianna Tymoczko (P 2003), Brent Doran (P 2003), Aravind Asok (P 2004) 3 and I suppose a good bunch of these eager critics are sitting in this room today. 4 5 Born — Lakewood, Ohio — May 25, 1944 B.A. — Swarthmore College 1966 Ph.D. — Harvard University 1970 Brown University — 1970 – 1987 M.I.T. — 1987 – 1994 I.A.S. — 1994 – 6 BA PhD Brown MIT IAS 1950 1960 1970 1980 1990 2000 7 Publication record (colour coded by the number of authors) 8 Part I. Test for random number generators [105] Fourier Analysis of Uniform Random Number Generators, with R. -
Fundamental Local Equivalences in Quantum Geometric Langlands
FUNDAMENTAL LOCAL EQUIVALENCES IN QUANTUM GEOMETRIC LANGLANDS JUSTIN CAMPBELL, GURBIR DHILLON, AND SAM RASKIN Abstract. In quantum geometric Langlands, the Satake equivalence plays a less prominent role than in the classical theory. Gaitsgory{Lurie proposed a conjectural substitute, later termed the fundamental local equivalence. With a few exceptions, we prove this conjecture and its extension to the affine flag variety by using what amount to Soergel module techniques. Contents 1. Introduction 1 2. Preliminary material 6 3. Fundamental local equivalences 10 Appendix A. Proof in the general case 27 References 29 1. Introduction 1.1. In the early days of the geometric Langlands program, experts observed that the fundamental objects of study deform over the space of levels κ for the reductive group G. For example, if G is simple, this is a 1-dimensional space. Moreover, levels admit duality as well: a level κ for G gives rise to a dual level1 κˇ for the Langlands dual group Gˇ. This observation suggested the existence of a quantum geometric Langlands program, deforming the usual Langlands program. The first triumph of this idea appeared in the work of Feigin{Frenkel [FF91], where they proved duality of affine W-algebras: WG,κ ' WG;ˇ κˇ. We emphasize that this result is quantum in its nature: the level κ appears. For the critical level κ = κc, which corresponds to classical geometric Langlands, Beilinson{Drinfeld [BDdf] used Feigin{Frenkel duality to give a beautiful construction of Hecke eigensheaves for certain irreducible local systems. 1.2. The major deficiency of the quantum geometric Langlands was understood immediately: the Satake equivalence is more degenerate. -
Philip Leverhulme Prize Winners 2012
Philip Leverhulme CLassiCs Professor Patrick Finglass Department of Classics, University of Nottingham Prize Winners 2012 In nine years since the award of his DPhil, Professor Patrick Finglass has established himself as one of the world’s leading scholars of Greek lyric and tragic poetry. His monumental commentaries on Sophoclean Philip Leverhulme Prizes, with a value of £70,000 each, are awarded tragedy (Electra, 2007; Ajax, 2011) are already standard works of to outstanding scholars who have made a substantial and recognised reference, a remarkable achievement for a scholar of his age; yet Finglass contribution to their particular field of study, recognised at an has produced not only these, but also a third major commentary (on international level, and where the expectation is that their greatest Pindar’s Pythian 11, 2007) and a substantial number of magisterial achievement is yet to come. articles, especially on the text and interpretation of Sophocles. His work The Prizes commemorate the contribution to the work of the Trust to date is characterized by an extraordinary combination of the energy made by Philip Leverhulme, the Third Viscount Leverhulme and and ambition of youth and the erudition and judgement that normally grandson of the Founder. come only with a lifetime’s experience. Commentaries on the remaining plays of Sophocles and on the fragments of the lyric poet, Stesichorus, The broad fields of research covered by this year’s awards were: are eagerly awaited; these will further cement what is already a towering • Classics international reputation. • Earth, Ocean and Atmospheric Sciences www.nottingham.ac.uk/classics/people/patrick.finglass • History of Art • Law Professor Miriam Leonard • Mathematics and Statistics Department of Greek and Latin, University College London • Medieval, Early Modern and Modern History.