Barry Mazur

Gerhard Gade University Professor

Office: Science Center 512
Tel: (617) 495-3352
Fax: (617) 495-5132
Email mazur@math.harvard.edu

Postal Address:
Department of Mathematics
Harvard University
One Oxford Street
Cambridge, MA 02138-2901

Research Interests

Number theory, Automorphic forms, and related issues in algebraic geometry. See the list of current projects, the page pointing to older material, MathSci Reviews and Google scholar links.

News

October 25 2022: Here are the notes for a lecture I gave (via Zoom) at the conference at the IAS in honor of Andrew Ogg (Thoughts about Andrew Ogg's (Torsion) Conjecture)

This semester Eric Maskin, Amartya Sen gave a course Phil248R Rationality in the Philosophy Department, cross-listed in Mathematics and Economics. (What does it mean for behavior to be `rational'? What is the relationship among rationality, logic, consistency, proof, and emotion? These questions and others were explored in this seminar course from the standpoint of philosophy, economics, mathematics, psychology, biology, and other disciplines. Students were expected to participate fully in classroom discussions. There were no formal prerequisites except for facility with formal reasoning and high school mathematics.) Here are the handouts I distributed for the sessions that I conducted (Organizing Rational Thought) August 16 , 2022: Here are the slides for a lecture I gave (via Zoom) at the Mordell 2022 Conference (Thoughts about Mordell and uniformity of finiteness bounds) and here is the corresponding text (Mordell.Text.from.Slides)
May 19 , 2022: I gave a lecture (Hilbert's Hotel and other encounters with infinity) and participated in a Q&A session in a Colloquium for Undergraduate Math Majors in the UK organized jointly by the ICMS and the London Mathematical Society. Here are the slides for that session, and here is the link to the video: https://media.ed.ac.uk/media/Hilbert%E2%80%99s+Hotel+and+other+encounters+with+infinity/1_rlqsdpj8

April 8 , 2022: I took part in a Q&A session of the course Morals and Mathematics taught in B.U. by Avner Ash and Ryan Hanley. I want to thank the students for bringing such a broad range of interests and experience into our conversation. Here ( Musings about Mathematics) is a slightly edited version of the handout for that session.

December 28, 2021: Amartya Sen, Eric Maskin and I co-taught a seminar course Phil 248 this past Fall semester. Here is a slightly edited version of the handouts for my part of the course, and here are the quotations taken from the reading list that was discussed in one of the sessions.

December 11, 2021: Karl Rubin and I have just published an article in the journal Experimental Mathematics, entitled: Arithmetic Conjectures Suggested by the Statistical Behavior of Modular Symbols. As I understand it, for a limited amount of time, the following link will work to download that article: https://www.tandfonline.com/eprint/DDUUPDV2DEK8XV2D3IPT/full?target=10.1080/10586458.2021.1982424

December 8, 2021: Heuristics in Arithmetic---and randomness (and ``Heuristics") are the slides and notes for lecture I gave and will give via Zoom (Dec.2, 9) at Hebrew University in Jerusalem.

October 29, 2021: Bridges between Geometry and ... Number Theory (and Chebotarev Questions) are the notes for a lecture I gave at the conference "Unifying Themes In Geometry" at the Lake Como School of Advanced Studies, September 2021.

July 6, 2021: I was asked to prepare a `five-minute-question" for the question session of the online workshop "Rational Points and Galois Representations" hosted by the University of Pittsburgh May 10-12, 2021. I thank Jennifer Balakrishnan, Netan Dogra, Brian Lawrence, and Carl Wang-Erickson for organizing the workshop; and David Zureick-Brown for organizing the question session. A question about quadratic points on X_0(N) are the revised background notes I put together (for myself) to frame my question.

May 13, 2021: Here are the slides of a lecture I gave at the Swinnerton-Dyer Memorial Conference (held --via Zoom---at the Newton Institute)

January 4, 2021: Lucia Caparoso, Joe Harris and I have just posted a paper Corrections to "Uniformity of rational points" and further comments on ArXiv: https://arxiv.org/abs/2012.14461 This consists of a correction, as the title indicates, to our paper published almost a quarter of a century ago.

July 28, 2020: Vasily Golyshev asked me to give a short general lecture (via Zoom, of course) about Main Conjectures in a summer seminar (focusing on motivic gamma functions) in Paris. Vasily wrote that the seminar is more of a basic notions seminar focusing on "grand design" so that "we understand what it all is about" (rather than technical reports, or reports about new work). Here are my notes.

July 22, 2020: Math in the Time of Plague is an essay that I wrote. It will be published in the Mathematical Intelligencer. Here is an excerpt of this with further thoughts (it is entitled The Consolation of Math in Plague Time and will be published by Springer in the volume of essays Math in the Time of Corona edited by Elizabeth Loew).

July 6, 2020: Karl Rubin and I have posted a paper entitled Big fields that are not large on ArXiv: https://arxiv.org/abs/2004.08989. It will appear in the Proceedings of the AMS.

November 17, 2019: Karl Rubin and I have posted a paper entitled Arithmetic conjectures suggested by the statistical behavior of modular symbols on ArXiv: https://arxiv.org/abs/1910.12798 It will appear in the journal Experimental Mathematics.

November 17, 2019: Amartya Sen and Eric Maskin and I gave a seminar-course (Phil273o: Objectivity and Subjectivity) in the Philosophy Department. Here are the readings and handouts for three sessions our seminar.

July 10, 2019: Here are the rough notes for the lecture I gave at the conference at the IHES on July 5. Here is a link to the video of the lecture: https://www.youtube.com/watch?v=TDkcOgdeQ-U

March 30, 2019: Here are the slides for the AMS Einstein Public Lecture in Mathematics ( On the arithmetic of curves) that I gave at the AMS regional conference in Hawaii (March 2019)

March 30, 2019: The statistical behavior of modular symbols and arithmetic conjectures Here are three versions of this talk (on joint work with Karl Rubin) that I gave: (1) at the Hawaii University March 2019 Number Theory Conference (2) as a Q & A (mostly Q) about statistics regarding diophantine stability. It consists of skeletal notes for a talk that I gave at the "Number Theorist's" seminar at Harvard (Oct. 6, 2017). The talk was meant to be a lead-in to this collection of slides; and: (3) here is a link to a lecture I gave at the Fields Institute in a symposium in honor of Manjul Bhargava. (November 16, 2016)

February 14, 2019: Here are links to videos for presentations during the last three days of the conference at Harvard for my 80-th birthday: Poetry Panel: https://www.youtube.com/watch?v=JBvbeJf6Nig Haruzo Hida: https://www.youtube.com/watch?v=kzvJnsUdY2I Jordan Ellenberg: https://www.youtube.com/watch?v=PUbUSE3OoN8 Wei Zhang: https://www.youtube.com/watch?v=EzJKiS8nJSE Philosophy/Law/Physics: https://www.youtube.com/watch?v=yM3YKQwAeYA Glenn Stevens: https://www.youtube.com/watch?v=uCeIbe4JOIk Persi Diaconis: https://www.youtube.com/watch?v=-nijw3vuRfg Alexandra Shlapentokh: https://www.youtube.com/watch?v=DRtiJQTZZN8 History of Science panel: https://www.youtube.com/watch?v=kQqXSn4DEeA Akshay Venkatesh: https://www.youtube.com/watch?v=konJw-8G64s Manjul Bhargava placekeeper: https://www.youtube.com/watch?v=f96pBanoNoE

February 12, 2019: I am currently running a "topics course" L-Functions and Arithmetic Statistics Math 258 which concentrates on modular symbols (and their connection with L-function special values), Selmer groups, and the method of Chabauty-Coleman-Kim. Here is the schedule of presentations, and I , II , III are handouts related to material already covered. The url for the course site is: https://canvas.harvard.edu/courses/50675

January 29, 2019: Eric Maskin, Amartya Sen and I gave a seminar-course last semester: `Axiomatic Reasoning' (PHIL 273B). Introduction to Axiomatic Reasoning gives a general sense of the aim of the seminar, and what we covered.

October 20, 2018: I gave The Paul Bernays Lectures at ETH In September, the overall title of this series being The Unity and Breadth of Mathematics: from Diophantus to today. The background notes for the three lectures are: What is it that unifies Mathematics? , New issues, and expectations, in the study of rational points , and Diophantine stability and the vanishing of L-functions at the central points. Here is a link to a video of these lectures.

July 8, 2018: Karl Rubin and I are about to run a workshop at the Fondation Les Treilles July 23-28 on Arithmetic Statistics and Diophantine Stability. This follows the workshop we ran in May 2017 at Harvard on the statistics of modular symbols, `theta-elements', and special values of $L$-functions. Here are links describing that conference: http://www.math.harvard.edu/conferences/modular17/index.html and https://www.math.uci.edu/~krubin/workshop_background.pdf. These workshops include themes discussed in these notes for the lecture I gave in honor of Manjul Bhargava at the 2016 Fields Medal Symposium in Toronto. Here is the set of slides for that lecture.

Nov. 19, 2017: Eric Maskin, Amartya Sen and I taught a course `Utility' (PHIL 273A). Usefulness is a set of skeletal notes describing the general nature of this seminar-course and includes the handouts for the sessions that I chaired.

Sept. 27, 2017: Here (and repeated on the ``Teaching" page) is a link to the archived collection of my recent course web-sites

May 23, 2017: Here is a link to an interview I gave that appears on Anthony Bonato's blog.

January 7, 2017: I was asked (by Vasily Golyshev) to post on my web-page a copy of my essay Bernoulli Numbers and the Unity of Mathematics, this being a write-up of the Daniel Bartlett Memorial Lecture I gave at the University of Arizona six years ago. Here it is.

December 10 2016: Here is a link to Diophantine Stability, an article I wrote with Karl Rubin (with an appendix by Michael Larsen) to appear in the American Journal of Mathematics.

I taught a seminar-course with Mark Schiefsky in the History of Science Department in the Fall semester 2016 (HISTSCI 206R: Science, Mathematics, and Explanation: What Does it Mean to Explain Something?). We studied different answers to this question by means of a close examination of examples drawn from ancient, early modern, and contemporary mathematics, science, and philosophy. Related to this are the notes "On the word 'because' in mathematics, and elsewhere"

January 15, 2016: Here is a link to "Thinking about Grothendieck" which was published in the NOTICES of the AMS, vol, 63 no. 4, pp. 403, 404 (2016)

August 17, 2015: Here is a link to "A Mathematical Fable" I made a few years ago for NumberPhile.

August 17, 2015: Here are rough notes for my lecture at the Model Theory Seminar at MSRI on April 29, 2014 Questions about isogenies, automorphisms and bounds ([PDF]).

March 14, 2015: Here is Rational families of 17-torsion points of elliptic curves over number fields, an article Maarten Derickx, Sheldon Kamienny, and I have written in honor of Fumiyuki Momose.

November 9, 2014: Here (I) (II) (III) are slides for Logic, Elliptic curves, and Diophantine stability, a series of lectures that I gave in Princeton (The Minerva Lectures; October 14, 15, 17; 2014). Here are my informal notes to these three lectures.

August 3, 2014: Here is a link to a video regarding the book Prime Numbers and the Riemann Hypothesis I am writing with William Stein. Here is a link to the book.

July 16, 2014: Here is Descending cohomology, geometrically, which was presented at the session of "open problems" at the 60th birthday conference for Joe Harris at Harvard University and has been published in the July 2014 issue of the Notices of the ICCM.

July 16, 2014: I co-taught the seminar-course History of Science 206r: "It's Only a Hypothesis" with Professor Mark Schiefsky.
(Harvard College/GSAS: 2410 Meeting Time: Th., 2-4 Exam Group: 14)

An examination of the notions of hypothesis and hypothetical method in science and mathematics, with attention to issues in the philosophy of science such as the realism/instrumentalism debate, Bayesian formulations in the empirical sciences, axiom systems in mathematics (including the transition from Euclid to the system of axioms as reconfigured by Hilbert, and modern foundational issues) and the role of models. Readings drawn from ancient Greek philosophy and mathematics, ancient and early modern astronomy, and a (small) bit of contemporary mathematics and physics." Open to graduates and qualified undergraduates with permission of the instructors. Here is its syllabus and its provisional reading list. (This requires a Harvard ID)

June 13 , 2014: Here (A Celebration of the Mathematics of Glenn Stevens) are the notes for my lecture (June 2, 2014) at the B.U. Conference in honor of Glenn Stevens' 60th birthday.

February 16, 2014: I recently co-taught Math265x, a seminar-course Reasoning via Models with Eric Maskin and Amartya Sen. The aim was, in effect, to live with the concept of "modeling" for a semester, and to acquaint ourselves with a broad span of types of models that appear in different disciplines. Even though our presentations were meant to be general introductions---in contrast---the final papers of the students in our class were often original, and specific to (and illuminating for) the field of interest of the student: a great reward of this course was to read those final papers. Here (Various takes on Mathematics and Modelling) are some notes to my presentations regarding models related to the subject of mathematics. Here is the webpage for the course (which requires a Harvard ID). Here (The authority of the incomprehensible) is an essay on related issues (published in the online journal Berfrois ([PDF]).

January 15, 2014: Here are rough notes for a short presentation entitled The Faces of Evidence (in Mathematics) ([PDF]) I gave to the group Museion at MSRI on February 5, 2014.

January 7, 2014: Karl Rubin and I have posted two (linked) papers on arXiv: Controlling Selmer groups in the higher core rank case: [PDF] and Refined class number formulas for G_m:[PDF]). In the second of these papers we formulate a generalization of a `refined class number formula' of Darmon and we prove a large part of this conjecture when the order of vanishing of the corresponding complex L-function is 1. These are now published (in the same issue of J. Th. Nombres Bordeaux): volume 28 (2016), no. 1, 185-211 and 145-183.

October 12, 2013: Here are notes for the Erdos Memorial Lecture, given at Temple University on {Arithmetic statistics: elliptic curves and other things} ([PDF]). These notes also relate to the lecture I gave the week before at the Quebec/Maine Number Theory Conference.

October 5, 2013: Some comments on elliptic curves over general number fields and Brill-Noether modular varieties ([PDF]) are rough notes for a lecture I gave at the Quebec/Maine Number Theory Conference.

March 2013: Arithmetic statistics of Selmer Groups of quadratic twist families of elliptic curves My co-authors, Zev Klagsbrun and Karl Rubin (and I) have posted two papers having to do with the above subject on the arXiv. The first, Disparity in Selmer ranks of quadratic twists of elliptic curves, [PDF], has appeared in the Annals of Mathematics 178 (2013), 287-320. The sequel to that paper, A Markov model for Selmer ranks in families of twists is here: [PDF]. It has appeared: Compos. Math. 150 (2014), no. 7, 1077-1106. A related paper that Karl Rubin and I wrote discussing elliptic curves whose quadratic twist families have exactly the same Selmer data, let alone statistics is "Selmer Companion Curves" ([PDF]) has appeared: Trans. Amer. Math. Soc. 367 (2015), no. 1, 401-421.

March 2013: The notes entitled How explicit is the Explicit Formula? ([PDF]) were written jointly with William Stein to provide material for some unusually short lectures on arithmetic phenomenology that we gave. Specifically we used it for each of our 20 minute lectures at the San Diego Conference this January, and I replayed it (in synopsized form) in a 6 minute and 40 second lecture at Harvard. (See the slides). This article is only meant to be light reading; it contains no new results and no proofs of anything. But it may be a good source of student projects, and---at the very least---might be a companion to any reader of the beautiful expository articles on classical Chebyshev bias and more recent studies of bias in the arithmetic theory of elliptic curves.

March 2013: The Nature of Evidence (HLS 2480) was a seminar-course I co-taught with Professor Noah Feldman in the Harvard Law School ([PDF]). For a published account (in the Simons Foundation Science News Letter) of some of the notes to my seminar course, see Shadows of Evidence , [PDF]. Here are pdf files of three other handouts for the seminar: a beginning reading list, [PDF], some preliminary comments on the mission of the seminar, [PDF], and some notes on the Bayesian view of statistics, [PDF].

February 2013: Here [PDF] is the write-up the lecture History of Mathematics as a tool I gave in the History of Mathematics seminar organized by Anthony Liu and Lucia Mocz.

September 2012: Questions about powers of numbers [PDF] are some notes to an expository lecture I once gave on the ABC conjecture. Paper in pdf format: [PDF].

August 2012: Here are notes to my lecture "A brief introduction to the work of Haruzo Hida" for the conference in celebration of the 60th birthday of Haruzo Hida, held at UCLA, June 18-23, 2012 ([PDF])

July 2012: Here are notes to my lecture "Primes, knots and Po" for the conference in celebration of the 80th birthday of Valentin Poenaru, held in Autrans, France July 1-6, 2012 ([PDF])

February 27, 2012: Here [PDF] are rough notes to my Basic Notions Seminar talk "Elliptic curves and their statistics" on February 28, 2012. Earlier versions of this talk for different audiences are these: Here [PDF] are rough notes to the talk "Disparity in the statistics of elliptic curves" I gave on joint work with Zev Klagsbrun and Karl Rubin< in the April 2011 workshop of the program "Arithmetic Statistics" at MSRI, and here are notes [PDF] for a joint MSRI-UC Berkeley colloquium for graduate students entitled "Elliptic curves, their companions, and their statistics" given in conjunction with the MSRI program "Arithmetic Statistics."

Visions, Dreams, and Mathematics [PDF] was published (Princeton University Press) in a collection of essays ( Circles Disturbed: The interplay of Mathematics and Narrative) that I coedited with Apostolos Doxiadis. Here is an interview I gave to the journal NATURE about this volume: [PDF] and here is a link to brief description I gave of this volume to the online journal Berfrois.

January 2012: "Is it Plausible?" ([PDF]) are rough notes related to a talk I gave at the JMM Conference, January 5, 2012. This has been published in the January 2014 issue of The Mathematical Intelligencer.

"What is the surface area of a hedgehog?" ([PDF]) are rough notes related to the Steiner Lecture I gave at St. John's College in Annapolis, last Fall. It was published in the St. John's Review (Vol. {\bf 54}, Fall 2012> Here is a link to that (entire) issue ([PDF]).

December 13 2011: Here is a link to a paper The spin of prime ideals with John Friedlander, Henryk Iwaniec, and Karl Rubin, where we study spin densities of prime ideals with applications to Selmer ranks. It has appeared in the journal Inventiones Mathematicae, 193, Issue 3 (2013) 697-749