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2024
  • February 1, 2024: Here are notes for “Various Equalities,” a talk I gave to the Cambridge Scientific Club.
  • March 11, 2024: Here is a file containing the (compactified) slides for the Clay Lectures that I gave at the 2024 Arizona Winter School (Abelian Varieties)
2023
2022
  • 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).
  • 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.
  • 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.
2021
  • 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.
  • 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. This consists of a correction, as the title indicates, to our paper published almost a quarter of a century ago.
2020
  • 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. It will appear in the Proceedings of the AMS.
2019
2018
2017
  • 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.
2016
  • 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).
2015
2014
  • November 9, 2014: Here (first lecture) (second lecture) (third lecture) 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.
  • 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.
  • March 4, 2014: Here are the slides for the lectures I gave as Clay Lecturer at the 2024 Arizona Winter School (on Abelian Varieties).
  • 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).
  • January 15, 2014: Here are rough notes for a short presentation entitled “The Faces of Evidence (in Mathematics)” 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” and “Refined class number formulas for G_m”. 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.
2013
2012
2011
  • 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.