Recent Expository Articles

• February 2013: Here is the write-up the lecture “History of Mathematics as a tool” I gave in the History of Mathematics seminar organized this semester by Anthony Liu and Lucia Mocz.
• 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.
• 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.
• Visions, Dreams, and Mathematics was published in the Spring of 2012 by 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, and here is a link to brief description I gave of this volume to the online journal Berfrois.
• Here are rough notes entitled “Is it Plausible?” related to a talk I gave at the JMM Conference, January 5, 2012.
• Here are rough notes entitled “What is the surface area of a hedgehog?” related to the Steiner Lecture I gave at St. Johns College in Annapolis, in the Fall of 2012.
• Here are the notes for a joint MSRI-UC Berkeley colloquium for graduate students entitled “Elliptic curves, their companions, and their statistics” given in the winter of 2011 in conjunction with the MSRI program “Arithmetic Statistics.”
• June 17 2010: “The Language of Explanation” notes is an essay I wrote for the University of Utah Symposium in Science and Literature (November, 2009) and in anticipation of a panel discussion on Mathematics, Language, and Imagination with Alice Fulton and Fred Lerdahl.
• Here are some notes on my Number Theory Seminar lecture “Ranks of twists of elliptic curves” on Feb. 3, 2010.
• Here are the notes to my Basic Notions Talk (Feb 1, 2010) on Hilbert’s Tenth Problem and Elliptic Curves.
• I wrote a brief response to a question posed by Phil Davis and David Mumford: “What should a professional mathematician know?“
• I participated in a panel on Number Theory and Logic at the MAMLS@Harvard conference (May 9 2009). Here are my notes, entitled “Connections between Logic and Number Theory.”
• I gave a brief talk “Arithmetic and some of its aspirations” to the “Friends of Harvard Mathematics” on May 5 2009 about Number Theory. Here are my notes for that talk.
• I was asked to give a short (twenty-five minute) talk on the subject of “Time” in an up-coming conference in Comparative Literature (to be held at Harvard, March 2009). Here is a first draft of my text, entitled “On time (in mathematics and literature).”
• May 16, 2008: Here are some notes about Imagination and Mathematics that I wrote to prepare myself for a public event at the Philoctetes Center in New York City on May 13, 2008.
• January 11, 2008: the article “Mathematical Platonism and its Opposites” was published in the June 2008 European Mathematical Newsletter (pp.19-21). For a discussion of this article and related ones see here.
• I wrote a “math news” article for the journal Nature ([NATURE Vol 443, 7 September 2006) about the recent work on the Sato-Tate Conjecture. (Also, for this article, William Stein did some beautiful calculations. A more complete account of them is available on William Stein’s website.) Here is a longer version of this, entitled “Finding meaning in error terms” which started as the text of a lecture I gave in the Current Developments in Mathematics section of the AMS winter meeting in New Orleans in January 2007, but in its present form is now an expository article that I am submitting to the Bulletin of the American Mathematical Society.”
• Sep. 17, 2007: A version, dated Jul 29, 2006 of the article “When is one thing equal to some other thing?”. It has been published in the volume: Proof and Other Dilemmas: Mathematics and Philosophy B. Gold and R, Simons, Editors (2008) Series: Spectrum.
• I wrote a short piece “About the cover: Diophantus’s Arithmetica” that appears in the July 2006 issue of the Bulletin of the American Mathematical Society, for which I want to mention an erratum: I erroneously identified the figure on the cover as Erato, muse of erotic poetry, but it seems, rather, to be Orpheus.”
• I was asked by the French journal “Les Dossiers de la Recherche” to write an expository article, intelligible to a general audience, hinting why prime numbers are exciting and important. The article “Pourquoi les nombres premiers?” appeared in the August-October 2005 issue of that journal.
• I had a conversation with Peter Pesic about “imagination and mathematics” has appeared in the Spring 2005 issue of the journal “Daedalus” (pp. 124-130). Here is a provisional version of the text. Here, also, are some notes about Imagination and Mathematics: The Geometry of Thought, a conversation with Eva Brann that I wrote to prepare myself for a public event at the Philoctetes Center in New York City on May 13, 2008.
• Here is an expository article on the theme of deformations entitled “Perturbations, deformations, and variations (and ‘near-misses’) in geometry, physics, and number theory” for the issue of the Bulletin of the A.M.S. in honor of Ren� Thom (Bull. Amer. Math. Soc. 41 (2004), 307-336).
• A short article I wrote entitled “What is a Motive?” appeared in the Notices: Notices of the AMS 51, no 10, (2004)1214-1216).
• The journal “For the learning of Mathematics” reprinted a section of my book Imagining numbers (particularly the square root of minus fifteen) that dealt with the concept of time in mathematics, and asked me to write an article expanding on that theme. My article, “On the absence of time in mathematics”, was published in the November 2004 issue of that journal.
• I wrote an expository article for the French journal ‘Pour la Science’ entitled “Plus symétrique que la sphere” (Pour la Science Oct/Dec (2003) 78-85), [PDF].
• My article “How did Theaetetus prove his Theorem?” has now been published in “The Envisoned Life: Essays in honor of Eva Brann” (eds: P. Kalkavage, E. Salem) Paul Dry Books, Philadelphia, 2007. The article discusses the curious fact that although Theaetetus’ Theorem is beautifully formulated in Plato’s dialogue (“The Theaetetus”), no proof of it seems to have been preserved in what has come down to us from the ancient literature, nor does there seem to be much discussion (in the secondary literature) of what Theaetetus’ proof might have looked like.
• This expository piece about Algebraic Numbers has appeared as a chapter in the book The Princeton Companion for Mathematics, by Timothy Gowers, June Barrow-Green, Imre Leader (Editors), American Mathematical Society (2008).

Forewords to Books

• A second edition of the classic text “Number Fields” by Daniel A. Marcus has been published by Springer. Here is the foreword I wrote for it.
• The classic expository text “Number: The Language of Science” by Tobias Dantzig has been republished by pi-Press. It is edited by my brother Joseph Mazur who also wrote an afterword for this republication, and I wrote a foreword for it.
• I wrote a foreword for “Fearless Symmetry: Exposing the Hidden Patterns of Numbers” by Avner Ash & Robert Gross. It was published this past year by Princeton University Press.
• I wrote forewords for two classic popular books on mathematics written by Lillian R. Lieber, with drawings by Hugh Gray Lieber . These are “The Education of T.C. MITS” published last year by Paul Dry Books and “Infinity:Beyond the beyond the beyond” to be published shortly by Paul Dry Books.
• I wrote a foreword to a book of poems by Emily Galvin (“Do the Math” to be published shortly by Tupelo Press). Download it here [PDF].