Bjorn Poonen

Bjorn Poonen Department of Mathematics   MIT, 2-244   77 Massachusetts Avenue   Cambridge, MA 02139-4307, USA

Office: Building 2, Room 244 Phone: (617)-258-8164  Fax: (617)-253-4358  email: Office hours: M 10:15-11:30am, Tu 1:20-2:20pm, W 10:15-11:30am.

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Introduction to my research
Curriculum vitae and publication list (.pdf) (.dvi.gz)
MathSciNet search for Poonen  (this link works only if your institution subscribes)
My research is supported by the National Science Foundation.
Students, postdocs, and other researchers supervised or sponsored
There are currently three Ph.D. students working with me.

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Courses and seminars

• 18.02, Multivariable calculus, Spring 2013.
• 18.726, Algebraic Geometry II, Spring 2013.
• 18.787, Rational points on varieties, Fall 2013.
• Past courses

• MIT number theory seminar (co-organized with Sug Woo Shin)
• BC-MIT number theory seminar (co-organized with Sol Friedberg, Ben Howard, and Sug Woo Shin)
• STAGE (co-organized with Gregory Minton)

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Conferences I am helping to organize

• Explicit methods in number theory, an Oberwolfach workshop (co-organized with Karim Belabas and Don B. Zagier), July 14-20, 2013.
• Past conferences

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Editorial positions

• Algebra & Number Theory, managing editor, 2006-
• Involve (new journal for high-quality research involving students), editorial board member, 2007-
• A K Peters Research Notes in Mathematics book series, editorial board member, 2005-
• Past editorial positions

Before sending me any papers about the Goldbach Conjecture, the Riemann Hypothesis, the infinitude of twin primes, the nonexistence of odd perfect numbers, Fermat's Last Theorem, or generalizations of any of these, please read the advice by Henry Cohn at this website.

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Other professional activities

• AMS Committee to Select the Winner of the E. H. Moore Research Article Prize, 2012-2018.
• MIT Pure Mathematics Committee, 2008-.
• MIT Mathematics Department Executive Committee, 2009-.
• MIT Mathematics Department Nominations Committee, 2008-.
• MIT Mathematics Department Teaching Assignment Area Captain for Logic, 2008-.
• MIT Committee on Graduate Admissions, 2011-.
• Museum of Mathematics Advisory Council, 2008-.
• Girls' Angle Advisory Board, 2009-.
• Past professional activities

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Some of my papers   (An MIT website bug sometimes gives a "Forbidden" error; if you get this, try again after a few seconds.) (disclaimer)

• Research articles
  • Chabauty's method proves that most odd degree hyperelliptic curves have only one rational point (.pdf) (with Michael Stoll)
  • Modeling the distribution of ranks, Selmer groups, and Shafarevich-Tate groups of elliptic curves (.pdf) (with Manjul Bhargava, Daniel M. Kane, Hendrik W. Lenstra jr., and Eric Rains)
  • Berkovich spaces embed in Euclidean spaces (.pdf) (with Ehud Hrushovski and François Loeser)
  • Computing Néron-Severi groups and cycle class groups (.pdf) (with Damiano Testa and Ronald van Luijk)
  • Uniform boundedness of rational points and preperiodic points (.pdf)
  • Generalized explicit descent and its application to curves of genus 3 (.pdf) (with Nils Bruin and Michael Stoll)
  • Extending self-maps to projective space over finite fields (.pdf)
  • Convergence of the restricted Nelder-Mead algorithm in two dimensions (.pdf) (with Jeffrey C. Lagarias and Margaret H. Wright)
  • Self cup products and the theta characteristic torsor (.pdf) (.dvi.gz) (with Eric Rains)
  • Random maximal isotropic subspaces and Selmer groups (.pdf) (.dvi.gz) (with Eric Rains)
  • Néron-Severi groups under specialization (.pdf) (.dvi.gz) (with Davesh Maulik)
  • Automorphisms mapping a point into a subvariety (.pdf) (.dvi.gz) (with an appendix by Matthias Aschenbrenner)
  • Multivariable polynomial injections on rational numbers (.pdf) (.dvi.gz)
  • Curves over every global field violating the local-global principle (.pdf) (.dvi.gz)
  • Insufficiency of the Brauer-Manin obstruction applied to etale covers (.pdf) (.dvi.gz)
  • Existence of rational points on smooth projective varieties (.pdf) (.dvi.gz)
  • The set of nonsquares in a number field is diophantine(.pdf) (.dvi.gz)
  • The p-adic closure of a subgroup of rational points on a commutative algebraic group (.pdf) (.dvi.gz)
  • • Independence of points on elliptic curves arising from special points on modular and Shimura curves, I: global results (.pdf) (.dvi.gz) (with Alexandru Buium)
    • Independence of points on elliptic curves arising from special points on modular and Shimura curves, II: local results (.pdf) (.dvi.gz) (with Alexandru Buium)
  • Characterizing integers among rational numbers with a universal-existential formula (.pdf) (.dvi.gz)
  • The Brauer-Manin obstruction for subvarieties of abelian varieties over function fields (.pdf) (.dvi.gz) (with José Felipe Voloch)
  • Smooth hypersurface sections containing a given subscheme over a finite field (.pdf) (.dvi.gz)
  • The moduli space of commutative algebras of finite rank (.pdf) (.dvi.gz)
  • Isomorphism types of commutative algebras of finite rank over an algebraically closed field (.pdf) (.dvi.gz)
  • First-order characterization of function field invariants over large fields (.pdf) (.dvi.gz) (with Florian Pop)
  • Gonality of modular curves in characteristic p (.pdf) (.dvi.gz)
  • Uniform first-order definitions in finitely generated fields (.pdf) (.dvi.gz)
  • Twists of X(7) and primitive solutions to x^2+y^3=z^7 (.pdf) (with Edward F. Schaefer and Michael Stoll)
  • Heuristics for the Brauer-Manin obstruction for curves (.pdf) (.dvi.gz)
  • Diophantine definability of infinite discrete nonarchimedean sets and Diophantine models over large subrings of number fields (.pdf) (.dvi.gz) (with Alexandra Shlapentokh)
  • Multiples of subvarieties in algebraic groups over finite fields (.pdf) (.dvi.gz)
  • Unramified covers of Galois covers of low genus curves (.pdf) (.dvi.gz)
  • Orbits of automorphism groups of fields (.pdf) (.dvi.gz) (with Kiran S. Kedlaya)
  • Random diophantine equations (.pdf) (.dvi.gz) (with José Felipe Voloch, and containing appendices by Jean-Louis Colliot-Thélène and Nicholas M. Katz)
  • Sums of values of a rational function (.pdf) (.dvi.gz)
  • Hilbert's Tenth Problem and Mazur's Conjecture for large subrings of Q (.pdf) (.dvi.gz)
  • Finiteness results for modular curves of genus at least 2 (.pdf) (.dvi.gz) (with Matt Baker, Enrique Gonzalez-Jimenez, and Josep Gonzalez)
  • Curves of every genus with many points, II: asymptotically good families (.pdf) (.dvi.gz) (with Noam Elkies, Everett Howe, Andrew Kresch, Joseph Wetherell, and Michael Zieve)
  • The Grothendieck ring of varieties is not a domain (.pdf) (.dvi.gz)
  • Using elliptic curves of rank one towards the undecidability of Hilbert's Tenth Problem over rings of algebraic integers (.pdf) (.dvi.gz) (Springer LNCS link)
  • Bertini theorems over finite fields (.pdf) (.dvi.gz)
  • Squarefree values of multivariable polynomials (.pdf) (.dvi.gz)
  • The conjugate dimension of algebraic numbers (.pdf) (.dvi.gz) (with N.~Berry, A.~Dubickas, N.~Elkies, and C.~Smyth)
  • Zeros of sparse polynomials over local fields of characteristic p (.pdf) (.dvi.gz)
  • Points having the same residue field as their image under a morphism (.pdf) (.dvi.gz)
  • Néron-Tate projection of algebraic points (.pdf) (.dvi.gz)
  • Computing torsion points on curves (.pdf) (.dvi.gz)
  • Random polynomials having few or no real zeros (.pdf) (.dvi.gz) (with Amir Dembo, Qi-Man Shao, and Ofer Zeitouni)
  • The following three papers are the renamed and updated versions of the preprints "Curves over finite fields without extra automorphisms" and "Hypersurfaces over finite fields without extra automorphisms" referred to in Katz and Sarnak, "Random matrices, Frobenius eigenvalues, and monodromy":
  • Varieties without extra automorphisms I: curves (.pdf) (.dvi.gz)
  • Varieties without extra automorphisms II: hyperelliptic curves (.pdf) (.dvi.gz)
  • Varieties without extra automorphisms III: hypersurfaces (.pdf) (.dvi.gz)
  • An explicit algebraic family of genus-one curves violating the Hasse principle (.ps) (.dvi.gz + planecubics.eps)
  • Algebraic families of nonzero elements of Shafarevich-Tate groups (with Jean-Louis Colliot-Thélène)
  • Mordell-Lang plus Bogomolov
  • The Cassels-Tate pairing on polarized abelian varieties (with M. Stoll)
  • Large torsion subgroups of split Jacobians of curves of genus two or three (.pdf) (.dvi.gz) (with Everett Howe and Franck Leprévost)
  • The number of intersection points made by the diagonals of a regular polygon(.pdf)(.ps) (with M. Rubinstein. Auxiliary programs:  ngon.c, ngon.m)
  • Drinfeld modules with no supersingular primes (.pdf) (.dvi.gz)
  • Explicit descent for Jacobians of cyclic covers of the projective line (.pdf) (.dvi.gz) (with E. Schaefer)
  • The worst case in Shellsort and related algorithms (.pdf) (.dvi.gz)
• Expository articles
  • Lattice polygons and the number 12 (with F. Rodriguez-Villegas)
  • A brief summary of the main statements of class field theory (.pdf) (.dvi.gz)
  • Elliptic curves (.pdf) (.dvi.gz)
  • Computing rational points on curves (.pdf) (.dvi.gz)
  • Computational aspects of curves of genus at least 2 (.pdf) (.dvi.gz)
  • Hilbert's Tenth Problem over rings of number-theoretic interest (.pdf) (.dvi.gz)
  • Sieve methods for varieties over finite fields and arithmetic schemes (.pdf) (.dvi.gz)
  • The method of Chabauty and Coleman (.pdf) (.dvi.gz) (with William McCallum)
  • Undecidability in number theory (.pdf) (.dvi.gz) (awarded the 2011 Chauvenet Prize)
  • Average rank of elliptic curves (.pdf)
  • Undecidable problems: a sampler (.pdf)
• Books
  • The William Lowell Putnam Mathematical Competition 1985-2000: Problems, Solutions, and Commentary, with Kiran Kedlaya and Ravi Vakil.
  • Arithmetic of Higher-Dimensional Algebraic Varieties, edited with Yuri Tschinkel.
• Slides
  • Introduction to rational points
  • Hilbert's tenth problem
  • The projective line minus three fractional points
  • Undecidability everywhere (the third of the 2008 Cantrell Lectures given at the University of Georgia)
  • The first-order theory of finitely generated fields
  • Uniform boundedness of rational points

Remarks and errata (.pdf) (.dvi.gz)