Education
o Ph.D. in Mathematics, New York University, 1988.
o B.A., M.A. in Mathematics, B.S.E. in Computer Science, University of Pennsylvania, 1983.
Publications
· Book: Hilbert’s Tenth Problem: Diophantine Classes and Other Extensions to Global Fields, Cambridge University Press, 2006. To see a preliminary draft, please click on the following link. Linked PDF file of the book All reports of errors are very much appreciated.
· Defining the integers in large rings of number fields using one universal quantifier to appear in Zapiski Nauchnykh Seminarov POMI (with Gunther Cornelissen)
· “Diophantine Definability and Decidability in Extensions of Degree 2 of Totally Real Fields”, Journal of Algebra, volume 313(2), 2007, pages 846-896. Click here for a preliminary version.
· “Turing Degrees of Isomorphism Types of Algebraic Objects” (with Wesley Calvert and Valentina Harizanov), Journal of the London Mathematical Society, 75 (2), 2007, pages 273-286
· “Elliptic Curves Retaining Their Rank in Finite Extensions and Hilbert’s Tenth Problem” , to appear in Transactions of AMS
· “First Order Definitions of Rational Functions and $S$-integers over Holomorphy Rings of Algebraic Functions of Characteristic 0”, Annals of Pure and Applied Logic, pages 267-283, Volume 136, Issue 3, November 2005 (Click here for a preliminary version)
· “Diophantine Definability of Infinite Discrete non-Archimedean Sets and Diophantine Models for Large Subrings of Number Fields” (with Bjorn Poonen), Journal für die Reine und Angewandte Mathematik, Volume 588, pages 27-47 (Click here for a preliminary version)
· “Diophantine undecidability for some function fields of infinite transcendence degree and positive characteristic”, (in Russian) in Zapiski Nauchnykh Seminarov POMI, volume 304, (2003), pages 141-167. (Click here for the Russian version) English translation of the paper can be found in Journal of Mathematical Sciences, Volume 130, Number 2, pages 4631 – 4642. (Click here for the link to the English version)
· “On Diophantine definability and decidability in some infinite totally real extensions of Q”, Transactions of AMS, volume 356, number 8, 3189-3207. Click here for the link to the article
· “Existential Definability with Bounds on Archimedean Valuations”, Journal of Symbolic Logic, volume 68, Issue 3, September 2003, 860-878. Click here for the article
· “A Ring Version of Mazur’s Conjecture on Topology of Rational Points”, IMRN, 2003:7, 2003, pages 411-423. Click here for the link to the article.
· “On Diophantine definability and decidability in large subrings of totally real number fields and their totally complex extensions of degree 2”, Journal of Number Theory (95), 2002, pages 227-252. Click here for the link to the article.
· “Diophantine undecidability of function fields of characteristic greater than 2 finitely generated over a field algebraic over a finite field”, Compositio Mathematica, volume 132(1), May 2002, 99-120. Click here for a link to the article.
· “Generalized Weak Presentations”, Journal of Symbolic Logic, volume 67, number 2, 2002, pages 787-819
· “On Diophantine decidability and definability in some rings of algebraic functions of characteristic 0”, Journal of Symbolic Logic, volume 67, number 2, 2002, pages 759-786
· “Defining Integrality at Prime Sets of High Density over Function Fields”, Monatshefte fuer Mathematik, vol 135 (2002), 59-67
· “Hilbert’s Tenth Problem over Number Fields, a Survey”, Proceedings of Workshop on Hilbert’s Tenth Problem: Relation with Arithmetic and Algebraic Geometry, Ghent, Belgium, November 1999. Contemporary Mathematics, volume 270, pages 107-137.
· “Diophantine Definability over Non-finitely Generated Non-degenerate Modules of Algebraic Extensions of Q”, Archive for Mathematical Logic, volume 40, number 4, (2001), pages 297-328. Click here for the link to the article.
· “Towards Hilbert’s Tenth Problem for Algebraic Number Fields”, Proceedings of joint AMS-IMS-SIAM Summer Conference on Computability Theory and Applications, June 1999, Boulder, Co, Contemporary Mathematics, Volume 257, 2000, AMS, Providence, RI
· “Defining Integrality at Prime Sets of High Density in Number Fields”, Duke Mathematical Journal, Volume 101, Number 1, 2000, 117-134
· “Hilbert’s Tenth Problem for Algebraic Function Fields over Infinite Fields of Constants of Positive Characteristic”, Pacific Journal of Mathematics, Volume 193, Number 2, 2000, pages 463-500
· “Weak Presentations of Fields Not Extendible to Recursive Presentations”, Recursion Theory and Complexity, Proceedings of the Kazan 97 Workshop, Kazan, July 1997, Walter de Gruyter, New York, 1999, pages 131-156.
· “Weak Presentations of Non-finitely Generated Fields”, Annals of Pure and Applied Logic, Volume 94, (1998), 223-252.
· “Diophantine Definability Over Holomorphy Rings of Algebraic Function Fields with Infinite Number of Primes allowed as Poles”, International Journal of Mathematics, Volume 9, No. 8, (1998), 1041-1066.
· “Holomorphy Modules of Number Fields and Function Fields”,with Heather Ries, Algebra Colloquium, vol. 5(2), 1998, 219-240.
· “Diophantine Definability over Some Rings of Algebraic Numbers with Infinite Number of Primes Allowed in the Denominator”, Inventiones Mathematicae, vol 129(3), 1997, 489-507.
· “The Logic of Pseudo S-integers”, Israel Journal of Mathematics, vol 101, (1997), 229-254.
· “Polynomials with a Given Discriminant over Fields of Functions of Positive Characteristic”, Pacific Journal of Mathematics, volume 173, Number 2, April 1996.
· “Algebraic and Turing Separability of Rings”, Journal of Algebra, volume 185, 229-257.
· “Diophantine Undecidability over Algebraic Function Fields over Finite Fields of Constants”, Journal of Number Theory, volume 58, Number 2, June 1996, 317-342.
· “Rational Separability over a Global Field”, Annals of Pure and Applied Logic , vol 79, 1996, p. 93-108.
· “Non-Standard Extensions of Weak Presentations”, Journal of Algebra, vol 176, 1995, p. 735-749
· “Weak Presentations of Computable Fields”, Journal of Symbolic Logic, Volume 60, Number 1, March 1995, pp. 199-208 (with Carl Jockusch).
· “Diophantine Classes of Holomorphy Rings of Global Fields”, Journal of Algebra, 169(1), October 1994, pp. 139-175.
· “Diophantine Equivalence and Countable Rings”, Journal of Symbolic Logic, 59(3), September 1994, pp. 1068-1095.
· “Diophantine Undecidability for Some Holomorphy Rings of Algebraic Functions of Characteristic O”, Communications in Algebra, 22(11), August 1994, pp. 4379-4404.
· “Diophantine Undecidability in Some Rings of Algebraic Numbers of Totally Real Infinite Extensions of Q”, Annals of Pure and Applied Logic, 68 (1994), pp. 299-325.
· “Diophantine Relations Between the Rings of S-integers of Fields of Algebraic Functions in One Variable Over Constant Fields of Positive Characteristic”, Journal of Symbolic Logic, 58(1), March 1993, pp. 158-192.
· “A Diophantine Definition of Rational Integers in Some Rings of Algebraic Numbers”, Notre Dame Journal of Formal Logic, 33(3), Summer 1992, pp. 299-321.
· “Hilbert’s Tenth Problem for Rings of Algebraic Functions of Positive Characteristic”, Transactions of American Mathematical Society, 333(1) September 1992, pp. 275-298.
· “Hilbert’s Tenth Problem for Rings of Algebraic Functions of Characteristic Zero”, Journal of Number Theory, 40(2), (February 1992), pp. 218-236.
· “Diophantine Definitions for Rings of Rational Numbers”, Communications on Pure and Applied Mathematics, Vol. XLIV, (1991), pp. 853-867.
· “Diophantine Definitions for Some Polynomial Rings”, Communications on Pure and Applied Mathematics, Vol. XLIII, (1990), pp. 1055-1066.
· “Diophantine Relations Between Algebraic Number Fields”, (with Harold Shapiro), Communications on Pure and Applied Mathematics, Vol. XLII (1989), pp. 1113- 1122.
· “Extension of Hilbert’s Tenth Problem to Some Algebraic Number Fields”, Communications on Pure and Applied Mathematics, Vol. XLII (1989), pp. 939-962.
o Rational Inseparability of Integral Closure
o Rings of Algebraic Numbers in Infinite Extensions of $\Q$ and Elliptic Curves Retaining Their Rank
o Undecidability in function fields of positive characteristic (with Kirsten Eisentraeger)
Grants
o John Templeton Foundation, Project Title ““First-Order Models and Definitions of Z over Large Subrings of Number Fields”, September 2008 – May 2010. Principal Investigator
o National Science Foundation, Project Title “Definability and Decidability over Algebraic Extensions of Product Formula Fields”, June 2007 – May 2010, Principal Investigator
o National Science Foundation, Project Title "Existential Definability over Product Formula Fields", June, 2004 - May, 2007, Principal Investigator
o National Science Foundation, Project Title "Diophantine Definability and Decidability over Algebraic Extensions of Global Fields", August, 2000 - July, 2003, Principal Investigator
o National Security Agency Mathematical Sciences Research Program, Project Title: "Some Problems Concerning Logic of Algebraic Fields", 1998-2000, Principal Investigator.
o National Security Agency Mathematical Sciences Research Program, Project Title: "Some Problems Concerning Logic of Global Fields" 1996-1998, Principal Investigator.
o National Security Agency Mathematical Sciences Research Program, Project Title: "Extensions of Hilbert's Tenth Problem", 1992-94, Principal Investigator.
Selected Talks and Conferences
· MASS Colloquium Speaker, Penn State, State College, PA, November 2007. Slides of my talk can be found here
· Algebra and Number Theory Seminar Speaker, Penn State, State College, PA, November 2007. Slides of my talk can be found here
· Colloquium Speaker, Department of Mathematics, University of Calgary, Calgary, Canada, October 2007. Slides of my talk can be found here.
· Colloquium Speaker, Department of Mathematics, Physics and Engineering, Mount Royal College, Calgary, Canada, October 2007. Slides of my talk can be found here.
· Invited Speaker, Number Theory and Computability Workshop, ICMS, Edinburgh, UK, June 2007. Click here for more information on this workshop. Slides of my first talk can be found here. Slides of my second talk can be found here. You can find some pictures I took at the meeting here.
· Invited Speaker, Clay Institute Conference on Hilbert’s Tenth Problem, March 2007, Cambridge, MA. Click here for more information on this conference. Slides of my talk can be found here.
· Invited Speaker, Workshop on Model Theory and Computable Model Theory, Gainesville, Florida, February 2007. Click here for more information about the workshop and here for the PDF slides of my talk.
· VIGRE and Number Theory Seminar Speaker, University of Georgia, October 2006. Click here for the PDF slides of my VIGRE talk and Click here for the PDF slides of my Number Theory seminar talk
· Colloquium Speaker. University of Utrecht, Netherlands, October 2006
· Speaker at the Field Arithmetic meeting, Oberwolfach, Germany, February 2006. Click here for the PDF slides of my talk
· Member of MSRI for the Rational Points Semester, Spring 2006, Berkeley, CA. Click here for more information on this program.
· Co-organizer (with Bjorn Poonen, Karim Zahidi, and Xavier Vidaux) of Workshop on Hilbert’s Tenth Problem at American Institute of Mathematics, Palo Alto, Ca, March 2005. Click here for some pictures I took at the meeting
· Program Committee Member, ASL Winter Meeting (joint with AMS), Atlanta, January 2005
· Logic Seminar Speaker, Mathematics Institute of Oxford University, Oxford, UK, March 2004
· Number Theory Seminar Speaker, Imperial College, London, UK, March 2004
· Colloquium Speaker, Department of Mathematics, University of East Anglia, Norwich, UK, March 2004
· Seminar Speaker, Department of Mathematics, Tel Aviv University, Tel Aviv, Israel, February 2003
· Co-organizer (with Gunther Cornelissen, Yurii Matiyasevich and Maxim Vsemirnov) of a Workshop on Hilbert’s Tenth Problem, Mazur’s Conjectures and Divisibility Sequences, Oberwolfach, Germany, January 2003. Click here for some pictures I took at the meeting.
· Participant and speaker at the workshop on rational and integral points on higher-dimensional varieties, American Institute of Mathematics, Palo Alto, California, December 2002. Click here for the workshop web page.
· Logic Seminar speaker, Department of Mathematics, University of Athens, Athens, Greece, November, 2002.
· Colloquium speaker, Department of Mathematics, University of Crete, Heraklion, Greece, November 2002.
· Speaker at 21st Days of Weak Arithmetic, St. Petersburg, Russia, June 2002. Click here for the conference home page.
· Participant in the Field Arithmetic Meeting, Oberwolfach, Germany, February 2002.
· Participant in the Recursion Theory Meeting, Oberwolfach, Germany, January 2001
· Participant in the Workshop on Arithmetic Geometry, MSRI, Berkeley, CA, December 2000. Click here for the Workshop Homepage, here for slides of my talk and here for video of my talk.
· Speaker, Special Session on Recursion Theory, ASL Winter Meeting, Washington DC, January 2000
· Invited Speaker, Workshop on Hilbert's Tenth Problem and Algebraic Geometry, Gent, Belgium, November, 1999. Click here for Conference Homepage.
· Invited speaker, AMS Summer Research Conference on Computability Theory and Applications, Boulder, Colorado, June 1999.
· Speaker at the AMS Special Session on Computability Theory, AMS Meeting, Gainsville, Florida, March 1999.
· Participant in the Model Theory, Algebra and Arithmetic Workshop, Mathematical Sciences Research Institute, Berkeley, California, June 1998.
· Colloquium speaker, Mathematics Department, University of Arizona, Tucson, February 1998.
· Participant in the Introductory Model Theory Workshop, Mathematical Sciences Research Institute, Berkeley, California, January 1998.
· Speaker at the Logic Seminar, Graduate School, City University of New York, December 1997.
· Invited speaker at the National Security Agency Grants Day, Fort Meade, Maryland, July 1997.
· Participant and speaker at the Recursion Theory Workshop, Kazan, Russia, July 1997.
· Colloquium speaker, Mathematics Department, Notre Dame University, May 1997.
· Colloquium speaker, Mathematics Department, George Washington University, April 1997.
· Participant and speaker at the Recursion Theory Meeting, Oberwolfach, Germany, 1996.
· Invited Speaker at the winter meeting of the Association for Symbolic Logic, San Francisco, 1995.
This page was last modified on 3/19/08