Papers written or cowritten by Thomas Scanlon
Thesis
Model Theory of Valued D-Fields, PhD thesis,
Harvard University, May 1997.
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Published papers
- The abc-theorem for commutative algebraic groups
in characteristic p, International Mathematics
Research Notices 1997, no. 18, 881 -- 898.
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- p-adic distance from torsion points of semi-abelian
varieties, J. reine angew. Math. 499 (1998), 225 --
236. Postscript or
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- Supersimple fields and division rings, with
Anand Pillay
and
Frank Wagner , Mathematics Research Letters 5
(1998), 473 -- 483.
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- Difference algebraic subgroups of commutative algebraic
groups over finite fields, with
J. F. Voloch,
Manuscripta Math. 99 (1999), no. 3, 329--339.
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- The conjecture of Tate and Voloch on p-adic
proximity to torsion, International Mathematics Research Notices
1999 , no. 17, 909 -- 914.
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- Lascar and Morley ranks differ in differentially closed
fields, with Ehud Hrushovski, Journal of Symbolic Logic 64 ,
no. 3, Sept. 1999, 1280 -- 1284.
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- Combinatorics with definable sets: Euler characteristics
and Grothendieck rings, with
Jan Krajicek, Bulletin of
Symbolic Logic 6 , no 3, September 2000, 311 -- 330.
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- A model complete theory of valued D-fields, Journal
of Symbolic Logic, 65 , no. 4, December 2000, 1758 - 1784.
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- Diophantine geometry from model theory, Bulletin
of Symbolic Logic, 7 , no. 1, March 2001, 37 - 57.
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- Public key cryptosystems based on Drinfeld modules are insecure,
Journal of Cryptology, 14, no. 4, 2001, 225--230.
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- Compact complex manifolds with the DOP and other
properties,
with Anand Pillay,
Journal of Symbolic Logic, 67, no. 2, June 2002, 737 - 743.
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- Model theory and differential algebra, in Differential Algebra
and Related Topics Proceedings of the International Workshop
(Newark, November 2000), eds. L. Guo, P. J. Cassidy, W. F. Keigher
& W. Y. Sit, World Scientific 2002, pp. 125 - 150.
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- Stable Groups (a review of the English translation of
B. Poizat's Groupes Stables), Bull. AMS
39 no. 4 (2002), 573-579.
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- Diophantine geometry of the torsion of a Drinfeld module,
Journal of Number Theory 97 no. 1 (November 2002),
10 - 25.
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- Meromorphic groups, with
Anand Pillay,
Trans. AMS, 355 (2003), 3843-3859.
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- Quantifier elimination for the relative Frobenius, in Valuation
Theory and Its Applications Volume II, conference
proceedings of the International Conference on Valuation Theory (Saskatoon,
1999), Franz-Viktor Kuhlmann, Salma Kuhlmann, and Murray Marshall, eds.,
Fields Institute Communications Series, (AMS, Providence), 2003, 323 - 352.
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- F-structures and integral points on semiabelian varieties over
finite fields, with Rahim Moosa,
Amer. J. Math. 126 (2004), 473 - 522. Postscript or PDF
- Automatic uniformity, International Mathematics
Research Notices 2004, no. 62,
3317 - 3326. Postscript or
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- The Mordell-Lang conjecture in positive characteristic revisited,
with Rahim Moosa,
in Model Theory and Applications (eds. L. Bélair, P. D'Aquino,
D. Marker, M. Otero, F. Point, & A. Wilkie), 2003, 273 - 296. Postscript or PDF
- Local André-Oort conjecture for the universal abelian variety,
Inventiones mathematicae, (Online first, 19 October 2005), 163, no. 1, January 2006, 191 - 211.
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- Positive characteristic Manin-Mumford theorem, Compositio mathematica 141, no. 6 (November 2005), 1351-1364.
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- Nonstandard meromorphic groups, Electronic Notes in Theoretical Computer Science,
Volume 143, (6 January 2006) Proceedings of the 12th Workshop on Logic, Language,
Information and Computation (WoLLIC 2005) Florianópolis, Santa Catarina, Brazil 19-22 July 2005,
185 - 196.
- Groups in the theory of compact complex analytic spaces, with
Matthias Aschenbrenner and Rahim Moosa, J. Symbolic Logic 71 (2006), 529-552.
- Analytic difference rings, Proceedings of the International Congress of Mathematicians,
Madrid, Spain, 2006, Volume II, 71 -- 92.
- Model theory of the Frobenius on the Witt vectors, with Luc Bélair and Angus Macintyre, American Journal of Mathematics,
Volume 129, Number 3, June 2007, 665 - 721.
- Differential arcs and regular types in differential fields, with
Rahim Moosa and
Anand Pillay, J. Reine Angew. Math. 620 (2008), 35 - 54.
- Differentially valued fields are not differentially closed, Model theory with applications to algebra and analysis. Vol. 1, 111--115, London Math. Soc. Lecture Note Ser., 349, Cambridge Univ. Press, Cambridge, 2008.
- Infinite finitely generated fields are biinterpretable with
N, J. Amer. Math. Soc. 21 (2008) 893 - 908. (Erratum, J. Amer. Math. Soc. 24 (2011), no. 3, 917.
There is a serious mistake in the proof of the definability of valuations. As it stands, Pop's Conjecture is still open.)
- Jet and prolongation spaces, with Rahim Moosa, Journal of the Institute of Mathematics Jussieu 9 (2010), no. 2, 391 - 430.
- Counting special points: Logic, diophantine geometry, and transcendence theory.,
Bull. Amer. Math. Soc. (N.S.) 49 (2012), no. 1, 51 - 71.
- A Euclidean Skolem-Mahler-Lech-Chabauty method, Math. Res. Lett. 18 (2011), no. 5, 833 - 842.
- Artin-Schreier extensions in nip and simple fields, with Itay Kaplan
and Frank Wagner, Israel J. Math. 185 (2011), 141 - 153.
- Generalized Hasse-Schmidt varieties and their jet spaces, with Rahim Moosa,
Proc. Lond. Math. Soc. (3) 103 (2011), no. 2, 197 - 234.
- Algebraic equations on the adèlic closure of a Drinfeld module, with Dragos Ghioca, Israel
J. Math., on-line first, 2012.
- Periods of rational maps modulo primes, with R. Benedetto, D. Ghioca, B. Hutz, P. Kurlberg, and T. Tucker, Math. Ann., on-line first, 2012.
- A proof of the André-Oort conjecture using mathematical logic [after Pila, Wilkie and Zannier], Astérisque, Séminaire Bourbaki, Exposé 1037, 2010 - 2011.
- Exponential-polynomial equations and dynamical return sets, with Yu Yasufuku, International Mathematics Research Notices, published on-line April 30, 2013 doi:10.1093/imrn/rnt081
- Invariant varieties for polynomial dynamical systems, with Alice Medvedev, Annals of Mathematics, 179 (2014), no. 1, 81 - 177.
- Model theory of fields with free operators in characteristic zero, with Rahim Moosa, Journal of Mathematical Logic 14 (2014), no. 2, 1450009, 43 pp.
- O-minimality, Gazette des Mathématiciens,
no. 149, Julliet 2016.
- Density of orbits of
endomorphisms of abelian varieties, with Dragos Ghioca,
Trans. AMS, 14 April 2016, (to appear).
- Strong minimality and the j-function,
with James Freitag, Journal of the European Mathematical
Society, 2015, (to appear).
- Differential
Chow varieties exist, with James Freitag and Wei Li (and an
appendix by William Johnson), 27 pages with a 4 page appendix,
Journal of the London Mathematical Society, 2016, (to appear).
- O-minimality as an approach to the André-Oort conjecture, in Around the Zilber-Pink conjecture, Panoramas et Synthèses, no. 52 (2017), 111 - 165.
- The logical complexity of finitely generated commutative rings, with Matthias Aschenbrenner, Anatole Khélif, and Eudes Naziazeno, International Mathematics Research Notices, to appear.
Papers under review
- Communication lower bounds and optimal algorithms for programs that references arrays - part I, with F. Micahel Christ, James Demmel, Nicholas Knight, and Katherine Yelick, 2013.
- Algebraic differential equations from covering maps, 25 pages, 2014.
- Effective difference elimination and Nullstellensatz, with Alexey Ovchinnikov and Gleb Pogudin, 25 pages, 2017.
- The dynamical Mordell-Lang conjecture for endomorphisms of semiabelian varieties defined over fields of positive characteristic with Pietro Corvaja, Dragos Ghioca and Umberto Zannier, 27 pages, 2018.
Preprints and Notes not intended for formal publication
- The absolute Mordell-Lang conjecture in positive characteristic, 1998.
- Infinite stable fields are Artin-Schreier closed, 1999.
- Fields admitting ordered nontrivial strong Euler characteristics
are quasifinite, 2000.
- An elementary proof that
order one sets living on curves of general type have no structure, 2000.
- Uniformity in the Mordell-Lang theorem, 2001.
- Superstability for F-structures, with Rahim Moosa, 2003.
- Algebraic relations amongst periodic points of a lifting
of Frobenius, 2007.
- Analytic relations on a dynamical orbit, 2008.
Slides from recent talks
- Higher dimensional dynamical Mordell-Lang problems, Conference on Diophantine Problems and Arithmetic
Dynamics, Academia Sinica, Taipei, Taiwan, 27 June 2013.
- Lectures on Model Theory and Algebraic Dynamics, CUNY 7 - 11 June 2010 Day 1 Day 2 Day 3 Day 4 Day 5
- "Dynamical Mordell-Lang problems, Będlewo, Poland, 13 August 2009
- "Motivic Integration: An outsider's tutorial, Durham, England, 21 & 22 July 2009
- Definability in fields, Computable model theory, University of Florida at
Gainesville, 5-6 February 2007. Lecture 1
Lecture 2 Lecture 3
- Pop's conjecture on the elementary theory of finitely generated fields,
Paris VII, 8 June 2006.
- Groups in the theory of compact complex manifolds, III Encuentro de la
Teoría de Modelos, (Villa de Leyva, Colombia), 18 August 2005.
- Categoricidad relativa para campos finitamente generados,
XV Congreso Nacional de Matemáticas, (Bogotá), 12 August 2005.
- Additive groups, Model Theory and Its Applications to Algebra and Analysis, Isaac Newton
Institute, (Cambridge, UK), 15 July 2005.
- A local André-Oort theorem via ACFA,
Model Theory and Its Applications to Algebra and Analysis, Isaac Newton
Institute, (Cambridge, UK), 31 March 2005.
- Geometric stability theory,
Logic Colloquium 2004 (Torino), 27, 28 & 30 July 2004.
- Model theory of valued difference fields, Maurice Boffa Lecture (Mons, Belgium),
27 May 2004.
- Differential arcs, Interactions between model theory and geometry, Banff,
15 March 2004.
- Model Theory of Enriched Valued Fields, Logic Colloquium 2002 (Münster),
4 August 2002.
- A Drinfeld module version of the Mordell-Lang conjecture,
Arizona Winter School 2003, 17 - 19 March 2003.
- p-Jets and Uniform Unramified Manin-Mumford,
Joint AMS-SMF Meeting, Lyon 19 July 2001.
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- Model Theory and Differential Algebra, Newark meeting on
differential algebra and related topics, 3 November 2000.
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- Grothendieck Rings, Euler Characteristics, and
Schanuel Dimensions, Berkeley Logic Colloquium, 29 September 2000.
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Lecture notes
Math 225a, Autumn 2013
Note that most of the work described on this page has been supported by grants of the National Science Foundation.