Rahim Moosa's Homepage

Rahim N. Moosa

Professor and Math Faculty Research Chair
Department of Pure Mathematics
University of Waterloo, MC 5006
200 University Avenue West, Waterloo, Ontario, N2L 3G1
Phone number : (519) 888-4567 ext. 42453
email: rmoosa at uwaterloo dot ca

Teaching

I am teaching the newly restructured logic courses this year: Mathematical Logic (PMATH 432/632) in Fall 2024 and Model Theory (PMATH 433/733) in Winter 2025.

Editorial work

I am an editor-in-chief (along with Martin Hils) of Model Theory.

I am an associate editor for the Canadian Journal of Mathematics and the Canadian Mathematical Bulletin.

Research interests

Model theory (a branch of mathematical logic). Especially interactions with algebra, geometry and number theory.

Here is a link to the logic group homepage at Waterloo.

Preprints

Binding groups for algebraic dynamics (with Moshe Kamensky). Submitted, 40 pages.

Differential-algebraic permutation groups (with James Freitag and Leo Jimenez).
To appear in the JIMJ, 22 pages.

On invariant rational functions under rational transformations (with Jason Bell and Matthew Satriano).
To appear in Selecta, 20 pages.

Six lectures on model theory and differential-algebraic geometry.
Submitted, 21 pages.

Abelian reduction in differential-algebraic and bimeromorphic geometry (with Remi Jaoui).
To appear in the Annales de l'Institut Fourier, 32 pages.

Effective isotrivial Mordell-Lang in positive characteristic (with Jason Bell and Dragos Ghioca).
To appear in the American Journal of Mathematics, 43 pages.

Bounding nonminimality and a conjecture of Borovik-Cherlin (with James Freitag).
To appear in JEMS, doi 10.4171/JEMS/1384, 25 pages.

Model theory and the DME: a survey.
To appear in Research Trends in Contemporary Logic (Eds. M. Fitting et al).

Publications

Model theory of compact complex manifolds

A model theory for meromorphic vector fields.
Model Theory (Special issue dedicated to Boris Zilber), volume 3 (2024), number 2, 259--283.

Model theory of compact complex manifolds with an automorphism (with M. Bays and M. Hils).
Transactions of the AMS, volume 369 (2017), number 6, 4485--4516.
Correction to the statement of Theorem C.2.

A note on subvarieties of powers of OT-manifolds (with M. Toma).
Bulletin Mathematiques de la Societe des Sciences Mathematiques de Roumanie, Tome 58 (2015), number 3, 311--316.

Countably categorical strongly minimal compact complex manifolds (with A. Pillay).
Proceedings of the American Mathematical Society, volume 140 (2012), number 5, 1785--1801.

Model theory and complex geometry
Notices of the American Mathematical Society, volume 57 (2010), number 2, 230--235.

An essentially saturated surface not of Kaehler-type (with R. Moraru and M. Toma)
Bull. Lond. Math. Soc., volume 40 (2008), number 5, 845--854.

Model theory and Kaehler geometry (with A. Pillay)
In Model Theory with Applications to Algebra and Analysis Volume 1 (Eds. Z. Chatzidakis, D. Macpherson, A. Pillay, A. Wilkie),
London Mathematical Society Lecture Note Series Nr 349, Cambridge University Press 2008, 167--195.

K-analytic versus CCM-analytic sets in nonstandard compact complex manifolds (with S. Starchenko)
Fundamenta Mathematicae, volume 198 (2008), number 2, 139--148.

Strongly minimal groups in the theory of compact complex spaces (with M. Aschenbrenner and T. Scanlon)
The Journal of Symbolic Logic, volume 71 (2006), number 2, 529--552.

On saturation and the model theory of compact Kaehler manifolds
J. reine angew. Math., volume 586 (2005), 1--20.

The model theory of compact complex spaces
Logic Colloquium '01 , Lecture Notes in Logic , vol. 20 (M. Baaz, S. Friedman, and J. Krajicek, eds.),
Association for Symbolic Logic, 2005, pp. 317--349.

A nonstandard Riemann existence theorem
Transactions of the American Mathematical Society, volume 356 (2004), number 5, 1781--1797.

Model theory of fields

The degree of nonminimality is at most two (with James Freitag and Remi Jaoui).
Journal of Mathematical Logic, volume 23 (2023), number 3, Paper No. 2250031, 6 pp.

When any three solutions are independent (with James Freitag and Remi Jaoui).
Inventiones Mathematicae, volume 230 (2022), number 3, 1249--1265.

Commutative bidifferential algebra (with Omar Leon Sanchez).
Journal of Algebra, volume 609 (2022), 764--791.

Invariant hypersurfaces (with Jason Bell and Adam Topaz).
Journal de l'Institut de Mathématiques de Jussieu, volume 21 (2022), number 2, 713--739.

Internality of logarithmic-differential pullbacks (with Ruizhang Jin).
Transactions of the AMS, volume 373 (2020), number 7, 4863--4887.

D-groups and the Dixmier-Moeglin equivalence (with J. Bell and O. Leon Sanchez).
Algebra & Number Theory 12 (2018), number 2, 343--378.
Corrigendum (with R. Jaoui) filling a gap in the proof of Proposition 3.5 on twisted D-groups.

Finiteness theorems on hypersurfaces in partial differential-algebraic geometry (with J. Freitag).
Advances in Mathematics 314 (2017), 726--755.

Poisson algebras via model theory and differential-algebraic geometry (with J. Bell, S. Launois, and O. Leon Sanchez).
Journal of the European Math Society 19 (2017), issue 7, 2019--2049.

The model companion of differential fields with free operators (with O. Leon Sanchez).
Journal of Symbolic Logic 81 (2016), number 2, 493--509.

Differential-algebraic jet spaces preserve internality to the constants (with Z. Chatzidakis and M. Harrison-Trainor).
Journal of Symbolic Logic, 80 (2015), number 3, 1022--1034.

Model theory of fields with free operators in characteristic zero (with T. Scanlon).
Journal of Mathematical Logic. 2014, doi: 10.1142/s0219061314500093.

Nonstandard methods for bounds in differential polynomial rings (with M. Harrison-Trainor and J. Klys).
Journal of Algebra, 360 (2012), 71--86

Generalised Hasse-Schmidt varieties and their jet spaces (with T. Scanlon)
Proceedings of the London Mathematical Society, volume 103 (2011), number 2, 197--234.

Jet and prolongation spaces (with T. Scanlon)
Journal de l'Institut de Mathematiques de Jussieu, volume 9 (2010), number 2, 391--430.
Corrigendum to Proposition 2.1 on Weil restrictions.

Differential arcs and regular types in differential fields (with A. Pillay and T. Scanlon)
J. Reine Angew. Math., volume 620 (2008), 35--54.

On difference fields with quantifier elimination
Bulletin of the London Mathematical Society, volume 33 (2001), number 6, 641--646.

A note on uniform definability and minimal fields of definition
The Journal of Symbolic Logic, volume 65 (2000), number 2, 817--821.

Diophantine geometry in positive characteristic

F-sets and finite automata (with Jason Bell).
Journal de Théorie des Nombres de Bordeaux, volume 31 (2019), number 1, 101--130.
Addendum by Christopher Hawthorne

Division points on subvarieties of isotrivial semiabelian varieties (with D. Ghioca)
International Mathematics Research Notices, volume 2006 (2006), Article ID 65437, 23 pages.

F-structures and integral points on semiabelian varieties over finite fields (with T. Scanlon)
American Journal of Mathematics, volume 126 (2004), number 3, 473--522.

The Mordell-Lang conjecture in positive characteristic revisited (with T. Scanlon)
Model Theory and Applications,
Quaderni di matematica, vol. 11 (L. Belair, Z. Chatzidakis, P. D'Aquino, D. Marker, M. Otero, F. Point, and A. Wilkie, eds.),
Seconda Universita di Napoli, 2002, p. 273--296.

Geometric Stability Theory

Isolated types of finite rank: an abstract Dixmier-Moeglin equivalence (with Omar Leon Sanchez).
Selecta Mathematica, volume 25 (2019), number 1, Art.10, 10pp.

Functoriality and uniformity in Hrushovski's groupoid-cover correspondence (with Levon Haykazyan).
Annals of Pure and Applied Logic, volume 169 (2018), number 8, 715--730.

Some model theory of fibrations and algebraic reductions (with A. Pillay).
Selecta Mathematica, volume 20 (2014), issue 4, 1067--1082.
Correction to the proof of Example 5.4.

A model-theoretic counterpart to Moishezon morphisms
In Models, Logics, and Higher-Dimensional Categories: A tribute to the work of Mihaly Makkai (Eds. B. Hart, T. Kucera, A. Pillay, P. Scott, R. Seely),
Centre de Recherches Mathematiques Proceedings & Lecture Notes, Volume 53, American Mathematical Society 2011, 177--188.

On canonical bases and internality criteria (with A. Pillay)
Illinois Journal of Mathematics, volume 52 (2008), number 3, 901--917.

Stable definablity and generic relations (with B. Kim)
Journal of Symbolic Logic, volume 72 (2007), number 4, 1163--1176.

Other writings

A differential analogue of the wild automorphism conjecture (with Jason Bell, Colin Ingalls, Matthew Satriano).
(E-print at http://arxiv.org/abs/2211.02122.) November 2022.

Three recent applications of model theory is a short exposition I wrote for MSRI's Spring 2014 newsletter, The Emissary.

Compactness of cycle spaces and definability in o-minimal expansions of R_an (with Patrick Speissegger and Sergei Starchenko).
(E-print at http://www.newton.cam.ac.uk/preprints/NI05040.pdf.) July 2005.

Jet spaces in complex analytic geometry: An Exposition.
(E-print at http://arxiv.org/abs/math.LO/0405563.) May 2004.

Ph.D. thesis: Contributions to the model theory of fields and compact complex spaces.
University of Illinois at Urbana-Champaign, 2001.

Students' work:

Model theory of higher derivations. Vincent Boldt's Master's Essay. 2023

A guide to F-automatic sets. Christopher Hawthorne's PhD Thesis. 2021

Not all F-sparse sets are F-sets. Christopher Hawthorne, 2019

The logarithmic derivative and model-theoretic analysability in differentially closed fields. Ruizhang Jin's PhD Thesis. 2019

Formal languages and the model theory of finitely generated free monoids. Christopher Hawthorne's Master's Essay. 2016

Contributions to the model theory of partial differential fields. Omar Leon Sanchez's PhD Thesis. 2013

Canonical bases in stable theories. Ruizhang Jin's Master's Essay. 2013

A categorical generalization of Hrushovski's limit structure. Christopher Hawthorne's USRA report. 2013

An introduction to equations and equational theories. Allen O'Hara's Master's Essay. 2011

The Mordell-Lang theorem from the Zilber dichotomy. Chris Eagle's Master's Thesis. 2010

A proof of omega-stability for m-DCF₀. By Omar Leon Sanchez and Atul Sivaswamy. 2009

Notes on "Groupoids, imaginaries and internal covers". Oleg Chterental's USRA report. 2009

2022 Vita