My research interests lie in the field of model theory, a branch of mathematical logic. More specifically, I am interested in:
- Pure and applied model theory
- Nonstandard analysis
- Model theory for metric structures
- Lie theory
- Geometric group theory
- Operator algebras
- Combinatorial number theory
Here is a link to my Curriculum Vitae.
Bezout domains and elliptic curves (with Marc Masdeu), Communications in Algebra, Volume 36 (2008) Issue 12, 4492-4499.
Nonstandard hulls of locally exponential Lie algebras, Journal of Logic and Analysis, Volume 1:5 (2009), 1-25.
Locally compact contractive local groups (with Lou van den Dries), Journal of Lie Theory, Volume 19 (2009), 685-695.
Hilbert's fifth problem for local groups, Annals of Mathematics, Volume 172 (2010), Number 2, 1269-1314.
Globalizing locally compact local groups (with Lou van den Dries), Journal of Lie Theory, Volume 20 (2010), 519-524.
Ends of groups: a nonstandard perspective, Journal of Logic and Analysis, Volume 3:7 (2011), 1-28.
Definable functions in Urysohn's metric space Illinois Journal of Mathematics, Volume 55, Number 4 (2011), 1423-1435.
Thorn-forking in continuous logic (with Clifton Ealy) Journal of Symbolic Logic, Volume 77 (2012), 63-93.
Definable operators on Hilbert spaces Notre Dame Journal of Formal Logic, Volume 53, Number 2(2012), 193-201.
An approximate Herbrand's theorem and definable functions in metric structures Mathematical Logic Quarterly, Volume 58, Issue 3 (2012), 208-216.
Nonstandard hulls of locally uniform groups Fundamenta Mathematicae, Volume 220 (2013), 93-118.
Further thoughts on definability in the Urysohn sphere Proceedings of the 12th Asian Logic Conference, World Scientific (2013), 144-154.
The theory of tracial von Neumann algebras does not have a model companion (with Bradd Hart and Thomas Sinclair) Journal of Symbolic Logic, Volume 78, Number 3 (2013), 1000-1004.
Transseries and Todorov-Vernaeve's asymptotic fields (with Matthias Aschenbrenner) Archive for Mathematical Logic, Volume 53 (2014), 65-87.
An approximate logic for measures (with Henry Towsner) Israel Journal of Mathematics, Volume 199, Number 2 (2014), 867-913.
Definable closure in randomizations (with Uri Andrews and H. Jerome Keisler) Annals of Pure and Applied Logic, Volume 166, Issue 3 (2015), 325-341.
High density piecewise syndeticity of sumsets (with Mauro Di Nasso, Renling Jin, Steven Leth, Martino Lupini, and Karl Mahlburg) Advances in Mathematics, Volume 278 (2015), 1-33.
On Kirchberg's embedding problem (with Thomas Sinclair) Journal of Functional Analysis, Volume 269 (2015), 155-198.
On a sumset conjecture of Erdős (with Mauro Di Nasso, Renling Jin, Steven Leth, Martino Lupini, and Karl Mahlburg) Canadian Journal of Mathematics, Volume 67 (2015), 795-809.
Pseudofinite and pseudocompact metric structures (with Vinicius Cifú Lopes) Notre Dame Journal of Formal Logic, Volume 56 (2015), 493-510.
Games and elementary equivalence of II_1 factors (with Thomas Sinclair) Pacific Journal of Mathematics, Volume 278 (2015), 103-118.
Dividing and weak quasi-dimensions in arbitrary theories (with Henry Towsner) Archive for Mathematical Logic, Volume 54 (2015), 915-920.
Hilbert's fifth problem (with Lou van den Dries). L'Enseignement Mathématique, Volume 61/62 (2015), 3-43.
The fundamental group of a locally finite graph with ends: a hyperfinite approach (with Alessandro Sisto) Fundamenta Mathematicae, Volume 232 (2016), 21-39.
Existentially closed II_1 factors (with Ilijas Farah, Bradd Hart, and David Sherman) Fundamenta Mathematicae, Volume 233 (2016), 173-196.
Papers accepted for publication
High piecewise syndeticity of product sets in amenable groups (with Mauro Di Nasso, Renling Jin, Steven Leth, Martino Lupini, and Karl Mahlburg) To appear in Journal of Symbolic Logic.
Omitting types in operator systems (with Thomas Sinclair) To appear in the Indiana University Mathematics Journal.
A computability-theoretic reformulation of the Connes Embedding Problem (with Bradd Hart) To appear in the Bulletin of Symbolic Logic.
A monad measure space for logarithmic density (with Mauro Di Nasso, Renling Jin, Steven Leth, Martino Lupini, and Karl Mahlburg) To appear in Monatshefte für Mathematik
Model theory and the QWEP conjecture To appear in the Illinois Journal of Mathematics.
The pseudoarc is a co-existentially closed continuum (with Christopher Eagle and Alessandro Vignati) To appear in Topology and its applications.
Papers submitted for publication
Independence in randomizations (with Uri Andrews and H. Jerome Keisler)
Model-theoretic aspects of the Gurarij operator system (with Martino Lupini)
Approximate polynomial structure in additively large sets (with Mauro Di Nasso, Renling Jin, Steven Leth, Martino Lupini, and Karl Mahlburg)
Hindman's theorem and idempotent types (with Uri Andrews)
On the theories of McDuffs II_1 factors (with Bradd Hart)
On the axiomatizability of C* algebras as operator systems (with Thomas Sinclair)
Amalgamating R^\omega-embeddable von Neumann algebras (with Ilijas Farah and Bradd Hart)
Seminar notes on Hilbert's Fifth Problem (with Lou van den Dries)
An erratum to "Globalizing locally compact local groups"
A short note on Hilbert's fifth problem and Ind/Type-definable groups (with Anand Pillay).
A note on ends of groups.
Two small nonstandard group-theoretic observations.
A note on Property (T).
Slides from talks
Monad measures spaces and combinatorial number theory, given at the Canadian Mathematics' Society Winter Meeting on 12/6/14
Progress on a sumset conjecture of Erdos, given at the UIC Logic Seminar on 8/27/13
The fundamental group of a locally finite graph with ends: a hyperfinite approach, given at the University of Pennsylvania Logic Seminar on 10/16/12.
The theory of tracial von Neumann algebras does not have a model companion, given at the University of Illinois at Urbana Champaign Logic Seminar on 10/12/12.
Nonstandard hulls of locally uniform groups, given at the special session in Nonstandard Analysis at the AMS Spring Western Sectional Meeting at the University of Hawaii at Manoa on 3/4/12
Definable functions in Urysohn's Metric Space, given at the UC Irvine Logic Seminar on 2/6/12
The Urysohn space is rosy, given at the UC Irvine Logic Seminar on 2/27/12
Nonstandard methods in Lie theory, given at the UIC Math Department Colloquium on 1/20/12.
Calculus with Infinitesimals, given at the Loyola Marymount Math Department Seminar on 10/7/11.
Rosiness in Continuous Logic, given at the Notre Dame Model Theory Seminar on 9/1/09.
Thorn-forking and Rosiness in Continuous Logic, given at ASL North American Annual Meeting in Notre Dame on 5/20/09.
Ends of Finitely Generated Groups from a Nonstandard Perspective, given at McMaster on 10/23/08.
Hilbert's Fifth Problem for Local Groups, given at OSU on 10/30/08.
Nonstandard Hulls of Locally Exponential Lie Groups and Algebras, given at Seminar Sophus Lie in Paderborn, Germany on 1/17/09.