Research Interests

My research interests lie in the field of model theory, a branch of mathematical logic. More specifically, I am interested in:

Here is a link to my Curriculum Vitae.

Publications

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.
Model theory and the QWEP conjecture, Illinois Journal of Mathematics, Volume 59 (2015), 795-799.
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.
The pseudoarc is a co-existentially closed continuum (with Christopher Eagle and Alessandro Vignati), Topology and its applications, Volume 207 (2016), 1-9.
A computability-theoretic reformulation of the Connes Embedding Problem (with Bradd Hart), Bulletin of Symbolic Logic, Volume 22 (2016), 238-248.
Approximate polynomial structure in additively large sets (with Mauro Di Nasso, Renling Jin, Steven Leth, Martino Lupini, and Karl Mahlburg), Integers, Volume 16 (2016), Article A49.
High piecewise syndeticity of product sets in amenable groups (with Mauro Di Nasso, Renling Jin, Steven Leth, Martino Lupini, and Karl Mahlburg), Journal of Symbolic Logic, Volume 81 (2016), 1555-1562.
A monad measure space for logarithmic density (with Mauro Di Nasso, Renling Jin, Steven Leth, Martino Lupini, and Karl Mahlburg), Monatshefte für Mathematik, Volume 181 (2016), 577-599.
Robinson forcing and the quasidiagonality problem (with Thomas Sinclair), International Journal of Mathematics, Volume 28 (2017), Article 1750008.
Omitting types in operator systems (with Thomas Sinclair), Indiana University Mathematics Journal, Volume 66 (2017), 821-844.
On the theories of McDuffs II_1 factors (with Bradd Hart), International Mathematics Research Notices, Volume 27, Issue 18 (2017), 5609-5628.
Model-theoretic aspects of the Gurarij operator system (with Martino Lupini), Israel Journal of Mathematics, Volume 226 (2018), 87–118.
Explicit sentences distinguishing McDuff's II_1 factors (with Bradd Hart and Henry Towsner), Israel Journal of Mathematics, Volume 227 (2018), 365–377.
Hindman's theorem and idempotent types (with Uri Andrews), Semigroup Forum, Volume 97, 471-477.
Definable sets containing productsets in expansions of groups (with Uri Andrews and Gabriel Conant), Journal of Group Theory, Volume 22 (2019), 63-82.
Independence in randomizations (with Uri Andrews and H. Jerome Keisler), Journal of Mathematical Logic, Volume 19 (2019), Article 1950005.
On the axiomatizability of C* algebras as operator systems (with Thomas Sinclair), Glasgow Journal of Mathematics, Volume 61 (2019), 629-635.
An invitation to nonstandard analysis and its recent applications (with Sean Walsh), Notices of the AMS, Volume 66 (June/July 2019), 842-851.
Boundary amenability of groups via ultrapowers (with Stephen Avsec), Houston Journal of Mathematics, Volume 45 (2019) 731-741.
On supra-SIM sets of natural numbers (with Steven Leth), chapter in the book Combinatorial and Additive Number Theory III (2020), 123-132.
Scattered sentences have few separable randomizations (with Uri Andrews, Sherwood Hachtman, H. Jerome Keisler, and David Marker), Archive for Mathematical Logic, Volume 59 (2020), 743-754.
Properties expressible in small fragments of the theory of the hyperfinite II_1 factor (with Bradd Hart), Confluentes Mathematici, Tome 12 (2020), 37-47.
On Popa's factorial commutant embedding problem, Proceedings of the American Mathematical Society, Volume 148 (2020), 5007-5012.
Enforceable operator algebras, Journal of the Institute of Mathematics of Jussieu, Volume 20 (2021), 31-63.
Operator algebras with hyperarithmetic theory (with Bradd Hart), Journal of Logic and Computation, Volume 31 (2021), 612-629.
Ultrafilter methods in combinatorics, Snapshots of Modern Mathematics from Oberwolfach, 2021-006.
A nonstandard proof of the spectral theorem for unbounded self-adjoint operators, Expositiones Mathematicae, Volume 39 (2021), 590-603.
Oracle computability of conditional expectations onto subfactors, New York Journal of Mathematics, Volume 27 (2021), 1085-1095.
The almost sure theory of finite metric spaces (with Bradd Hart and Alex Kruckman), Bulletin of the London Mathematical Society, Volume 53 (2021), 1740-1748.
A tribute to Lou van den Dries, Celebratio Mathematica (2021).
Factorial relative commutants and II_1 factors with the generalized Jung property (with Scott Atkinson and Srivatsav Kunnawalkam Elayavalli), Advances in Mathematics Volume 396 (2022), Paper No. 108107.

Papers accepted for publication

Spectral gap and definability, to appear in the book Beyond First Order Model Theory Volume 2.
Continuous sentences preserved under reduced products (with H. Jerome Keisler), to appear in the Journal of Symbolic Logic.
A nonstandard take on central sets, to appear in the volume Research Trends in Contemporary Logic.
Non-embeddable II_1 factors resembling the hyperfinite II_1 factor, to appear in the Journal of Noncommutative Geometry.
The Connes Embedding Problem: a guided tour, to appear in the Bulletin of the American Mathematical Society.
Existentially closed W^*-probability spaces (with Cyril Houdayer), to appear in Mathematische Zeitschrift.
Everettian mechanics with hyperfinitely many worlds (with Jeffrey Barrett), to appear in Erkenntnis.

Papers submitted for publication

Correspondences, ultraproducts, and model theory (with Bradd Hart and Thomas Sinclair)
The universal theory of the hyperfinite II_1 factor is not computable (with Bradd Hart)
On the complexity of the theory of a computably presented metric structure (with Caleb Cambrud and Timothy McNicholl)
Generic algebraic properties in spaces of enumerated groups (with Srivatsav Kunnawalkam Elayavalli and Yash Lodha)
Unitary representations of locally compact groups as metric structures (with Itaï Ben Yaacov)

Preprints

The undecidability of having QWEP (with Jananan Arulseelan and Bradd Hart)
A quantum complexity approach to the Kirchberg Embedding Problem (with Bradd Hart)

Books

Ultrafilters throughout mathematics. To appear in the American Mathematical Society's Graduate Studies in Mathematics series.
Nonstandard methods in Ramsey theory and combinatorial number theory (with Mauro Di Nasso and Martino Lupini). Lecture Notes in Mathematics, Volume 2239 (2019).

Research notes

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).

Lecture notes

A gentle introduction to von Neumann algebras for model theorists (UIC, Fall 2012)
Nonstandard Analysis (UCLA Summer School in Logic, 2009 & 2012)

Slides from talks

Scattered sentences have few separable randomizations, given at the special session on Continuous model theory at the 2017 North American Annual Meeting of the ASL at Boise State University on 3/20/17.
Model theory and the weak expectation property, given at the special session on Classification problems in operator algebras at the 2016 Joint Meetings in Seattle
The model-theoretic content of some conjectures about C* algebras, given at the Bogota Logic Colloquium on 12/11/15
Omitting types in C* algebras, given at the UCSD Operator Algebras Seminar on 10/22/15
High piecewise syndeticity of product sets in amenable groups, given at the Combinatorics meets ergodic theory workshop in Banff on 7/23/15
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.