# Research

I am interested in model theory, a subject lying on the boundary of mathematical logic and geometry.  My works explore interactions between model theory and other areas of mathematics like combinatorics, field arithmetic, and number theory.

My Erdõs number is 3.

Coauthors: Jinpeng An, Abdul Basit, Neer Bhardwaj, Artem Chernikov, Tigran Hakobyan, Daniel Hoffmann, Alex Kruckman, Yifan Jing (5), Will Johnson, Souktik Roy, Sergei Starchenko, Terence Tao, Ruixiang Zhang (2), Erik Walsberg (3), Jinhe Ye (2).

(A complete list of papers and preprints is available HERE. Most of them are available on arxiv.)

Below are four main research directions, each represented by a paper/preprint with informal description.

1. Small expansions in nonabelian groups

The Freiman’s type of results in arithmetic combinatorics tells us that if a finite subset $$A$$ of an abelian group $$G$$ has small expansion (i.e. $$|A+A|< K|A|$$ with $$A+A =\{a_1+a_2: a_1, a_2 \in A\}$$ and $$K>0$$ a fixed constant), then $$A$$ must be “convexly structured” (more precisely, commensurable to a coset progression). Using ideas from model theory, this was generalized to nonabelian groups and locally compact groups (with a left Haar measure instead of cardinality) by Hrushovski, Breuillard–Green–Tao, and Carolino. Among other goals, we want to obtain versions of their results with effective bounds by a machinery mirroring ideas in model theory. This was achieved for connected $$G$$ and “very” small measure expansion via the recent solution of the inverse Kemperman problem by An, Jing, Zhang and me.

Minimal and nearly minimal measure expansions in connected unimodular groups (with Yifan Jing), arXiv:2006.01824, preprint 2020.

Let $$G$$ be a connected locally compact groups with a left Haar-measure $$\mu$$, and let $$A, B\subseteq G$$ be nonempty and compact. In 1964, Kemperman showed that if $$\mu$$ is also a right Haar-measure (equivalently, $$G$$ is unimodular, a condition holds when $$G$$ is compact or a semisimple Lie group), then $\mu(AB) \geq \min \{\mu(A)+\mu(B), \mu(G)\} .$ The inverse Kemperman problem (proposed by Griesmer, Kemperman, and Tao) asks when the equality happens or nearly happens. This paper answer the equality case and the near equality with compact $$G$$ case of the problem. As an application, we obtain the first measure expansion gap result for connected compact simple Lie groups.

2. Canonical topologies

Many structures come with natural topologies (e.g., algebraically closed fields with the Zariski topologies, real closed fields with the Euclidean topologies). We want a common method to define such topologies. This was achieved for the class of large fields by Johnson, Walsberg, Ye, and me with the new notion of etale-open (system of) topologies. Using this, we resolved the stable fields conjecture for this class.

Étale-open topology and the stable field conjecture (with Will Johnson, Erik Walsberg, and Jinhe Ye), arXiv:2101.07782, accepted to Journal of the European Mathematical Society.

We introduce the étale-open system of topologies for a given field, show that this recover the canonical system of topologies in many situations (algebraically closed fields, real closed fields, p-adically closed field), and obtain characterization of field-theoretic properties (e.g. largeness) in term of the étale-open system of topologies. As an application, we confirm the special case of the longstanding stable fields conjecture for the class of large fields which contains all known model-theoretically tame fields.

3. Combinatorial consequences of modularity

We would like to study combinatorial consequences of the absence-of-complexity assumption of modularity (i.e., being “module-like”). Under nonmodularity, Jing, Roy, and I proved a generalized sumproduct result. Toward the other direction, for the modular structure $$(\mathbb{R}; +, <)$$, Basit, Chernikov, Starchenko, Tao, and I obtained a strong incidence bound for $$n$$ points and $$n$$ shapes in a definable family.

Zarankiewicz’s problem for semilinear hypergraphs (with Abdul Basit, Artem Chernikov, Sergei Starchenko, and Terence Tao), arXiv:2009.02922, Forum of Mathematics, Sigma, volume 9, 2021.

For a $$K_{k,k}$$-free bipartite graph $$(V_1, V_2, E)$$ is  with $$V_1, V_2 \subseteq \mathbb{R}^n$$ and $$E$$ the subset of $$V_1 \times V_2$$ satisfying a fixed system of linear inequalities of complexity $$s$$, we show that  $|E| \leq \alpha|V|( \log|V| )^ \beta$ with $$\alpha =\alpha(k,s)$$ and $$\beta=\beta(s)$$. We also obtain a version of this result for hypergraph, show that the appearance of log is necessary in general, and connect this to o-minimal trichotomy in model theory.

4. Model theory of partially random structures

Finally, Bhardwaj, Kruckman, Wasberg, and I investigated structures built by combining simpler structures interacting in a random/generic fashion with one another. We showed that many natural examples in model theory can be put under this framework, and properties of definable sets in the combined structures can be understood from those of the component structures.

Interpolative Fusions (with Alex Kruckman and Erik Walsberg), Journal of Mathematical Logic, volume 21, no 2.

We define the interpolative fusion of multiple theories over a common reduct, a notion that aims to provide a general framework to study model-theoretic properties of structures with randomness. In the special case where the theories involved are model complete, their interpolative fusion is precisely the model companion of their union. Several theories of model-theoretic interest are shown to be canonically bi-interpretable with interpolative fusions of simpler theories. We initiate a systematic study of interpolative fusions by also giving general conditions for their existence and relating their properties to those of the individual theories from which they are built.