# Research

My research interest includes combinatorics and model theory. My Erdős number is at most 3. Below are some selected papers and preprints.

(A complete list of papers and preprints is available here.)

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

Answering a question asked by Kemperman in 1964, we classify connected and compact groups $$G$$  and nonempty compact $$A, B \subseteq G$$ with $$\mu_G(AB) =\mu_G(A)+\mu_G(B)$$.  We also get a near equality version of the above result with sharp bound (up to a constant factor) confirming conjectures asked by Griesmer and by Tao. As an application, we obtain a measure expansion gap result for connected compact simple Lie groups.

A nonabelian Brunn-Minkowski inequality (with Yifan Jing and Ruixiang Zhang), arXiv:2101.07782, submitted, 2021

In 1953, Henstock and Macbeath asked whether there is a Brunn–Minkowski inequality in all locally compact groups. In this paper, we obtain such an inequality and prove it is sharp for helix-free groups (including linear algebraic groups, semisimple groups with a finite center, solvable groups, Nash groups, etc). This also answers questions asked by Hrushovski and by Tao.

Étale-open topology and the stable field conjecture (with Will Johnson, Erik Walsberg, and Jinhe Ye), arXiv:2101.07782, submitted, 2021

We introduce the étale-open system of topologies for a given field, show that this recover the canonical system of topologies in many situations, and obtain characterization of field-theoretic properties 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.

Zarankiewicz’s problem for semilinear hypergraphs (with Abdul Basit, Artem Chernikov, Sergei Starchenko, and Terence Tao), arxiv:2009.02922, submitted, 2020

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.

Interpolative Fusions (with Alex Kruckman and Erik Walsberg), Journal of Mathematical Logic. 2020 Nov 28:2150010

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.