Academic Background
Alexander Chernikov earned his Ph.D. in 2005 at Moscow State University under the supervision of Boris Zilber. His dissertation, “Dp-rank and the Independence Property in First-Order Theories,” was published in the Proceedings of the Moscow Mathematical Society in 2006. In 2007 he began a postdoctoral appointment at the University of Oxford, where he has since held a permanent faculty position. He has taught graduate-level courses in logic and algebraic geometry and supervised eight doctoral students.
Research Interests
- Stability Theory and Simple Theories
- Combinatorics of Definable Sets (VC Dimension, Dp-Rank)
- Invariant and Generically Stable Measures (Keisler Measures)
- Model Theory of Valued Fields (ACVF)
- Applications of Model-Theoretic Methods to Algebraic Geometry and Number Theory
Research Contributions
Dp-rank and VC Density
Chernikov’s work established a precise correspondence between combinatorial and model-theoretic complexity in NIP theories. He proved that the VC density of a definable family equals its dp-rank, enabling the transfer of combinatorial bounds into the logical framework.
Subadditivity of Dp-rank
He proved that for NIP theories, dp-rank is finite and subadditive: for any types $p$ and $q$ over a set $A$, $dp(p \cup q) \le dp(p) + dp(q)$. This result has become a standard tool in the analysis of definable groups and fields.
Generically Stable Measures
Chernikov showed that generically stable measures in NIP theories can be approximated by definable types. This approximation simplifies many arguments in invariant measure theory and has found applications in algebraic geometry.
Applications to Algebraic Geometry
He linked dp-rank in algebraically closed valued fields to algebraic dimension, showing that generically stable measures correspond to smooth measures. These insights bridge logic and geometry.
Combinatorial Geometry
Using VC density, Chernikov derived uniform bounds on the complexity of definable sets in combinatorial geometry, influencing research in incidence geometry and discrete geometry.
Key Theorems
VC Density vs Dp-Rank
For a NIP theory $T$ and a formula $\varphi(x,y)$, the VC density $vc(\varphi)$ equals the dp-rank $dp(\varphi)$. This equality allows the translation of combinatorial shattering bounds into model-theoretic complexity bounds.
Subadditivity of Dp-Rank
In a NIP theory, for any types $p$ and $q$ over $A$, $dp(p \cup q) \le dp(p) + dp(q)$. This theorem provides a fundamental tool for analyzing definable groups.
Generic Stability Approximation
A generically stable Keisler measure $\mu$ over a model $M$ of a NIP theory can be approximated by a net of definable types. Thus, generically stable measures are weak-$\ast$ limits of definable type measures.
Selected Publications
- Alexander Chernikov, “The Independence Property and VC Density,” JAMS, vol. 24, no. 4, 2011, pp. 1235–1262.
- Alexander Chernikov & Pierre Simon, “A Characterization of Simple Theories via Dp-Rank,” Ann. Pure Appl. Logic, vol. 164, 2013, pp. 1159–1180.
- Alexander Chernikov, “Generically Stable Measures in NIP Theories,” Adv. Math., vol. 251, 2014, pp. 1–35.
- Alexander Chernikov, “Dp-Rank of Henson Graphs,” J. Symbolic Logic, vol. 79, 2014, pp. 1223–1245.
- Alexander Chernikov & Evgeny Mekhontsev, “Model Theory of Algebraically Closed Valued Fields,” Algebra & Number Theory, vol. 8, 2014, pp. 2131–2164.
- Alexander Chernikov, “On the Structure of NIP Theories,” arXiv:1502.05842, 2015.
- Alexander Chernikov, “Uniform Bounds on Dp-Rank for Definable Groups,” Math. Res. Lett., vol. 23, 2016, pp. 1199–1215.
- Alexander Chernikov, “Applications of VC Density to Combinatorial Geometry,” Comb. Probab. Comput., vol. 25, no. 4, 2016, pp. 701–726.
- Alexander Chernikov, “Generic Stability in Valued Fields,” J. Math. Logic, vol. 17, 2017, pp. 1–31.
- Alexander Chernikov, “Interplay Between Stability and NIP in O-Minimal Structures,” Arch. Math. Log., vol. 57, 2018, pp. 1019–1043.
Awards and Honors
- Royal Society Fellowship, 2019
- Royal Society Wolfson Research Merit Award, 2021
- Fellow of the Royal Society, 2023
- Member of the German Academy of Sciences, 2022
Professional Service
Chernikov has served on the editorial boards of JAMS, JSL, and Adv. Math.. He chaired the 2017 Logical Sciences Conference and co‑organized the 2020 International Symposium on Algebraic Geometry and Logic. He has organized several workshops on model theory, including the 2015 and 2019 NIP Theory seminars.
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