Time：14:00-15:00, Friday, March 20 2026
Venue：E14-212
Speaker：Will Johnson, Fudan University
Biography：Professor Will Johnson received his PhD in mathematics from UC Berkeley in 2016. Following graduation, he worked as a software engineer at Niantic, Inc. in San Francisco from 2016 to 2018. and then as a mathematics postdoc at Fudan University in Shanghai from 2018 to 2020. Since 2020, he has worked in the mathematical logic group of the Fudan philosophy department.
Professor Will Johnson's research area is model theory, a branch of logic with close ties to algebra. Within model theory, his focus is on the model theory of fields and commutative rings, especially topological fields, pseudo-algebraically closed fields, and valuation rings. He has been invited to speak at the coming ICM 2026.
Title：Definable sets, tame topology, and VC dimension
Abstract：A set in ℝ is "definable" in the field (ℝ,+,*) if it is defined using a combination of polynomial equations and logical operators (including the quantifiers ∀ and ∃). Definable sets and functions turn out to be "topologically tame" in several ways. For example, definable functions are piecewise smooth, and definable sets only have finitely many connected components. None of the pathological counterexamples in topology can be definable. Many of these tameness properties persist if we extend the class of definable sets to allow definitions using the exponential function e^x. In model-theoretic parlance, the structure (ℝ,+,*,exp) is "o-minimal". An unexpected property of o-minimal structures is that definable families of sets have finite "VC-dimension" in the sense of statistical learning theory. These properties of tame topology and finite VC-dimension generalize from o-minimal structures to a larger family of structures including the field of p-adic numbers ℚ_p from number theory. In this talk, I will sketch the classical results described above, as well as some of my own results in this area.
