2025 Fudan Logic Summer School
Time: Jul 21 - Aug 1, 2025
Location: Fudan University (Handan Road Campus)
Enroll now! (until June 1)
The first week (Jul 21 - Jul 25): 喻良
The second week (Jul 28 - Aug 1):Ben Castle
Room: TBA
Hours:
Lecture 1: 9:15 - 10:30 (GMT+8)
Lecture 2: 11:00 - 12:15 (GMT+8)
Section: 15:00 - 17:00 (GMT+8)
Strong Minimality and Geometric Stability Theory
A subfield of model theory, geometric stability theory refers to a body of results and methods aimed at recovering familiar algebraic structures from purely logical data (here ‘stability’ refers to model-theoretic stability, which provides many of the tools used; and ‘geometric’ refers to the use of combinatorial ‘geometries’ to sort out the different types of algebraic structures). For example, one of the earliest results along these lines is a theorem of Zilber on totally categorical theories (theories with the ‘fewest’ models possible): in any model of such a theory, there is an infinite interpretable set with either no structure at all, or precisely the structure of a vector space over a finite field. The remarkable feature of this result is that an assumption seemingly far removed from algebra (namely a restriction on the number of models of a theory) is ultimately intrinsically tied to vector spaces. This course will provide an introduction to geometric stability through the single case of strongly minimal theories: a complete theory T is strongly minimal if for every M ⊧ T, we have (i) M is infinite, but (ii) every definable subset of M is finite or cofinite. The main examples of such theories are pure sets, vector spaces, and algebraically closed fields. The main goals of the course will be: 1. Develop the basic machinery of strongly minimal theories, particularly dimension theory of definable sets and types, and the related notions of canonical bases and Zilber’s trichotomy. 2. As a main goal, prove the locally modular case of the group configuration theorem (one of the most influential theorems of stability theory) and the associated structure theorems for locally modular groups. These theorems will demonstrate the main ideas model theorists use to construct groups and vector spaces out of abstract data. On the one hand, strong minimality is a very strong restriction on a theory; on the other hand, fundamental results of Baldwin and Lachlan, and later Buechler, show that much larger classes of theories can be understood in terms of strongly minimal ones. For these reasons, strongly minimal theories continue to play a fundamental role in modern model theory, and many modern definitions are abstracted from the strongly minimal case. Thus, this course can also be treated as a hands-on tour of some of the core ideas of modern model theory (namely stability and its generalizations).
Program (subject to change):
Day 1: The Basics.
Historical overview of uncountably categorical and totally categorical theories, focusing on the finite axiomatizability problem.
Basic properties and examples of strongly minimal theories.
The dimension of a definable set, and the decomposition into stationary components.
Day 2: Dimension via Pregeometries
Review of saturated models
The algebraic closure operator; pregeometries and geometries.
Dimension theory for tuples and types, and the duality with dimension theory for definable sets.
Proof that strongly minimal theories are uncountably categorical.
Day 3:
Review of imaginary elements and canonical parameters, and weak elimination of imaginaries in the strongly minimal case.
Definition and construction of canonical bases; relation to ‘normal’ families of sets.
Definition and examples of 1-based strongly minimal theories.
Families of plane curves, and the three levels of Zilber’s trichotomy.
Day 4:
The groupoid of germs.
Construction of an infinite family of germs in a non-trivial 1-based strongly minimal theory.
Interpreting a group from a family of germs.
If time, survey the more general group configuration theorem.
Day 5:
Abelianity of the group, and the structure theorem for definable sets in 1-based groups
Complete classification of totally categorical strongly minimal theories up to finite covers (assuming Zilber’s theorem that such theories are 1-based)
Informal description of Hrushovski’s proof of Zilber’s theorem above on 1-basedness
Informal account of how and when one can interpret a field rather than a group.
Informal discussion of applications to other areas of mathematics.
Lecturer:
Ben Castle is a mathematical logician at Department of Mathematics, University of Illinois, Urbana-Champaign. He is specializing in model theory and its interactions with algebra, geometry, and combinatorics. Before coming to Illinois, Castle held postdoctoral positions at the Fields Institute, Notre Dame University, Ben-Gurion University of the Negev, and the University of Maryland; during this time, his work focused on studying reconstruction theorems in geometry from a model-theoretic lens. He holds a PhD in mathematics from UC Berkeley.
Links：Organizers - Mathematical Logic at Fudan