Spring 2026
Dima Sinapova
Subtitle:
Combinatorial Set Theory
Course Description:
We will cover recent developments in infinitary combinatorics, and applications of forcing and large cardinals. We will introduce combinatorial principles such as stationary reflection, Aronszajn trees, and how they interact with cardinal arithmetic (e.g. CH, SCH). We will discuss the combinatorics at successors of regulars, like $\aleph_2$; and successors of singulars, like $\aleph_{\omega+1}$. Prior knowledge of forcing is helpful, but nor required. I will define the necessary concepts and terminology.
Text:
N/A
Prereq:
Graduate standing or Math 561 or approval from the course instructor
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Spring 2025
Filippo Calderoni
Subtitle:
Groups, Logic, and Dynamics
Course Description:
This topics course is divided into two parts. First we will discuss the theory of orderable groups. We will present it as a subject in its own right and discuss its connections with geometric topology. The second part is an introduction to countable Borel equivalence relations, a major research topic in modern descriptive set theory. The two parts of the course are characterized by the same high interdisciplinary nature, and as we will see they are surprisingly connected.
Text:
No textbook is required. The instructor will share some notes.
Prerequisites:
Some background knowledge of group theory (MATH 451 or equivalent) and general topology (MATH 441 or equivalent) will be helpful.
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Fall 2023
Filippo Calderoni
Subtitle:
Modern descriptive set theory
Course Description:
Descriptive set theory was developed at the beginning of the 20th century as the study of well-behaved subsets of the real line. Since then the classical analysis on subsets of reals has been extended to those of any separable and completely metrizable topological space.After a crash course on classical descriptive set theory, we will discuss recent results that use descriptive set theory and impact other areas. This will include zero-one laws in group theory, (elementary) amenable groups, anticlassification results, and countable Borel equivalence relations. This course assumes no prerequisites in logic, and aims at stimulating a transversal interest between different subjects such as mathematical logic, group theory, and ergodic theory.
Text:
None
Prerequisites:
Mathematical maturity and curiosity
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Spring 2021 - Grigor Sargsyan
Subtitle:
Determinacy
Course Description:
The Axiom of Determinacy (AD) says that all two player games of perfect information in which players play integers are determined. AD contradicts the Axiom of Choice but has many appealing consequence. Moreover, over the last 60 years it has become increasingly clear that determinacy of definable games is consistent with the usual axioms of set theory. The begining of such results was the original Gale-Steart theorem that all open games are determined, which was then extended by Martin to all Borel games. The Determinacy of more complicated games cannot be shown in ZFC alone. -- In this course, we will study the consequence of the Axiom of Determinacy. In a further course, we will study its connections with areas of mathematics.
Prerequisites:
basic graduate level mathematical maturity
Text:
None
Schedule of Sections:
Previous Semesters:
- Fall 2019 Prof. Thomas
- Fall 2017 Prof. Thomas
Fall 2017 - Simon Thomas
Subtitle: Countable Borel Equivalence Relations
Course Description:
This course will be an introduction to countable Borel equivalence relations, a very active area of classical descriptive set theory which interacts nontrivially with such diverse areas of mathematics as model theory, computability theory, group theory and ergodic theory. The topics to be covered will include applications of superrigidity theory to countable Borel equivalence relations, as well as some recent applications of Borel determinacy. No prior knowledge of superrigidity or determinacy will be assumed.
Text:
None
Prerequisites:
Familiarity with the basic theory of complete separable metric spaces and their Borel subsets