Past Semesters

Thu · Sep 25, 2025
Why dimension 4?
Yikai Teng
Abstract: In this talk, we introduce the field of manifold topology and its motivation questions.
Furthermore, we explain why manifolds behave differently in different dimensions, and why dimension 4 is the most mysterious of all.

Thu · Oct 2, 2025
Handle decompositions, Heegaard diagrams, and Kirby diagrams
Yikai Teng
Abstract: In this talk, we introduce a combinatorial way to study geometric topology, i.e., handle decompositions. We focus on handle decompositions in low dimensions, namely, Heegaard diagrams in dimension 3, and Kirby diagrams in dimension 4.

Thu · Oct 9, 2025
Ultraproducts in Logic

Jose Alfredo Urrea

Abstract: Mathematicians frequently use limiting procedures and completions, such as forming the real numbers from Cauchy sequences or gluing groups to create universal objects. In model theory, these processes are generalized by the method of ultraproducts. The power of this soft construction, illuminated by Łoś’s Theorem, explains its wide utility throughout mathematics. I hope this talk serves as an introduction to first order logic and it’s applications to analysis and algebra.

Thu · Oct 16, 2025
An Introduction to Stable, O-minimal, and Simple Theories

Jose Alfredo Urrea

Abstract: Ultraproducts are a powerful but “wild” method for building models of a first-order theory. While they preserve the theory, their internal structure can be difficult to control. In this talk, we will explore families of theories where this wildness is tamed, where saturated models are well-behaved and admit a classification. We will introduce the key properties of stable, o-minimal, and simple theories, showing how they lead to a rich and orderly model-theoretic structure. This talk will also explain why model theory can be useful in other areas of math.

Thu · Oct 23, 2025

Combinatorial Games, Zermelo’s Theorem, and The Determinacy Game

Seth Eisenberger

Abstract: In this talk, we explore a certain axiomization of two-player, perfect information, no-collaboration, turn-based games and prove that all such finite games are determined for one of the two players (Zermelo’s Theorem). We discuss games of countably infinite duration, and the consequences of the Axiom of Determinacy, an axiom about the determinacy of a certain infinite game.

Thu · Oct 30, 2025

Basic Topics in Set Theory: Ordinals, Continuum Hypothesis, and Axiom of Choice

Seth Eisenberger

Abstract: We give the definitions of ordinal and cardinal numbers in ZF, as well as basic ordinal arithmetic, transfinite induction/recursion, and the alephs. We briefly discuss the Continuum Hypothesis, and then discuss the main equivalent forms of the Axiom of Choice and prove their equivalency. In ZFC, cardinals serve as a representation of isomorphism classes of sets by the Well-Ordering Principle.

Thu · Nov 6, 2025

Mapping Class Group via First Homology Action

Jia Biao Too

Abstract: Mapping Class Group lies in the important intersection of geometric topology, low-dimensional topology, and geometric group theory. The definition of the group is deceptively simple, but it is subtle to comprehend the group structure. A good compromise is to give a criterion when an element in the group is trivial first, and the second is to reduce the homotopical information of free homotopy classes of loops to homological information, which are first homology classes.

Thu · Nov 13, 2025

Braids and Braid Groups

Jia Biao Too

Abstract: Why does an algebraist like equivalent definitions? It shall be answered via braid groups. Braid groups and braids lie in the important intersection of geometric topology, low-dimensional topology, category theory, combinatorial group theory and hopefully eventually geometric group theory. But deep down, every equivalent definition have the same fundamental idea. There is no surprise, it is about braids indeed.

Thu · Nov 20, 2025

Introduction to A-infinity structures, I

Qixuan Fang

Abstract: We introduce A-infinity algebras, by starting with the traditional definition of associative algebras. Then we introduce the idea of “associative up to higher homotopies” and give an example from Hochschild cohomology of associative algebras.

Thu · Nov 27, 2025

NO TALKS, HAPPY THANKSGIVING!

Thu· Dec 4, 2025

Introduction to A-infinity structures, II

Qixuan Fang

Abstract: We introduce the morphisms of A-infinity algebras, then analogously, A-infinity categories and functors between them. Then we introduce the bar construction to see how A-infinity categories translate to dg-categories. If time allows, we will briefly introduce derived categories of A-infinity categories and examples including Fukaya categories and homological mirror symmetry conjecture.