Bahar Acu (Pitzer College)
From Morse Theory to Lefschetz Fibrations: Building Geometry from Topology
This series of lectures will explore how topological structures give rise to geometric ideas through the lens of fibrations. We begin by revisiting cellular homology to explore the topological foundations of manifolds and their decomposition into simpler building blocks. Building on this, we introduce Morse and Floer theory as tools for probing manifold topology via smooth functions and critical points, setting the stage for Morse handlebody decompositions and their role in constructing and gluing submanifolds. The series culminates with a topological introduction to Lefschetz fibrations and open book decompositions, highlighting how these structures organize the topology of manifolds and naturally connect to geometric settings such as Weinstein handlebody diagrams. We conclude with a forward-looking discussion of how these ideas appear in symplectic topology and mirror symmetry, offering a broad view of the dialogue between topology and geometry.
Daniel Groves (University of Illinois at Chicago)
Introduction to hyperbolic spaces and groups
In this minicourse, we will introduce delta hyperbolic spaces and groups, with a particular focus on how they relate to topology in dimensions 2 and 3. Examples of delta hyperbolic spaces are trees and the hyperbolic plane, and we will investigate their metric geometry, and groups acting on them by isometries. The focus will be on examples and pictures, as well as aiming towards describing current research directions. No background in infinite group theory, or hyperbolic geometry will be assumed.
Miriam Kuzbary (Amherst college)
Knots, Links, and the Topology of 3- and 4-manifolds
Talk 1: Introduction to Knot Theory
Abstract: Knot theory is a rich and active area of research involving questions of interest both to mathematicians and to researchers outside of mathematics. Many of these questions boil down to a single essential query: how can we tell when two knots are different? In this talk, we will discuss why this is a difficult question to answer and start to develop some of the tools mathematicians use to address this question.
Talk 2: Cell complexes and homology computations
Abstract: In this workshop-style talk, we will learn about some of the fundamental algebraic tools used to study n-dimensional spaces which look locally like n-dimensional Euclidean space. These spaces are called n-manifolds, and we will learn how to decompose them into standard pieces in order to calculate their algebraic properties.
Talk 3: Dehn Surgery, Kirby Calculus, and 4-manifolds
Abstract: We will continue the mini-course by developing our understanding of 3 and 4 manifolds specifically and we will discuss what exactly is so interesting about these dimensions. We will define a special operation on a knot (or link) in a 3-manifold called Dehn surgery, and see that this simple operation on links in the 3-sphere is enough to construct all orientable 3-manifolds. After that, we will explore a related 4-dimensional construction which captures diffeomorphisms between 4-manifolds using a diagrammatic calculus with prescribed moves.
Tutorial\Problem session on knot theory, homology, and Kirby calculus
Talk 4: Bridging dimensions 3 and 4 with Knots and Links
Abstract: It is a common theme in topology to study n-manifolds based on the (n-1)-manifolds forming their boundaries, or the (n+1)-manifolds bounded by them. We’ll explore together why this is an interesting and useful thing to do in dimensions 3 and 4! We will talk about knots in the 3-sphere which are secretly related in 4-dimensional ways and how this can help us think about 3-manifolds that are similarly mysteriously connected.
Caglar Uyanik (University of Wisconsin-Madison)
Flat surfaces, SL(2, R) action, and Veech groups.
It will be an introduction to the study of translation surfaces, which are Riemann surfaces that are locally Euclidean outside of finitely many cone points. These surfaces play an important role in low dimensional topology and Teichmuller dynamics and can be studied from different perspectives. After the introduction of main tools and objects, we will focus on Veech surfaces, which are translation surfaces with “maximal” symmetries.