Informal Geometry and Topology Seminar (Summer 2025)

This is the page for the Informal geometry and topology seminar for summer 2025 at LSU, organised by my fellow colleague  Matthew and me. 

Topic: Topological K-Theory

Venue: Lockett 233

Time: Mondays and Thursdays, 1-2 pm (CST)

Zoom Information:   

Zoom link: https://lsu.zoom.us/j/98838338874?pwd=dlBYTVJ0WmdlaU0yall3SXVONHNhUT09

Meeting ID: 988 3833 8874

Passcode: LSUMath

Resources:

• K-theory and Characteristic Classes: A Homotopical Perspective, Inna Zakharevich. (The main resource for this seminar. These notes are very detailed and elaborated.)

• K-Theory and Characteristic Classes, David Merhle. (These notes are easy to follow and go through, and we will draw our outline of the talks from here, but are not free from typos and LaTeX errors.)

• Vector Bundles and K-Theory, Allen Hatcher. (Very elaborated notes written in typical `Hatcher' fashion. We will refer to it for details accordingly.)

• What on earth is... topological K-theory? (For a quick overview of topological K-theory.)

• Topological K-Theory and some of its application. (A good reference for some application.)

• K-Theory lecture notes, Atiyah. (Honourable mention.)

• Characteristic Classes, Milnor. (Honourable mention.)

Structure:

To ensure a balanced and effective learning experience, the seminar schedule is designed to align with the depth and complexity of the material in Inna Zakharevich's and David Merhle’s notes on Topological K-Theory. As discussed, we will meet twice a week, for 1 hour, to present the topics.

Ahead of each session, the speaker will receive a suggested outline for their talk. This outline is intended as a guiding framework to help structure the presentation and emphasize key ideas. Speakers are welcome to follow it closely or adapt it to their own understanding and insights.

Our goal is to maintain a balance between rigour and accessibility, creating a collaborative environment where participants can explore both the foundational ideas and modern homotopical perspectives in K-theory.

Here is the overall plan for the chapters of Inna's notes.

Chapters 1–3: Foundational Concepts (Week 1-2)

We begin with the core building blocks of K-theory: vector bundles, Grassmannians, Pullback bundles and classification of vector bundles. These chapters are conceptually lighter and largely algebraic or categorical.

Chapters 4–6: Conceptual Core (Week 3-5)

These chapters introduce major conceptual shifts: omega spectra, generalised cohomology, Euler class, and characteristic class.

Since these ideas are central to the subject and often subtle, we allow more time per chapter to ensure clarity and depth.

Chapters 7–10: Topological K theory (Week 6-10)

This is the soul of this seminar: Bott periodicity, Topological K-theory, Adams operation

These chapters are rich and abstract, so we slow down here to allow for examples, discussion, and synthesis of earlier material.