Topology and Geometry Seminar

Title: Introduction to topological K-theory

Abstract: Topological K-theory is an invariant of spaces based on vector bundles. We will review vector bundles and their representability by Grassmannians. We will then introduce the group K^0(X), which measures to what extent vector bundles on X can be non-trivial. Finally, Bott periodicity will allow us to extend the group K^0(X) to a cohomology theory.

Title: Introduction to topological K-theory, part 2

Abstract: In this second part, we will further discuss Stiefel manifolds and Grassmannians, as well as stabilization of vector bundles. Next, we will introduce the group K^0(X) along with its reduced version, and compute some examples. Time permitting, we will use Bott periodicity to extend the group K^0(X) to a cohomology theory.

Title: Introduction to topological K-theory, part 3

Abstract: Now that we have the K-theory group K(X) at our disposal, our next goal will be to compute it for the circle S^1 and the sphere S^2. We will then discuss Bott periodicity and extend the group K(X) to a cohomology theory.

Title: The Chern character and its Applications

Abstract: In this talk we will first review some of the properties of Chern classes and Steifel Whitney classes. Next, we will introduce the Chern character, now that we have made K(X) into a cohomology theory using Bott periodicity, in the earlier talk. Following that, we will see some interesting applications of the Chern character map and hence compute the unreduced K-theory of CP^n.

Title: The Chern character: Epilogue and Prelude

Abstract: At this point we have two cohomology theories in hand – HZ (singular cohomology) and KU (complex K-theory) – and the Chern character ch establishes a comparison between them. Even the first properties of ch can already give rise to interesting consequences, as we saw last time. 

The story, however, does not simply stop here. Both HZ and KU are “very geometric”, as they support Thom classes for complex vector bundles. This talk will invite us to take a further (geometric) look at the comparison ch: how does it compare the K-theoretic Thom class with the ordinary cohomological one? Exploring this question will lead us to define the Todd class as the correction term, and to introduce a topological version of the Grothendieck-Riemann-Roch theorem.

Additional advertisement for topologists: This talk will open the doors to a number of important future topics, including the e-invariant and the J-homomorphism.

Additional advertisement for number theorists: Bernoulli numbers will appear (multiple times) in the talk.

Title: Adams operations

Abstract: In this talk, we will explore cohomology operations on K-theory, called the Adams operations, that arise from exterior power constructions. We will discuss some of their properties and see how Adams and Atiyah used them to give a simple solution to the Hopf invariant one problem, which has connections to parallelizability of spheres as well as algebraic structures on spheres and Euclidean spaces.

Title: Holomorphic Lie categories and monoidal C-sheaves

Abstract: In this talk I will define a category internal to complex manifolds which I call holomorphic Lie categories. Given a holomorphic Lie category, there is a category of complex sheaves associated to it which can be used to construct various algebras. I will use holomorphic Lie categories to construct a known class of noncommutative algebras called the Feigin-Odesskii algebras. This talk is based on work from my PhD thesis.

Title: Bridging Geometry and Algebra: The Serre-Swan Theorem

Abstract: The Serre-Swan theorem provides a profound and foundational dictionary between the geometric world of vector bundles and the purely algebraic realm of projective modules. In this lecture, we will explore how local geometric triviality translates into global algebraic structure, ultimately paving the way for algebraic K-theory.

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