fall 2025: index theory

Schedule and abstracts 

09/19 (Note unusual time/place: Friday 5 pm @ Harvard SC 232)

Title: The Atiyah-Singer Index Theorem: Statement and Examples

Speaker: Ollie Thakar

Abstract: The Atiyah-Singer Index Theorem is one of the most beautiful theorems in all of mathematics and one of the greatest cultural achievements of the last century. We will unpack the statement and ingredients of the index theorem and then discuss several examples of how to specialize or apply it, such as to the Gauss-Bonnet Theorem, the (Hirzebruch)-Riemann-Roch Theorem, the Hirzebruch Signature Theorem, Rokhlin’s Theorem, Lichnerowicz’s Theorem, and (if time) a theorem of Joyce on 8-manifolds with exceptional holonomy.

Notes: drive.google.com/file/d/119cOjyUiRM1Jb2SBB9VYUxazzCvAJOj3/view?usp=sharing 

09/24

Title: Pseudodifferential operators and the cobordism proof of the index theorem 

Speaker: Joye Chen

Abstract: I will present Atiyah and Singer's first proof of the index theorem which is based on a cobordism argument. There are two major steps I will focus on. The first is to show the analytic index is defined on topological K-theory: this involves constructing a sufficiently large class of operators containing differential operators (the Seeley algebra). The second is to show the analytic index is a cobordism invariant. We will show that a Dirac-type operator on a nullcobordant manifold has vanishing index. The key input comes from understanding which boundary values arise from solutions to Du = 0 over a bounding manifold.

Notes: drive.google.com/file/d/1uwodOAsVC2PHHUbolMTD6Mp73oRm7rNS/view?usp=sharing 

Reference: Palais, Richard S., et al. Seminar on Atiyah-Singer Index Theorem. (AM-57). Princeton University Press, 1965, Chapters I, XV-XVII. 

10/01

Title: The K-theory formulation and proof of the Atiyah-Singer index theorem

Speaker: Yonghwan Kim

Abstract: Although the Atiyah-Singer index theorem was originally proved using cobordism methods, the argument in the published proof is formulated in K-theory. This perspective is more amenable to generalizations to families and equivariant settings. In this talk, I will present the K-theory formulation of the Atiyah-Singer index theorem. I will explain how the topological and analytic indices are defined in this framework, and how the uniqueness of an index function from topological K-theory to Z satisfying covariant functoriality ensures that they agree. Since the previous talk covered much of the analytic input, this talk will focus more on the K-theoretic ideas, emphasizing a comparison between the cobordism based proof of the Hirzebruch-Riemann-Roch theorem and the functorial proof of the Grothendieck-Riemann-Roch theorem.

Notes: https://drive.google.com/file/d/1eoVeKXNVNlUKgrEryRonHtwk0Fwmp5jh/view?usp=sharing 

Reference: Landweber, K-theory and elliptic operators. 

10/08

Title: Index Density of Dirac Operators

Speaker: Maya Chande

Abstract: We present a local index theorem which implies the usual Atiyah-Singer Index Theorem for a Dirac operator D on a Clifford module. An approach suggested by Atiyah-Bott and McKean-Singer and carried out by Patodi and Gilkey is to extract relevant information from a small-time asymptotic expansion of the heat kernel of D^2. We provide a slightly refined version from which the A-hat genus and Chern character locally emerge more naturally. This relies on an approximation of D^2 as a formal harmonic oscillator and an application of an appropriate version of Mehler's formula.

Notes: https://drive.google.com/file/d/1TPdoW10b44nd14qKJIdclyUw7pUuzhCD/view?usp=drive_link 

Reference: Berline-Getzler-Verne, Heat Kernels and Dirac operators.

10/17 (Note unusual time/place: Friday 5 pm @ Harvard SC 232)

Title: The A-hat Genus from the Exponential Map

Speaker: Owen Brass

Abstract: Following the book of Berline, Getzler, and Vergne, we present another proof of the local index theorem for Dirac operators D on Clifford modules. This proof relies on small-time analysis of the asymptotic expansion of the scalar heat kernel on the principal frame bundle, which when averaged over the fibers yields the heat kernel of D^2. The A-hat genus emerges naturally from the averaging process via a general formula for the Jacobian of the exponential map on a Lie group.

Reference: Berline-Getzler-Verne, Heat Kernels and Dirac operators.

10/22

Title: The families index theorem

Speaker: Daniel Santiago

Abstract: We discuss the extension of the Atiyah-Singer index theorem to families of differential operators. We will follow the K-theory formulation and proof which is a direct generalization of the one given in a previous talk. The proof itself is mostly analogous with some technical differences. As an application, we discuss a result of Atiyah that the signature is not multiplicative in general fiber bundles.

Notes: https://drive.google.com/file/d/1O-OXwU4dxaEwNutyDQyI1NLq2zBhBWBN/view?usp=sharing 

Reference: Atiyah-Singer. "The index of elliptic operators IV.", Atiyah, "The signature of fiber bundles."

10/29

Title: The Real Atiyah-Singer Index Theorem

Speaker: Runze Yu 

Abstract: I will develop an index theory for operators between real vector bundles. The argument largely follows the usual proofs but requires a refined K-theory. I will also discuss examples of real operators such as skew-adjoint operators and Dirac operators on Riemann surfaces.

11/05

Title: The Atiyah-Patodi-Singer Index Theorem

Speaker: Yike He

Abstract: We will discuss the extension of the Atiyah–Singer index theorem to manifolds with boundary. The proof of this index formula combines heat kernel methods on closed manifolds and cylindrical ends. We will also present some applications, including the index formula for Dirac operators and the signature formula on manifolds with nontrivial boundary.

Reference: Atiyah-Patodi-Singer I. 

11/14 (Note unusual time/place: Friday 5 pm @ Harvard SC 232)

Title: Relative eta-invariants 

Speaker: Joye Chen 

Abstract: Last time, we defined the eta-invariant for a given smooth Riemannian manifold (Y, g) and self-adjoint first-order elliptic operator A. It is metric-dependent; however, one can extract topological invariants by taking a suitable difference of eta-invariants. We define these relative eta-invariants and prove metric dependence. I will also explain how to interpret the Chern-Simons invariant as a relative eta-invariant. Finally, if there is time, I will discuss a curious map from a variant of the integer homology cobordism group to algebraic K-theory and explain how relative eta-invariants fit into this story. 

Reference: Atiyah-Patodi-Singer II, Adams "On the groups J(X), IV", Jones-Westbury "Algebraic K-theory, homology spheres, and the eta-invariant". 

11/19

Title: Spectral flow and the eta invariant

Speaker: Owen Brass 

Abstract: In this talk, we will discuss the eta invariant for arbitrary elliptic self-adjoint operators, which intuitively acts as a measure of “spectral asymmetry." Following the third part of the seminal series of papers of Atiyah, Patodi, and Singer, we will show that it is well-defined for any self-adjoint elliptic operator on an odd-dimensional manifold. Furthermore, we will discuss the relationship between the eta invariant and spectral flow, and provide some applications.

11/26 (No meeting; happy Thanksgiving!)

12/03 (No meeting due to MIT Winter Social)

12/10

Title: Applications of index theory 

Speaker: Everyone! 

We will have an informal index-theory themed party at the usual time/place. Please bring your favorite application of index theory or index-theoretic computation to share (5-10 minutes). Ollie and Joye will bring miscellaneous baked goods. :)) 

Sign-up here: https://docs.google.com/spreadsheets/d/1vJkIWqANnFqymvWZ7F3LM4VLz8M1tPODMieZ3ueV358/edit?usp=sharing