The 2022 Summer Tutorial Program

The summer tutorial program offers some interesting mathematics for mathematically minded Harvard people this summer. Each tutorial will run for roughly six weeks, meeting two or three times per week in the evenings (so as not to interfere with day time jobs). The tutorials will start at some point in June or early July and run for six weeks. The precise starting dates and meeting times will be arranged for the convenience of the participants once the tutorial rosters are set.

The format will be much like that of the term-time tutorials, with the tutorial leader lecturing in the first few meetings and students lecturing later on. Unlike the term-time tutorials, the summer tutorials have no official Harvard status: you will not receive either Harvard or concentration credit for them. Moreover, enrollment in the tutorial does not qualify you for any Harvard-related perks (such as a place to live if you are in Boston for the summer). However, the Math Department will pay each student participant a stipend of $700, and primary Mathematics concentrators can hand in the final paper from the tutorial for their junior 5-page paper requirement. (Students in either Harvard’s PRISE program or the Herchel Smith program can not receive a stipend.)

The topics and leaders of the two tutorials this summer are:

A description of each topic appears below. You can sign up for a tutorial by emailing Cliff Taubes. Please sign up by Sunday, May 15. You can sign up for both tutorials if you like; or sign up for both but list one as a preference over the other. If you have further questions about a given topic, contact the tutorial leader via the email. Please contact Cliff Taubes if you have questions about administration of the tutorials.


Topological K-theory: In this tutorial we’ll learn about topological K-theory. K-theory is an algebraic gadget which associates a ring to each compact topological space. The recipe which does this requires surprisingly little background: some basic topology, linear algebra, and abstract algebra. These ingredients will let us cook up lots of exciting mathematics: The final reward will be a simple proof of the classification of real normed division algebras, i.e. that the only possible dimensions are 1, 2, 4, and 8.

We’ll meet three times a week, twice to talk about sections of Alan Hatcher’s free book Vector Bundles and K-theory, and in the third meeting we’ll do problems together. If all participants are in the Boston area, we will meet in person. Otherwise, this will be a remote tutorial. There are many possible final paper ideas which can be tailored to your background.

SYLLABUS: We’ll start out by learning about topological vector bundles, and then we’ll use them to construct the K-groups of a space. We’ll study these K-groups and prove the famous Bott periodicity theorem, which (in the real case) says that the groups repeat infinitely with period 8. As a final punch line we’ll prove Adams’ theorem on the Hopf invariant and see some of its famous applications—including the nonexistence of division algebras after the octonions. If we have time and depending on background we can also talk about other topics, including characteristic classes and the image of the J-homomorphism.

PREREQUISITES: If you know about groups, rings, ideals, and kernels and images of homomorphisms, then you know enough algebra. If you have seen a definition of a compact metric space (or especially a general topological space) and of continuous maps between them, then you should know enough topology. If you have seen inner products, duals, and direct sums (and maybe tensor products) of vector spaces, then you definitely know enough linear algebra.

Taught by Keeley Hoek

Symplectic and contact geometry: A symplectic manifold is a smooth manifold equipped with a closed, non-degenerate 2-form. The study of such objects is symplectic geometry; it is a central branch of differential geometry and topology. The original motivation came from classical physics where the primary examples are phase spaces of mechanical systems. Symplectic geometry is still relevant in modern physics; for example, it plays critical roles in string theory and mirror symmetry.

Recent decades have seen enormous developments in the field, with many results proved and tools developed. A notable landmark in symplectic geometry is M. Gromov’s nonsqueezing theorem. Roughly, it states that a symplectic elephant (no matter how squishy) can’t squeeze through the eye of a needle. To prove this theorem, Gromov invented pseudo-holomorphic curves which are now indispensable tools in the field. Symplectic geometry has important interactions with other fields of mathematics such as complex geometry, integrable systems, low-dimensional topology, homotopy theory, gauge theory and dynamical systems.

SYLLABUS: The tutorial will provide an introduction to the world of symplectic manifolds and their odd dimensional counterpart, contact manifolds. We will start with examples from Hamiltonian mechanics to motivate the definition of the symplectic structures. After that, we will work on the linear algebra of symplectic vector spaces, these being the simplest examples of symplectic manifolds. With the linear notions understood, we will introduce the concept of a symplectic manifold, study simple examples and introduce some of the basic tools (the notion of a symplectomorphism and Darboux’s theorem). We will also introduce an important class of submanifolds, the Lagrangian submanifolds. At the end, we will talk about complex structures, almost complex structures and contact structures; and then give many examples.

PREREQUISITES: A solid knowledge of manifolds, cotangent bundles, vector fields and differential forms is needed (the material in Math 132). Physics background is not necessary.

Taught by Clair Dai and Jiakai Li.

(This tutorial will be taught remotely.)