# The 2021 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 twice 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. Note that all of this will be remote this summer, via Zoom. 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 you can hand in your final paper from the tutorial for you 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 three tutorials this summer are:

**Morse theory**(Nathaniel Yang)**Quantum mechanics for the math minded**(Grant Barkley)**Modular forms**(Kai Xu)

A description of each topic appears below. You can sign up for a tutorial by emailing Cliff Taubes. Please sign up by Sunday, May 16. When you sign up, please list at least one other choice in case your preferred tutorial is either over-subscribed or under-subscribed. If you have further questions about any given topic, contact the tutorial leader via the email. Please contact Cliff Taubes if you have questions about the administration of the tutorials.

**Morse theory:** Rumor has it that topologists cannot tell the difference between coffee mugs and donuts but analysts can. In this course, we will bring these two groups of people together, by introducing Morse theory — a way of understanding the topology of smooth manifolds using analysis. We will give it a fairly comprehensive treatment, and then we will continue with some cool applications, including finding shortest paths, understanding basic Lie groups, and/or the proof of Poincaré conjecture in higher dimensions. Prerequisites include basic-level algebra, topology, and analysis. Knowledge of smooth manifolds is a plus, but I’ll include a crash course on the basics. This tutorial is especially beneficial for those who are familiar with algebraic topology and smooth manifolds, as we can see how the two fields interact with each other. (Taught by Nathaniel Yang)

**Quantum mechanics for the math minded:** An introduction to quantum theory from a mathematical perspective. On the physics side we’ll cover quantum computing, time-dependent and -independent quantum systems, wave-particle duality, harmonic oscillators, quantum algorithms, spin, particle statistics, and path integral formalism. To discuss these formally, we’ll introduce Hilbert spaces, L^{2} space, unitary and Hermitian operators, functional analysis, the spectral theorem, Fourier transforms, Lie algebras, spin groups, and more. The exact topics covered will depend on the background and interests of the participants. No physics background required! The only prerequisites are introductory courses in linear algebra and real analysis, and some exposure to differential equations. (Taught by Grant Barkley)

**Modular forms:** In mathematics, a modular form is a complex analytic function on the upper half-plane satisfying a certain kind of symmetry. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections with number theory. Modular forms appear in other areas as well, such as algebraic topology and string theory. This tutorial is essentially an elementary introduction to the theory of modular forms, we will start from the basic theory of modular forms and present some of its first applications. Then we will also venture into more advanced topics, which, depending on the interests of participants, might include number theory, topology and mathematical physics. Though this topic has many advanced applications, I wish to present everything at a very basic level. The only prerequisites needed will be a good understanding of algebra at the level of 122 and 123, as well as basic knowledge of complex analysis at the level of 113. Some knowledge of topology and geometry would be helpful but not required, as we will provide all the knowledge needed as we proceed. I believe that the algebraic aspects of modular forms, necessary to understand their role in applications, can be made accessible to students without previous backgrounds in algebraic number theory and algebraic geometry, and it is in fact a really good place to see the first examples of these complicated theories. So please join us if you are interested! (Taught by Kai Xu)