Harvard Mathematics Department : Summer Tutorials 2016

Archived Summer Tutorials:

Welcome Message

The summer tutorial program offers some interesting mathematics to those of you who will be in the Boston area during July and August. The tutorial will run for six weeks, meeting twice or three times per week in the evenings (so as not to interfere with day time jobs). The tutorial will start early in July or late in June, and run to mid August. The precise starting dates and meeting times will be arranged for the convenience of the participants once the tutorial roster is 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 presenting 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). However, the Math Department will pay each Harvard College student participant a stipend of approximately $700, and you can hand in your final paper from the tutorial for you junior paper requirement for the Math Concentration.
The topics and leaders of the tutorials this summer are:

Category Theory (Meng Guo)
Complex and Arithmetic Dynamics (Zijian Yao and Jake Marcinek)
Knots and Links (Yi Xie and Boyu Zhang)

A description of the topics are below. You can sign up for a tutorial only by emailing me at [email protected]. Please indicate in your email a first and second choice of the tutorials you are interested in taking. Please email me by noon, Friday, May 5. Places are filled on a first-come, first-served basis, but with priority being given to math concentrators. In the past, some tutorials have filled up quickly. If you have further questions about any given topic, contact the tutorial leaders via e-mail. Please contact me if you have questions about the administration of the tutorials.

Jacob Lurie

The topic and leader of tutorial this summer are:

Meng Guo: Category Theory:

Developed in the 40ties, category theory has developed rapidly and become a fundamental part in modern mathematics. It provides a convenient language to almost all of mathematics and has become a common toolbox to study analogies between different theories. While many of useful subjects lie outside of the standard undergraduate curriculum, this tutorial will provide a systematic study of them. We will introduce the subject by building on many familiar examples from algebra and topology. We will explain the meaning of categories, functors, natural transformations, Yoneda's lemma, limits, colimits, representability, and adjunctions. At every point, we will do enough examples in a category that the students are acquainted with. Once the basis of the subject has been throughly developed, we will venture into more advanced topics. Topics may include, but are not limited to:

Abelian categories and homological algebra
Model categories
Derived categories
Kan extensions
Higher categories

Students' backgrounds and interests will be the main parameters taken into consideration when choosing topics.

Prerequisites: Math 122 and Math 131. Contact: Meng Guo ([email protected])

(Zijian Yao and Jake Marcinek): Complex and Arithmetic Dynamics:

The subject of dynamics, or dynamical systems, is the study of an evolving system over time. This "time" is often seen as discrete steps (parameterized by Z) or a continuous flow (parameterized by R). We will be exploring the former (which can approximate the latter) for specific types of systems. Roughly speaking, given a map f from X to X, where X is a set, a topological space, or other geometric/arithmetic objects, one studies the limiting behavior of its iterations fⁿ, looking at periodic points, orbit closures, etc. In complex dynamics, the objects we are interested in are iterated self-maps of a geometric object respecting a complex structure and our main goal is to understand the fine line between structure and chaos. This dichotomy presents itself rather quickly. For instance, take a rational map f(z) = Q(z)/Q(z) from the Riemann sphere to itself. The Fatou set is the subset F(f) whose iterates tend towards infinity and the Julia set J(f) is its compliment. We have already arrived at fascinating fractals. Further, take f_c (z) = z² + c. The collection of c with 0 in J(f_c ) is the Mandelbrot set. Changing the base space allows us to use the inherent geometry of the surface to further investigate possible dynamical phenomena. Aside from its inherent beauty, complex dynamics uses results from and gives deep insights to many key ideas in different areas of mathematics including algebraic topology, complex analysis, hyperbolic geometry, and algebraic geometry. In this course, we will outline some preliminary connections to other areas of geometry. On the other hand, arithmetic dynamics deals with questions of a similar type over number fields. For example, we study iterations of a rational function f(x) where P and Q are polynomials with integer coefficients. This can be thought of a map from rational points of the Riemann sphere, and in general we could consider self-maps defined over the rational numbers of an elliptic curve (defined over Q) or other objects arising from algebraic geometry. Arithmetic dynamics is a great example of the marriage between dynamics and number theory - it uses techniques from various types of dynamical systems (in particular complex dynamics) and geometry to study number theoretic properties, and on the other hand, uses number theoretic techniques to give surprising dynamics statements. For example, a beautiful theorem of Northcott states that a rational function f(x) of degree greater than 1 on P¹(Q) has only finitely many periodic points. We will prove the statement in the course using elementary theory of height, which is useful in many other situations in arithmetics. The material are designed to be accessible to students who haven't seen these areas before, but also motivational for those who have seen some of the materials.

Prerequisites: linear algebra, elementary abstract algebra (group, rings, field), (very basic) topology. It would help (but not required) if you know the following: basic number theory, complex analysis.

Contact: Zijian Yao and Jake Marcinek

Geoffrey Smith: Coding Theory

Suppose we want to communicate with each other, but only have available a slow, noisy channel that may corrupt some of the information sent. By adding some redundant bits to the transmission, we want some way of detecting the presence of an error, or, better yet, correcting the error automatically. Coming up with good ways of encoding redundancy to allow error detection and correction is one of the central problems of coding theory. Though this subject is young, it has attracted large amounts of attention: mathematicians have come up with elaborate structures that produce codes (my second favorite type of code, the Goppa code, is derived from the study of divisors on algebraic curves, for instance); engineers have incorporated these codes into applications as diverse as communicating with Mars rovers and ensuring CDs with scratches still play.
In this course, we will discuss the systematic theory of error-correcting codes, and explore the menagerie of codes mathematicians have come up with, all while spending some time considering the practicalities of coding-how do we decode, how do we deal with other types of errors, and so on. These codes have connections to a number of beautiful mathematical subjects, including algebraic geometry, linear programming, sphere packings, and combinatorial designs, and depending on what interests the participants we can discuss these topics or some of many others. Moreover, the abundance of examples and applications in coding theory will provide several engaging topics for participant projects.
Contact: Geoffrey Smith.

Archive: Old Summer Tutorials, since 2001

Summer Tutorials:

Last update, 4/28/2016