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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. Each tutorial will run for six weeks, meeting twice per week in the evenings (so as not to interfere with day time jobs). The tutorials will start early in July and run to mid August. 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). 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.

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

A description of each topic is appended below. You can sign up for a tutorial by emailing me at [email protected]. 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 me if you have questions about the administration of the tutorials.

Yours,

Peter Kronheimer

Ramsey Theory (Thomas Barnet-Lamb, [email protected])

Ramsey theory is a branch of pure mathematics that seeks to prove results showing that under certain circumstances, it's impossible to arrange things, however hard you try, without certain kinds of structure popping up.

For example, suppose there are six people in a room at a party. For any two of them, they'd either met each other before this evening or not (no shades of gray!). Then, we can always either find three people who were all strangers to one another before this evening, or find three people all of whom knew each other before this evening. Maybe that's not so much of a big deal, but it's also the case that if there are `enough' people in the room, you'd always be able to either find 100 people that were mutual strangers before the party or 100 people all of whom knew each other.

Another result says that say that whenever you try to divide the natural numbers into three classes, you can't help but have at least one of the classes having arbitrarily long arithmetic progressions in it. Yet another (Hindman's theorem) tells us that if our opponent splits the natural numbers into a finite number of classes, then we will be able to choose one of the classes and find an infinite sequence a₁, a₂, ... of numbers such that all the finite sums of elements from our sequence belong to our chosen color class.

Some of these results are proved using ingenious and beautiful elementary arguments, while others will draw on deeper themes from modern mathematics, including some elementary point-set topology, and the theory of filters and ultra-filters (which we will develop in this course, and which provides an excuse to prove Arrow's theorem that 'democracy is impossible'). All of the proofs will be combinatorial in flavor. (It is sometimes said that if you're only going to learn one piece of combinatorics before you die, it should be Ramsey theory!) And many of the numbers that come out of Ramsey theory (for example, what it means to have `enough' people in the room example described above) are enormously, mind bogglingly large. So whatever else happens, you'll be able to Impress Your Friends with that.

Prerequisites: The majority of the course will be completely elementary, and the only prerequisite there will be some familiarity with the language of graph-theory. The latter part will require a little elementary point-set topology, in particular the notions of compactness and Hausdorffness. Math 23 or 25 together with some familiarity with point-set topology as covered in Math 131 will suffice.

Enumerative geometry (Dawei Chen, [email protected])

Enumerative geometry has been an active and attractive research subject in math for a long time. Many enumerative problems which originated in classical geometry can now be solved by ad hoc techniques in modern algebraic geometry. The purpose of this tutorial is to give an introduction to this fascinating subject. We will mostly focus on curves in low dimensional projective spaces. On the one hand, we do not need too much complicated theory to deal with them. On the other hand, those examples can still provide us some clue about how to play with enumerative problems. We hope that after this class, students will be familiar with some basic techniques in enumerative geometry and be able to pursue more advanced knowledge in this field. Some possible topics are:

A lot more topics are available and up to student interests. If you really appreciate the beauty of math, this tutorial is the right one for you!

Prerequisites: Familiarity with some ideas and techniques in Math 137 or equivalent would be helpful.

Elliptic curves (David Geraghty, [email protected] and Jeechul Woo, [email protected])

The theory of elliptic curves is a fundamental subject in number theory and algebraic geometry. Historically, interest in elliptic curves arose naturally from the study of elliptic integrals and the congruent number problem - finding a rational integer that is the area of a right triangle whose side lengths are rational numbers. In the modern language, an elliptic curve is a one dimensional smooth projective group variety but luckily for us they can be studied in a down to earth fashion. Indeed, any elliptic curve can be regarded as the solution set of an equation of the form y2=x3+ax+b. Elliptic curves play a central role in many areas of number theory, for example, the construction of abelian extensions of certain number fields (via Galois representations on torsion points of elliptic curve), the conjecture of Birch and Swinnerton-Dyer (aka the BSD conjecture - a Clay Math problem about the rank and L-series of elliptic curves) and Wiles' proof of Fermat's Last Theorem. In this tutorial we aim to introduce students to some of these advanced topics and to give an indication of the main applications of elliptic curves. However, our main focus will be on the basics of the theory. Our main reference will be the book of Silverman [S]. For the final projects, we will offer a range of topics varying in flavor from the more algebraic to the more analytic and varying in nature from the more concrete to the more theoretical. Students who take this tutorial will learn the basic algebraic, analytic and geometric tools used in the study of elliptic curves and will hopefully be inspired to study some more advanced topics later. We will also take time to introduce many topics which may not be familiar to undergraduate students such as algebraic curves, p-adic numbers and group cohomology.

Prerequisites: Math 123 or equivalent. Some knowledge of complex analysis (Math 113), algebraic curves (Math 137) and algebraic number theory (Math 129) would all be helpful, but not required.

Introduction to L-functions and Arithmetic (Stefan Patrikis, [email protected])

One of the deepest mysteries of number theory is the connection between the analytic behavior of L-functions and basic arithmetic phenomena. Non-vanishing properties of L-functions -- generalizations of the Riemann zeta function -- accounted for the two greatest achievements of 19th century number theory, the prime number theorem and Dirichlet's theorem on primes in arithmetic progressions. During the twentieth century, number theorists realized the power of systematically attaching L-functions to a wide variety of objects arising from algebra, analysis, and geometry, and today they are at the heart of many of the subject's deepest problems.

The great themes of the subject are already present in the study of the Riemann zeta function, the most basic example. Euler used the "Euler product," the factorization of the zeta function into an infinite product over primes, to give a new proof that there are infinitely many primes. Riemann proved that the zeta function could be meromorphically continued to the entire complex plane, and that it satisfied a functional equation; this led him to formulate a precise connection between zeroes of the zeta function and the distribution of the prime numbers, a very weak form of which is the classical prime number theorem. Riemann also related special values of the zeta function to the Bernoulli numbers, work that has very deep connections with class numbers of cyclotomic fields. The first goal of this tutorial will be to introduce these basic themes in the context of the classical L-functions of Riemann and Dirichlet.

The tutorial's second main goal will be to illustrate the diversity of interesting L-functions that number theorists study; this "bestiary" will draw examples from algebra (Dirichlet L-functions), analysis (L-functions of modular forms), and geometry (L-functions of elliptic curves). We will introduce those elements of the theory of modular forms necessary to understand the basic properties of their L-functions; in this case the functional equation is straightforward -- it is formally very similar to one proof for the zeta function, where it turns out there is a modular form lurking in the background! -- but the Euler product requires a serious innovation due to Hecke. With what time remains, we will introduce L-functions of elliptic curves and explain the (now proven) connection with L-functions of modular forms given by the Shimura-Taniyama conjecture. These examples are just the tip of the iceberg in the study of L-functions, but the fact that their study underlies Fermat's Last Theorem, the Riemann Hypothesis, and the Birch and Swinnerton-Dyer conjecture gives some sense of the subject's depth.

Prerequisites: This can be quite a lofty subject, but we will only need some complex analysis (Math 113) and basic algebra (Math 122). Some Galois theory (Math 123) is also recommended to help understand generalizations or more conceptual explanations of some of the things we study, but it will not be strictly necessary.

Archive: Old Summer Tutorials, since 2001