Clay Mathematics Institute

Dedicated to increasing and disseminating mathematical knowledge

2004 Academy Colloquium Series

Evolutionary Game Dynamics

Martin Nowak (Harvard University)

Darwinian evolution requires populations of reproducing individuals. Evolutionary change occurs by mutation and selection. Game dynamics are relevant whenever the success of one individual depends on its own strategy and the strategy of others in the population. Game dynamics are a generic description of evolution. I will discuss the evolution of cooperation and fairness. Lecture notes

The Number Theory of the Upper Half Plane

Kiran Kedlaya (MIT)

The complex upper half plane is a perfectly good place to do geometry; it gives a natural model of the hyperbolic plane. But it also has surprisingly strong connections to number theory, via the theory of modular forms. We will explain some of the geometry that gives rise to these beautiful objects, and briefly highlight two connections to elliptic curves: the "moduli space of elliptic curves" and the "modularity of elliptic curves" (the latter being key to the proof of Fermat's Last Theorem). Lecture notes


How Many Integral Solutions to Polynomial Equations

Akshay Venkatesh (CMI/MIT)

Let f(x,y) be a polynomial in two variables. How many solutions could the equation f(x,y) = 0 have in integers, if we restrict x and y to lie in a large box? This question is very far from understood. I will discuss various methods of getting information, and try emphasize the idea that solutions repel each other. Lecture notes

Physics and Math of Crystal Melting

Cumrun Vafa (Harvard University)

There has been a deep connection discovered recently between string theory, classical crystals and quantum gravity. In this talk aspects of crystal melting and its mathematical properties are discussed. The interpretation of cystal melting as a "quantum gravitational foam" is also presented.

The Mandelbrot Set, the Farey Tree, and the Fibonacci Sequence

Robert Devaney (Boston University)

In this lecture we describe several folk theorems concerning the Mandelbrot set. While this set is extremely complicated from a geometric point of view, we will show that, as long as you know how to add and how to count, you can understand this geometry completely. We will encounter many famous mathematical objects in the Mandelbrot set, like the Farey tree and the Fibonacci sequence. And we will find many soon-to-be-famous objects as well, like the "Devaney" sequence. There might even be a joke or two in the talk. Lecture notes

The largest positive integer you have ever contemplated

Timothy Gowers (Cambridge University, UK)

Consider the following game. Two players are given ten minutes and several sheets of paper, and each must use the time to define a positive integer. No cheating allowed, so e.g. "one plus the other person's number" is against the rules. I will explain to you how to win (unless of course your opponent has also been to the talk).

Quantum Computing

Peter Shor (MIT)

Quantum computers are hypothetical devices which use the principles of quantum mechanics to perform computations. For some difficult computational problems, including the cryptographically important problems of prime factorization and finding discrete logarithms, the best algorithms known for classical computers are exponentially slower than the algorithms known for quantum computers. Although they have not yet been built, quantum computers do not appear to violate any fundamental principles of physics. I will explain how quantum mechanics provides this extra computational power, and outline the factoring algorithm. One of the main difficulties in building quantum computers is in manipulating quantum states without introducing errors or losing coherence.

This problem can be alleviated by the use of quantum error correcting codes; if a quantum computer can be built with only moderately reliable hardware, software can be used to make it extremely reliable. Lecture notes

Distributions - Generalized Functions

András Vasy (CMI/MIT)

One of the achievements of 19th century analysis was to carefully examine notions such as continuity and differentiability, and to show that there are many continuous functions that are not differentiable, i.e. their difference quotients do not have limits. However, this does not prevent one from extending differentiation to continuous functions if one is willing to generalize the class in which the result (i.e. the derivative) will lie. As I will explain, a suitable notion of generalized functions is that of distributions. With this generalization, every continuous function, indeed every distribution, can be differentiated as many times as desired, with the result being yet another distribution. Of course, there is a downside: one has to give up pointwise multiplication! I will also give various examples, including that of the Dirac delta `function', and explain some applications.  Lecture notes

The Mathematics of Google

Jim Carlson (CMI)

Two Stanford Graduate Students, Sergei Brin and Larry Page developed a way of searching the world wide web that was far more effective than the methods used by other search engines. Thus Google was born. Their method was based on an elegant piece of mathematics which we will explain.  Lecture notes