UW Math: Milliman Lectures Archive

Milliman Lecture Archives

2008-2009 Milliman Lectures

Andrei Okounkov

Professor of Mathematics, Princeton University

A series of three lectures entitled Applied Noncommutative Geometry

Abstract: Noncommutative projective geometry deals with noncommutative algebras that, in a certain technical sense, are just as good as polynomials.

Their study was begun by M. Artin and W. Schelter over twenty years ago and grew into a rich and fascinating subject since. In these lectures, I will explain some applications of this theory to such basic mathematical objects as linear difference equations. These difference equations are of direct probabilistic interest (which will be also explained in the lectures) and thus noncommutative algebraic surfaces shed light on the behavior of random stepped surfaces.

View a streaming video of Lecture I using the free RealPlayer.

View a streaming video of Lecture II using the free RealPlayer.

View a streaming video of Lecture III using the free RealPlayer.

2007-2008 Milliman Lectures

Terence Tao

Professor of Mathematics, UCLA

A series of three lectures entitled Recent Developments in Arithmetic Combinatorics

Lecture I: Additive Combinatorics and the Primes
Given a set A of N integers, how many additive patterns (e.g. arithmetic progressions of length three) does A have? The answer depends of course on the nature of A. At one extreme are sets with very strong additive structure, such as the arithmetic progression {1,..., N}, which tend to have many additive patterns. At the other extreme are "random" sets A, which tend to have relatively few additive patterns. Understanding and classifying these two extremes, and the extent to which an arbitrary set lies between these extremes, is a major topic of additive combinatorics today. Recent progress on these questions has led to applications in number theory, particularly regarding the question of detecting additive patterns in the primes; in particular, I will discuss my result with Ben Green that the primes contain arbitrarily long arithmetic progressions.
View a streaming video of Lecture I using the free RealPlayer.

Lecture II: Additive Combinatorics and Random Matrices
The theory of random continuous matrices (such as the gaussian unitary ensemble or GUE), and in particular the study of their eigenvalues, is an intensively studied and well-understood subject. In contrast, the theory of random discrete matrices, such as the Bernoulli ensemble in which each entry of the matrix equals +1 and -1 with equal probability, is only just now being developed. Much of the recent progress relies on an understanding of the distribution of discrete random walks, and in particular on solving the inverse Littlewood-Offord problem. This in turn requires the use of tools from additive combinatorics, such as the geometry of multidimensional arithmetic progressions, and Freiman's inverse theorem. I will survey these developments, which include joint work of myself with Van Vu.
View a streaming video of Lecture II using the free RealPlayer.

Lecture III: Sum-product Estimates, Expanders, and Exponential Sums
Consider a finite set A of elements in a ring or field. This set might be "almost closed under addition", which for instance occurs when A is an arithmetic progression such as {1,..., N}. Or it may be "almost closed under multiplication", which for instance occurs when A is a geometric progression such as {1, a,..., a^N}. But it is difficult for A to be almost closed under addition and multiplication simultaneously, unless it is very close to a subring or subfield. The emerging field of sum-product estimates in arithmetic combinatorics seeks to establish strong inequalities to quantify this phenomenon. Recent breakthroughs in this area by Bourgain and coauthors, based primarily on Fourier analysis and additive combinatorial tools, have yielded new exponential sum and sieve estimates in number theory, new randomness extractors in computer science, and new constructions of expander graphs in combinatorics. This lecture will be a survey of several of these developments.
View a streaming video of Lecture III using the free RealPlayer.

2006-2007 Milliman Lectures

Richard Hamilton

Professor of Mathematics, Columbia University

Lecture I: Geometrization of Three-Dimensional Spaces
View a streaming video of Lecture I using the free RealPlayer.

Lecture II: The Ricci Flow on Kahler Manifolds
View a streaming video of Lecture II using the free RealPlayer.

Lecture III: Estimates for the Ricci Flow
View a streaming video of Lecture III using the free RealPlayer.

2005-2006 Milliman Lectures

Timothy Gowers

Cambridge University (Rouse Ball Professor of Mathematics)

Lecture I: Some Theorems and Open Problems in Arithmetic Combinatorics
Arithmetic combinatorics is the name that has been given to a thriving new area of mathematics, or, to be more accurate, a new and very interesting fusion of old areas. I shall give an overview of the area and try to explain what combinatorics, harmonic analysis, and number theory have to offer each other.

Lecture II: Discrete Fourier Analysis: Its Uses and Limitations
This lecture will be about one of the main techniques in arithmetic combinatorics: the use of the discrete Fourier transform. I shall outline how it can be used to prove two central theorems in the subject, namely Roth's theorem on arithmetic progressions, and Freiman's theorem on the structure of sets with small sumsets. Both these theorems result in major open problems, and I shall try to explain their difficulty by highlighting what it is that we do not yet understand about Fourier analysis.

Lecture III: Discrete Fourier Analysis: Its Uses and Limitations
In this lecture I shall try to give some idea about the proofs of two celebrated results about arithmetic progressions: Szemerédi's theorem, which states that every dense set of integers contains arbitrarily long arithmetic progressions, and the Green-Tao theorem, which tells us that the same is true of the set of primes. Both these results involve situations where conventional Fourier analysis is inadequate, but recently discovered "polynomial" generalizations can be used instead. Much remains to be understood: if time permits, I shall speculate about what arithmetic combinatorics may look like in twenty years' time.

2004-2005 Milliman Lectures

Luis Caffarelli

University of Texas at Austin (Sid Richardson Chair holder, Professor in Mathematics) and the Texas Institute for Computations and Applied Mathematics

A series of three lectures entitled Free Boundary Problems of Obstacle Type

Abstract: Free boundary problems of obstacle type became of interest during the development of the theory of Variational Inequalities in the late 1960s, and applications surfaced in many areas of applied math, probability and geometry.

From the variational point of view, the problem consists of minimizing some variational integral , among those configurations staying above a given graph. From the probabilistic point of view, it concerns an optimal stopping time problem that gives rise to a Hamilton-Jacobi-Bellman equation.

Typical examples of the variational integrals involve those giving rise to second order linear or non linear equations (Laplacian, minimal surfaces, p-Laplacian), fourth order problems like the bi-Laplacian (clamped plates), or the fractional Laplacian for instance for boundary control problems or optimal stopping times for Levi processes. We will discuss matters of regularity, stability, and the geometric properties of the contact set and its free boundary.

Lecture I: We will discuss what is a free boundary problem, in particular one of obstacle type, describe different areas, from geometry to fluid dynamics to probability in which they appear, and what type of information we seek.

Lecture II: We will give a more detailed discussion of regularity and geometric behavior of solutions to these types of problems, as well as its interpretations, in particular in connection with recent work on non-local variational inequalities.

Lecture III: We will discuss in greater detail the main techniques to tackle this type of problem, in particular "convexity" properties of solutions, and the role of monotonicity formulas.

2003-2004 Milliman Lectures

Alain Connes

Collège de France, Institut des Hautes Études Scientifiques

A series of three lectures entitled Non-Commutative Geometry

Abstract: Non-commutative geometry has its roots in the following theorem of Gelfand: if X is a (locally) compact Hausdorff space and C(X) is the C^*-algebra of continuous complex-valued functions on X, then X may be recovered as the space of maximal ideals of C(X). More precisely, the category of commutative C^*-algebras is equivalent to the category of locally compact Hausdorff spaces.

Gelfand's Theorem allows one to transport the usual geometric notions to corresponding algebraic properties and features of the ring C(X). For example, complex vector bundles on X correspond to finitely generated projective modules over C(X), and the topological K-theory of X is isomorphic to the algebraic K-theory of C(X).

The basic idea behind non-commutative geometry is allow C^*-algebras that are not necessarily commutative to play the role of "functions" on spaces that are not necessarily Hausdorff.

Among the naturally occurring spaces with obvious geometric meaning that fail to be Hausdorff are:

the space of leaves of a foliation
the space of irreducible representations of a discrete group
the space of Penrose tilings of the plane
the Brouillon zone in the quantum Hall effect
phase space in quantum mechanics
space-time
the space of Q-lattices in Rⁿ

Far-reaching non-commutative extensions of classical geometric concepts such as measure theory, topology, differential geometry, and Riemannian geometry have been obtained with varying degrees of perfection. These extensions allow the usual geometric notions to be used to analyze spaces such as those above.

These three talks will provide an introduction to, and overview of, non-commutative geometry. Connections with other branches of mathematics and physics will be emphasized.

2002-2003 Milliman Lectures

János Kollár

Princeton University

Lecture I: What is the Biggest Multiplicity of a Root of a Degree d Polynomial?

Lecture II: What are the Simplest Algebraic Varieties?

Lecture III: Rational Connected Varieties over Finite Fields

2001-2002 Milliman Lectures

Peter Sarnak

Institute for Advanced Study, Princeton and New York University

A series of three lectures entitled Familes of L-functions and Applications

Abstract: L-functions, starting from Riemann's zeta function and continuing to the modern automorphic L-function have played a central role in number theory. These lectures will focus on recent developments in the analytic theory of such functions. A key technique which is at the heart of these advances is based on the formation and analysis of families of L-functions. There are numerous applications of these developments to number theory, combinatorics and mathematical physics. We will describe two in detail. One is the solution of Hilbert's eleventh problem which asks about the representability of integers in a number field by an integral quadratic form. The other is to eigenfunctions on an arithmetic surface and in particular problems in quantum chaos.

2000-2001 Milliman Lectures

Charles Fefferman

Princeton University

A series of three lectures entitled Unsolved Problems of Fluid Dynamics

Abstract: The talks will state some of the main unsolved problems on the Euler and Navier-Stokes equations for incompressible fluids, and sketch the proofs of some of the main results known so far on these and related equations. I hope to get through the main ideas in the proofs of the Beale-Kato-Majda theorem and results of Constantin, Majda and me on breakdown of Euler solutions, the work of Sheffer, Caffarelli-Kohn-Nirenberg and F. H. Lin on Navier-Stokes, and recent results by D. Cordoba and me on breakdown scenarios. Also, I hope to state precisely some problems arising from Kolmogorov's ideas on turbulence.

1999-2000 Milliman Lectures

Peter Shor

AT&T Research

Lecture I: Quantum Algorithms
Quantum computers are hypothetical devices that 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 ones 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 show how Fourier transforms lets quantum computers find periodicity in situations classical computers cannot. Finally, I show how large integers can be factored by finding the periodicity of an exponentially long sequence.

Lecture II: Quantum Error Correction
One of the main difficulties in building quantum computers is in manipulating coherent quantum states without introducing errors or losing coherence. This problem can be alleviated by the use of quantum error correcting codes. Until these codes were discovered, it was widely believed that quantum computers could not be made resistant to errors - the argument was that to detect an error, a quantum computer's state would have to be measured, which by the laws of quantum mechanics would perturb the computer's state and thus disrupt the computation. Quantum error correcting codes get around this by encoding the data so that errors can be measured without disturbing the computational state. I will explain how these codes work, and briefly sketch how they can be used to make quantum computes extremely reliable via software if they can be built with only moderately reliable hardware.

Lecture III: Quantum Information Theory
One of the most remarkable papers of this century was the 1948 paper in which Shannon laid out the fundamentals of information theory, giving formulae for the compressibility of data and for the information-carrying capacity of a channel. Only in the last ten years, however, have researchers started to investigate what happens to the theorems of information theory in a quantum mechanical setting. We will define quantum channels, and present what is known about the analogs of Shannon's theorems in the quantum setting. Aside from knowledge of basic quantum mechanics (the necessary part of which I plan to cover in the first lecture), these lectures should be independent.

1998-1999 Milliman Lectures

Cliff Taubes

Harvard University

A series of three lectures titled New Geometry in Four Dimensions

Abstract: The more we learn about the classification of smooth (even simply connected) four dimensional manifolds, the less we understand; to the point where today, we have no viable conjectures. My lectures will focus first on how we got to this amusing state (the Seiberg-Witten invariants helped) - here you will see lots of 4-dimensional manifolds which sit just across the boundary of what we understand. Then, I will discuss a particular approach for moving across the frontier, this being a curious four dimensional generalization of Morse theory in dimension 3 which mixes Riemannian, symplectic and holomorphic geometry.

1997-1998 Milliman Lectures

Robert MacPherson

Institute for Advanced Study

A series of three lectures entitled:

Counting Faces of Polyhedra

Fixed Points in Spaces of Lattices

Topology of Spaces with Torus Actions

1996-1997 Milliman Lectures

Ingrid Daubechies

Princeton University

Lecture I: Wavelets: An Overview, with Recent Applications
Wavelets have emerged in the last decade as a synthesis from many disciplines, ranging from pure mathematics (where forerunners were used to study singular integral operators) to electrical engineering (quadrature mirror filters), borrowing in passing from quantum physics, from geophysics and from computer aided design. The first part of the talk will present an overview of the ideas in wavelet theory, in particular wavelet bases. The second part of the talk will discuss some recent applications, ranging from loss-less as well as lossy image compression or speech analysis to nonlinear approximation methods.

Lecture II: Subdivision Schemes
Subdivision schemes are used in computer aided design for the construction of locally adaptable curves and surfaces. A "good" scheme can lead to surfaces or curves that look quite smooth. This talk will discuss mathematical techniques to study the regularity of these curves and surfaces. In the "equally spaced" case, this is all well understood. The corresponding techniques in the non-equally spaced case are under development now; the talk will end with a presentation of some work in progress on this topic.

Lecture III: Using Transfer Operators to Find the Regularity of Refinable Functions
A function $f(x)$ is refinable if it can be written as a linear combination of the dilated and translated copies $f(2x-n)$ of $f$ . Such functions come up naturally in the construction of wavelet bases and in equally spaced subdivision schemes. Many techniques exist to find the regularity of such functions; for most, the complexity of the computation depends crucially on the number of coefficients is the refinement equation. This talk will present a link with transfer operators for the doubling operator, used in an estimation method for the regularity that can handle even infinite numbers of coefficients.

1995-1996 Milliman Lectures

John Conway

Princeton University

A series of three lectures entitled On Games, Codes, and Balls

Games and Numbers: How adding up games helps to understand numbers.

Codes and Numbers: The mysterious arithmetic of lexicographic codes.

Spheres and Clusters: How to pack a big box with small balls.

1994-1995 Milliman Lectures

Gil Kalai

Hebrew University of Jerusalem

A series of three lectures entitled:

Lecture I: Combinatorics and Convexity

Lecture II: Simple Polytopes

Lecture III: The Combinatorics of the Simplex Algorithm

1993-1994 Milliman Lectures

Louis Nirenberg

NYU Courant

A series of three lectures entitled Variational Methods, the Maximum Principle and Related Topics

1992-1993 Milliman Lectures

Raoul Bott

Harvard University

A series of three lectures entitled On some Physics-Inspired Aspects of Geometry and Topology

Abstract: General relativity and the Quantum Theory have been two of the great theoretical driving forces of this century - not only of Physics but also of Mathematics. Relativity is by its very nature geometric and, dealing as it does with the very distant, it has natural links to topology. The Quantum theory on the other hand is traditionally more associated with questions in functional analysis. Still this theory has profound and rather surprising relations with topology and in these lectures I hope to explore some of these.

1991-1992 Milliman Lectures

Elias M. Stein

Princeton University

A series of three lectures entitled Oscillatory Integrals in Analysis

1990-1991 Milliman Lectures

Barry Mazur

Harvard University

A series of three lectures entitled In Search of the Monster

1989-1990 Milliman Lectures

Richard Stanley

Massachusetts Institute of Technology

A series of three lectures entitled Some Applications of Algebra to Combinatorics

Abstract: We will survey a number of applications of algebra to some problems in combinatorics dealing with extremal set theory, partially ordered sets, enumeration, and convex polytopes. No special knowledge of algebra or combinatorics will be necessary to understand these talks. We begin with teh simplest linear algebra and build up more and more sophisticated machinery as it is required. Some of the topics covered are the following:

the use of linear algebra to prove the classical Sperner theorem (which determines the largest collection A of subsets of an n-element set such that no member of A is a subset of another) and related results,
the use of finite group theory to prove some variations of the Sperner theorem, including the unimodality of the q-binomial coefficients,
a proof of a number-theoretic conjecture of Erdos and Moser using the representation theory of the Lie algebra sl(2,C),
a proof of the Upper Bound Conjecture for Spheres using the theory of Cohen-Macaulay rings, and
a proof of McMullen's g-conjecture for convex polytopes based on the theory of toric variaties and the hard Lefschetz theorem.