A series of three lectures entitled Aspects of Conformal Invariant Randomness
Lecture I: The Global Picture
In this lecture, I will try to give an overview of some of the main ideas and results in the general area of two-dimensional
random models - including the discussion of conformal invariance of discrete models, Schramm's SLE processes and their
generalizations, planar maps, and some open questions.
Lecture II: Introduction to the Gaussian Free Field
In this lecture, I will give an introduction to the Gaussian Free Field, its properties and how it can be related to
SLE processes, and Conformal Loop Ensembles.
Lecture III: Conformal Loop Ensembles
In this third lecture, I will focus more on some properties of the Conformal Loop Ensembles, including their precise
relation to SLE and Brownian Loops.
Lecture I:
Lecture II:
Lecture III:
2010-2011 Milliman Lectures
Richard Taylor
Herchel Smith Professor of Mathematics, Harvard University
Lecture I: Reciprocity Laws and Density Theorems
If one fixes a polynomial (or a system of polynomials) in one or more variables, one can ask how the number of solutions
modulo a prime number p varies with the prime p. Reciprocity laws give a formula involving completely different areas of
mathematics (discrete subgroups of Lie groups). Density theorems give statistical information on how the number of
solutions varies with p.
In the first lecture I will present a leisurely historical introduction to reciprocity laws and density theorems, starting
with Gauss' law of quadratic reciprocity and finishing with the Sato-Tate conjecture. This lecture will also sketch Serre's
approach to deducing density theorems from reciprocity laws.
View video of the lecture here.
Lecture II: Galois Representations
In the second lecture I will talk about a more general framework for discussing reciprocity laws. I will introduce Galois
representations, L-functions and automorphic forms, and discuss their relevance in number theory. I will describe Langlands'
very general reciprocity conjecture and the Fontaine-Mazur conjecture.
View video of the lecture here.
Lecture III: Automorphy
I will describe what we currently know about general reciprocity theorems. I will give some indication of the techniques we
have and what I see as the main stumbling blocks to further progress. View video of the lecture here.
2009-2010 Milliman Lectures
Nick Trefethen
Professor of Numerical Analysis, Oxford University
Lecture I: Four Bugs on a Rectangle (and the Biggest Numbers You've Ever Seen)
Suppose four bugs at the corners of a 2 x 1 rectangle start chasing each other at speed 1. Bug 1 chases bug 2, bug 2
chases bug 3, and so on. What happens next will amaze you -- or at least it amazed me! As we follow the bugs to their
eventual collision at the center, we will encounter the biggest numbers you've probably ever seen and confront some
fundamental questions about what it means to try to understand our world through mathematics.
View video of the lecture here.
Lecture II: Approximation Theory in One Hour
My first love was approximation theory, and
during my current sabbatical I have returned to this subject to write a book
called Approximation Theory and Approximation Practice. This talk will be a
fast tour of about twenty of the main theorems of this book (with a handout),
each illustrated by chebfun computations on the computer. We will see vividly
how approximation ideas are at the heart of all kinds of practical computational
problems including quadrature, rootfinding, and solution of differential
equations. View video of the lecture here.
A handout from this lecture.
Lecture III: You Can't Beat Gibbs and Runge
Suppose you sample an analytic function f
in n equispaced points in [-1,1]. Can you use this data to approximate
f in a manner that converges exponentially as n
→ ∞? Many algorithms have been proposed that
are effective for moderate values of n, but we prove that they must all
fail in the limit n → ∞: exponential
convergence implies exponential instability. This is joint work with Rodrigo
Platte and Arno Kuijlaars. View video of the lecture here.
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.
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,..., aN}. 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.
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
Collčge 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.
1988-1989 Milliman Lectures
Michael Artin
Massachusetts Institute of Technology
A series of three lectures entitled Deformation of Polynomial Rings
1987-1988 Milliman Lectures
Peter Lax
New York University Courant Institute of Mathematical Sciences
A series of three lectures entitled Wave Propagation
1986-1987 Milliman Lectures
Stephen Smale
A series of three lectures entitled Structure and Complexity Theory of
Algorithms for Nonlinear Equations
1985-1986 Milliman Lectures
Kyosi Itô
A series of four lectures entitled Diffusions in Infinite Dimensional Spaces
The lectures were in conjunction with the Workshop in Markov Processes which
took place at the University of Washington from June 23 to July 2, 1986.
Hillel Furstenberg
Hebrew University
A series of three lectures entitled Ergodic Theory and Combinatorics
