Joint Mathematics Meetings

(For updated locations, see the timetable; All locations are subject to change)

AMS Short Courses

Registration

Two Short Course proposals have been selected for presentation just before the Joint Mathematics Meetings begin. These Short Courses will take place on January 2 and 3, 2012 (Monday and Tuesday), Sheraton Boston. Topics will be Random Fields and Random Geometry and Computing with Elliptic Curves using Sage.

The cost to participate is the same for both courses. Advance registration fees are: member of the AMS or MAA, US$102; nonmembers are US$145; students, unemployed, or emeritus are US$50. These fees are in effect until December 15. If you choose to register at the meeting, the fees are US$136 for members of the AMS or MAA, US$175 for nonmembers, and US$71 for students, unemployed, or emeritus.

Advance registration will begin on September 1, 2011. Onsite registration will take place on Monday, January 2, 2012, 8:00 a.m. - noon, Back Bay Ballroom D, Sheraton.

Random Fields and Random Geometry

Location: Back Bay Ballroom A, 2nd Floor, Sheraton
Organizers: Robert Adler, Technion - Israel institute of Technology robert@ee.technion.ac.il, Jonathan Taylor, Stanford University jonathan.taylor@stanford.edu

Introduction

The main theme of the lectures will be to describe the geometric aspects of smooth Gaussian and related random processes over parameter spaces of dimension 2 and higher. The study of such processes -known as random fields - has seen significant theoretical advances over the past decade. While these developments have occurred primarily with the framework of Probability and Stochastic Geometry, the new results and techniques are feeding ideas back into the mathematical world of topology and are finding new applications in areas which have traditionally used random fields as stochastic models. Among these are:

  1. Physical oceanography:  Here the random field is typically water pressure or surface temperature, and interest lies in spatio-temporal pattern analysis.
  2. Cosmology:  This includes the analysis of COBE and WMAP microwave data as well as galactic density data. The main applications of random field theory here lie in the analysis of patterns in the data in order to differentiate between competing cosmological theories.
  3. Theoretical physics:  In quantum chaos random planar waves replace deterministic (but unobtainable) solutions of Schrödinger equations, with primary interest centered on the behavior of nodal lines. In spin glasses random fields are a basic tool in modeling low energy states and can be used to study replica symmetry breaking.
  4. Medical imaging:  This application is one of the most developed. Here random field geometry is used as a statistical tool for establishing (or refuting) hypotheses of relations between physical or intellectual activity and brain morphology.

The lectures and tutorials of this short course will be designed to give a broad introduction to the modern theory of smooth random fields as well as describing some of its more exciting and important applications.

Lectures

We have the following confirmed speakers with preliminary titles and abstracts for their talks.

Gaussian Fields and Kac-Rice formulae
Robert Adler, Technion
In the first half of his lecture Adler will give a general introduction to the modern theory of Gaussian and Gaussian related random fields, describing their construction and basic sample path properties.  In the second half, he will discuss various versions of the Kac-Rice formula, which allows the computation of moments of many point set random variables generated by (not necessarily) Gaussian fields. These formulae will appear, in one form or another, in most of the other lectures.

Course Materials

Lecture notes can be found here.

The Gaussian kinematic formula
Jonathan Taylor, Stanford
Taylor will review some concepts from integral geometry, focusing on Kinematic formulae and tube formulae on Euclidean space and the sphere. The key quantities involved are integral geometric invariants known as curvature measures. These quantities, along with some Gaussian analogues, appear in the Gaussian Kinematic Formula (GKF) which describes average integral geometric quantities of paths of smooth Gaussian random fields.

Course Materials

Lecture notes can be found here.

Random matrices and Gaussian analytic functions
Balint Virag, Toronto
Virag will discuss Gaussian analytic functions and their zero sets, as well as their relationship to random matrix eigenvalues. This will include describing phenomena such as the local repulsion of zeros, laws of large numbers and central limit theorems. Probabilistic aspects of these processes will be emphasized.

Gaussian models in fMRI image analysis
Jonathan Taylor, Stanford
In this lecture Taylor will review the application of some of the results of smooth random fields to medical image analysis, particularly functional MRI (fMRI). The main mathematical tool is the expected Euler characteristic heuristic, which links a geometric quantity (the Euler characteristic) to the distribution of the maximum of a smooth random field. Time permitting, other applications, such as approximations to the size of the largest component of an excursion set, will also be considered. 

Course Materials

The slides can be found at http://webee.technion.ac.il/people/adler/jonathan2-boston.pdf.

Random fields in physics
Mark Dennis, Bristol
Dennis will describe uses of random field theory in areas of physics including quantum chaos, statistical optics, cosmology and quantum turbulence. In all of these cases, the topology of the nodes (preimages of zero) play an important role in understanding the physics behind the random geometry. Particular focus will be placed on open problems in statistical topology, such as the distribution of knot types in zeros of Gaussian random functions from 3-space to the complex plane.

Course Materials

Lecture notes can be found here.

Random metrics
Dmitry Jakobson, Montreal
Jakobson will discuss two models of random Riemannian metrics on compact manifolds: random metrics in a fixed conformal class (with applications to the study of scalar curvature) and, more generally, all metrics on a given manifold.  From a technical point of view, this amounts to introducing Gaussian measures on appropriate spaces of metrics. He will describe connections to conformal field theory, quantum gravity, random wave model in quantum chaos, and the theory of Gaussian random fields on manifolds.

Course Materials

Lecture notes can be found here.

The slides can be found at http://webee.technion.ac.il/people/adler/dima-boston.pdf.

Schedule

It is planned that each lecture will be 75 minutes long. There will be two lectures each morning and one following lunch. On the first day the remainder of the afternoon will be devoted to tutorial sessions aimed at getting hands on experience with the basic results of Gaussian random field theory. On the second day there will be discussion groups on each of the application areas treated in the main lectures.

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Computing with Elliptic Curves Using Sage

Location: Back Bay Ballroom B, 2nd Floor, Sheraton
Organizers: William Stein, University of Washington wstein@gmail.com
Pdf version here

Course Materials

A current snapshot of the Sage worksheets, slides, etc. for the short course on elliptic curves can be found at http://wiki.sagemath.org/jmm12?action=AttachFile&do=get&target=JMM2011-notes.pdf

Links to individual course materials such as Sage worksheets under Course Materials can be found at http://wiki.sagemath.org/jmm12 .

1 Introduction

This short course will explore computing with elliptic curves using the free open source mathematical software system Sage. Half of the lectures will be accessible to a general mathematical audience with little prior exposure to elliptic curves, and will provide a good way for mathematicians to learn about Sage in the context of strikingly beautiful mathematics.

diagram

An elliptic curve is a curve defined by a cubic equation of the form

y2 = x3 + Ax + B

in two variables x and y. The extent to which elliptic curves play a central role in both pure and applied modern number theory is astounding. Deep problems in number theory such as the congruent number problem--which integers are the area of a right triangle with rational side lengths?--translate naturally into questions about elliptic curves. Other questions, such as the famous unsolved Birch and Swinnerton-Dyer conjecture, propose startling relationships between algebra and analysis. Elliptic curves also play a starring role in Andrew Wiles's proof of Fermat's Last Theorem by arising naturally from any counterexample to the assertion. In a more applied direction, the abelian groups attached to elliptic curves over finite fields are extremely advantageous in the construction of public-key cryptosystems. In particular, elliptic curves are widely believed to provide good security with small key sizes, which is useful in applications--if we are going to print an encryption key on a postage stamp, it is helpful if the key is short!

Sage (see http://sagemath.org) is a free open-source mathematics software system licensed under the GNU Public License. It has extensive capabilities for computing with elliptic curves. Sage is built out of around 100 open-source packages and features a unifed interface. Sage can be used to study elementary and advanced, pure and applied mathematics. This includes a huge range of mathematics, including basic algebra, calculus, elementary to advanced number theory, cryptography, numerical computation, commutative algebra, group theory, combinatorics, graph theory, exact linear algebra and much more. It combines various software packages and seamlessly integrates their functionality into a common experience. It is well suited for education and research. The user interface is a notebook in a web browser or the command line. Using the notebook, Sage connects either locally to your own Sage installation or to a Sage server on the network. Inside the Sage notebook you can create embedded graphics, beautifully typeset mathematical expressions, add and delete input, and share your work across the network.

Topics covered in the course may include:

  1. How to program Sage using Python.
  2. How to use databases of elliptic curves with Sage.
  3. How to construct and work with public-key cryptosystems using elliptic curves over finite fields. How to count points on elliptic curves over finite fields.
  4. How to compute quantities appearing in the Birch and Swinnerton-Dyer conjecture: torsion points, Tamagawa numbers, Mordell-Weil groups, L-series, etc.
  5. How to compute p-adic L-series and p-adic regulators, and use them to bound Shafarevich-Tate groups

2 Lecture Series

2.1 Introduction to Python and Sage

Kiran Kedlaya, UC San Diego

Kedlaya will give an overview of how to use Sage using the Python programming language. No prior knowledge of Python or Sage will be assumed.

2.2 Computing with elliptic curves over finite fields using Sage

Ken Ribet, UC Berkeley

Ribet will explain how to use Sage to perform the sort of computations with elliptic curves over finite fields that are needed to follow along with a cryptography book and do exercises that involve long computations.

2.3 Computing with elliptic surfaces

Noam Elkies, Harvard University

Elkies's lectures will link some of the same arithmetical ideas that appear in the other lectures with other mathematical topics including algebraic geometry of surfaces, Euclidean and hyperbolic lattices, etc., that either are already or should be in Sage. This topic will also touch on elliptic curves of high rank, since every rank record is obtained by specialization from an elliptic surface (or possibly an elliptic curve over P^n for some n > 1).

2.4 Computing with Shafarevich-Tate Groups using Sage

Jared Weinstein, Boston University

The Shafarevich-Tate group is the most mysterious invariant attached to an elliptic curve. Weinstein will explain how to use Sage to compute information about Shafarevich-Tate groups. His talk may touch on work of Heegner, Kolvyagin, Kato, Schneider, Mazur and others.

2.5 Using Sage to explore the Birch and Swinnerton-Dyer conjecture

William Stein, University of Washington

Stein will introduce the Birch and Swinnerton-Dyer conjecture via the congruent number problem. He will then discuss how to use Sage to compute Mordell-Weil groups, values of L-functions, regulators, heights, and Tamagawa numbers. He will also talk about computing the Birch and Swinnerton-Dyer invariants for various tables of elliptic curves.

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