Fast matrix multiplication

Yuval Filmus
Institute for Advanced Study; Member, School of Mathematics
February 25, 2014
How many arithmetic operations does it take to multiply two square matrices? This question has captured the imagination of computer scientists ever since Strassen showed in 1969 that \(O(n^{2.81})\) operations suffice. We survey the classical theory of fast matrix multiplication, starting with Strassen's algorithm and culminating with the state-of-the-art Coppersmith-Winograd algorithm and its recent improvements. We also describe Coppersmith's \(O(n^2 log^2 n)\) algorithm for multiplying rectangular matrices, used by Ryan Williams in his separation of ACC and NEXP.

Nearly time-periodic water waves

Jon Wilkening
University of California, Berkeley
February 25, 2014
We compute new families of time-periodic and quasi-periodic solutions of the free-surface Euler equations involving extreme standing waves and collisions of traveling waves of various types. A Floquet analysis shows that many of the new solutions are linearly stable to harmonic perturbations. Evolving such perturbations (nonlinearly) over tens of thousands of cycles suggests that the solutions remain nearly time-periodic forever.

Almost Global Solutions for Incompressible Elasticity in 2D

Zhen Lei
Fudan University; Member, School of Mathematics
February 25, 2014
The systems of elasticity in 2D are wave-type equations with two different propagation speeds at a linear level. Due to the incompressibility, the system is nonlocal and is not Lorentz invariant, but it is inherently linear degenerate. We talk about its long time existence of classical solutions.

Paul Klee, Wilhelm Hausenstein, and the "Problem of Style"

Charles Mark Haxthausen
Robert Sterling Clark Professor of Art History, Williams College
February 25, 2014
In the art of Paul Klee (1879-1940), we find an unmatched pluralism of styles--figurative as well as abstract, geometric as well as biomorphic, linear as well as painterly, severe styles alongside more fluid ones, often within the production of a single year.

Quantum Hall Phases, plasma analogy and incompressibility estimates

Jakob Yngvason
University of Vienna
February 27, 2014
When a 2D many-particle system with a repulsive interaction is subject to a sufficiently strong magnetic field, that can also be produced by rapid rotation, strongly correlated many-body states in the lowest Landau level LLL may emerge. In the talk conditions for the ground state to include the Laughlin state as a fact or will be presented, together with an analysis of the particle density in such states. This is joint work with Sylvia Serfaty and Nicolas Rougerie.

Many-body Anderson localization

David Huse
Princeton University
February 28, 2014
I will review some aspects of many-body Anderson localization. Many-body localized systems have a type of integrable Hamiltonian, with an extensive set of operators that are localized in real-space that each commute with the Hamiltonian. The eigenstates of the Hamiltonian within the localized phase may exhibit symmetry-breaking long-range order (or topological order), even in low dimensions and at high "temperature", where such order can not occur at thermal equilibrium.