The problem of solving linear equations in these matrices motivates a new notion of what it means for one graph to approximate another. This leads to the problem of approximating graphs by sparse graphs. Our algorithms for solving Laplacian linear equations will exploit surprisingly strong approximations of graphs by sparse graphs, and even by trees.

We will survey the roles that spectral graph theory, random matrix theory, graph sparsification, low-stretch spanning trees and local clustering algorithms play in the design of fast algorithms for solving Laplacian linear equations.

Daniel Alan Spielman received his B.A. in Mathematics and Computer Science from Yale in 1992, and his Ph.D in Applied Mathematics from M.I.T. in 1995. He spent a year as a NSF Mathematical Sciences Postdoc in the Computer Science Department at U.C. Berkeley, and then taught in the Applied Mathematics Department at M.I.T. until 2005. Since 2006, he has been a Professor at Yale University. He is presently the Henry Ford II Professor of Computer Science, Mathematics, and Applied Mathematics.

He has received many awards, including the 1995 ACM Doctoral Dissertation Award, the 2002 IEEE Information Theory Paper Award, the 2008 Godel Prize, the 2009 Fulkerson Prize, the 2010 Nevanlinna Prize, an inaugural Simons Invesigator Award, and a MacArthur Fellowship. He is a Fellow of the Association for Computing Machinery and a member of the Connecticut Academy of Science and Engineering. His main research interests include the design and analysis of algorithms, graph theory, machine learning, error-correcting codes and combinatorial scientific computing.