MIT

One of the big ideas in linear algebra is {\em eigenvalues}. Most matrices become in some basis {\em diagonal} matrices; so a lot of information about the matrix (which is specified by $n^2$ matrix entries) is encoded by by just $n$ eigenvalues. The fact that lots of different matrices can have the same eigenvalues reflects the fact that matrix multiplication is not commutative.

I'll look at how to make these vague statements (``lots of different matrices...") more precise; how to extend them from matrices to abstract symmetry groups; and how to relate abstract symmetry groups to matrices.

We introduce and develop a theory of limits for sequences of sparse graphs based on $L^p$ graphons, which generalizes both the existing $L^\infty$ theory of dense graph limits and its extension by Bollob\'as and Riordan to sparse graphs without dense spots. In doing so, we replace the no dense spots hypothesis with weaker assumptions, which allow us to analyze graphs with power law degree distributions. This gives the first broadly applicable limit theory for sparse graphs with unbounded average degrees.

Joint work with Christian Borgs, Jennifer T. Chayes, and Henry Cohn.

We consider spacetimes arising from perturbations of the interior of Kerr

black holes. These spacetimes have a null boundary in the future such that

the metric extends continuously beyond. However, the Christoffel symbols

may fail to be square integrable in a neighborhood of any point on the

boundary. This is joint work with M. Dafermos

I will describe a version of the definition of stability conditions on a triangulated category to which we were led by the study of quantization of symplectic resolutions of singularities over fields of positive characteristic. Partly motivated by ideas of Tom Bridgeland, we conjectured a relation of this structure to equivariant quantum cohomology; this conjecture has been verified in some classes of examples. The talk is based on joint projects with Anno, Mirkovic, Okounkov and others

I will describe a version of the definition of stability conditions on a triangulated category to which we were led by the study of quantization of symplectic resolutions of singularities over fields of positive characteristic. Partly motivated by ideas of Tom Bridgeland, we conjectured a relation of this structure to equivariant quantum cohomology; this conjecture has been verified in some classes of examples. The talk is based on joint projects with Anno, Mirkovic, Okounkov and others

Studies of the ocean circulation and climate have come to be dominated by the results of complex numerical models encompassing hundreds of thousands of lines of computer code and whose physics may be more difficult to penetrate than the real system. Some insight into the large-scale ocean circulation can perhaps be gained by taking a step back and considering the gross time scales governing oceanic changes. These can derived from a wide variety of simple considerations such as energy flux rates, signal velocities, tracer equilibrium times, and others. At any given time, observed changes are likely a summation of shifts taking place over all of these time scales.

Tridiagonal matrices and three term recurrences and second order equations appear amazingly often, throughout all of mathematics. We won't try to review this subject. Instead we look in two less familiar directions.

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Here is a tridiagonal matrix problem that waited surprisingly long for a solution. Forward elimination factors T into LDU, with the pivots in D as usual. Backward elimination, from row n to row 1, factors T into U_D_L_. Parlett asked for a proof that diag(D + D_) = diag(T) + diag(T^-1).^-1. In an excellent paper (Lin Alg Appl 1997) Dhillon and Parlett extended this four-diagonal identity to block tridiagonal matrices, and also applied it to their "Holy Grail" algorithm for the eigenproblem. I would like to make a different connection, to the Kalman filter.

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The second topic is a generalization of tridiagonal to "tree-diagonal". Unlike the interval, the tree can branch. In the matrix T, each vertex is connected only to its neighbors (but a branch point has more than two neighbors). The continuous analogue is a second order differential equation on a tree. The "non-jump" conditions at a meeting of N edges are continuity of the potential (N-1 equations) and Kirchhoff's Current Law (1 equation). Several important properties of tridiagonal matrices, including O(N) algorithms, survive on trees.