Princeton University

The joint spectral radius (JSR) of a set of matrices characterizes the maximum growth rate that can be achieved by multiplying them in arbitrary order. This concept, which essentially generalizes the notion of the "largest eigenvalue" from one matrix to many, was introduced by Rota and Strang in the early 60s and has since emerged in many areas of application such as stability of switched linear systems, computation of the capacity of codes, convergence of consensus algorithms, tracability of graphs, and many others. The JSR is a very difficult quantity to compute even for a pair of matrices. In this talk, we present optimization-based algorithms (e.g., via semidefinite programming or dynamic programming) that can either compute the JSR exactly in special cases or approximate it with arbitrary prescribed accuracy in the general case.

Based on joint work (in different subsets) with Raphael Jungers, Pablo Parrilo, and Mardavij Roozbehani.
 

It follows from the ellipsoid method and results of Grotschel, Lovasz and Schrijver that one can find an optimal colouring of a perfect graph in polynomial time. But no ''combinatorial'' algorithm to do this is known.

Here we give a combinatorial algorithm to do this in an n-vertex perfect graph in time O(n^{k+1}^2) where k is the clique number; so polynomial-time for fixed k. The algorithm depends on another result, a polynomial-time algorithm to find a ''balanced skew partition'' in a perfect graph if there is one.

Joint work with Maria Chudnovsky, Aurelie Lagoutte, and Sophie Spirkl.

For a positive measure set of Klein-Gordon masses mu^2 > 0, we construct one-parameter families of solutions to the Einstein-Klein-Gordon equations bifurcating off the Kerr solution such that the underlying family of spacetimes are each an asymptotically flat, stationary, axisymmetric, black hole spacetime, and such that the corresponding scalar fields are non-zero and time-periodic. An immediate corollary is that for these Klein-Gordon masses, the Kerr family is not asymptotically stable as a solution to the Einstein-Klein-Gordon equations. This is joint work with Otis Chodosh.

We establish sharp Sobolev inequalities of order four on Euclidean $d$-balls for $d$ greater than or equal to four. When $d=4$, our inequality generalizes the classical second order Lebedev-Milin inequality on Euclidean $2$-balls. Our method relies on the use of scattering theory on hyperbolic $d$-balls. As an application, we charcaterize the extremals of the main term in the log-determinant formula corresponding to the conformal Laplacian coupled with the boundary Robin operator on Euclidean $4$-balls. This is joint work with Alice Chang. 

Stochastic optimization for sequential decision problems under uncertainty arises in many settings, and as a result as evolved under several canonical frameworks with names such as dynamic programming, stochastic programming, optimal control, robust optimization, and simulation optimization (to name a few).  This is in sharp contrast with the universally accepted canonical frameworks for deterministic math programming (or deterministic optimal control).  We have found that these competing frameworks are actually hiding different classes of policies to solve a single problem which encompasses all of these fields.  In this talk, I provide a canonical framework which, while familiar to some, is not universally used, but should be.  The framework involves solving an objective function which requires searching over a class of policies, a step that can seem like mathematical hand waving.  We then identify four fundamental classes of policies, called policy function approximations (PFAs), cost function approximations (CFAs), policies based on value function approximations (VFAs), and lookahead policies (which themselves come in different flavors).  With the exception of CFAs, these policies have been widely studied under names that make it seem as if they are fundamentally different approaches (policy search, approximate dynamic programming or reinforcement learning, model predictive control, stochastic programming and robust optimization).  We use a simple energy storage problem to demonstrate that minor changes in the nature of the data can produce problems where each of the four classes might work best, or a hybrid.  This exercise supports our claim that any formulation of a sequential decision problem should start with a recognition that we need to search over a space of policies.

We will present recent results regarding conservation laws for the wave equation on null hypersurfaces.  We will show that an important example of a null hypersurface admitting such conserved quantities is the event horizon of extremal black holes. We will also show that a global analysis of the wave equation on such backgrounds implies that certain derivatives of solutions to the wave equation asymptotically blow up along the event horizon of such backgrounds. In the second part of the talk we will present a complete characterization of null hypersurfaces admitting conservation laws. For this, we will introduce and study the gluing problem for characteristic initial data and show that the only obstruction to gluing is in fact the existence of such conservation laws.

We derive an exact implied volatility expansion for any model whose European call price can be expanded analytically around a Black-Scholes call price. Two examples of our framework are provided (i) exponential Levy models and (ii) CEV-like models with local stochastic volatility and local stochastic jump-intensity.

Please note that this seminar has been cancelled due to unforeseen circumstances.