I present a new method for optimizing quadratic functions subject to simple bound constraints.  If the problem happens to be strictly convex, the algorithm reduces to a highly efficient method by Dostal and Schoberl.  Our algorithm, however, is also able to efficiently solve nonconcex problems. During this talk I will present the algorithm, a sketch of the convergence theory, and numerical results for convex and nonconvex problems.

We will begin by discussing the risk parity portfolio selection problem, which aims to find  portfolios for which the contributions of risk from all assets are equally weighted. The risk parity may be satisfied over either individual assets or groups of assets. We show how convex optimization techniques can find a risk parity solution in the nonnegative  orthant, however, in general cases the number of such solutions can be anywhere between zero and  exponential in the dimension. We then propose a nonconvex least-squares formulation which allows us to consider and possibly solve the general case. 

Motivated by this problem we present several alternating direction schemes for specially structured nonlinear nonconvex problems. The problem structure allows convenient 2-block variable splitting.  Our methods rely on solving convex subproblems at each iteration and converge to a local stationary point. Specifically, discuss approach  alternating directions method of multipliers and the alternating linearization method and we provide convergence rate results for both classes of methods. Moreover, global optimization techniques from polynomial optimization literature are applied to complement our local methods and to provide lower bounds.

In this talk, we use Carleman type techniques to address uniqueness of self-shrinkers of mean curvature flow with given asymptotic behaviors.

Eigenvectors of square matrices are central to linear algebra. Eigenvectors of tensors are a natural generalization. The spectral theory of tensors was pioneered by Lim and Qi around 2005. It has numerous applications, and ties in closely with optimization and dynamical systems.  We present an introduction that emphasizes algebraic and geometric aspects

I will review Bott's classical periodicity result about topological K-theory (with period 2 in the case of complex K-theory, and period 8 in the case of real K-theory), and provide an easy sketch of proof, based on the algebraic periodicity of Clifford algebras. I will then introduce the `higher real K-theory' of Hopkins and Miller, also known as TMF. I'll discuss its periodicity (with period 576), and present a conjecture about a corresponding algebraic periodicity of `higher Clifford algebras'. Finally, applications to physics will be discussed.