Georgia Institute of Technology College of Sciences

We discuss results from classical weighted theory and give a characterization of the two-weight inequality for a simple vector-valued operator.

Consider self-adjoint operators $A, B, C : \mathcal{H} \to \mathcal{H}$ on a finite-dimensional Hilbert space such that $A + B + C = 0$. Let $\{\lambda_j (A)\}$, $\{\lambda_j (B)\}$, and $\{\lambda_j (C)\}$ be sequences of eigenvalues of $A, B$, and $C$ counting multiplicity, arranged in decreasing order. In 1962, A. Horn conjectured that the relations of $\{\lambda_j (A)\}$,$\{\lambda_j (B)\}$, and $\{\lambda_j (C)\}$ can be characterized by a set of inequalities defined inductively. This problem was eventually solved by A. Klyachko and Knutson-Tao in the late 1990s. Recently together with H. Bercovici, Collins, Dykema, and Timotin, we are able to find a proof to show that the inequalities are valid for self-adjoint elements that satisfies the relation $A+B+C=0$,  and the proof can be applied to finite von Neumann algebra. The major difficulty in our argument is to show that certain generalized Schubert cells have nonempty intersection. In the finite dimensional case, it follows from the classical intersection theory. However, there is no readily available intersection theory for von Neumann algebras. Our argument requiresa good understanding of the combinatorial structure of honeycombs, and produces an actual element in the intersection algorithmically, and it seems to be new even in finite dimensions.

Motivated by applications to vehicular traffic, supply chains and others, various continuous models for traffic flow on networks were recently proposed. We first present some results for theory of conservation laws on graphs. Then we focus on recent mixed models, involving continuous-discrete spaces and ode-pde systems. Then a time evolving measures approach is showed, with applications to crowd dynamics.

Given a knot, a simple Lie algebra L and an irreducible representation V of L one can construct a one-variable polynomial with integer coefficients. When L is the simplest simple Lie algebra (sl_2) this gives a sequence of polynomials, whose sequence of degrees is a quadratic quasi-polynomial. We will discuss a conjecture for the degree of the colored Jones polynomial for an arbitrary simple Lie algebra, and we will give evidence for sl_3. This is joint work with Thao Vuong.