Georgia Institute of Technology College of Sciences

Consider Hermitian matrices A, B, C on an n-dimensional Hilbert space such that C=A+B. Let a={a_1,a_2,...,a_n}, b={b_1, b_2,...,b_n}, and c={c_1, c_2,...,c_n} be sequences of eigenvalues of A, B, and C counting multiplicity, arranged in decreasing order. Such a triple of real numbers (a,b,c) that satisfies the so-called Horn inequalities, describes the eigenvalues of the sum of n by n Hermitian matrices. The Horn inequalities is a set of inequalities conjectured by A. Horn in 1960 and later proved by the work of Klyachko and Knutson-Tao. In these two talks, I will start by discussing some of the history of Horn's conjecture and then move on to its more recent developments. We will show that these inequalities are also valid for selfadjoint elements in a finite factor, for types of torsion modules over division rings, and for singular values for products of matrices, and how additional information can be obtained whenever a Horn inequality saturates. The major difficulty in our argument is the proof 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 requires a 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. If time permits, we will also discuss some of the intricate combinatorics involved here. In addition, some recent work and open questions will also be presented.

The Ricci-Stein theory of singular integrals concerns operators of the form \int e^{i P(y)} f (x-y) \frac {dy}y.The L^p boundedness was established in the early 1980's, and the weak-type L^1 estimate by Chanillo-Christ in 1987. We establish the weak type estimate for the maximal truncations. This method of proof might well shed much more information about the fine behavior of these transforms. Joint work with Ben Krause.

In this talk we discuss two weight estimates for well-localized operators acting on vector-valued function spaces with matrix weights. We will show that the Sawyer-type testing conditions are necessary and sufficient for the boundedness of this class of operators, which includes Haar shifts and their various generalizations. More explicitly, we will show that it is suficient to check the estimates of the operator and its adjoint only on characteristic functions of cubes. This result generalizes the work of Nazarov-Treil-Volberg in the scalar setting and is joint work with K. Bickel, S. Treil, and B. Wick.

Recently, Awasthi et al proved that a semidefinite relaxation of the k-means clustering problem is tight under a particular data model called the stochastic ball model. This result exhibits two shortcomings: (1) naive solvers of the semidefinite program are computationally slow, and (2) the stochastic ball model prevents outliers that occur, for example, in the Gaussian mixture model. This talk will cover recent work that tackles each of these shortcomings. First, I will discuss a new type of algorithm (introduced by Bandeira) that combines fast non-convex solvers with the optimality certificates provided by convex relaxations. Second, I will discuss how to analyze the semidefinite relaxation under the Gaussian mixture model. In this case, outliers in the data obstruct tightness in the relaxation, and so fundamentally different techniques are required. Several open problems will be posed throughout.This is joint work with Takayuki Iguchi and Jesse Peterson (AFIT), as well as Soledad Villar and Rachel Ward (UT Austin).