In this expository talk we will discuss aspects of spectral theory of the complex Laplacian, revolving around
the notion of positivity.  We will discuss geometric/potential theoretic characterizations
for positivity of the complex Neumann Laplacian and explain some applications of the theory in complex geometry.

Of concern to quantum chemists and solid state physicists is the approximate numerical computation of the ground state wave function, and the ground state energy and density for molecular and other quantum mechanical systems. Since the number of molecules in bulk matter is of the order of 10e26 , direct computation is too cumbersome or impossible in many situations. In 1927, L. Thomas and E. Fermi proposed replacing the ground state wave function by the ground state density, which is a function of only three variables. Independently, each found an approximate expansion for the energy associated with a density. (The wave function uniquely determines the density, but not conversely.)

A computationally better approximate expansion was found in the 1960’s by W. Kohn and his collaborators; for this work Kohn got the Nobel Prize in Chemistry in 1998. A successful attempt to put Thomas-Fermi theory into a rigorous mathematical framework was begun in the 1970’s by E. Lieb and B. Simon and was continued and expanded by Ph. Benilan, H. Brezis and others. Very little rigorous mathematics supporting Kohn density functional theory is known. In this talk I will present a survey of rigorous Thomas-Fermi theory, including recent developments and open problems (in the calculus of variations and semilinear elliptic systems).

We will discuss Viale-Weiss paper ``On the consistency strength of the proper forcing axiom".

One way to classify bounded domains of several complex variables is to determine the boundary behavior.  It is a conjecture of Greene and Krantz that if an automorphism orbit accumulates at the boundary of a smooth domain, then that point is of finite type in the sense of D'Angelo.  We will discuss some supporting results and a special case of the conjecture. 

One of the most straightforward formulations of a feature selection problem boils down to the linear algebraic problem of selecting good columns from a data matrix.  This formulation has the advantage of yielding features that are interpretable to scientists in the domain from which the data are drawn, an important consideration when machine learning methods are applied to realistic scientific data.  While simple, this problem is central to many other seemingly nonlinear learning methods.  Moreover, while unsupervised, this problem also has strong connections with related supervised learning methods such as Linear Discriminant Analysis and Canonical Correlation Analysis.  We will describe recent work implementing Randomized Linear Algebra algorithms for this feature selection problem in parallel and distributed environments on inputs of size ranging from ones to tens of terabytes, as well as the application of these implementations to specific scientific problems in areas such as mass spectrometry imaging and climate modeling.