we give a survey on recent work of bao, chan, gaddis, he, kirkman, moore, walton, won, and others in noncommutative invariant theory.

i will speak on a recent joint study with j. marzuola and d. tataru which proves low regularity local well-posedness for quasilinear schroedinger equations. similar results were previously proved by kenig, ponce, and vega in much higher regularity spaces using an artificial viscosity method. our techniques, and in particular the spaces in which we work, are motivated by those used by bejenaru and tataru for semilinear equations.

free araki-woods factors (fawf) were introduced by shlyakhtenko in 1996. in some sense they are free probability analogs of the hyperfinite factors. shlyakhtenko showed that they are typically von neumann algebras of type $iii_1$, and moreover he constructed a one parameter family of non isomorphic type $iii_1$ fawf. in this talk i will discuss about the complexity of the classification problem of fawf from the descriptive set theory point of view.

i present recent work on the design of scalable optimization algorithms for aiding in the big data task of subspace clustering. in particular, i will describe three approaches that we recently developed to solve optimization problems constructed from the so-called self-expressiveness property of data that lies in the union of low-dimensional subspaces. sources of data that lie in the union of low-dimensional subspaces include multi-class clustering and motion segmentation. our optimization algorithms achieve scalability by leveraging three features: a rapidly adapting active-set approach, a greedy optimization method, and a divide-and-conquer technique. numerical results demonstrating the scalability of our approaches will be presented.