Applications of Wenzl's Formula
Zhengwei Liu, Vanderbilt University
Location: Stevenson 1404
We will talk about Wenzl's formula and its general form. We will see its applications to other subfactor concepts, like Jellyfish, spoke subfactors, the embedding theorem, normalizers.
Progress on Hard Thresholding Pursuit
Jean-Luc Bouchot, Drexel University
Location: Stevenson 1307
The Hard Thresholding Pursuit algorithm for sparse recovery is revisited using a new theoretical analysis. The main result states that all sparse vectors can be exactly recovered from incomplete linear measurements in a number of iterations at most proportional to the sparsity level as soon as the measurement matrix obeys a restricted isometry condition. The recovery is also robust to measurement error The same conclusions are derived for a variation of Hard Thresholding Pursuit, called Graded Hard Thresholding Pursuit, which is a natural companion to Orthogonal Matching Pursuit and runs without a prior estimation of the sparsity level. In two extreme cases of the vector shape, it is also shown that, with high probability on the draw of random measurements, a fixed sparse vector is robustly recovered in a number of iterations precisely equal to the sparsity level. These theoretical findings are experimentally validated, too.
A Survey of Algebraic Geometry and Model Theory for Free and Hyperbolic Groups
Olga Kharlampovich, Hunter College, CUNY
Location: Stevenson 5211
I will survey results of Kharlampovich--Miasnikov and Sela on first-order theories of free and hyperbolic groups. I will show that in the presence of ``negative curvature'' in groups, there exists a robust algebraic geometry and the principal Tarski-type problems are decidable. In particular, there is an algorithm for the elimination of quantifiers (to boolean combinations of AE-formulas). I will also give a description of definable sets in free and hyperbolic groups (joint result with Miasnikov). This solves Malcev's problem from 1965. Tea at 3:30 pm in SC 1425.
Character Rigidity for Lattices and Commensurators
Darren Creutz, Vanderbilt University
Location: Stevenson 1432
Characters on groups (positive definite conjugation-invariant functions) arise naturally both from probability-preserving actions (the measure of the set of fixed points) and unitary representations on finite factors (the trace). I will present joint work with J. Peterson showing the nonexistence of nontrivial characters for irreducible lattices in semisimple groups and for their commensurators. Consequently, any finite factor representation of such a group generates either the left regular representation or a finite-dimensional representation, generalizing our earlier result that every nonatomic probability-preserving action of such groups is essentially free. The key new idea is to use the contractive nature of the Poisson boundary to bring it in operator algebraic setting and along with it the rigidity behavior of lattices in their ambient groups.
Global Analysis of an Age-Structured Multi-Strain Virus Model
Cameron Browne, Vanderbilt University
Location: Stevenson 1307
We consider a general model of a within-host viral infection with multiple virus strains and explicit age-since-infection structure for infected cells. Existence and uniqueness of solutions, along with asymptotic smoothness of the nonlinear semigroup generated by the family of solutions, are established. For each viral strain, a quantity called the reproduction number is defined. The main result is that the single-strain equilibrium corresponding to the virus strain with maximal reproduction number is a global attractor (provided that this maximal reproduction number is greater than unity and all reproduction numbers are distinct). In other words, the virus strain with maximal reproduction number competitively excludes all other strains. As an application of the model, HIV evolution is considered and simulations are conducted.