In this work, we further extend the recently developed adaptive data analysis method, the Sparse Time-Frequency Representation (STFR) method. This method is based on the assumption that many physical signals inherently contain AM-FM representations. We propose a sparse optimization method to extract the AM-FM representations of such signals. We prove the convergence of the method for periodic signals under certain assumptions and provide practical algorithms specifically for the non-periodic STFR, which extends the method to tackle problems that former STFR methods could not handle, including stability to noise and non-periodic data analysis. This is a significant improvement since many adaptive and non-adaptive signal processing methods are not fully capable of handling non-periodic signals. In particular, we present a simplified and modified version of the STFR algorithm that is potentially useful for the diagnosis and monitoring of some cardiovascular diseases.

A prevailing question in the study of von Neumann algebras asks to what extent certain algebras constructed from groups and their actions "remember" the original group and action. Pursuing this question led naturally to the study of von Neumann algebras coming from certain equivalence relations as well. Though a large class of groups and actions which produce "forgetful" algebras have been known since the 1970s (due to Connes and Zimmer), very little progress was made outside of this class until a breakthrough by Sorin Popa some 30 years later. We will give an overview of Popa's powerful deformation/rigidity theory, state a recent result for von Neumann algebras of equivalence relations, and discuss future directions of research.

In this talk I'll discuss how Lagrangian particle methods are being used to study the dynamics of fluid vortices. These methods use the Biot-Savart integral to recover the velocity from the vorticity and they track the flow map using adaptive particle discretizations. I'll present computations of vortex sheet motion in 2D flow, with reference to Kelvin-Helmholtz instability, the Moore singularity, spiral roll-up, and chaotic dynamics. Other examples include vortex rings in 3D flow, and vortex dynamics on a rotating sphere.

Inside the moduli space of curves of genus 2 with 2 marked points, the loci of curves admitting a map to P1 of degree d totally ramified at the two marked points have codimension two. In this talk I will show how to compute the classes of the compactifications of such loci in the moduli space of stable curves. I will also discuss the relation with the related work of Hain, Grushevsky-Zakharov, Chen-Coskun, Cavalieri-Marcus-Wise.