The works of F. Murray and J. von Neumann outlined a natural method to associate a von Neumann algebra to a group. Since then, an active area of research seeks to investigate which structural aspects of the group extend to its von Neumann algebra.  The difficulty of this problem is best illustrated by Conne's landmark result which states all ICC amenable groups give rise to isomorphic von Neumann algebras.  In essence, standard group invariants are not typically detectable for the resulting von Neumann algebra.  When the group is non-amenable, the situation may be strikingly different. 

This talk surveys advances made in this area, with an emphasis on the results stemming from Popa's deformation/rigidity theory.  I present several instances where elementary group theoretic properties, such as direct products, can be recovered from the algebra.  We will also discuss recent progress made by Ben Hayes, Dan Hoff, Thomas Sinclair and myself in the case where the underlying group has positive first $\ell^2 $-Betti number.  We will explore the relationship between s-malleable deformations of von Neumann algebras and $\ell^2 $ co-cycles which lays the foundation for our work. 

In their breakthrough 2011 paper, Chatterjee and Varadhan established a large deviations principle (LDP) for the Erdös-Rényi graph G(N,p), viewed as a measure on the space of graphons with the cut metric topology. This yields LDPs for subgraph counts, such as the number of triangles in G(N,p), as these are continuous functions of graphons. However, as with other applications of graphon theory, the LDP is only useful for dense graphs, with p ϵ (0,1) fixed independent of N. 

Since then, the effort to extend the LDP to sparse graphs with p ~ N^{-c} for some fixed c>0 has spurred rapid developments in the theory of "nonlinear large deviations". We will report on recent results increasing the sparsity range for the LDP, in particular allowing c as large as 1/2 for cycle counts, improving on previous results of Chatterjee-Dembo and Eldan. These come as applications of new quantitative versions of the classic regularity and counting lemmas from extremal graph theory, optimized for sparse random graphs. (Joint work with Amir Dembo.)

The scalable optimization of sensor and actuator placements remains an open challenge in estimation and control. In general, determining optimal sensor locations amounts to an intractable brute-force search among all candidate locations. My works exploits linear dimensionality reduction tools, including SVD and balanced model reduction, to bypass this combinatorial search. Sensor and actuator locations are computed using efficient matrix pivoting operations on the resulting low-dimensional representations, allowing runtime to scale only linearly with the number of candidate locations. Results are demonstrated on high-dimensional examples from imaging, fluids and control with thousands of candidate locations, and are comparable to placements computed using more expensive methods. If time permits, recent directions in nonlinear dimensionality reduction for forecasting will be discussed.

I will give a proof of S.Y.  Cheng's conjecture  that a bounded strongly pseudoconvex domain in C^n has  its Bergman metric being Einstein if and only if it is holomorphically equivalent to the ball.