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.

One of the major results in the study of 3-manifolds is the fact that most 3-manifolds have a finite cover that fibers over S1. One may ask what is the counterpart of this result for other classes of manifolds. In this talk we will discuss the case of smooth projective varieties (or more generally Kaehler manifolds) and present some geometric and group-theoretic aspects of  "virtual algebraic fibrations" of their fundamental groups.