Linear regression on network-linked observations has been essential in modeling the relationships between responses and covariates with additional network structures. Many approaches either lack inference tools or rely on restrictive assumptions of social effects. More importantly, these methods usually assume that networks are error-free. I introduce a regression model with nonparametric network effects based on subspace assumptions. This model does not assume the network structure to be precisely observed and is provably robust to network observational errors. An inference framework is established under the general requirement of network observational errors, and corresponding robustness is studied in detail when observational errors arise from random network models. Results reveal a phase-transition phenomenon of inference validity in relation to network density when no prior knowledge of the network model is available. I also show that significant improvements can be achieved when the network model is known. I then briefly discuss an ensemble network estimation strategy, network mixing, which can improve the adaptivity of the proposed method. The regression model is applied to investigate social impacts on students' perceptions of school safety based on observed friendship relations. It enables reliable analysis thanks to the nonparametric network effects and the robustness to network observational errors.

 Projected-search methods for bound-constrained optimization are based on performing a search along a piecewise-linear continuous path obtained by projecting a search direction onto the feasible region. A potential benefit of a projected-search method is that the direction of the search path may change multiple times at the cost of computing a single direction.
In this talk, we present a new interior method for general nonlinearly constrained optimization that combines a shifted primal-dual interior method with a projected-search method for bound-constrained optimization. The method is based on the formulation of a primal-dual penalty-barrier function that incorporates shifts on both primal and dual variables.  A modified Newton direction is used in conjunction with a new projected-search algorithm that employs a non-monotone flexible quasi-Armijo line search for the minimization of the penalty-barrier function. Computational results indicate that the proposed method requires fewer iterations than a conventional interior method, thereby reducing the number of times that the search direction need be computed.

The notion of Strong Approximate Transitivity (SAT) for group actions on probability measure spaces was introduced by Jaworski in the early 90's. A canonical example of an SAT group action is provided by a group acting on its Poisson boundary (with respect to some "nice" probability measure on the group).   
In this talk, I will discuss a noncommutative analogue of the SAT property, and its connection with noncommutative Poisson boundary inclusions. 

The celebrated Brunn-Minkowski inequality states that for compact subsets $X$ and $Y$ of $\Bbb{R}^d$, $m(X+Y)^{1/d} \geq m(X)^{1/d}+m(Y)^{1/d}$ where $m(\cdot)$ is the Lebesgue measure. We will introduce a conjecture generalizing this inequality to every locally compact group where the exponent is believed to be sharp. In a joint work with Yifan Jing and Chieu-Minh Tran, we prove this conjecture for a large class of groups (including e.g. all real linear algebraic groups). We also prove that the general conjecture will follow from the simple Lie group case. For those groups where we do not know the conjecture yet (one typical example being the universal covering of $SL_2(\Bbb{R})$), we also obtain partial results. In this talk I will discuss this inequality and explain important ingredients, old and new, in our proof.

The algebraic K-theory of the category of finite type $n$ spectra is a fundamental object containing structural information about the stable homotopy category. However, until recently almost nothing was known about it for $n>1$, primarily because it is not the K-theory of a connective ring. In this talk, I will explain how, for $n=2$, it can be computed in terms of K-theory of discrete rings and topological cyclic homology. In particular, we can read off the K groups in low degrees and find that there is an infinite family of 2-torsion classes in $K_0$ at the prime 2. I will also explain how to construct type 2 spectra representing these $K_0$ classes.

In this talk, we introduce the dynamic stochastic variational inequality (DSVI). The DSVI is an ODE whose right hand side is defined by the natural mapping of a VI (referred to as the first-stage VI) and is coupled with another stochastic VI (referred to as the second-stage SVI). The DSVI provides a unified modeling framework for various applications involving equilibrium/optimality conditions (VI), dynamics (ODE), and uncertainties (stochasticity). We establish solution existence and uniqueness for two classes of DSVIs: the first class is defined by a strongly monotone SVI in the second stage, and the second class pertains to a box-constrained stochastic linear VI with the P-property in the second stage. Preliminary results on switching dynamics of the DSVI are presented. We develop sample average approximation (SAA) and time-stepping schemes to compute the DSVI. The uniform convergence and exponential convergence are established for the SAA under suitable conditions. A time-stepping EDIIS (energy direct inversion on the iterative subspace) method is proposed to solve the differential VI arising from the SAA of the DSVI. Our results are illustrated by an instantaneous dynamic user equilibrium  problem in transportation engineering. This is a joint work with Dr. Xiaojun Chen of the Hong Kong Polytechnic University.