Recently, we generalize the known DDVV-type inequalities for real (skew-)symmetric and complex (skew-)Hermitian matrices into arbitrary real, complex and quaternionic matrices. Inspired by the Erd/H{o}s-Mordell inequality, we establish DDVV-type inequalities for matrices in the subspaces spanned by a Clifford system or a Clifford algebra. We also generalize the B/"{o}ttcher-Wenzel inequality to quaternionic matrices. In this talk, we will briefly introduce the background, the new progress above, and some open questions.

Einstein metrics are natural Riemannian metrics on differentiable manifolds. In dimensions 2 and 3, they must have constant sectional curvature, while in dimension 4, they are much more complicated. For the complex setting, in 1990 Tian classified Kahler-Einstein four-manifolds with positive scalar curvature, and in 2012 LeBrun classified Hermitian, Einstein four-manifolds with positive scalar curvature. For the real setting, however less is known, even assuming a (strong) condition of positive sectional curvature.

Let ，let be a sequence of independent copies of a real-valued random variable , and set , . Motivated by a Theorem of Mikosch (1984), this talk is devoted to establishing a strong law of large numbers for the sequence . More specifically, necessary and sufficient conditions are given for where

The coverage control problem of a network of heterogeneous mobile sensors is addressed in this talk, where the goal is to minimize a coverage cost function which is defined to be the largest arrival time from the mobile sensor network to the points on a cricle. A necessary and sufficient condition for the global minimization of the coverage cost function is firstly derived. Then, distributed coverage control schemes with input saturation are developed to drive the sensors to the optimal configuration such that the necessary and sufficient condition is satisfied. Under the distributed coverage control schemes, the mobile sensors’ spatial order on the circle is preserved throughout the network’s evolution and thus collision between mobile sensors is avoided. Finally, simulation results are presented to illustrate the effectiveness of the proposed distributed control schemes.

In this paper, we pose two kinds of Minkowski problems involving the $p$-Laplacian operator. The Hadamard variational formulas for some $p$-Laplacian functionals are obtained. A good application is to prove symmetry results for solutions to some overdetermined problems of $p$-Laplacian equations.

T-duality is an equivalence of two physical theories (quantum field theories or string theories) with different spacetime geometries. It originates from string theories, finds applications in condensed matter physics and has profound connections with topology/geometry. In this talk, I will first briefly review the elegant mathematical formulation of T-duality in an H-flux by Bouwknegt-Evslin-Mathai. Then I will describe our joint work with Mathai about developing exotic twisted /mathbb{T}-equivariant cohomology theories in order to enhance the T-duality by capturing the exchange of winding and momentum, a fundamental feature of T-duality.

I would present the recent progress in the construction of singular constant scalar curvature Kahler (cscK) metrics. The singular cscK metric is an extension of the singular Kahler-Einstein metric.I would also present the application of such singular metrics to Yau-Tian-Donaldson conjecture.

In this work, we study the finite element approximation of a free boundary plasma problem, which is related to some problems in electromagnetics. Using a mixed approach (which resembles an optimal control problem with control constraints), we formulate a weak formulation and study the existence and uniqueness of a solution to the continuous model problem. Using the same setting, we formulate and analyze the discrete problem. We derive optimal order energy norm a priori error estimates proving the convergence of the method.Further,

This talk concerns the minimization problem of $L^2$ norm of curl of vector fields in multiconnected bounded domains prescribed full trace on the boundary. The existence and $H^2$ estimate of the minimizers are established.

In this seminar, a novel approach is presented for control design with guaranteed transient performance for multiple-input multiple-output discrete-time linear polytope difference inclusions. We establish a theorem that gives necessary and sufficient conditions for the state to evolve from one polyhedral subset of the state-space to another. Then we present an algorithm which, given a time-varying polyhedral set called the target tube, which may be specified from typical transient performance specifications, constructs an output feedback law that guarantees that the state evolves in this tube.