We introduce a recent (on-going) work joint with Fagui Li on minimal hypersurfaces in unit spheres, where some integral inequalities and their equality characterizations are obtained.

We present some work on analysis of elliptic PDEs of non-local types. And we give a brief introduction on: some basic maximum principles, existence and uniqueness, classi_cations and Liouville type theorems, Bocher type theorems and the related maximum principles for singular solutions. The focus is on the ideas and related basic estimates behind these research work. We will also discuss some basic estimates for linear equations.

I will survey some recent progress on differential calculus of quivers with potentials and give some applications of it to algebraic geometry and representation theory.

We propose an inexact proximal augmented Lagrangian framework with explicit inner problem termination rule for composite convex optimization problems. We consider arbitrary linearly convergent inner solver including in particular stochastic algorithms, making the resulting framework more scalable facing the ever-increasing problem dimension. Each subproblem is solved inexactly with an explicit and self-adaptive stopping criterion, without requiring to set an a priori target accuracy. When the primal and dual domain

In this talk, we present a novel nonlocal nonlinear coarse grid approximation using a machine learning algorithm. Multiscale models for complex nonlinear systems require nonlocal multicontinua approaches. These rigorous techniques require complex local computations, which involve solving local problems in oversampled regions subject to constraints. The solutions of these local problems can be replaced by solving original problem on a coarse (oversampled) region for many input parameters (boundary and source terms) and computing effective properties derived by nonlinear nonlocal multicontinua approaches. The effective properties depend on many variables (oversampled region and the number of continua), thus their calculations require some type of machine learning techniques.

The semiclassical Schrodinger equation with multiscale and random potentials often appearswhen studying electron dynamics in heterogeneous quantum systems. As time evolves, the wavefunction develops high-frequency oscillations in both the physical space and the random space, which poses severe challenges for numerical methods. We propose a multiscale reduced basis method, where we construct multiscale reduced basis functions using an optimization method and the proper orthogonal decomposition method in the physical space and employ the quasi-Monte Carlo method in the random space.

We provide error analysis of a mixed method with stabilization along mesh interface for the Quad-Curl problem. We present a new discrete embedding inequality. With the help of this embedding inequality, we show the mixed method converges optimally in energy norm even when the solution has only low regularity.

Let H_q be the semisimple Hecke algebra of type D_n. To each simple module V over H_q(D_n), we explicitly construct a quasi-idempotent whose associated idempotent is a primitive idempotent associated to V. A matrix unit basis of H_q(D_n) is also obtained. We generalize some earlier work of Dipper, James, Murphy and Mathas to the type D_n setting.