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.

We study small covers and the real moment-angle manifold over a simple polytope that admit Riemannian metrics of positive scalar curvature. We first explain some general facts on the existence of positive scalar curvature on a smooth manifold.Then we show a class of simple polytopes over which the small covers and the real moment-angle manifolds

In this talk, I will introduce a possible way to consider the cabling conjecture from the view of Heegaard splitting.

Let M be a unitary closed manifold that admits an action of a circle preserving the unitary structure and fixing some isolated points. Kosniowski conjectured in 1980 that if M is not a boundary, then the number of isolated points is at least [dim M/4]+1. This talk will discuss different statements of Kosniowski conjecture and state some recent progresses.