We shall explore consequences of (anti-) self-duality for solutions of Yang-Mills equations in dimension 4.

Following Chapter 7 of the HoTT Book, I will present some basic properties of Homotopy n-Types.

Calibrated submanifolds are special kinds of minimal submanifolds (vanishing mean curvature) that are defined by first order conditions on the immersion. The most studied examples are complex submanifolds of Kahler manifolds, special Lagrangian submanifolds in Calabi-Yau manifolds, and certain special types of submanifolds in $G2$ and $Spin(7)$ manifolds. By imposing a certain amount of symmetry, one can sometimes reduce the nonlinear elliptic first order equations defining such submanifolds to simpler equations on lower-dimensional manifolds. For example, a result of Harvey-Lawson is that the conormal bundle of an austere submanifold of $\mathbb{R}^n$ is special Lagrangian in $\mathbb{C}^n$. This "bundle construction" was generalized in 2004 by Ionel-Karigiannis-Min-Oo to other calibrations, and then extended in 2012 by Karigiannis-Leung to a "twisted" version. Thus, in particular, we obtain many more examples of special Lagrangian submanifolds of $\mathbb{C}^n$ by considering "twisted austere submanifolds" of $\mathbb{R}^n$. I will describe these constructions and review the earlier results. Then I will state several theorems that give a partial classification of twisted austere submanifolds of dimension 3. These new theorems are joint work with Tom Ivey at Charleston College.

By an annulus we mean a domain in $\mathbb{C}^n$ obtained by removing a compact set from a pseudoconvex domain. We study when the $L^2$ $\overline\partial$ operator has closed range from functions to $(0,1)$-forms. In particular, we show that the Chinese Coin problem, i.e. to prove $L^2$-estimates on a domain in $\mathbb{C}^2$ obtained by removing a bidisc from a ball, has a positive solution.

The natural numbers are the first thing one learns in mathematics. Because they lack some desirable properties, one soon extends them to the integers, the rational, real and complex numbers. In this talk I want to focus on other numbers systems that are less often encountered in undergraduate mathematics courses, for instance $p$-adic numbers, quaternions, octonions and cardinal numbers.

We try to investigate a generalization of the Heisenberg commutation relation ${[p,q]=-i \hslash}$, introduced by Chamseddine, Connes and Mukhanov as ``the one-sided and the two-sided quantization equations'', which captures the geometry. The momentum variable $p$ is encoded by the Dirac operator and the analogue of the position variable $q$ is the Feynman slash of real scalar fields over a closed even-dimensional spin manifold. Existence of a solution of the one-sided equation implies that the manifold decomposes into a disconnected sum of spheres of unit volume which represent quanta of geometry. The two-sided equation, as the refined version of the one-sided equation by involving the real structure on a spin manifold, implies the quantization of the volume of the spin manifold.