Posts

Fall 2022

The regular time and place for the Seminar will be Monday at 4:30 – 5:30 p.m in Math Tower 154. If you are interested, please contact me.

date speaker institution title
Sep 12 Liding Yao OSU Sharp Hölder Regularity for Nirenberg’s Complex Frobenius Theorem
Oct 3 Adam Christopherson OSU TBD
Oct 10 Xianghong Gong UW-Madison On the regularity of d-bar solutions on domains in complex manifolds satisfying condition a_q
Nov 14 Ziming Shi Rutger TBD


Liding Yao

Title: Sharp Hölder Regularity for Nirenberg’s Complex Frobenius Theorem

Abstract: Nirenberg’s famous complex Frobenius theorem gives necessary and sufficient conditions on a locally integrable structure for when it is equal to the span of some real and complex coordinate vector fields. In the talk I will discuss some differential complexes and how some of the notions make sense in the non-smooth setting. For a $C^{k,s}$ complex Frobenius structure, we show that there is a $C^{k,s}$ coordinate chart such that the structure is spanned by coordinate vector fields which are $C^{k,s-\varepsilon}$ for all $\varepsilon>0$. Here the $\varepsilon>0$ loss in the result is optimal.

Adam Christopherson

TBD

Xianghong Gong

Title: On the regularity of d-bar solutions on domains in complex manifolds satisfying condition a_q

Abstract: We will start with some recent regularity results for the d-bar equation on strictly pseudoconvex domains with C^2 boundary in the complex Euclidean space C^n. These results have been proved by using homotopy formulas and estimates hold for forms that are not necessarily d-bar closed.  

We will then describe new regularity results for the d-bar equation on a domain in a complex manifold when the boundary of the domain has either a sufficient number of positive or negative Levi eigenvalues. We will prove the same regularity result for given forms of type (0,1). For forms of type (0,q) with q>1, the same regularity for the d-bar solutions holds when the boundary is sufficiently smooth.

Ziming Shi

TBD