Abstract: The Cauchy-Riemann problem, also known as the $\overline\partial$-problem, is a central problem in several complex variables. It concerns the regularity estimates to the equation $\overline\partial u=f$ on forms in a bounded domain $\Omega\subset\mathbb C^n$. We will talk about some background of the $\overline\partial$-regularity theory, and the obstructions on solving the $\overline\partial$ equation when f is a generic distributions, and our recent works using new technique from extension operators. We use the so-called Rychkov's extension operator, which extends functions on a bounded Lipschitz domain and has boundedness on all Besov spaces and Triebel-Lizorkin spaces. This is based on multiple works which are in part joint with Ziming Shi and Yuan Zhang. 

Location and Zoom link: SCEN 322 and at https://uark.zoom.us/j/87503165233?pwd=S7QdLHdFaJYEkZL3at0OV4aE2rRPwh.1

Abstract: An important problem in several complex variables is to classify all proper holomorphic maps that send the unit ball in C^n to the unit ball in C^N. When such mappings extend continuously to the boundary, they necessarily send the unit sphere to the unit sphere and are called sphere maps. Many open questions remain, even for monomial sphere maps. In this talk, we focus on group-invariant monomial sphere maps, which satisfy some extra symmetry properties, and seek to identify the target ranks for such maps. Rather than give a detailed proof of one specific theorem, we give an overview of the recent results of several undergraduate and graduate student researchers at BYU.  

Location and Zoom link: SCEN 322 and at https://uark.zoom.us/j/87503165233?pwd=S7QdLHdFaJYEkZL3at0OV4aE2rRPwh.1

Abstract: 

The sequence  Besov space $b_p$, $p>1$, is the space of analytic functions $f(z)=\sum_{n=0}^{\infty} a_n\, z^n$ on the open unit disk 

$\mathbb D$ with $$\sum_{n=0}^{\infty} n^{p-1}\, |a_n|^p<\infty\, .$$

We will give a brief introduction to the space $b_p$, as well as explore how certain operators behave on the space. For example, let $\varphi$ be an analytic self-map of $\mathbb D$. We study the multiplication operator $M_\varphi$ on $b_p$ and look at examples including when $\varphi$ is a polynomial. We also study the composition operator $C_\varphi$ on $b_p$, and in more depth on $b_2$ the Dirichlet space. The culmination will be an analogue of the Littlewood Theorem on Hardy spaces for $b_2$.

Location and Zoom link: SCEN 322 and at https://uark.zoom.us/j/87503165233?pwd=S7QdLHdFaJYEkZL3at0OV4aE2rRPwh.1

Abstract: TBA

Location and Zoom link: SCEN 322 and at https://uark.zoom.us/j/87503165233?pwd=S7QdLHdFaJYEkZL3at0OV4aE2rRPwh.1

Abstract: TBA

Location and Zoom link: SCEN 322 and at https://uark.zoom.us/j/87503165233?pwd=S7QdLHdFaJYEkZL3at0OV4aE2rRPwh.1