## Function Spaces

#### Location

Cupples I Room 115

#### Start Date

7-21-2016 2:30 PM

#### End Date

21-7-2016 2:50 PM

#### Description

The set $\Gamma {\stackrel{\rm def}{=}} \{(z+w,zw):|z|\leq 1,|w|\leq 1\} \subset {\mathbb{C}}^2$ has intriguing complex-geometric properties; it has a 3-parameter group of automorphisms, its distinguished boundary is a ruled surface homeomorphic to the M\"obius band and it has a special subvariety which is the only complex geodesic that is invariant under all automorphisms. We exploit this geometry to develop an explicit and detailed structure theory for the rational maps from the unit disc to $\Gamma$ that map the boundary of the disc to the distinguished boundary of $\Gamma$. The talk is based on joint work with Jim Agler and Nicholas Young. \begin{itemize} \item[{[1]}] Jim Agler, Zinaida A. Lykova and N. J. Young: Algebraic and geometric aspects of rational $\Gamma$-inner functions, (arXiv: 1502.04216 [math.CV] 17 Febr. 2015) 22 pp. \end{itemize}

#### Share

COinS

Jul 21st, 2:30 PM Jul 21st, 2:50 PM

Algebraic and geometric aspects of rational $\Gamma$-inner functions

Cupples I Room 115

The set $\Gamma {\stackrel{\rm def}{=}} \{(z+w,zw):|z|\leq 1,|w|\leq 1\} \subset {\mathbb{C}}^2$ has intriguing complex-geometric properties; it has a 3-parameter group of automorphisms, its distinguished boundary is a ruled surface homeomorphic to the M\"obius band and it has a special subvariety which is the only complex geodesic that is invariant under all automorphisms. We exploit this geometry to develop an explicit and detailed structure theory for the rational maps from the unit disc to $\Gamma$ that map the boundary of the disc to the distinguished boundary of $\Gamma$. The talk is based on joint work with Jim Agler and Nicholas Young. \begin{itemize} \item[{[1]}] Jim Agler, Zinaida A. Lykova and N. J. Young: Algebraic and geometric aspects of rational $\Gamma$-inner functions, (arXiv: 1502.04216 [math.CV] 17 Febr. 2015) 22 pp. \end{itemize}