Abstract

Problems labelled as Jacobian Conjectures seek information about maps or flows by looking at the appropriate Jacobian matrix. For maps, the classical Keller Jacobian Conjecture (1939) asks whether polynomial maps $f:Cn-->Cn$ with $Det(f´)=1$ must be bijective with a polynomial inverse. This problem is still open, even in dimension two. Several other related conjectures have been posed. For flows, the Markus-Yamabe Conjecture (1960) considers a system of differential equations $x´=\; f(x)$, where $x\in \; Rn$ and $f(0)=0$. If the eigenvalues of $f´(x)$ all have negative real part for all $x$, must $x=0$ be globally asymptotically stable? This question was completely settled in the last decade. Related to both these problems are unipotent maps, maps whose Jacobian eigenvalues are all one.
This talk introduces the listener to these Jacobian conjectures and explains recent work in studying $C1$ maps whose Jacobian eigenvalues are constant. Several open problems will be given.