Algebraic Kohno-Pajitnov jump loci
16/02/2018, 14:00 - 15:00 in MAP/0G/018
Hüttemann, Thomas (Queen’s University Belfast)
The concept of jump loci has seen many developments in topology and algebra in recent years. The idea is to analyse a certain family of objects, parametrised by points in a suitable space, and say at what points the properties of the objects change or “jump”.
Specifically, Kohno and Pajitnov examined jump loci of a purely algebraic nature. A point in the parameter space is collections of directions in n-space; such data determines a map of Laurent polynomial rings, and hence an extension-of-scalar construction for modules over Laurent polynomial rings. Roughly speaking, Kohno and Pajitnov proved that the jump loci for the “dimension” of a module under extension-of-scalars have a surprisingly simple structure.
I will describe joint work with Zuhong Zhang on extending the result to more general notions of “dimension”. The set-up includes jump loci of the so-called McCoy rank of matrices (which is related to solvability of systems of linear equations over commutative rings).