48,402 research outputs found
Topology of the Stokes phenomenon
Several intrinsic topological ways to encode connections on vector bundles on smooth complex algebraic curves will be described. In particular the notion of Stokes decompositions will be formalised, as a convenient intermediate category between the Stokes filtrations and the Stokes local systems/wild monodromy representations. The main result establishes a new simple characterisation of the Stokes decompositions
Computing Linear Matrix Representations of Helton-Vinnikov Curves
Helton and Vinnikov showed that every rigidly convex curve in the real plane
bounds a spectrahedron. This leads to the computational problem of explicitly
producing a symmetric (positive definite) linear determinantal representation
for a given curve. We study three approaches to this problem: an algebraic
approach via solving polynomial equations, a geometric approach via contact
curves, and an analytic approach via theta functions. These are explained,
compared, and tested experimentally for low degree instances.Comment: 19 pages, 3 figures, minor revisions; Mathematical Methods in
Systems, Optimization and Control, Birkhauser, Base
On the classification of rank two representations of quasiprojective fundamental groups
Suppose is a smooth quasiprojective variety over \cc and \rho : \pi
_1(X,x) \to SL(2,\cc) is a Zariski-dense representation with quasiunipotent
monodromy at infinity. Then factors through a map with
either a DM-curve or a Shimura modular stack.Comment: minor changes in exposition, citation
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