GAUGE THEORY AND WILD RAMIFICATION

Gauge Theory and Wild Ramification

The gauge theory approach to the geometric Langlands program is extended to the case of wild ramification. The new ingredients that are required, relative to the tamely ramified case, are differential operators with irregular singularities, Stokes phenomena, isomonodromic deformation, and, from a physical point of view, new surface operators associated with higher order singularities.

Deformations of Covers, Brill-noether Theory, and Wild Ramification

In this paper, we give a simple description of the deformations of a map between two smooth curves with partially prescribed branching, in the cases that both curves are fixed, and that the source is allowed to vary. Both descriptions work equally well in the tame or wild case. We then apply this result to obtain a positive-characteristic Brill-Noether-type result for ramified maps from general...

Models of Curves and Wild Ramification

Dedicated to John Tate Abstract: Let K be a complete discrete valuation field with ring of integers OK and residue field k of characteristic p ≥ 0, assumed to be algebraically closed. Let X/K denote a smooth proper geometrically connected curve of genus g ≥ 1, and let X/OK denote its minimal regular model. When g ≥ 2, or g = 1 and X(K) 6= ∅, there exists a finite Galois extension L/K minimal wi...

Wild ramification of schemes and sheaves

We discuss recent developments on geometric theory of ramification of schemes and sheaves. For invariants of -adic cohomology, we present formulas of Riemann-Roch type expressing them in terms of ramification theoretic invariants of sheaves. The latter invariants allow geometric computations involving some new blow-up constructions. Mathematics Subject Classification (2000). Primary 14F20; Seco...

Lectures on Field Theory and Ramification Theory