Isoresidual fibration and resonance arrangements
Organizer
Speaker
Time
Wednesday, March 13, 2024 1:30 PM - 3:30 PM
Venue
A3-1-101
Online
Zoom 928 682 9093
(BIMSA)
Abstract
Meromorphic 1-forms on the Riemann sphere with prescribed orders of singularities form strata endowed with period coordinates. Fixing residues at the poles defines a fibration of any stratum to the vector space of configurations of residues. In a joint work with Quentin Gendron, it has been proved that for strata of 1-forms with only one zero, the isoresidual fibration is a cover of the space of configurations of residues ramified over an arrangement of complex hyperplanes called the resonance arrangement. Using combinatorics of decorated tree and the dictionary between complex analysis and flat geometry, we give a formula to compute the degree of this cover and investigate its monodromy. In a more recent work with Dawei Chen, Quentin Gendron and Miguel Prado, we investigate the case of strata with two zeroes where isoresidual fibers are complex curves endowed with a canonical translation structure. Singularities of this structure provide topological invariants of the fibers that refine the Euler characteristic and still lack an interpretation in terms of enumerative geometry.
Speaker Intro
Guillaume Tahar obtained his Ph.D from Université Paris Diderot, under the supervision of Anton Zorich. He was a senior postdoctoral fellow in Weizmann Institute of Science and joined BIMSA as an assistant professor in 2022. He contributed to the study of moduli spaces of various flavours of geometric structures on surfaces. His results include proving the existence of closed geodesics in dilation surfaces and the complete characterization of configurations of local invariants realized by a differential on a Riemann surface. His recent research interests include linear differential operators, simplicial arrangements of lines and quantum invariants of knots.