On the topology of isoresidual fibers
Organizer
Speaker
Time
Thursday, March 13, 2025 1:30 PM - 2:30 PM
Venue
A3-2a-201
Abstract
Meromorphic 1-forms on the Riemann sphere, with prescribed orders of singularities, form strata equipped with period coordinates. Fixing the residues at the poles defines the isoresidual fibration of any given stratum onto the vector space of residue configurations.
In a joint work with Dawei Chen, Quentin Gendron and Miguel Prado, we show that for strata with two zeroes, isoresidual fibers are complex curves endowed with a canonical translation structure. The singularities of these fibers encode, through their local invariants, the corresponding degenerations of the parametrized objects in the multi-scale boundary. In particular, computing the genus of the underlying complex curve is reduced to a purely combinatorial problem.
As an application, we provide the classification of connected components of generic isoresidual fibers for strata with an arbitrary number of zeroes.
Speaker Intro
Guillaume Tahar obtained his Ph.D from Université Paris Diderot, under the supervision of Anton Zorich. He was a senior postdoctoral fellow at the Weizmann Institute of Science and joined BIMSA as an Assistant Professor in 2022. His research focuses on geometric structures on surfaces, with applications to moduli spaces and dynamical systems. He contributed to the study of various flavours of geometric structures, including translation surfaces, polyhedral metrics, cone spherical metrics and complex affine structures. His approach typically involves a mix of complex analysis, geometric constructions, and combinatorial reasoning.
His key results include the proof of the existence of closed geodesics in dilation surfaces, the complete characterization of configurations of local invariants realized by a differential on a Riemann surface and the establishment of Grünbaum's asymptotic classification for simplicial line arrangements with few double points.
His recent research interests include the topological interpretation of quantum invariants of knots, the counting of BPS states in quantum field theory and holomorphic dynamics in higher dimensions.
His key results include the proof of the existence of closed geodesics in dilation surfaces, the complete characterization of configurations of local invariants realized by a differential on a Riemann surface and the establishment of Grünbaum's asymptotic classification for simplicial line arrangements with few double points.
His recent research interests include the topological interpretation of quantum invariants of knots, the counting of BPS states in quantum field theory and holomorphic dynamics in higher dimensions.