Stiff connections on pseudo-Euclidean spaces
Organizers
Kotaro Kawai
, Chao Qian
Speaker
Time
Wednesday, May 17, 2023 3:20 PM - 4:20 PM
Venue
A3-1-103
Online
Zoom 928 682 9093
(BIMSA)
Abstract
Unless it is a flat connection, an affine connection cannot be at the same time projectively flat (geodesics being straight lines) and conformal (conformal structure being preserved by parallel transport). This impossibility is examplified by the two standard models of hyperbolic geometry: Beltrami and Poincaré models.
We define stiff connections by weakening the conformality hypothesis to the requirement that the first order infinitesimal holonomies are infinitesimal isometries. In a given pseudo-Euclidean space, stiff connections are characterized by the choice of a potential and form a continuous family of non-flat connections with surprising properties.
In particular, we prove the existence of a unique affine connection on the disk that is geodesically complete, infinitesimally conformal and projectively flat. This uniquely characterized connection achieves a compromise between properties of Beltrami and Poincaré models of the disk.
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.