3:00pm - 5:00pmRiemann Surfaces
Chair(s): Daniele Agostini (Humboldt-Universität), Türkü Özlüm Çelik (Max Planck Institute for Mathematics in the Sciences), Christian Klein (Institut de Mathématiques de Bourgogne), Emre Can Sertöz (Max Planck Institute for Mathematics in the Sciences)
In the past decades, the central role played by Riemann surfaces in pure mathematics has been strengthened with their surprising appearance in string theory, cryptography and material science. This minisymposium is intended for the curve theorists and the avant-garde applied mathematician. Our emphasis will be on the computational aspects of Riemann surfaces that are prominent in pure mathematics but are not yet part of the canon of applied mathematics. Some of the subjects that will be touched upon by our speakers are integrable systems, Teichmüller curves, Arakelov geometry, tropical geometry, arithmetic geometry and cryptography of curves.
(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)
Real soliton lattices of KP-II equation and desingularization of spectral curves
Simonetta Abenda
Università di Bologna
Planar bicolored (plabic) networks in the disk were originally introduced by A.Postnikov to parametrize positroid cells in totally nonnegative Grassmannians and used by Y. Kodama and L.Williams to explain the asymptotic behavior of real regular multiline soliton solutions (rrss) of Kadomtsev-Petviashvili II (KP) equation.
In this talk based on recent papers in collaboration with P.G. Grinevich (arXiv:1801.00208, arXiv:1803.10968, arXiv:1805.05641) we explain a different relation of plabic networks with KP theory based on the spectral theory for degenerate finite-gap solutions on reducible curves by Krichever where the rrss play the role of potentials.
In our construction the plabic graph is dual to a reducible curve which is the rational degeneration of a smooth M-curve of genus equal to the number of faces of the graph diminished by one. The boundary of the disk corresponds to the rational curve associated to the soliton data in the direct spectral problem and each internal vertex is a rational component. Edges are the double points where two such components are glued. We then introduce and characterize systems of edge vectors on plabic networks and use them to uniquely associate to generic soliton data a Krichever divisor satisfying the reality and regularity conditions of Dubrovin-Natanzon.
Our approach is constructive and may be used to effectively desingularize curves. The case of soliton data in the positive part of Gr(2,4) is shown in detail.
Conformal patterns on closed surfaces via discrete conformal maps and holomorphic differentials
Alexander I. Bobenko
Technische Universität Berlin
Using uniformization of discrete Riemann surfaces we construct conformal patterns on closed surfaces without cuts and overlapping.
Arakelov invariants in the tropical limit
Robin de Jong
University of Leiden
In this talk we are interested in semistable degenerations of compact Riemann surfaces. We consider the asymptotic behavior, under such degenerations, of certain canonical metrics, Green's functions and related invariants as studied in Arakelov theory and string perturbation theory. We obtain precise expressions for the asymptotics under consideration in terms of potential theory on metric graphs, aka tropical curves. In particular, non-archimedean and tropical geometry appear naturally when studying degenerations of Riemann surfaces.
Siegel modular forms and classical invariants
Christophe Ritzenthaler
University Rennes 1
For abelian varieties of dimension 2 and 3, Siegel modular forms for the full symplectic group can be reinterpreted as classical invariants for the action of GL2 or GL3. There are many applications and consequences of this dictionary. We will show in particular that one can decrease the number of generators for the ring of Siegel modular forms in dimension 3 obtained by Tsuyumine (1986). Joint work with Reynald Lercier.
