The upcoming talk


At the next session of the TCS,

Łukasz Rudnicki

(Uniwersytet Gdański)

shall talk about

Basically all linear second order ordinary differential equations can be “solved” by geodesic curves in two dimensional hyperbolic geometry“.


Abstract

I will present two preprints [arXiv:2503.17816] and [arXiv:2503.19415], devoted to a basic equation of the form u”(x) + h(x) u(x) = 0. I will start with a discussion about the generality of this equation and then introduce a family of metrics on an upper half plane, defined in terms of the function h(x). The sectional curvature of all these metrics is equal to -1, thus, they locally describe the same hyperbolic geometry. As the main result, I will present a general solution of the equation in question expressed in terms of geodesic curves in the proposed two dimensional hyperbolic model. I will also show that an arbitrary pair of linearly independent solutions gives a diffeomorphism between the introduced geometry and the Poincaré upper half-plane. In the second part, I will discuss an equivalent approach in which the Riemannian geometry is replaced by an analogous Lorentzian geometry. Interestingly, it turns out that solutions of the associated Ricatti equation are at the same time geodesics in this second model. In the last part, I will resort to complex Riemannian geometry with a holomorphic metric, in which I generalize previous results to the complex case with a holomorphic function h(z). Interestingly, this last geometry is locally diffeomorphic to a complex sphere with imaginary radius.


(All seminar talks are given on the online zoom platform. Unless requested otherwise by the speaker or required by the circumstances, and then specifically indicated in the announcement, the talks consist of two halves, 45 minutes each, separated by a 15-minute break. The web link for every session is circulated via the TCS mailing list. In order to be enlisted, please contact the Organisers.)


Calendar 2024/2025


DateSpeaker (affiliation)Title (with a link to the slides, whenever available)
October 9th, 2024Thomas Strobl (ICJ, Université Claude Bernard Lyon 1)From Gauge Theories to the octonionic Lie groupoid
October 16th, 2024Calin I. Lazaroiu (IFIN-HH, București & UNED, Madrid)Consistency conditions and fiducial 2-field models for SRRT inflation
October 23rd, 2024Alexander Schmeding (NTNU, Trondheim) Linking infinite-dimensional and higher differential geometry
October 30th, 2024Konrad Waldorf (Universität Greifswald)Buscher rules in general topology
November 6th, 2024Mikołaj Rotkiewicz (MIMUW, Warsaw)Exploring the structure of higher algebroids
November 13th, 2024No seminar on this date.
November 20th, 2024Participation in an online lecture by Alain Connes (Collège de France & IHES, Paris) at IM PASFrom class field theory to ζ spectral triples
November 27th, 2024Marian Aprodu (IMAR, București)Resonance and vector bundles
December 4th, 2024Carlos Shahbazi (UNED, Madrid)A class of differential-geometric problems in supergravity
December 11th, 2024No seminar on this date.
December 18th, 2024Anton Alekseev (Université de Genève)Jackiw-Teitelboim gravity, Teichmüller spaces, and Virasoro algebra
January 8th, 2025Ingo Runkel (Universität Hamburg)Topological symmetries and their gaugings in 2dCFT and 3dTFT
January 15th, 2025No seminar on this date.
January 22nd, 2025Peter Kristel (Universität Greifswald)An Eclectic Excursion into 2-Categorical Geometry
February 26th, 2025No seminar on this date.
March 5th, 2025Katarzyna Grabowska (KMMF WFUW)Dirac algebroids in Hamiltonian and Lagrangian dynamics – theory and examples
March 12th, 2025Asier López-Gordón (IM PAN)Homogeneous symplectic manifolds and integrable contact systems
March 19th, 2025Leonid Ryvkin (ICJ, Université Claude Bernard Lyon 1)Introduction to singular foliations
March 26th, 2025Witold Respondek (INSA Rouen Normandie) Linearization of mechanical control systems
April 2nd, 2025Bartosz Prech (FUW)Symplectic approach to contact integrators
April 9th, 2025Arthemy V. Kiselev (BIM,
Rijksuniversiteit Groningen)
The differential graded Lie algebra of undirected graphs
April 16th, 2025Tilmann Wurzbacher (IECL Metz)Geometric prequantisation: from symplectic to two-plectic manifolds
April 23rd, 2025Rafał R. Suszek (KMMF WFUW)(Natural Constructions with) Principaloid Bundles
April 30th, 2025Casey A. Blacker (Augusta University)Geometric and algebraic reduction of multisymplectic manifolds
May 7th, 2025No seminar on this date.
May 14th, 2025Calin I. Lazaroiu (IFIN-HH, București & UNED, Madrid)Dynamical consistency conditions for strong rapid turn inflation
May 21st, 2025Nils Carqueville (Universität Wien)Higher structures in topological quantum field theory
May 28th, 2025Łukasz Rudnicki (Uniwersytet Gdański)Basically all linear second order ordinary differential equations can be “solved” by geodesic curves in two dimensional hyperbolic geometry
June 4th, 2025
June 11th, 2025

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