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