The upcoming talk
At the next session of the TCS,
on the 9th of April 2025,
at 2:15 – 4:00 p.m. CET / 3:15 – 5:00 p.m. EET
Arthemy V. Kiselev
(Bernoulli Institute of Mathematics, Rijksuniversiteit Groningen)
shall talk about
“The differential graded Lie algebra of undirected graphs“.
Abstract
Like matrices or vector fields, undirected graphs with wedge ordering of edges form a Lie algebra. In [arXiv:1710.00658] we recall this intuitive construction (designed by Kontsevich in 1993-4 in the context of mirror symmetry), and we establish that it is well defined [arXiv:1811.10638]. Indeed, a graph can equal minus itself so that the entire calculus goes modulo zero graphs.
A structure of differential graded Lie algebra (dgLa) on the vector
space of graphs modulo equivalence relation from the edge count is provided by the vertex-expanding differential [*-*,.]. The study of cocycles located on and near the ray (#V,#E)=(n,2n-2) is a domain of active research (Willwacher et al., Merkulov et al., Kontsevich) because countably many cocycles on the ray (n,2n-2) stem from the generators of Grothendieck-Teichmüller Lie algebra g (introduced by Drinfeld). Every such cocycle at n=(2l+1)+1 contains a (2l+1)-gon wheel with nonzero coefficient. Knowing these cocycles (and their iterated commutators, which generate a free Lie algebra) is important: the graph orientation morphism [arXiv:1811.07878, and arXiv:1904.13293 in Banach Center Publ.] takes them to universal infinitesimal symmetries of Poisson brackets on arbitrary finite-dimensional affine Poisson manifolds: e.g., see [arXiv:1608.01710]. We illustrate the calculus of graphs by using the tetrahedron (l=1), pentagon-wheel cocycle (l=2), and heptagon-wheel cocycle (l=3), see [arXiv:1710.00658].
Examples and substantiation proofs are joint work with R. Buring and N.J. Rutten [arXiv:1811.10638]. (This material can be used as student exercises in general algebra courses, e.g., group theory.)
(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) | TBA |
| April 23rd, 2025 | Rafał R. Suszek (KMMF WFUW) | (Constructions with) Principaloid Bundles |
| April 30th, 2025 | ||
| May 7th, 2025 | ||
| May 14th, 2025 | ||
| May 21st, 2025 | Nils Carqueville (Universität Wien) | Higher structures in topological quantum field theory |
| May 28th, 2025 | ||
| June 4th, 2025 | ||
| June 11th, 2025 |