BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Marian Mrozek (Jagiellonian University\, Poland) DTSTART:20210622T140000Z DTEND:20210622T150000Z DTSTAMP:20260423T024743Z UID:CRM-CAMP/47 DESCRIPTION:Title: Combinatorial Topological Dynamics\nby Marian Mrozek (Jagiellonian U niversity\, Poland) as part of CRM CAMP (Computer-Assisted Mathematical Pr oofs) in Nonlinear Analysis\n\n\nAbstract\nSince the publication in 1998 o f the seminal work by Robin Forman on combinatorial Morse theory there has been growing interest in dynamical systems on finite spaces. The main mot ivation to study combinatorial dynamics comes from data science. But\, the y also provide very concise models of dynamical phenomena and show some po tential in certain computer assisted proofs in dynamics.\n\nIn the talk I will present the basic ideas of Conley theory for combinatorial dynamical system\, particularly for a combinatorial multivector field which is a gen eralization of combinatorial vector field introduced by Forman. The theory is based on concepts which are analogous to the concepts of classical the ory: isolating neighborhood\, isolated invariant set\, index pair\, Conley index\, Morse decomposition\, connection matrix. The concepts are analogo us but in some cases surprisingly different in details. This may be explai ned by the non-Hausdorff nature of combinatorial topological spaces.\n\nDe spite the differences there seem to be strong formal ties between the comb inatorial and classical dynamics on topological level. A Morse decompositi on of a combinatorial vector field on an abstract simplicial complex induc es a semiflow on the geometric realization of the complex with a Morse dec omposition exhibiting the same Conley-Morse graph. Actually\, this corresp ondence of Morse decompositions and Conley-Morse graphs applies to every s emiflow which is transversal to the boundaries of top dimensional cells of a certain cellular decomposition of the phase space associated with the c ombinatorial vector field.\n\nThere is also a formal relation in the oppos ite direction. Given a smooth flow and a cellular decomposition of its pha se space which is transversal to the flow\, there is an induced combinator ial multivector field on the cellular structure of the phase space. Moreov er\, if the induced combinatorial multivector field admits a periodic traj ectory with an appropriate Conley index\, a periodic orbit exists also for the original smooth flow.\n\nThe formal ties seem to provide a natural fr amework for a rigorous global analysis of the dynamics of a flow: the deco mposition into the gradient and recurrent part together with the computati on of the Conley-Morse graph\, connection matrix and revealing the interna l structure of the recurrent part.\n\nBased on joint work with J. Barmak\, T. Dey\, M. Juda\, T. Kaczynski\, T. Kapela\, J. Kubica\,  M. Lipiński   R. Slechta\, R. Srzednicki\, J. Thorpe and Th. Wanner.\n LOCATION:https://researchseminars.org/talk/CRM-CAMP/47/ END:VEVENT END:VCALENDAR