Date:

28 January 2019

Location:

Department of Mathematics, University of York, room G/N/020 on the ground floor.

Local organiser

Emilie Dufresne, University of York

Schedule

13:10 - 14:00 Felipe Rincón (Queen Mary University of London)

14:00 - 14:30 Coffee/tea break

14:30 - 15:20 Elizabeth Mansfield (University of Kent)

15:30 - 16:20 Evita Nestoridi (University of Cambridge)

Abstracts

Felipe Rincón: Tropical Geometry.

Tropical geometry is geometry over the tropical semiring, where multiplication is replaced by addition and addition is replaced by minimum. One can "tropicalise" algebraic varieties in this way and get polyhedral complexes as a result, which can be combinatorially studied. In this talk I will give a gentle introduction to this topic, and present a couple of applications in different fields of mathematics.

Elizabeth Mansfield: On Poisson Structures arising from a Lie group action.

Let M be a manifold on which a Lie group G acts, and let L be its Lie algebra. In this talk I will discuss two Lie algebras of sections of a bundle M x L which appeared in the paper [1]. Since these Lie algebras of sections are in fact Lie algebroids, we may use a theorem of Marle [2], to obtain finite dimensional Poisson structures on the space of smooth real-valued functions on M x L*, where L* is the dual of L. The Poisson bracket associated to the second Lie bracket extends and generalises the Lie Poisson bracket acting on the space of smooth real-valued functions on L*. Further, if the Lie group and the manifold are both the n-dimensional real vector space acting by translations on itself, then the standard (Darboux) symplectic structure is obtained. Results include a study of the associated Hamiltonian flows and their invariants, canonical maps induced by the Lie group action, and compatible Poisson structures. Plots of examples of Hamiltonian flows will be given and I will show how the orbits compare to those of the associated Lie Poisson Hamiltonian flow. The approach is computational and example led. The Lie brackets from which our results derive, arose from a consideration of connections on bundles with zero curvature and constant torsion. An alternate derivation of the second Lie bracket will be shown. By examining central extensions of the Lie algebra of sections, infinite dimensional Poisson structures can be written down. Time permitting, I will discuss the infinite dimensional case.

This is joint work with Gloria Mari Beffa.

[1] H. Munthe–Kaas, A. Lundervold, On post-Lie algeras, Lie–Butcher series and Moving Frames, Foundations of Computational Mathematics, 13 (2013), 583–613.

[2] C–M. Marle, Differential calculus on a Lie algebroid and Poisson manifolds. In: The J.A. Pereira Birthday Schrift, Textos de matematica 32, Departamento de matematica da Universidade de Coimbra, Portugal (2002), 83–149. Available from https://arxiv.org/abs/0804.2451.

Evita Nestoridi: Using algebra to bound mixing times.

Markov chains are random processes that retain no memory of the past. The mixing time of a Markov chain is the time it takes for it to reach equilibrium. Each entry of the transition matrix of the Markov chain captures the probability of moving form one state to the other after one step. We will explain how one can use the spectrum of the transition matrix to bound mixing times and how representation theory is useful in order to find the eigenvalues of a random walk on a group. We will present recent results concerning famous problems, that have been proven using these algebraic techniques.

Visitor information

The talks will be in room G/N/020 in the Mathematics Building, which is part of James College (here is a link to the interactive campus map). Refreshments will be served in the same room. Bus 66 from the Railway station will take you to the University campus in about 20 minutes. Here is a map indicating the walking path from the bus top to the Department as well as the precise location of the room G/N/020 and a walking path to the John Kirk Centre where you'll find one of the Campus restaurants and a Costa Cafe.

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Sponsors:

We are grateful for the financial support from the Glasgow Mathematical Journal Learning and Research Support Fund, from the Edinburgh Mathematical Society, the London Mathematical Society.