Monday 28 January 2019, 1.10PM to 4:20pm

This research network brings together UK academics who are interested in applications of algebra and geometry, and related algebraically-minded fields, be it commutative algebra, representation theory, group theory, process algebras, as well as algebraic geometry, category theory, and algebraic topology. The scope extends to computational algebra for applications including data-science, biology, medicine, engineering, physics, etc.

The activities of the Applied Algebra and Geometry research network are funded by the London Mathematical Society, the Edinburgh Mathematical Society, and the Glasgow Mathematical Journal Trust. Plesae click here for the research networkds webpage.

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

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)

**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.

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

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 common room.

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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**.

**Location:** Department of Mathematics, University of York, room G/N/020