British and Irish Geometry Meeting (BI-GeM) 2018
The British and Irish Geometry Meeting will take place at the Mathematical Sciences Research Centre, School of Mathematics and Physics, Queen’s University Belfast, on 1^{st} and 2^{nd} of June 2018.
The conference programme (schedule, titles and abstracts) can be found below.
Invited speakers:
- Barbara Baumeister, Universität Bielefeld (Germany), The smallest non-abelian quotient of Aut(F_{n})
- Jürgen Berndt, King’s College London (UK), Symmetries in Riemannian geometry
- Tom Brady, Dublin City University (Ireland), Triangulating the permutahedron
- John Burns, National University of Ireland in Galway (Ireland), Graded Lie Algebras – their representations and applications
- Lynn Heller, Universität Hannover (Germany), Recent progress in integrable surface theory
- Jarek Kędra, University of Aberdeen (UK), On qualitative counting of closed geodesics
- Mary Rees, University of Liverpool (UK), An example of puzzles and parapuzzles in complex dynamics
Registration is now open: https://knock.qub.ac.uk/ecommerce/bigm/
The registration fee is £20 (for PhD students: £15), payable by credit card.
Travel to Belfast: There is helpful information on the QUB travel web page. Flying to Dublin is sometimes easier (and cheaper) than flying to Belfast. There are regular bus connections from Dublin Airport to Belfast and back (travel time around two hours; see for example Aircoach or Translink buses).
Accommodation:Participants are required to make their own arrangements.
Schedule
Abstract can be found further down the page.
Friday | 09:00-09:30 | Registration | |
| 09:30-10:30 | John Burns | Graded Lie Algebras - their representations and applications |
| 10:30-10:50 | Coffee break | |
| 10:50-11:20 | Tasha Montgomery | On the projective line associated with a Z-graded ring |
| 11:30-12:30 | Barbara Baumeister | The smallest non-abelian quotient of Aut(F_{n}) |
| 12:30-15:00 | Conference Lunch | |
| 15:00-15:30 | Sungkyung Kang | A transverse knot invariant from Z/2-equivariant Heegaard Floer cohomology |
| 15:40-16:10 | Ittay Weiss | A metric formalism for topology with a view to persistence theory |
| 16:20-17:20 | Tom Brady | Triangulating the permutahedron |
| 17:30-18:00 | Danny Sugrue | Rational G-Mackey functors for G profinite |
Saturday | 09:00-09:30 | Qays Shakir | Contacts of Circular Arcs Representation of Certain Torus Graphs |
| 09:40-10:40 | Jürgen Berndt | Symmetries in Riemannian geometry |
| 10:40-11:00 | Coffee break | |
| 11:00-12:00 | Lynn Heller | Recent progress in integrable surface theory |
| 12:00-13:30 | Lunch break | |
| 13:30-14:00 | Florian Pausinger | On lattices and their shortest vectors |
| 14:10-15:10 | Jarek Kędra | On qualitative counting of closed geodesics |
| 15:10-15:30 | Coffee break | |
| 15:30-16:30 | Mary Rees | An example of puzzles and parapuzzles in complex dynamics |
Ttitles and abstracts (click on title to see abstract)
Barbara Baumeister, Universität Bielefeld (Germany): The smallest non-abelian quotient of Aut(F_{n})
The non-abelian finite simple group L_{n}(2) is a quotient of Aut(F_{n}): factor out F’_{n} and then reduce modulo 2. In the talk I will confirm the conjecture by Mecchia-Zimmermann that this is the smallest non-abelian finite quotient of Aut(F_{n}). On the way some other nice and new results will appear. This is joint work with Dawid Kielak and Emilio Pierro.
Jürgen Berndt, King's College London (UK): Symmetries in Riemannian geometry
Symmetry is one of the fundamental concepts in geometry. In the talk I plan to give a survey about some old and new results in Riemannian geometry involving continuous symmetries. In the first part of the talk I will motivate some concepts involving symmetry. This will lead us to homogeneous spaces and symmetric spaces, which were studied thoroughly by Felix Klein and Élie Cartan respectively. The modern and quite general question I plan to discuss is: What spaces are close to homogeneous spaces and symmetric spaces, and why might they be of interest?
Tom Brady, Dublin City University (Ireland): Triangulating the permutahedron
For an Artin group A(W) of finite type W, we construct a homotopy equivalence from the A(W) classifying space of Salvetti to the one defined by noncrossing partitions. The construction involves the type-W asssociahedron. This is joint work with Emanuele Delucchi and Colum Watt.
John Burns, NUI Galway (Ireland): Graded Lie Algebras - their representations and applications
Let M=G/P be a rational homogeneous manifold, with P a maximal parabolic subgroup of a complex simple Lie group G. Viewing the Lie algebra of G as a graded Lie algebra in a natural way, we use some simple representation theory to give uniform (for all complex simple G) formulae for the dimension of M, the dimensions of the irreducible factors of the restriction of the isotropy representation to a Levi subgroup of P and the nef values of homogeneous line bundles on M. We also give a selection of applications of our results. This is joint work with Adib Makrooni.
Lynn Heller, Universität Hannover (Germany): Recent progress in integrable surface theory
I consider surfaces in 3-space which are critical with respect to certain geometric variational problems, such as CMC and minimal surfaces and (constrained) Willmore surfaces. In this talk I want to give an overview on recent results on the construction of new examples of higher genus CMC surfaces and on the identification of constrained Willmore minimizers in the class of conformal tori. Moreover, by viewing minimal surfaces in different space forms within the constrained Willmore integrable system, counterexamples to a question of Simpson are constructed. This suggests a deeper connection between Willmore surfaces, i.e., rank 4 harmonic maps theory, and the rank 2 theory of Hitchin’s self-duality equations. This talk is based on joint work with Cheikh Birahim Ndiaye, Sebastian Heller and Nicholas Schmitt.
Sungkyung Kang, University of Oxford (UK): A transverse knot invariant from Z/2-equivariant Heegaard Floer cohomology
A Z/2-equivariant Heegaard Floer cohomlogy of based double coverings of S^{3} along a based knot, defined by Lipshitz, Hendricks and Sarkar, is a well-defined isomorphism class of F_{2}[θ]-modules. In this talk, we will see why this invariant is a natural invariant, and is functorial under based cobordisms. Then we will observe that, given a based transverse knot K in the standard contact S^{3}, we have a well-defined element in the Z/2-equivariant Heegaard Floer cohomology, which depends only on the tranverse isotopy class of K, and this element is functorial under certain class of symplectic cobordisms.
Jarek Kędra, University of Aberdeen (UK): On qualitative counting of closed geodesics
Let (X,d) be a geodesic metric space, e.g., a complete Riemannian manifold. We consider closed geodesics passing through the basepoint * in X and ask the following questions: Do they generate the fundamental group of X? If yes, then how fast? We measure the speed of generation as follows. If there exists a number C>0 such that every element of the fundamental group of X is a concatenation of at most C closed geodesics then we say that this is fast generation. On the other hand, if no such number C exists then the generation is slow. I will present various examples and show how to answer the above questions in certain cases. This is a joint work with Michał Marcinkowski.
Tasha Montgomery, Queen's University Belfast: On the projective line associated with a Z-graded ring
It is known that the K-theory of the projective line over an arbitrary commutative ring splits into two copies of the K-theory of the ground ring. This was generalised, by Bass and Quillen, to non–commutative rings. My aim for this talk is to give a further generalisation, by considering a projective line associated to a graded ring. The process, perhaps surprisingly, works much like in the “classical” case, however new phenomena are quickly encountered. For example, the familiar family of twisting sheaves from algebraic geometry now depends on a two-parameter construction as opposed to just one. This work is part of my ongoing PhD thesis project under the supervision of Dr Thomas Hüttemann.
Florian Pausinger, Queen's University Belfast: On lattices and their shortest vectors
The hexagonal lattice gives the highest density circle packing among all lattices in the plane. In this talk I first recall the basic notions about lattices in the plane, before I construct a sequence of lattices with integer bases that approximate the hexagonal lattice. The construction uses elementary number theory and is based on particular palindromic continued fraction expansions. As an application I obtain lattices modulo $N$ with longest possible shortest distances.
Mary Rees, University of Liverpool (UK): An example of puzzles and parapuzzles in complex dynamics
Topologically, a puzzle (in complex dynamics) is a sequence of successively larger finite graphs on the Riemann sphere. Dynamics, and the iterative definition of the graohs is given by a holomorphic map f which, for present purposes, we will take to be a rational map of the Riemann sphere. Then the first graph in the sequence, say G_{0}, satisfies G_{0}⊂ f^{-1}(G_{0}). We then define G_{n}=f^{-n}(G_{0}), so that G_{n}⊂ G_{n+1} for all n. Puzzles tend to be locally persistent. For instance if f=f_{0} is in a family of maps f_{λ} parametrised by an open subset Λ of the complex plane with 0∈ Λ then it often happens (and can be proved) that G_{0} can be isotoped to a graph G_{0}(λ ) with G_{0}(λ )⊂ f_{λ}^{-1}(G_{0}(λ)), at least for λ near 0. We can then define G_{n}(λ)=f_{λ}^{-n}(G_{0}(λ)). It is then not usually true that the graphs G-n(λ) are all isotopic. But the way in which the graphs change can often be recorded in a parapuzzle.
The most famous puzzles and parapuzzles are the Yoccoz puzzles and parapuzzles, so-called after J-C Yoccoz made important advances in a conjecture called MLC using them, in the 1980’s. The aim is to look at these famous examples brefly and discuss how the ideas can aply to other situations and more generally.
Qays Shakir, NUI Galway (Ireland): Contacts of Circular Arcs Representation of Certain Torus Graphs
We will discuss representations of surface graphs as contact graphs of configurations of circular arcs. In this representation, vertices of the graphs are represented by circular arcs in surfaces of constant curvature while their edges are represented by the contacts of circular arcs. We first review some previously known results for contact representations in the plane. Then we show that every (2,2)-tight torus graph can be represented by a circular arc configuration in the flat torus. This work forms part of a joint project with James Cruickshank, Derek Kitson and Stephen Power.
Ittay Weiss, University of Portsmouth (UK): A metric formalism for topology with a view to persistence theory
Topological Data Analysis (TDA) employs topological techniques to identify geometric features in data involving, for instance, clustering problems and the persistence of phenomena to distinguish between relevant information and noise. The motivating philosophy behind topological approaches to geometric problems involves the inherent blindness of topology to certain metric issues such as dimensionality (of data). This apparent clash of philosophies embodied by TDA is present at the outset: the techniques of TDA are topological but require a metric. Based on work of Kopperman and Flagg from around 1990 it will be shown that there is a suitable generalisation of the concept metric space giving rise to a genuinely metrically flavoured formalism equivalent to topology. This metric formalism for topology will be explored with a heightened emphasis on metric geometrical aspects.
The conference is supported by the London Mathematical Society and the Irish Mathematical Society.
Conference organisers: Thomas Hüttemann and Brian McMaster.
Last updated: 2018-05-31