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Time and venue: Wednesday 3pm, Roger Stevens LT12. Coffee/Tea served in the Common Room after 4pm. Programme: 24 January 2018 Roger Moser (Bath) Geometric structures arising from a variational model for crystal surfaces Abstract. The free energy of a crystal surface can be modelled by an anisotropic area functional, but a model proposed by Herring adds a curvature term as well. The sum of the two can be regarded as a singular perturbation problem and an asymptotic analysis then leads to a limiting variational problem. Here the crystal surfaces are represented by (generalised) polyhedra, and the free energy gives rise to a functional depending on the lengths of the edges. We discuss aspects of this asymptotic analysis and some consequences for the geometric structures emerging thereby. 31 January 2018 Melanie Rupflin (Oxford) Flowing to minimal surfaces Abstract. For maps from surfaces there is a close connection between the area functional and Dirichlet energy and thus also between their critical points. As such, one way to try to find minimal surfaces is to consider a gradient flow of the Dirichlet energy, which not only evolves a map but also the domain metric in order to find a map that is not only harmonic but also (weakly) conformal and thus a (branched) minimal immersion. In this talk I will discuss the construction of such a flow, the Teichmueller harmonic map flow, and explain in particular how this flow decomposes any given initial map from a closed surface into minimal surfaces. This is joint work with Peter Topping. 7 February 2018 Weiyi Zhang (Warwick) Intersection of almost complex manifolds and pseudoholomorphic maps Abstract. An almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex or symplectic manifold is an almost complex manifold, but not vice versa. Transversality is the notion of general position in manifold topology. If two submanifolds intersect transversely in some ambient manifold, then their intersection is a manifold. We will discuss differential topology of almost complex manifolds, explain how to use transversality statements for smooth manifolds to formulate and prove corresponding results for an arbitrary almost complex manifold. The examples include intersection of almost complex manifolds, pseudoholomorphic maps and zero locus of certain harmonic forms. One of the main technical tools is Taubes' notion of "positive cohomology assignment", which plays the role of local intersection number. I will begin with explaining its motivation through multiplicities of zeros of a smooth function. 14 February 2018 Richard Webb (Cambridge) The conjugacy problem for the mapping class group in polynomial time Abstract. Fix S an orientable surface of finite type. We show that the conjugacy problem for the mapping class group of S can be solved in polynomial time. I will explain how we prove this and its context in low-dimensional geometry and geometric group theory. Joint work with Mark Bell. 2 March 2018 Yorkshire and Durham Geometry Day (to be held at the University of Durham) 18 April 2018 Thomas Waters (Portsmouth) Focusing geodesics -- geometry and topology Abstract. Geodesics are the straight lines of curved surfaces. If we consider a spray of geodesics emanating from a certain point then, due to the curvature of the surface, the geodesics can focus along curves. These focusing curves are variously called caustics, envelopes or the conjugate locus, and they can be terrifically complex. In particular they have spikes or cusps, and they can fold up on themselves. We will show there is a simple relationship between the number of cusps and the rotation index of the conjugate locus on convex surfaces, and how we can use this to then prove other strong results. Finally we will consider the extension to focusing of geodesics in 3-dimensional manifolds and some early results. 25 April 2018 Viveka Erlandsson (Bristol) Counting Curves on surfaces Abstract. Let S be a closed surface and consider all curves on S which self-intersect a bounded number of times. Due to Mirzakhani we know the asymptotic growth of the number of such curves whose hyperbolic length is bounded by L, as L grows. More specifically, if S has genus g and is equipped with a hyperbolic structure, she showed that the number of such curves on S (in each mapping class group orbit) is asymptotic to a constant times L^{6g-6}. In this talk I will explain, through the use of geodesic currents, why the same asymptotics hold for other metrics on the underlying topological surface, in particular for any Riemannian metric. 2 May 2017 Yorkshire and Durham Geometry Day at Leeds 23 May 2018 Nuno Romão (IHES) The geometry of the space of BPS vortex-antivortex pairs MALL1 (unusual room) Abstract. The vortex equations describe BPS configurations in gauged sigma-models on surfaces. Their moduli spaces support Kähler metrics that encode crucial information about the underlying field theories, at both classical and quantum level. An interesting setting is when the target of these field theories is nonlinear -- i.e. a Kähler manifold with holomorphic and Hamiltonian action which does not simply correspond to a group representation. This setup gives rise to interesting phenomena that are not present in more familiar field theory models that it interpolates, namely, the sigma-model (trivial group) and the gauged linear sigma-model (linear action). Examples of such phenomena are the coexistence of more than one type of solitonic "particle" within the same BPS configuration, and the emergence of boundaries on the moduli spaces that correspond to coalescence of different BPS particles. In my talk, I will report on joint work with Martin Speight, describing very concrete results for the asymptotics of the moduli space metrics, when the target is the 2-sphere with usual circle action. |