### Irish Geometry Conference 2019 - Titles and Abstracts

**Speaker:** Wilhelm Klingenberg

**Title:** On a conjecture of Toponogov on complete convex surfaces

**Abstract :**

In 1995, Victor Andreevich Toponogov authored the following conjecture: “Every smooth strictly convex and complete classical surface of the type of a plane has an umbilic point, possibly at infinity“. He proceeded to prove this in case the gradients of mean and Gauss curvatures on the surface under consideration are bounded. In our talk, we will outline a proof, in collaboration with Brendan Guilfoyle, of the general case. It proceeds indirectly by showing that a counterexample gives rise to a Riemann Hilbert boundary problem for holomorphic discs with negative index. However, Mean Curvature Flow serves to prove existence of a holomorphic disc in a geometrization of the space of oriented lines of R^3.

**Speaker:** Vladimir Dotsenko

**Title:** Homotopy type of the moduli space of stable rational curves

**Abstract: **

I shall show that the rational cohomology of the moduli space of stable rational curves is a Koszul algebra (answering a question of Yu. I. Manin, D. Petersen and V. Reiner), and explain how this allows one to compute the rational homotopy invariants of this space in a very explicit way. Time permitting, I shall talk about a few classes of spaces for which similar results are available, and a few other conjectural classes of spaces like that.

**Speaker:** Graham Ellis

**Title:** The cuspidal cohomology of an arithmetic group

**Abstract:**

I'll start with the definition of the cohomology of a group, and go on to explain what is meant by the cuspidal cohomology of an arithmetic group such as SL(n,O) with O the ring of integers of an algebraic number field. I'll propose a method, based on simple homotopy collapses, for calculating this cohomology on a computer.

**Speaker:** Madeeha Khalid

**Title:** Examples of Moduli of vector bundles on K3 surfaces

We define K3 surfaces and describe some examples of algebraic K3 surfaces. Mukai showed in his seminal works (1980’s) that, under certain conditions, moduli spaces of vector bundles on K3’s are also isomorphic to K3’s. We consider modifications of a classical example, first noted by Mukai, and produce concrete examples of moduli spaces that have slightly different features. Joint work with Colin Ingalls.

**Speaker:** Eduardo Mota Sanchez

**Title:** Constant Mean Curvature Surfaces and Heun's Differential Equations.

**Abstract:**

The generalised Weierstrass representation for surfaces with constant mean curvature allows to describe any conformal constant mean curvature immersion in R3, H3 or S3 with four ingredients: a Riemann surface, a base point, a holomorphic loop Lie algebra valued 1-form and the initial condition for a 2x2 linear system of ODEs. Associating to this linear system a second order differential equation from the class of Heun's Differential Equations, we can prescribe certain kind of singularities in the constructing method that appear in the resulting surface. Regular singularities produce asymptotically Delaunay ends in the surface and irregular singularities produce irregular ends.

We discuss global issues such as period problems and asymptotic behavior involved in the construction of this kind of surfaces. Finally we show how to construct new parametric families of constant mean curvature surfaces in R3 with genus zero that possess at least one irregular end using these methods.

**Speaker**: Arne Rueffer

**Title: ** Bridgeland stability conditions for the category of holomorphic triples

**Abstract:**

Stability conditions on triangulated categories have been introduced by Bridgeland in 2005. They generalise stability concepts on the abelian category Coh(C) of coherent sheaves on a smooth projective curve C to a triangulated category. A particular example of a triangulated category in which the stability space is completly understood, is the bounded derived category of Coh(C). Drawing on this knowledge we describe the stability space of the bounded derived category of the abelian category of holomorphic triples. This category has objects phi:E1→E2, where E1, E2 are coherent sheaves and phi is a morphism between them.

**Speaker:** Stephen Buckley

**Title:** Quasihyperbolic geodesics are hyperbolic quasigeodesics

**Abstract:**

The hyperbolic metric, known also as the Poincaré metric for simply connected domains, is an important tool in complex analysis and complex geometric function theory. The quasihyperbolic metric is a key tool in quasiconformal analysis, and has many other applications. Both of these metrics are defined in the setting of hyperbolic plane domains, but they are not in general bilipschitz equivalent. Nevertheless. we prove that a geodesic curve for either one of them is in a certain precise sense not far from being a geodesic curve for the other, regardless of the domain. More precisely, we prove that, as curves, hyperbolic and quasihyperbolic quasigeodesics are quantitatively the same (with no quantitative dependence on the domain). We also show that a domain is Gromov hyperbolic with respect to one of these metrics if and only if it is Gromov hyperbolic with respect to the other.

**Speaker:** Mark Walsh

**Title:** The Space of Positive Scalar Curvature Metrics on a Manifold with Boundary

**Abstract: **

The problem of whether or not a given smooth closed manifold admits a Riemannian metric with positive scalar curvature (psc-metric) has recieved considerable attention over the years. In particular, in the case of simply connected manifolds (of dimension not equal 4) this problem is completely understood. More recently, a good deal of progress has occurred in understanding the topology of the space of psc-metrics for a given manifold as well its various moduli spaces. Less is understood about the related problem for manifolds with boundary, where metrics are required to satisfy certain boundary constraints. In this talk I will provide some background to this problem before presenting some new results.