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Ara Basmajian
Extremal length, Geometric length, and Ergodicity of the Geodesic Flow on a Riemann Surface slides A Riemann surface having a dual existence as both a complex analytic surface as well as a (hyperbolic) geometric surface, opens a wealth of possible avenues for investigation. Our focus in this talk will be on finding geometric conditions (in terms of length-twist parameters) for when a Riemann surface is of so-called parabolic type. That is, it does not support a Green's function or equivalently the geodesic flow on the unit tangent bundle is ergodic. Much of the talk will concentrate on flute surfaces, arguably the simplest infinite type surface. This is joint work with Hrant Hakobyan and Dragomir Saric. |
Peter Buser
Energy distribution of harmonic 1-forms on thick and thin parts slides We estimate the energy distribution of a harmonic 1-form on a compact hyperbolic surface with respect to its thick and thin decomposition. Suppose the surface looks like a dumbbell: two thick parts connected by a thin tube. If on one of the thick parts a harmonic 1-form has all its cycles equal to zero, intuition says that the form ``lives on the other side,’’ mainly so because energy diffusion through thin tubes is difficult. We prove this quantitatively with only elementary tools: explicit formulas for cylinders and gradient estimates. It turns out that energy decays extremely rapidly along the tube when it is thin. The results apply to the Jacobian variety of a Riemann surface. This is joint work with Eran Makover, Bjoern Muetzel and Robert Silhol. |
Moira Chas
Tantalizing patterns of closed curves on surfaces which became theorems video on the CIRM website Consider an orientable surface S with negative Euler characteristic, a minimal set of generators of the fundamental group of S, and a hyperbolic metric on S. Each unbased homotopy class C of closed oriented curves on S determines three numbers: the word length (that is, the minimal number of letters needed to express C as a cyclic word in the generators and their inverses), the minimal geometric self-intersection number, and finally the geometric length. Also, the set of free homotopy classes of closed directed curves on S (as a set) is the vector space basis of a Lie algebra discovered by Goldman. This Lie algebra is closely related to the intersection structure of curves on S. These three numbers, as well as the Goldman Lie bracket of two classes, can be explicitly computed (or approximated) using a computer. We will discuss the algorithms to compute or approximate these numbers, and how these computer experiments led to counterexamples to existing conjectures, formulations of new conjectures and (sometimes) subsequent theorems. This talk means to be accessible to mathematically young people. These results are joint work with different collaborators; mainly Arpan Kabiraj, Steven Lalley and Rachel Zhang. |
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Nicolas Curien
Random Hyperbolic Surfaces and Maps slides video on the CIRM website We shall survey recent progress towards the understanding of the geometry of random hyperbolic surfaces and random maps (both in low and high genus). We will in particular try to highlight similarities between a few constructions on both theories. The talk will be based on joint works in progress with Timothy Budd on one side and Thomas Budzinski & Bram Petri on the other side. |
Jeff Erickson
Fun with Toroidal Spring Embeddings slides Tutte’s classical spring embedding theorem is the foundation of hundreds of algorithms for drawing and manipulating planar graphs. A somewhat less well-known generalization of Tutte’s theorem, first proved by Yves Colin de Verdière in 1990, applies to graphs on more complex surfaces. I will describe two recent applications of this more general theorem to graphs on the Euclidean flat torus. The first is a natural toroidal analogue of the Maxwell-Cremona correspondence, which relates equilibrium stresses, orthogonal dual embeddings, and weighted Delaunay complexes. The second is a simple and natural algorithm to continuously morph between geodesic torus graphs, generalizing a planar morphing algorithm of Floater and Gotsman. |
Myfanwy Evans
Constructing triply-periodic tangles Projects Using periodic surfaces as a scaffold is a convenient route to making periodic entanglements. I will present a systematic way of building new tangled periodic structures, using low-dimensional topology and combinatorics, posing the question of how to characterise the structures more completely. I will also give an insight into applications of these structures. |
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Federica Fanoni
Generating big mapping class groups Preprint video on the CIRM website The mapping class group of a surface is the group of its homeomorphisms up to homotopy. A natural question to ask is: what is a good set of generators? If the surface is compact (or more generally of finite type) there are multiple satisfactory answers. If the surface is of infinite type, the question is wide open. I will discuss this problem and present a partial (negative) result in this context. Joint work with Sebastian Hensel. |
Chris Judge
Translation structures, ideas and connections video on the CIRM website Though the term translation surface is relatively new, the notion is quite old (Abelian differential) and in fact predates the notion of hyperbolic surface. The term translation surface emphasizes the (G,X) point of view introduced to their study by Thurston in about 1980. Since then, ideas from both topology and dynamics have revolutionized the study of these structures. In this talk I will describe some of these ideas with preference given to those ideas that connect to both Delaunay triangulations and hyperbolic geometry. |
Stephan Tillmann
The things one finds in Fock-Goncharov coordinates video on the CIRM website Fock and Goncharov give parameterisations of two different types of moduli spaces of properly convex real projective structures. I'll discuss a number of observations made about these parameterisations, the geometric structures that are parameterised by them, their relationship with representations into SL(3,R), canonical cell decompositions, and compactifications. This includes joint work with Alex Casella, Robert Haraway, Robert Löwe and Dominic Tate. |