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XV Weihnachtsworkshop on Geometry and Number Theory 2017

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XV Weihnachtsworkshop on Geometry and Number Theory 2017 XV Weihnachtsworkshop on Geometry and Number Theory 2017 Universit¨atdes Saarlandes 18 - 20 December, 2017 Programme Monday 18 Tuesday 19 Wednesday 20 9:30-10:30 K. Vogtmann M. Walter J. K¨onig 10:30-11:30 J. Aramayona D. Torres-Teigell B. Peters 11:30-12:00 Coffee break 12:00-13:00 A. Martino M. Rennig R. Kucharczyk 13:00-15:00 L u n c h b r e a k 15:00-16:00 F. G¨ultepe J. Smillie B. K¨ock 16:00-16:30 Coffee break 16:30-17:30 V. Lazi´c A. Randecker 19:00- Weihnachtsfeier List of talks • Javier Aramayona: Integer-valued homomorphisms from mapping class groups • Funda G¨ultepe: A distance formula for the outer automorphism group of the free group • Bernhard K¨ock: Integrality of epsilon representations of wildly ramified Galois covers • Joachim K¨onig: Almost-regular dessins on a torus and sphere • Robert Kucharczyk: Notions of hyperbolicity • Vladimir Lazi´c: Rational curves on projective manifolds • Armando Martino: The Lipschitz Metric On Culler-Vogtmann Space • Benjamin Peters: Orbit Closures in a Hurwitz Space of Translation Surfaces of Genus Three • Anja Randecker: The ladder surface and its Veech group • Markus Rennig: The covering radius at CM-points of Teichm¨uller curves • John Smillie: Polygonal billiards, horocycle flows and universality • David Torres-Teigell: Cutting out Teichm¨uller curves with modular forms • Karen Vogtmann: Structure of Outer space near infinity • Michael Walter: Algorithms for geometric invariant theory Abstracts Author: Javier Aramayona (Universidad Aut´onomade Madrid) Title: Integer-valued homomorphisms from mapping class groups Abstract: We will start by giving a proof of a theorem of Powell, which asserts that mapping class groups of finite-type surfaces of genus at least 3 have trivial abelianization. We will then explain how to construct nontrivial integer-valued homomorphisms from infinite- type mapping class groups. Moreover, we will give a complete description of all the possible ways in which these arise. This is joint work with Priyam Patel and Nick Vlamis. Author: Funda G¨ultepe (University of Luxembourg) Title: A distance formula for the outer automorphism group of the free group Abstract: We will discuss estimating word distance between two geometric outer automor- phisms of the free group using the geometry of the hyperbolic simplicial complexes Out(Fn) acts on. We will present the construction of the obtained distance formula for Out(Fn) through its analogy with the similar formulas for the special linear group and the mapping class group. This is a joint work with Yulan Qing and Kasra Rafi. Author: Bernhard K¨ock (University of Southampton) Title: Integrality of epsilon representations of wildly ramified Galois covers. Abstract: Given a curve X over Fp and a finite group G acting on it, the "-representation ¯ E(G; X) is a virtual Qp-representation of G with rational coefficients which captures the p- adic valuation of the "-constant "(V ) for all Q¯ -representations V of G. From a joint theorem with H. Fischbacher-Weitz it follows that the coefficients of E(G; X) are in fact integral if π : X ! X=G is at most weakly ramified. In joint work with A. Marmora we determine which denominators may appear in E(G; X) if π is arbitrarily wildly ramified. Author: Joachim K¨onig (Universit¨atW¨urzburg) Title: Almost-regular dessins on a torus and sphere Abstract: The Hurwitz problem asks which ramification data are realizable, that is appear as the ramification type of a covering. We use dessins denfant to show that apart from four explicitly given series of examples, families of genus 1 ramification data which are sufficiently \close" to a regular ramification type, are realizable. More concretely, we show how to obtain dessins for such families via \small" changes to dessins of a regular type. We also present analogies (and problems) in the case of genus 0. Finally, we relate the above problems to a purely group theoretical property of \stability in permutations". This talk is based on joint work with Arielle Leitner and Danny Neftin. Author: Robert Kucharczyk (ETH Z¨urich) Title: Notions of hyperbolicity Abstract: In algebraic geometry and the surrounding fields, one often speaks of certain varieties being 'hyperbolic'. When pushed to give a precise definition, people come up with quite different ideas what hyperbolicity should mean. However, some varieties are universally agreed to be hyperbolic (e.g., smooth projective curves of genus > 1) or not to be hyperbolic (e.g., projective spaces), and some procedures are universally assumed to preserve hyperbolicity. In this talk, I will give some tentative criteria that notions of hyperbolicity should satisfy, and present a number of examples. This is joint work with Ariyan Javanpeykar. Author: Vladimir Lazi´c(Universit¨atdes Saarlandes) Title: Rational curves on projective manifolds Abstract: A projective manifold is rationally connected if through every two general points on it one can draw a rational curve. This is an important class of varieties which contains all Fano manifolds, i.e. manifolds with positive Ricci curvature. I will present the context and previous work on a conjecture of Mumford, which characterises this class of varieties, and will then report on a recent progress in a joint work with Thomas Peternell. Author: Armando Martino (University of Southampton) Title: The Lipschitz Metric On Culler-Vogtmann Space Abstract: (joint work with Stefano Francaviglia) In recent years there has been interest in the geometry of Culler-Vogtmann space - the natural space on which Out(Fn), the auto- morphism group of a free group, acts. A major result here is that of Bestvina, reproving the existence of train track representatives for so-called irreducible automorphisms. The idea here is to find good representatives for automorphisms by looking at the points of minimal displacement with respect to the Lipschitz metric on Culler-Vogtmann space. We will give an introduction to this metric, and explain some results concerning points of minimal displacement. Our main new result is a proof that the points of minimal dis- placement form a connected subset of Culler-Vogtmann space (a result that generalises for arbitrary automorphisms, to points displaced up to a fixed amount). As quick applications, we outline a couple of known results: a solution to the conjugacy problem for irreducible automorphisms, and algorithmic detectability of reducibility. We shall also discuss future directions for this work. Author: Benjamin Peters (KIT) Title: Orbit Closures in a Hurwitz Space of Translation Surfaces of Genus Three Abstract: In this talk, we will study a special Hurwitz space which is home to the Wollmilchsau. Every translation surface in this space is a covering of the torus with four ramification points. This enables us to compute all orbit closures in this space which are described by additional automorphisms. Finally, we will discuss generalizations of this construction and which difficulties may occur. Author: Anja Randecker (University of Toronto) Title: The ladder surface and its Veech group Abstract: (In)finite translation surfaces can be obtained by gluing the edges of (in)finitely many polygons via translations. The group SL(2; R) acts on all these translation surfaces by affine transformations and the stabilizer of a translation surface under this action is called Veech group. In this talk, I will introduce a family of infinite translation surfaces that is related to the infinite staircase, the Bouw{Mller surfaces, and the baker's map surface. Furthermore, I'll explain how to calculate the Veech group for a specific surface in this family. Author: Markus Rennig (Goethe-Universit¨atFrankfurt) Title: The covering radius at CM-points of Teichm¨ullercurves Abstract: We are considering hyperbolic algebraic curves X defined over the algebraic numbers together with an universal cover ' sending 0 2 B1(0) to a rational point. Serge Lang first asked the question whether the tangent map '0(0) of the universal cover is al- gebraic or transcendental. This question was answered by Wolfart and W¨ustholz for the orbifold points and cusps of triangle curves. We reproof the result for orbifold points with- out the use of explicit values of hypergeometric functions by extending it to CM-points on Teichm¨ullercurves. Author: John Smillie (University of Warwick) Title: Polygonal billiards, horocycle flows and universality Abstract: Billiard tables are studied in dynamics because they often give aproachable ex- amples of deeper and more technical dynamical phenomena. The study of rational polygonal billiards is an instance of this and it leads to the deep dynamical idea of renormalisation. Renormalisation dynamics has properties similar to homogeneous dynamics but it also has some surprising differences. I will describe some recent positive results joint with Barak Weiss and Matt Bainbridge and some recent negative results joint with Barak Weiss and Jon Chaika. Author: David Torres-Teigell (Universit¨atdes Saarlandes) Title: Cutting out Teichm¨uller curves with modular forms Abstract: Teichm¨ullercurves arise as the projection to the moduli space of certain orbits of the action of SL(2; R) on the space of flat surfaces. By results of M¨oller,the Jacobian of a point in C always contains a subvariety that admits real multiplication so, in particular, there exists certain Prym-Torelli map that allows us to see the curve C inside a Hilbert modular variety parametrising abelian varieties with real multiplication. In this talk we will introduce the Gothic Teichm¨ullercurves, discovered by McMullen- Mukamel-Wright, and describe their Prym-Torelli images inside a Hilbert modular surface. Our main objective is to cut this image out as the vanishing locus of some Hilbert modular form and use this description to calculate their Euler characteristics. This is joint work with M. M¨oller. Author: Karen Vogtmann (University of Warwick) Title: Structure of Outer space near infinity Abstract: Outer space is a contractible space on which the group Out(Fn) of outer au- tomorphisms of a free group acts properly.
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