ICTS

Title: Gap Distributions and Homogeneous Dynamics [Slides]

 

Abstract: The Farey sequence F(Q) is the collection of fractions between [0,1] whose denominator (when written in lowest terms) is at most Q. As Q grows, these points become uniformly distributed in the interval, so in some sense, look `random'. However, when you look at the gaps between them, they do not behave like those for uniformly distributed random variables, but instead follow an unusual law known as Hall's Distribution. We will explain a proof of this result that uses horocycle flow on the space of lattices SL(2,R)/SL(2, Z), and, time permitting discuss how this picture can be generalized to understanding gaps between directions of saddle connections on Veech surfaces. This talk will include elements from joint work with Y. Cheung, joint work with J. Chaika, and joint work with J. Chaika and S. Lelievre.

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Title: Groups of virtual links

Abstract: For every virtual link we define some group that is an invariant of this link. We prove that this group is the stronger invariant than the group which was defined by L. Kauffman.

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Kingshook Biswas

Title : On almost isometric extension of Moebius maps

Abstract : Motivated by the problem of marked length spectrum rigidity for closed negatively curved manifolds, we consider the problem of extending Moebius maps between the boundaries of CAT(-1) spaces to isometries between the spaces. We show that if $X,Y$ are proper, geodesically complete CAT(-1) spaces, then any Moebius homeomorphism
$f: \partial X \to \partial Y$ extends to a $(1, \log 2)$-quasi-isometry $F: X \to Y$. If $X,Y$ are in addition metric trees, then $F$ can be taken to be a surjective isometry.

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Waldemaar Barrera & J. P. Navarette

Title: KULKARNI LIMIT SET OF SUBGROUPS OF $PSL(3, \Bbb{C})$  [Slides]

Abstract: We recall the definition of Kulkarni limit set in a general setting. We compute Kulkarni limit set of cyclic subgroups of $PSL(3, \Bbb{C})$ acting on $P^2 _{\Bbb{C}}$. We give a sketch of the proof of the following theorem:

    If $G$ is a discrete subgroup of $PU(2,1)$ acting on $P^2 _{\Bbb{C}}$, then its Kulkarni limit set is the union of complex projective lines tangent to the 3-sphere at infinity at the points in the classical limit set of $G$ considered as a group of isometries of complex hyperbolic space $H^2 _{\Bbb{C}} \subset P^2 _{\Bbb{C}}$.
  Under some conditions on a discrete subgroup of $PSL(3, \Bbb{C})$, it is proved that its Kulkarni limit set is a union of complex projective lines.   We give an example of a discrete subgroup of $PSL(3, \Bbb{C})$ whose Kulkarni limit set consists of three complex projective line in general position.  We classify those subgroups of $PSL(3, \Bbb{C})$ such that the maximum number of complex projective lines in general position contained in its limit set is equal four. Finally, we give a construction of a subgroup of $PSL(3, \Bbb{C})$ not conjugate to any subgroup of $PU(2,1)$ such that its limit set is not all of $P^2 _{\Bbb{C}}$, but it contains five complex projective lines in general position, hence it contains infinitely many complex projective lines in general position.

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Title: Filling invariants: homological vs. homotpical [Slides]
 

Abstract:  Classical isoperimetric inequalities give bounds on the minimal-area fillings of loops by disks in Riemannian manifolds.  Thinking of finitely presented groups as metric spaces, one can use the idea of an isoperimetric inequality to obtain an invariant that distinguishes such groups up to "coarse isometry".  There are a number of variations: rather than sticking to loops and disks, one might consider fillings of spheres by balls, or cycles by chains.  This leads to a natural question: in a given group, are these functions the
same?  After describing previously known results on fillings of dimension 3 and higher, I will talk about recent work with A. Abrams, N. Brady and R. Young which addresses this question in the case when the fillings are 2-dimensional.

Title: Bifurcations of Complex Polynomial Vector Fields in C [Slides]

Abstract: The space of degree d single-variable monic and centered complex polynomial vector fields can be decomposed into loci consisting of vector fields with the same combinatorial structure. We study the topology and geometry of these loci, and describe some possible bifurcations. This work in progress aims to give a complete description of the possible bifurcations of these vector fields.

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Gopal Datt

Title: Transcendental dynamics: escaping sets and composite functions. [Slides]

Abstract: We will discuss our work on escaping sets of transcendental entire functions and domains associated with composite transcendental entire functions. We will also discuss our results concerning sharing values. This is a joint work with Sanjay Pant.

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Allan Edmonds

Title: Introduction to Haken n-Manifolds [Slides]

Abstract: Haken n-manifolds have recently been developed by B. Foozwell and H. Rubinstein of Melbourne University.  They generalize the familiar Haken manifolds of dimension 3, constituting a class of manifolds that can be cut open along essential codimension-one submanifolds until they become a disjoint union of cells. The resulting sequence of manifolds, along with the cutting hypersurfaces, is called a hierarchy. Foozwell has shown that Haken  n-manifolds are aspherical, with the interior of the universal covering homeomorphic to euclidean space. Their structure facilitates proofs by induction on dimension and on the length of a hierarchy.

We will apply this notion to the sign problem for the Euler characteristic of  closed aspherical manifolds of even dimension, proving that the Euler characteristic of a closed Haken 4-manifold is non-negative.

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Fred Gardiner

Title: AN EXTENSION OF SLODKOWSKI'S HOLOMORPHIC EXTENSION THEOREM

Abstract: Here

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Subhojoy Gupta

Title: Projective structures, grafting and Teichmüller rays [Slides]

Abstract: We shall discuss a new result that relates grafting, which are certain deformations of complex projective structures on a surface, to the Teichmuller geodesic flow in the moduli space of Riemann surfaces.  A consequence is that for any fixed Fuchsian holonomy, such geometric structures are dense in moduli space. The proofs involve construction of quasiconformal maps of low dilatation between surfaces.

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William Harvey

Title:  Discrete groups and the geometry of manifolds.

Abstract:  This article considers the evolving interaction between hyperbolic geometry and low-dimensional topology over the past 50 years, focussing in particular on the contributions of Ravi Kulkarni to surfaces and 3-manifolds.

 

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Quitzeh Morales Meléndez

Title: Noncommutative signature (and fixed points)

Abstract: It will be defined a signature of a cocompact manifold with proper action of a discrete group as an invariant of bordisms of algebraic Poincare complexes of projective modules over the group algebra. This signature has values in the K-theory of the reduced group $C*$-algebra. It will be shown that this signature is compatible with the usual one and it will be proposed an inductive way to compute this invariant using the Connor-Floyd fix-point constuction.  

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Tarakanta Nayak

Title: Omitted values and Herman rings of small periods. [Slides]

Title: Values of linear maps on quadratic surfaces. [Slides]
 

Abstract : I will explain how one can use the theory of unipotent flows to establish necessary conditions for the set of values of a linear map on a quadratic surface to be dense in its range. If there is time I will briefly discuss how one can obtain quantitative versions of these results in certain special cases.

Title : On $p$-groups of Gorenstein-Kulkarni Type

Title: Coxeter group actions on C^4 preserving quartic varieties: Dynamics and Identities. [Slides]

Abstract: We will discuss the dynamics of the action of the Coxeter group generated by 4 reflections on C^4 preserving the quartic varieties x^2+y^2+z^2+w^2= xyzw + k, where k is a constant. In particular we describe a (non-empty) domain of discontinuity for this action, and also prove some new identities for the orbits of an element in the domain of discontinuity under this group action. We will also describe generalizations of actions of Coxeter groups generated by n reflections on C^n, time permitting. This is joint work with Hengnan Hu and Ying Zhang.