Invariants in Low Dimensional Geometry

Gazi University, Ankara, Turkey

10-14 August 2015

Title and Abstracts

All the talks will be held at Mimar Kemaleddin Conference Hall. The first talk will start at 10:15am on Monday. Talks will be 50-55min, with 5-10min for questions (one hour total).

Time: Thursday 8/13, 5 pm
Name: Baris Coskunuzer

Affiliation: Koc University

Title: Minimal Surfaces with Arbitrary Topology in H^2xR

Abstract: In this talk, we show that any open orientable surface can be embedded in H^2xR as a complete area minimizing surface. Furthermore, we will discuss the asymptotic Plateau problem in H^2xR, and give a fairly complete solution.
Time : Tuesday 8/11, 2:15 pm
Name: Stefan Friedl

Affiliation: University of Regensburg

Title: The Thurston norm via Fox calculus

Abstract: Given a 3-manifold whose fundamental group admits a presentation with two generators and one relator we will show how one can easily obtain the Thurston norm via Fox calculus. This is based on joint work with Kevin Schreve and Stephan Tillmann.
Time : Thursday 8/13, 3:30 pm
Name: David Futer

Affiliation: Temple University

Title: From triangulated 3-manifolds to generic few-relator groups

Abstract: Given a presentation of a group G with many more generators than relations, where the relations are random long words, we construct a 2-dimensional complex with nice geometry whose fundamental group is G. This complex is built out of hyperbolic polygons, glued by isometry along the edges, with a negative curvature condition at the vertices. The entire construction is guided in a big way by the study of ideal triangulations of 3-manifolds. As a consequence of this ''geometric realization'' of the group, we learn that G is hyperbolic and enjoys several other pleasant group-theoretic properties. For instance, all finitely generated subgroups are undistorted, hyperbolic, and separable. This is joint work with Dani Wise.
Time : Friday 8/14, 11:45 am
Name: Stavros Garoufalidis

Affiliation: Georgia Institute of Technology

Title: Evaluation of state integrals via Grothendieck residues

Abstract: Joint work with Rinat Kashaev. We will make a friendly introduction to state-integral invariants of ideally triangulated manifolds, and illustrate our results with computations.
Time : Thursday 8/13, 10:15 am
Name: Eriko Hironaka

Affiliation: Florida State University

Title: Dilatations of pseudo-Anosov mapping classes

Abstract: I will talk about the minimum dilatation problem for pseudo-Anosov mapping classes and some recent results for pseudo-Anosov braid monodromies.
Time : Tuesday 8/11, 11:45
Name: BoGwang Jeon

Affiliation: Columbia University

Title: Hyperbolic 3-manifolds of bounded volume and trace field degree

Abstract: In this talk, I present my proof of the conjecture that there are only a finite number of hyperbolic 3-manifolds of bounded volume and trace field degree.
Time : Friday 8/14, 2:15 pm
Name: Mustafa Kalafat

Affiliation: Tunceli University

Title: Constructing special 4-manifolds via hyperbolic 3-manifolds

Abstract: We use hyperbolic 3-manifold geometry to produce 4-manifolds with special structures. These are locally conformally flat, self-dual and almost complex structures. We can construct these 4-manifolds by sketching their handşebody diagrams. If time permits, we prove that the connected sum of two self-dual Riemannian 4-manifolds of positive scalar curvature is again self-dual of positive scalar curvature, under a vanishing hypothesis. The proof involves Kodaira-Spencer-Freedman deformation theory and Leray Spectral Sequence. Again if time permits, we will discuss metrics on the Quotients of Enriques Surfaces, and applications of the Geometric Invariant Theory, Complex/Almost Complex and Kahler structures. This is joint work with S. Akbulut.
Time : Wednesday 8/12, 11:45 pm
Name: Effie Kalfagianni

Affiliation: Michigan State University

Title: Non orientable knot genus and the Jones polynomial

Abstract: I will talk on joint work with Christine Lee where we use the Jones polynomial to estimate (and often calculate) the non-orientable genus (i.e. the crosscut number) of alternating and other families of knots.
Time : Wednesday 8/12, 9 am
Name: Joanna Kania-Bartoszynska

Affiliation: National Science Foundation

Title: Structure of the Kauffman bracket skein algebra of a surface.

Abstract: The Kauffman bracket skein algebra of an orientable surface is formed by taking linear combinations of isotopy classes of links in the cylinder over that surface, and dividing by the Kauffman bracket relations. We will discuss the structure of this algebra for closed surfaces and for surfaces with boundary. The talk is based on joint work with Charles Frohman.
Time : Wednesday 8/12, 5 pm
Name: Rinat Kashaev

Affiliation: University of Geneva

Title: A hidden 4D structure of the Teichmüller TQFT

Abstract: The Teichmüller TQFT partition function of a shaped triangulation X of a pseudo 3-manifold can written as an evaluation of a shape independent tempered distribution, called kinematical kernel, on a test function given by the tensor product of (shaped) quantum dilogarithms.Remarkably, the kinematical kernel can be given an interpretation of a partition function of a simple (generalized) TQFT in four dimensions of the cone over X.
Time : Monday 8/10, 10:15 am
Name: Mustafa Korkmaz

Affiliation: Middle East Technical University

Title: Arbitrarily long factorizations in mapping class groups

Abstract: On a compact oriented surface of genus $g$ with $n\geq 1$ boundary components, $\delta_1, \delta_2,\ldots, \delta_n$, we consider positive factorizations of the boundary multitwist $t_{\delta_1} t_{\delta_2} \cdots t_{\delta_n}$, where $t_{\delta_i}$ is the positive Dehn twist about the boundary $\delta_i$. We prove that for $g\geq 3$, the boundary multitwist $t_{\delta_1} t_{\delta_2}$ can be written as a product of arbitrarily large number of positive Dehn twists about nonseparating simple closed curves, extending a recent result of Baykur and Van Horn-Morris, who proved this result for $g\geq 8$. This fact has immediate corollaries on the Euler characteristics of the Stein fillings of conctact three manifolds. This is a joint work with Elif Dalyan and Mehmetcik Pamuk.
Time : Monday 8/10, 5 pm
Name: Thang Le

Affiliation: Georgia Institute of Technology

Title: Kauffman bracket skein modules of 3-manifolds at roots of 1

Abstract: We extend the Kauffman bracket skein modules of 3-manifolds to marked 3-manifolds and show how the the Chebyshev-Frobenius homomorphism appears naturally in this theory.
Time: Thursday 8/13, 11:45 am
Name: Joseph Maher

Affiliation: CSI and GC, CUNY

Title: Random walks on weakly hyperbolic groups

Abstract: Let G be a group acting by isometries on a Gromov hyperbolic space, which need not be proper. If G contains two hyperbolic elements with disjoint fixed points, then we show that a random walk on G converges to the boundary almost surely. This gives a unified approach to convergence for the mapping class groups of surfaces, Out(Fn) and acylindrical groups. This is joint work with Giulio Tiozzo.
Time: Monday 8/10, 3:30 pm
Name: Julien Marche

Affiliation: Université Pierre et Marie Curie

Title: Singular intersections on character varieties

Abstract: We study the singular intersections between a curve in a 2-torus and the 1-dimensional sub-tori. Such a problem arises when looking at rigidity of representations of the Dehn fillings of a 3-manifold with toric boundary.
Time: Wednesday 8/12, 2:15 pm
Name: Sergei Matveev

Affiliation: Chelyabink State University

Title: Dijkgraaf-Witten Z_2 -invariants for orientable Seifert 3-manifolds

Abstract: We prove that all twice orientable Seifert 3-manifolds can be decomposed into two classes A and B such that DW(M)=0 if M is in the class A and DW(M) =1/2 dim H^1 (M;Z_2) if M is in the class B.
Time: Tuesday 8/11, 3:30 pm
Name: Hitoshi Murakami

Affiliation: Tohoku University

Title: The colored Jones polynomials, the Chern-Simons invariants, and the Reidemeister torsions of a knot

Abstract: We discuss a relation of the colored Jones polynomials of a knot to the Chern-Simons invariants and the Reidemeister torsions of its complement.
Time: Monday 8/10, 2:15 pm
Name: Kate Petersen

Affiliation: Florida State University

Title: (P)SL(2,C) Representations of knot groups

Abstract: Representations of knot groups into groups such as SU(2) and (P)SL(2,C) have many connections to 3-manifold invariants. I'll discuss representations of knot groups into (P)SL(2,C), focusing on two-bridge knots and an algorithm for determining 'geometric' representations for a large class of knots including alternating knots.
Time: Tuesday 8/11, 10:15 am
Name: Alan Reid

Affiliation: University of Texas, Austin

Title: Determining hyperbolic 3-manifolds by geometric spectra.

Abstract: This talk will discuss to what extent closed hyperbolic 3-manifolds can be determined by various geometric spectra such as length spectra, and the set of pi_1-injective surfaces.
Time: Thursday 8/13, 9 am
Name: Makoto Sakuma

Affiliation: Hiroshima University

Title: Mapping class group action on the space of geodesic rays of a punctured hyperbolic surface

Abstract: For a hyperbolic punctured surface S of finite area,consider the space, G, of geodesic rays emanating from punctures. Then the mapping class group of S naturally acts on the space G. In the first half of my talk, I will explain the role of the action in the variations of McShane's identities for (i) punctured surface bundles (Bowditch and Akiyoshi-Miyachi-Sakuma) and (ii) 2-bridge links (Lee-Sakuma). In the second half of my talk, I will explain the following theorem proved by Bowditch, answering to a my question. Theorem. The non-wandering set of the action of the mapping class group on the space G has measure 0. I also hope to discuss the mapping class group action on the space of "simple" geodesic rays.
Time: Monday 8/10, 11:45 am
Name: Saul Schleimer

Affiliation: University of Warwick

Title: End invariants of splitting sequences

Abstract: Thurston introduced train tracks and geodesic laminations as tools to study surface diffeomorphisms and Kleinian groups. We'll start the talk with a relaxed introduction to these. Then, in analogy with the end invariants of Kleinian groups and Teichmüller geodesics, we will define the end invariants of an infinite splitting sequence of train tracks. These end invariants determine the set of laminations that are carried by all tracks in the infinite splitting sequence. If there is time, we'll use these ideas to sketch a new proof of Klarreich's theorem, determining the boundary of the curve complex.
Time: Thursday 8/13, 2:15 pm
Name: Henry Segerman

Affiliation: Oklahoma State University

Title: Veering Dehn surgery

Abstract: This is joint work with Saul Schleimer. Veering structures on ideal triangulations of cusped manifolds were introduced by Ian Agol, who showed that every pseudo-Anosov mapping torus over a surface, drilled along all singular points of the measured foliations, has an ideal triangulation with a veering structure. Any such structure coming from Agol's construction is necessarily layered, although a few non-layered structures have been found by randomised search. We introduce veering Dehn surgery, which can be applied to certain veering triangulations, to produce veering triangulations of a surgered manifold. As an application we find an infinite family of transverse veering triangulations none of which are layered. Until recently, it was hoped that veering triangulations might be geometric, however the first counterexamples were found recently by Issa, Hodgson and me. We also apply our surgery construction to find a different infinite family of transverse veering triangulations, none of which are geometric.
Time : Tuesday 8/11, 9 am
Name: Mehmet Haluk Sengun

Affiliation: University of Sheffield

Title: Cycle complexity of arithmetic hyperbolic 3-manifolds

Abstract: In joint work with N. Bergeron and A. Venkatesh, we propose a conjecture which essentially says that homology classes in arithmetic hyperbolic 3-manifolds can be represented by cycles of small topological complexity. We prove the conjecture in special cases using sophisticated machinery and some standard, but deep, conjectures from number theory. As a corollary of the conjecture, which was our original motivation for the work, we show that torsion homology grows exponentially with respect to the volume.
Time: Wednesday 8/12, 10:15 am

Affiliation: SUNY BUffalo

Title: Skein Algebras of Surfaces

Abstract: For a surface F the space of links in F x [0,1] modulo Kauffman bracket skein relations is called the skein algebra of F, denoted by S(F). It is a non-commutative deformation of the SL(2,C)-character variety in F and, at roots of unity, it is (almost) the quantum Teichmuller space of F. Except for a few simplest surfaces F, not much is known about the algebraic properties of S(F). We are going to prove the following two fundamental properties of skein algebras: 1. S(F) has no zero divisors, 2. Away from roots of unity, the center of S(F) is composed of polynomials in knots parallel to boundary components of F. We state analogous results for relative skein algebras.
Time: Tuesday 8/11, 5 pm
Name: Yasar Sozen

Affiliation: Hacettepe University

Title: Reidemeister torsion of Anasov representations

Abstract: This talks consists of two parts. In this first part, using symplectic chain complex and pant-decompositon of closed orientable hyperbolic surfaces, we present a formula for computing the topological invariant Reidemeister torsion of such surfaces. The second parts aims to prove that Reidemeister torsion is well defined for PSL(n,R)-Anasov representations of closed orientable hyperbolic surfaces. Using symplectic chain complex, we will also present a formula to compute Reidemeister torsion of such representations.
Time: Friday 8/14, 3:30 pm
Name: Roland van der Veen

Affiliation: University of Amsterdam

Title: Quantum invariant theory

Abstract: Three-dimensional hyperbolic geometry is the geometry of the group SL(2,C). The aim of this talk is to explore what happens to the geometry when one passes to quantum group. This is relevant for understanding quantum knot invariants such as the Jones polynomial. We illustrate our point of view using simple examples coming from polyhedra and the Alexander polynomial.
Time: Wednesday 8/12, 3:30 pm
Name: Andrei Vesnin

Affiliation: Sobolev Institute of Mathematics and Chelyabinsk State University

Title: Complexity and Turaev - Viro invariants of 3-manifolds.

Abstract: We will present recent results which demonstrate how to use Turaev – Viro invariants to find Matveev’s complexity of 3-manifolds. The talk is based on joint results with E. Fominykh and V. Turaev.
Time: Friday 8/14, 9 am
Name: Tian Yang

Affiliation: Stanford University

Title: On type-preserving representations of the four-punctured sphere group

Abstract: We give counterexamples to a conjecture of Bowditch that if a non-elementary type-preserving representation $\rho:\pi_1(\Sigma_{g,n})\rightarrow PSL(2;\mathbb R)$ of a punctured surface group sends every non-peripheral simple closed curve to a hyperbolic element, then $\rho$ must be Fuchsian. The counterexamples come from relative Euler class $\pm1$ representations of the four-punctured sphere group. As a related result, we show that the mapping class group action on each non-extremal component of the character space of type-preserving representations of the four-punctured sphere group is ergodic. The main tool we use is Penner's lengths coordinates of the decorated character spaces defined by Kashaev.
Time: Friday 8/14, 10:15 am
Name: Christian Zickert

Affiliation: University of Maryland

Title: Coordinates for representations of 3-manifold groups

Abstract: We study the shape and Ptolemy varieties of a compact 3-manifold M with a topological ideal triangulation. The varieties give coordinates for representations of pi_1(M) in the sense that each point determines a representation (up to conjugation). We describe the varieties, how to compute them, and how to compute invariants such as trace fields and complex volume.

Organizers : Abhijit Champanerkar (CUNY), Baki Karliga (Gazi University), Ilya Kofman, (CUNY), Feng Luo (Rutgers), Walter Neumann (Barnard, Columbia), Murat Savas (Gazi University)

Local Organizers : Atakan T. Yakut (Nigde University), Mustafa Ozkan (Gazi University), Sabiha Dodurgali (Gazi University), Fatma Yilmaz (Gazi University), Tugba Tamirci (Gazi University)

Contact : Please email questions to gconference2015@gmail.com.

Funding is kindly provided by Gazi University and the National Science Foundation (NSF).