Student Workshop

Workshop Location:

  • Talks will be held in room G60 in the Mathematical Sciences Building: 9 Rainforest Walk, Monash University, Clayton.

Workshop Schedule:

  • The workshop will run two days, Friday 14 December 2018 to Saturday 15 December 2018. It will include expository mini courses, group work, and short research talks by the participants.
Monash conference: workshop schedule

Mini Courses:

  • Mini Course 1: Hyam Rubinstein: 3-manifolds
    • Lecture 1: Nuts and bolts of 3-manifolds. Topics are triangulations, normal surfaces, incompressible surfaces, hierarchies, Dehn's lemma and the loop theorem.
    • Lecture 2: Basics of classification of 3-manifolds: We will sketch Waldhausen's theory of Haken 3-manifolds and survey JSJ theory.
  • Mini Course 2: Paul Wedrich: Conjectures in quantum topology: I will discuss quantum invariants of knots and links, such as the colored Jones polynomials and their higher-rank analogues, and how they can be computed using diagrammatic calculi. I will then survey a range of deep conjectures that relate quantum link invariants to 3-manifold topology and other parts of mathematics
  • Mini Course 3: Craig Hodgson and Saul Schleimer: Hyperbolic geometry and the topology of 3-manifolds: Topics are hyperbolic geometry in dimensions two and three, hyperbolic 3-manifolds, geometric triangulations, Thurston's gluing equations, hyperbolic volume, cusp shapes; introduction to SnapPy and applications to "detecting" and "ruling out" homeomorphisms between knot and link complements.

Talks:

  • S. Gilles: Computing $\pi_1(M) \to G$ with quivers. For a collection of tetrahedra with gluing data, we can ask ``Do these glue together to form a (hyperbolic) 3-manifold?''. The answer, given by Thurston's gluing equations, is ``as long as the angles of the tetrahedra add to $2\pi$ around each edge''. We can reinterpret that question as asking whether the angles define a (nice) map from the fundamental group to $\PSL(2,\mathbb{C})$. By using quivers and Ptolemy coordinates, we can replace the angles of tetrahedra with more complicated shape coordinates, which lets us ask (and even answer) similar questions for Lie groups other than $\PSL(2, \mathbb{C})$.
  • Takuya Katayama: On virtual embeddings of the braid groups in the mapping class groups of surfaces. By the Birman-Hilden theory, the braid group $B_{2g}$ on $2g$ strands is embedded in the mapping class group $\mathrm{Mod}(S_{g})$ of the closed surface of genus $g$. In this talk, using the curve graphs of surfaces and right-angled Artin groups in the mapping class groups, we prove that no finite index subgroup of $B_{2g+1}$ is embedded in $\mathrm{Mod}(S_{g})$.
  • Seonhwa Kim: A classification of links along the complexity of Wirtinger presentation. A straightforward way to obtain representations of a knot group is to find solutions to the system of equations from Wirtinger presentation. We define a simple hierarchy of diagrams along a complexity by studying dependency of Wirtinger relations. For example, if a knot or link diagram is a two-bridge diagram of Conway form then the system of equations is solved by a direct substitution iteratively. We show that the converse is also true, i.e. the easiest case in the hierarchy always can isotope to allow two local maxima, and moreover characterize the second easiest case.
  • Ka Ho Wong: Asymptotic Behavior of the Colored Jones polynomials and Turaev-Viro Invariants of the figure eight knot. The volume conjecture of the Turaev-Viro invariant is a new topic in quantum topology. It has been shown that the (2N + 1)-th Turaev-Viro invariant for the knot complement S 3\K can be expressed as a sum of the colored Jones polynomial of K evaluated at exp(2πi/(N + 1/2)). That leads to the study of the asymptotic expansion formula (AEF) for the colored Jones polynomials of K evaluated at half-integer root of unity. When K is the figure eight knot, by using saddle point approximation, H.Murakami had already found out the AEF for the N-th colored Jones polynomial of K evaluated at exp(2πi/N). In this talk, I will first review the strategy Murakami used to prove the AEF of the colored Jones polynomial. Then, I will further discuss, for M with a fixed limiting ratio of M and (N + 1/2), how the AEF for the M-th colored Jones polynomial for the figure eight knot evaluated at (N + 1/2)-th root of unity can be obtained. As an application of the asymptotic behavior of the colored Jones polynomials mentioned above, we obtain the asymptotic expansion formula for the Turaev-Viro invariant of the figure eight knot. Finally, we suggest a possible generalization of our approach so as to relate the AEF for the colored Jones polynomials and the AEF for the Turaev-Viro invariants for general hyperbolic knots