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Geometry and Topology Seminars

Person walking on campus through green grass with their bike.

See below for upcoming seminars or access the seminar archive.


Bill Bogley, Christine Escher and Ren Guo


Meeting time

M 1200-1250

Drawing and Morphing Graphs on Surfaces

Kidder Hall 237

Speaker: Yanwen Luo

In his famous paper ``How to draw a graph" in 1962, Tutte proposed a simple method to produce a straight-line embedding of a planar graph in the plane, known as Tutte's spring theorem. This construction provides not only one embedding of a planar graph, but infinite many distinct embeddings of the given graph. This observation leads to a surprisingly simple proof of a classical theorem proved by Bloch, Connelly, and Henderson in 1984 stating that the space of geodesic triangulations of a convex polygon is contractible. In this talk, we will introduce spaces of geodesic triangulations of surfaces, review Tutte's spring theorem, and present this short proof. We will briefly report the recent progress in identifying the homotopy types of spaces of geodesic triangulations of more complicated surfaces. This is joint work with Tianqi Wu and Xiaoping Zhu. Read more.

Quantizing the hyperbolic volume

Kidder Hall 237

Speaker: Calvin McPhail-Snyder

Most prime knots are hyperbolic. A knot K is hyperbolic if its complement admits a complete metric of constant negative curvature, which by Mostow-Prasad rigidity is uniquely determined by K. The volume of this metric (the hyperbolic volume) is an important invariant of K and it admits a natural complexification called the complex volume. Another way to get invariants of knots is to use the Reshetikhin-Turaev construction to interpret a diagram of K as a morphism between representations of a quantum group; this leads to quantum knot invariants like the Jones polynomial. This seems to have little to do with hyperbolic volume, but there are a number of conjectured relationships such as the Volume Conjecture of Kashaev and Murakami-Murakami. Recently I (joint with N. Reshetikhin) have defined a new family of knot invariants that quantize the complex volume: in some ways they behave like the complex volume, and in others they behave like Jones polynomials. In my talk I will expand… Read more.

Serifert-Torres Formulas for the Alexander Polynomial of Links from Quantum sl2

Kidder Hall 237

Speaker: Matthew Harper

In this talk, I will recall how the Alexander polynomial, a classical knot invariant, can be constructed as a quantum invariant from quantum sl2 at a fourth root of unity. I will then discuss the development of a diagrammatic calculus based on further investigation of quantum sl2 representations. Applying this calculus in the context of the Alexander polynomial allows us to compute the invariant for certain families of links using quantum algebraic methods, rather than using methods of classical topology. Read more.