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Series: Geometry Topology Seminar

When visualising topological objects via 3D printing, we need athree-dimensional geometric representation of the object. There areapproximately three broad strategies for doing this: "Manual" - usingwhatever design software is available to build the object by hand;"Parametric/Implicit" - generating the desired geometry using aparametrisation or implicit description of the object; and "Iterative" -numerically solving an optimisation problem.The manual strategy is unlikely to produce good results unless the subjectis very simple. In general, if there is a reasonably canonical geometricstructure on the topological object, then we hope to be able to produce aparametrisation of it. However, in many cases this seems to be impossibleand some form of iterative method is the best we can do. Within theparametric setting, there are still better and worse ways to proceed. Forexample, a geometric representation should demonstrate as many of thesymmetries of the object as possible. There are similar issues in makingthree-dimensional representations of higher dimensional objects. I willdiscuss these matters with many examples, including visualisation offour-dimensional polytopes (using orthogonal versus stereographicprojection) and Seifert surfaces (comparing my work with Saul Schleimerwith Jack van Wijk's iterative techniques).I will also describe some computational problems that have come up in my 3D printed work, including the design of 3D printed mobiles (joint work withMarco Mahler), "Triple gear" and a visualisation of the Klein Quartic(joint work with Saul Schleimer), and hinged surfaces with negativecurvature (joint work with Geoffrey Irving).

Series: Geometry Topology Seminar

The commutator length of an element g in the commutator subgroup [G,G] of agroup G is the smallest k such that g is the product of k commutators. WhenG is the fundamental group of a topological space, then the commutatorlength of g is the smallest genus of a surface bounding a homologicallytrivial loop that represents g. Commutator lengths are notoriouslydifficult to compute in practice. Therefore, one can ask for asymptotics.This leads to the notion of stable commutator length (scl) which is thespeed of growth of the commutator length of powers of g. It is known thatfor n > 2, SL(n,Z) is uniformly perfect; that is, every element is aproduct of a bounded number of commutators, and hence scl is 0 on allelements. In contrast, most elements in SL(2,Z) have positive scl. This isrelated to the fact that SL(2,Z) acts naturally on a tree (its Bass-Serretree) and hence has lots of nontrivial quasimorphisms.In this talk, I will discuss a result on the stable commutator lengths inright-angled Artin groups. This is a broad family of groups that includesfree and free abelian groups. These groups are appealing to work withbecause of their geometry; in particular, each right-angled Artin groupadmits a natural action on a CAT(0) cube complex. Our main result is anexplicit uniform lower bound for scl of any nontrivial element in anyright-angled Artin group. This work is joint with Talia Fernos and MaxForester.

Series: Geometry Topology Seminar

I will give an elementary introduction to Majid's theory of braided groups and how this may lead to a more geometric, less quantum, interpretation of knot invariants such as the Jones polynomial. The basic idea is set up a geometry where the coordinate functions commute according to a chosen representation of the braid group. The corresponding knot invariants now come out naturally if one attempts to impose such geometry on the knot complement.

Series: Geometry Topology Seminar

We call a 3-manifold Platonic if it can be decomposed into isometric Platonic solids. Many key examples in 3-manifold topology are Platonic manifolds, e.g., the Poincar\'e homology sphere, the Seifert-Weber dodecahedral space and the complements of the figure eight knot, the Whitehead link, and the minimally twisted 5-component chain link. They have a strong connection to regular tessellations and illustrate many phenomena such as hidden symmetries.I will talk about recent work on a census of hyperbolic Platonic manifolds and some new techniques we developed for its creation, e.g., verified canonical cell decompositions and the isometry signature which is a complete invariant of a cusped hyperbolic manifold.

Series: Geometry Topology Seminar

Please not non-standard day for seminar.

A trisection is a decomposition of a four-manifold into three trivial pieces and serves as a four-dimensional analogue to a Heegaard decomposition of a three-manifold. In this talk, I will discuss an adaptation of the theory of trisections to the relative setting of knotted surfaces in the four-sphere that serves as a four-dimensional analogue to bridge splittings of classical knots and links. I'll show that every such surface admits a decomposition into three standard pieces called a bridge trisection. I'll also describe how every such decomposition can be represented diagrammatically as a triple of trivial tangles and give a calculus of moves for passing between diagrams of a fixed surface. This is joint work with Alexander Zupan.

Series: Geometry Topology Seminar

A trisection of a smooth, oriented, compact 4-manifold X is a decomposition into three diffeomorphic 4-dimensional 1-handlebodies with certain nice intersections properties. This is a very natural 4-dimensional analog of Heegaard splittings of 3-manifolds. In this talk I will define trisections of closed 4-manifolds, but will quickly move to the case of 4-manifolds with connected boundary. I will discuss how these "relative trisections" interact with open book decompositions on the bounding 3-manifold. Finally, I will discuss a gluing theorem which allows us to glue together relative trisections to induce a trisection on a closed 4-manifold.

Series: Geometry Topology Seminar

I will describe the results of a joint project with Mike Mandell on the algebraic K-theory of the sphere spectrum, focusing on recent work that describes the fiber of the cyclotomic trace using a spectral lift of Tate-Poitou duality.

Series: Geometry Topology Seminar

In this talk, first, I'll briefly go over my proof of the conjecture that there are only afinite number of hyperbolic 3-manifolds of bounded volume and trace field degree. Then I'lldiscuss some conjectural pictures to quantitative results and applications to other similarproblems.

Series: Geometry Topology Seminar

The Ptolemy variety is an invariant of a triangulated 3-manifoldM. It detects SL(2,C)-representations of pi_1(M) in the sense that everypoint in the Ptolemy variety canonically determines a representation (up toconjugation). It is closely related to Thurston's gluing equation varietyfor PSL(2,C)-representations. Unfortunately, both the gluing equationvariety and the Ptolemy variety depend on the triangulation and may missseveral components of representations. We discuss the basic properties ofthese varieties, how to compute invariants such as trace fields and complexvolume, and how to obtain a variety, which is independent of thetriangulation.

Series: Geometry Topology Seminar

The hyperelliptic Torelli group of a genus g reference surface S_g is the subgroup of the mapping class group whose elements both commute with a fixed hyperelliptic involution of S_g and act trivially on the integral homology of S_g . This group is an important object in geometric topology and group theory, and also in algebraic geometry, where it appears as the fundamental group of the moduli space of genus g hyperelliptic curves with a homology framing. In this talk, we summarize what is known about the (infinite) topology of these moduli spaces, describe a few open problems, and report on some recent partial progress.