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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.

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

The Grothendieck ring of varieties is defined to be the free abelian group generated by k-varieties, modulo the relation that for any closed subvariety Y of a variety X, we impose the relation that [X] = [Y] + [X \ Y]; the ring structure is defined by [X][Y] = [X x Y]. Last December two longstanding questions about the Grothendieck ring of varieties were answered: 1. If two varieties X and Y are piecewise isomorphic then they are equal in the Grothendieck ring; does the converse hold? 2. Is the class of the affine line a zero divisor? Both questions were answered by Borisov, who constructed an element in the kernel of multiplication by the affine line; coincidentally, the proof also constructed two varieties whose classes in the Grothendieck ring are the same but which are not piecewise isomorphic. In this talk we will investigate these questions further by constructing a topological analog of the Grothendieck ring and analyzing its higher homotopy groups. Using this extra structure we will sketch a proof that Borisov's coincidence is not a coincidence at all: that any element in the annihilator of the Lefschetz motive can be represented by a difference of varieties which are equal in the Grothendieck ring but not piecewise isomorphic.