I will give an introduction to surface bundles and will discuss several places where they arise naturally. A surface bundle is a fiber bundle where the fiber is a surface. A first example is the mapping torus construction for 3-manifolds, which is a surface bundle over the circle. Topics will include a construction of 4-manifolds as well as section problems related to surface bundles. The talk will be based on a forthcoming Notices survey article by Salter and Tshishiku.
When a topological object admits a group action, we expect that our invariants reflect this symmetry in their structure. This talk will explore how link symmetries are reflected in three generations of related invariants: the Jones polynomial; its categorification, Khovanov homology; and the youngest invariant in the family, the Khovanov stable homotopy type, introduced by Lipshitz and Sarkar.
Cutting a polyhedron along some spanning tree of its edges will yield an isometric immersion of the polyhedron into the plane. If this immersion is also injective, we call it an unfolding. In this talk I will give some general results about unfoldings of polyhedra. There is also a notion of pseudo-edge unfolding, which involves cutting on a pseudo edge graph, as opposed to an edge graph. A pseudo edge graph is a 3-connected graph on the surface of the polyhedron, whose vertices coincide with the vertices of the polyhedron, and whose edges are geodesics.
In this informal chat, I will introduce the braid group and several equivalent topological perspectives from which to view it. In particular, we will discuss the role that complex polynomials play in this setting, along with a few classical results.
A question going back to Serre asks which groups arise as fundamental groups of smooth, complex projective varieties, or more generally, compact Kaehler manifolds. The most basic examples of these are surface groups, arising as fundamental groups of 1-dimensional projective varieties. We will survey known examples and restrictions on such groups and explain the special role surface groups play in their classification. Finally, we connect this circle of ideas to more general questions about surface bundles and mapping class groups.
Kodaira, and independently Atiyah, gave the first examples of surface bundles over surfaces whose signature does not vanish, demonstrating that signature need not be multiplicative. These examples, called Kodaira fibrations, are in fact complex projective surfaces admitting a holomorphic submersion onto a complex curve, whose fibers have nonconstant moduli. After reviewing the Atiyah-Kodaira construction, we consider Kodaira fibrations with nontrivial holomorphic invariants in degree one.
A 2-knot is a smooth embedding of S^2 in S^4, and a 0-concordance of 2-knots is a concordance with the property that every regular level set of the concordance is just a collection of S^2's. In his thesis, Paul Melvin proved that if two 2-knots are 0-concordant, then a Gluck twist along one will result in the same smooth 4-manifold as a Gluck twist on the other. He asked the following question: Are all 2-knots 0-slice (i.e. 0-concordant to the unknot)? I will explain all relevant definitions, and mostly follow the paper by Nathan Sunukjian on this topic.
I will discuss the orderability of the fundamental groups of knot complements including known results, a useful technique using some ideas of Baumslag, and some interesting questions that have recently arisen from this study.
I will discuss how a graph theoretic construction used by Hirasawa and Murasugi can be used to show that the commutator subgroup of the knot group of a two-bridge knot is a union of an ascending chain of parafree groups. Using a theorem of Baumslag, this implies that the commutator subgroup of a two-bridge knot group is residually torsion-free nilpotent which has applications to the anti-symmetry of ribbon concordance and the bi-orderability of two-bridge knots. In 1973, E. J. Mayland gave a conference talk in which he announced this result.
This is a general audience Geometry-Topology talk where I will give a broad overview of my research interests and techniques I use in my work. My research concerns the study of link concordance using groups, both extracting concordance data from group theoretic invariants and determining the properties of group structures on links modulo concordance.
