Seminars and Colloquia by Series

Classical knot invariants and slice surfaces by Peter Feller

Series
Geometry Topology Seminar Pre-talk
Time
Wednesday, April 3, 2019 - 12:45 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Peter Feller – ETH Zurich –

In the setup of classical knot theory---the study of embeddings of the circle into S^3---we recall two examples of classical knot invariants: the Alexander polynomial and the Seifert form.

We then introduce notions from knot-concordance theory, which is concerned with the study of slice surfaces of a knot K---surfaces embedded in the 4-ball B^4 with boundary the knot K. We will comment on the difference between the smooth and topological theory with a focus on a surprising feature of the topological theory: classical invariants govern the existence of slice surfaces of low genus in a way that is not the case in the smooth theory. This can be understood as an analogue of a dichotomy in the study of smooth and topological 4-manifolds.

Doubly slice Montesinos links

Series
Geometry Topology Seminar Pre-talk
Time
Monday, April 1, 2019 - 12:45 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ahmad Issa – University of Texas, Austin

A link in the 3-sphere is doubly slice if it is the cross-section of an unknotted 2-sphere in the 4-sphere. The double branched cover of a doubly slice link is a 3-manifold which embeds in the 4-sphere. For doubly slice Montesinos links, this produces embeddings of Seifert fibered spaces in S^4. In this pre-talk, I'll discuss a construction and an obstruction to being doubly slice.

Spheres in 4-manifolds

Series
Geometry Topology Seminar Pre-talk
Time
Monday, March 11, 2019 - 12:45 for 1 hour (actually 50 minutes)
Location
Skiles 257
Speaker
Hannah Schwartz – Bryn Mawr
In this talk, we will examine the relationship between homotopy, topological isotopy, and smooth isotopy of surfaces in 4-manifolds. In particular, we will discuss how to produce (1) examples of topologically but not smoothly isotopic spheres, and (2) a smooth isotopy from a homotopy, under special circumstances (i.e. Gabai's recent work on the ``4D Lightbulb Theorem").

A partial order on nu+ equivalence classes

Series
Geometry Topology Seminar Pre-talk
Time
Monday, March 4, 2019 - 12:45 for 1 hour (actually 50 minutes)
Location
Skiles 257
Speaker
Kouki Sato – University of Tokyo
I will review the definition of nu+ equivalence, which is an equivalence relation on the knot concordance group, and introduce a partial order on the equivalence classes. This partial order is preserved by all satellite maps and some concordance invariants. We also consider full-twist operations and its relationship to the partial order.

Singularities of Lagrangian and Legendrian fronts

Series
Geometry Topology Seminar Pre-talk
Time
Monday, February 11, 2019 - 12:45 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Daniel Álvarez-Gavela – IAS
The semi-cubical cusp which is formed in the bottom of a mug when you shine a light on it is an everyday example of a caustic. In this talk we will become familiar with the singularities of Lagrangian and Legendrian fronts, also known as caustics in the mathematics literature, which have played an important role in symplectic and contact topology since the work of Arnold and his collaborators. For this purpose we will discuss some basic singularity theory, the method of generating families in cotangent bundles, the geometry of the front projection, the Legendrian Reidemeister theorem, and draw many pictures of the simplest examples.

Introduction to symplectic flexibility

Series
Geometry Topology Seminar Pre-talk
Time
Monday, December 3, 2018 - 12:45 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Oleg Lazarev – Columbia
I will describe the h-principle philosophy and explain some recent developments on the flexible side of symplectic topology, including Murphy's h-principle for loose Legendrians and Cieliebak and Eliashberg's construction of flexible symplectic manifolds in high-dimensions.

Models of unstable motivic homotopy theory

Series
Geometry Topology Seminar Pre-talk
Time
Monday, November 12, 2018 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Tom Bachmann – MIT
I will review various ways of modeling the homotopy theory of spaces: several model categories of simplicial sheaves and simplicial presheaves, and related infinity categorical constructions.

Introduction to Freedman's disk embedding conjecture

Series
Geometry Topology Seminar Pre-talk
Time
Monday, November 5, 2018 - 12:45 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Min Hoon Kim – Korea Institute for Advanced Study
In 1982, by using his celebrated disk embedding theorem, Freedman classified simply connected topological 4-manifolds up to homeomorphism. The disk embedding conjecture says that the disk embedding theorem holds for general 4-manifolds with arbitrary fundamental groups. The conjecture is a central open question in 4-manifold topology. In this introductory survey talk, I will briefly discuss Freedman's disk embedding conjecture and some related conjectures (the topological 4-dimensional surgery conjecture and the s-cobordism conjecture). I will also explain why the disk embedding conjecture implies that all good boundary links are freely slice.

Transverse links in the tight three sphere

Series
Geometry Topology Seminar Pre-talk
Time
Monday, October 15, 2018 - 00:45 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Lev Tovstopyat-Nelip – Boston College
We explain the (classical) transverse Markov Theorem which relates transverse links in the tight three sphere to classical braid closures. We review an invariant of such transverse links coming from knot Floer homology and discuss some applications which appear in the literature.

Higher Order Linking Numbers

Series
Geometry Topology Seminar Pre-talk
Time
Monday, September 24, 2018 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Miriam Kuzbary – Rice University
In this introductory talk I will outline the general landscape of Milnor’s invariants for links. First introduced in Milnor’s master’s thesis in 1954, these invariants capture fundamental information about links and have remained a fascinating object of study throughout the past half century. In the early 80s, Turaev and Porter independently proved their long-conjectured correspondence with Massey products of the link complement and in 1990, Tim Cochran introduced a beautiful construction to compute them using intersection theory. I will give an overview of these constructions and motivate the importance of these invariants, particularly for the study of links considered up to concordance.

Pages