In this talk, we'll sketch how one might hope to construct spaces (or spectra) from Floer theories, including framed flow categories and finite-dimensional approximation. If time allows, we'll talk about some questions Floer spaces (or spectra) can be useful for.
I will talk about various notions of equivalence for manifolds and morphisms and the relationships between them. Questions, interruptions, and detours are strongly encouraged!
This will be an introduction to Legendrian contact homology (LCH), a version of Floer homology that's important in contact topology, for the setting of Legendrian knots in R^3 with the standard contact structure. LCH is the homology of a differential graded algebra that can be defined combinatorially in terms of a diagram for the knot. We'll explore this combinatorial definition, with examples, and discuss some auxiliary invariants derived from LCH. No background about contact manifolds or Legendrian knots will be assumed.
A central theme in 4-dimensional topology is the search for exotic 4-manifolds, i.e. families of smooth manifolds that are homeomorphic not diffeomorphic. We will survey some basic results in this area.
This talk will include background information on contact structures and open book decompositions of 3-manifolds and the relationship between them. I will state the necessary definitions and include a number of concrete examples. I will also review some convex surface theory, which is the tool used in the main talk to investigate the contact structure – open book relationship.
Elliptic surfaces are some of the most well-behaved families of smooth, simply-connected four-manifolds from the geometric and analytic perspective. Many of their smooth invariants are easily computable and they carry a fibration structure which makes it possible to modify them by various surgical operations. However, elliptic surfaces have large Euler characteristics which means even their simplest handle-decompositions appear to be quite complicated. In this seminar, we will learn how to draw several different handle diagrams of elliptic surfaces which show explicitly many of their nice properties. This will allow us to see many useful properties of elliptic surfaces combinatorially, and gives insight into the constructions of their exotic smooth structures.
A CAT(0) space is a geodesic metric space where triangles are thinner than comparison triangles in a Euclidean plane. Prime examples of CAT(0) spaces are Cartan-Hadamard manifolds: complete simply connected Riemannian spaces with nonpositive curvature, which include Euclidean and Hyperbolic space as special cases. The triangle condition ensures that every pair of points in a CAT(0) space can be connected by a unique geodesic. A subset of a CAT(0) space is convex if it contains the geodesic connecting every pair of its points. We will give a quick survey of classical results in differential geometry on characterization of convex sets, such the theorems of Hadamard and of Chern-Lashof, and also cover other background from the theory of CAT(0) spaces and Alexandrov geometry, including the rigidity theorem of Greene-Wu-Gromov, which will lead to the new results in the second talk.
This talk will be an introduction to the theory of surfaces, some tools we use to study surfaces, and some uses of surfaces in "real life". In particular, we will discuss the mapping class group and the curve complex. This talk will be aimed at an audience with a minimal background in low-dimensional topology.
Legendrian knots are smooth knots which are compatible with an ambient contact structure. They are an essential object of study in contact and symplectic geometry, and many easily posed questions about these knots remain unanswered. In this talk I will introduce Legendrian knots, their properties, some of their invariants. Expect lots of pictures.
The finite index subgroups of a finitely presented group generate a topology on the group. We will discuss using examples how this relates to the organization of a group's finite quotients, and introduce the ideas of profinite rigidity and flexibility.
