TBA
- Series
- Geometry Topology Student Seminar
- Time
- Wednesday, October 12, 2022 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- TBA
TBA
TBA
Please Note: A three-manifold is a space that locally looks like the Euclidean three-dimensional space. The study of three-manifolds has been at the heart of many beautiful constructions in low dimensional topology. This talk will provide a quick tour through some fundamental results about three-manifolds that were discovered between the late nineteenth century and the Fifties.
Ivanov’s metaconjecture says that every object naturally associated to a surface S with a sufficiently rich structure has the mapping class group as its group of automorphisms. In this talk, I will present several cases of curve graphs that satisfy this metaconjecture and some extensions to even richer structures.
We will discuss Akbulut's construction of two smooth, contractible four-manifolds whose boundaries are diffeomorphic and extend to a homeomorphism but not to a diffeomorphism of the manifolds.
Topological methods have had a rich history of use in convex optimization, including for instance the famous Pataki-Barvinok bound on the ranks of solutions to semidefinite programs, which involves the Borsuk-Ulam theorem. We will give two proofs of a similar sort involving the use of some basic homotopy theory. One is a new proof of Brickman's theorem, stating that the image of a sphere into R^2 under a quadratic map is convex, and the other is an original theorem stating that the image of certain matrix groups under linear maps into R^2 is convex. We will also conjecture some higher dimensional analogues.
This is an expository talk about the slice-ribbon conjecture. A knot is slice if it bounds a disk in the four ball. We call a slice knot ribbon if it bounds a slice disk with no local maxima. The slice-ribbon conjecture asserts all slice knots arise in this way. We also give a very brief introduction to Greene, Jabuka and Lecuona's works on the slice-ribbon conjecture for 3-stranded pretzel knots.
Quantum Teichmüller space was first introduced by Chekhov and Fock as a version of 2+1d quantum gravity. The definition was translated over time into an algebra of curves on surfaces, which coincides with an extension of the Kauffman bracket skein algebra. In this talk, we will discuss the relation between the Teichmüller space and the Kauffman bracket, and time permitting, the quantized version of this correspondence.
Meeting URL: https://bluejeans.com/106460449/5822
Please Note: BlueJeans link: https://bluejeans.com/609527728/0740
The main goal of manifold theory is to classify all n-dimensional topological manifolds. For a smooth 4-manifold X, we aim to understand all of the exotic smooth structures there are to the smooth structure on X. Exotic smooth structures are homeomorphic but not diffeomorphic. Cork twists, Gluck twists, and Log transforms are all ways to construct possible exotic pairs by re-gluing embedded surfaces in the 4-manifold. In this talk, we define these three constructions.
Please Note: BlueJeans link: https://bluejeans.com/575457754/6776
Given a surface S, the Alexander method is a combinatorial tool used to determine whether two self-homeomorphisms of S are isotopic. This statement was formalized in the case of finite-type surfaces, which are surfaces with finitely generated fundamental groups. A version of the Alexander method was extended to infinite-type surfaces by Hernández-Morales-Valdez and Hernández-Hidber. We extend the remainder of the Alexander method to include infinite-type surfaces.
In this talk, we will talk about several applications of the Alexander method. Then, we will discuss a technique useful in proofs dealing with infinite-type surfaces and provide a "proof by example" of an infinite-type analogue of the Alexander method.
This will be practice for a future talk and comments and suggestions are appreciated.
Please Note: BlueJeans link: https://bluejeans.com/473141052/9784
Morse theory is a standard concept used in the study of manifolds. PL-Morse theory is a variant of Morse theory developed by Bestvina and Brady that is used to study simplicial complexes. We develop an extension of PL-Morse theory to simplicial complexes equipped with an action of a group G. We will discuss some of the basic ideas in this theory and hopefully sketch proofs of some forthcoming results pertaining to the homology of the Torelli group.