Topology Reading Group Spring 2021
TRG is an informal grad student seminar providing a forum for grad students to share what they have been learning about with other students.
Most weeks featured pre-talks for the Boston College Geometry/Topology/Dynamics seminar.
Here are some links to a few previous iterations of this seminar;
February 4th: Gage Martin
4-genus, concordance, and amphicheiral knots
This talk will begin by recalling the basics of 4-genus and concordance, providing motivation for studying these questions, and recalling some invariants used to study them. From this background, we will then turn our attention to amphicheiral knots in these contexts. If time permits, we will say a word about Casson-Gordon signatures.
Pre-talk for Allison Miller's talk in the GTD seminar
February 11th: Gage Martin
What is Khovanov homology and how can you define a 4-dimensional invariant with it?
In this talk we will seek to answer the question of how one can define a 4-dimensional invariant using Khovanov homology. To do this we will first review some basics of Khovanov homology, including a basic computation. Along the way we will try to provide some applications of Khovanov homology to low-dimensional topology.
Pre-talk for Ikshu Neithalath's talk in the GTD seminar
February 18th: Marius Huber
Rational homology cobordisms and how to obstruct them
In this pre-talk to Daniele Celoria's seminar talk, I will first go over what a rational homology cobordism between two 4-manifolds is, and how these cobordisms naturally arise in knot theory.
Then I will explain what intersection lattices are and how they -- combined with a powerful theorem of Donaldson -- can be used to obstruct the existence of rational homology cobordisms between certain 4-manifolds.
Pre-talk for Daniele Celoria's talk in the GTD seminar
February 25th: Fraser Binns
Knots Groups and Orderability
I will discuss connections between group orderability and low dimensional topology, with a particular focus on knot groups.
Pre-talk for Jonathan Johnson's talk in the GTD seminar
March 4th: Fraser Binns
Genus Problems and Knot Floer homology
I will survey some minimal genus problems, and their interaction with Knot Floer homology.
Pre-talk for Katherine Raoux's talk in the GTD seminar
March 11th: Gage Martin
Stable homotopy and khovanov homology
In this talk we will discuss the ideas of stable homotopy theory and connections with khovanov homology and low dimensional topology.
Pre-talk for Ross Akhmechet's talk in the GTD seminar
March 18th: Ian Montague
Pin(2)-Equivariant Seiberg-Witten Floer Homology and a Disproof of the Triangulation Conjecture
In this talk I will provide motivation for, and an outline of Ciprian Manolescu's proof from 2013 that there exist non-triangulable n-dimensional manifolds for every n>=5. Here's a nice Quanta article about the subject that may be of interest:
https://www.quantamagazine.org/triangulation-conjecture-disproved-20150113/
March 25th: Braeden Reinoso
Open Books (in High Dimensions), Lefschetz Fibrations, and Embedding Problems
I'll overview the ideas of Lefschetz fibrations, open book decompositions, and their relationship with embedding problems. This is a pretalk for Kuldeep Saha's talk in the virtual GTD seminar later in the afternoon.
Pre-talk for Kuldeep Saha's talk in the GTD seminar
April 1st: Ethan Farber
The curve complex and asymptotic translation length
The curve complex is a classical object one may associate to a compact surface with boundary. In this talk we will introduce the curve complex and through it take an alternate perspective on the theory of mapping class groups. In particular, we will investigate the notion of the asymptotic translation length of a mapping class acting on the curve complex. Time permitting, we will discuss a generalization of the curve complex to three dimensions. Chenxi Wu will further develop this last during his seminar talk at 4pm in the Boston College Geometry/Topology/Dynamics seminar.
Pre-talk for Chenxi Wu's talk in the GTD seminar
April 8th: Jacob Caudell
Two-bridge knots, concordance, and cobordism
This is a pretalk for Marius Huber's talk in the GTD seminar. We will discuss the relationship between (ribbon) concordance and (ribbon) rational homology cobordism in the context of two-bridge knots and lens spaces. We will briefly discuss Lisca's combinatorial proof of the slice-ribbon conjecture for two-bridge knots.
Pre-talk for Marius Huber's talk in the GTD seminar
April 15th: Gage Martin
Annular Khovanov homology and meridional disks
The wrapping conjecture of Hoste-Przytycki suggests a relationship between the maximum non-zero annular Khovanov grading and the minimal geometric intersection number with a meridional disk. Inspired by this conjecture, we exhibit infinite families of annular links for which the maximum non-zero annular Khovanov grading grows infinitely large but the maximum non-zero annular Floer-theoretic gradings are bounded. We also show this phenomenon exists at the decategorified level for some of the infinite families. Our computations provide further evidence for the wrapping conjecture and its categorified analogue. Additionally, we show that certain satellite operations cannot be used to construct counterexamples to the categorified wrapping conjecture.
April 22nd: Ethan Farber
Billiards and flat surfaces
How does a billiard ball bounce around the table? While pool enthusiasts have long been satisfied with the answer for a rectangular table, dynamicists are still trying to understand the dynamics of billiards on other tables. Via two primary examples of billiard tables, we will motivate the study of flat surfaces. Better yet, we will introduce a parameter space of such surfaces and, time permitting, investigate dynamics on this space itself.
Pre-talk for Karl Winsor's talk in the GTD seminar
April 29th: Braeden Reinoso
Knot Floer homology and homology concordance
I'll introduce the basics of knot Floer homology and of homology concordance, with an eye towards defining a collection of homomorphisms on the knot homology concordance group via knot Floer homology. This is a pretalk for Linh Truong's talk in the virtual GTD seminar later that day.
Pre-talk for Linh Truong's talk in the GTD seminar
May 6th: Ethan Farber
Shifting perspectives in dynamics
Complex dynamics is at the intersection of many fascinating subfields of mathematics. In this talk, we will introduce the Mandelbrot set and explicate the role it plays in unifying several mathematical narratives by providing an alternate perspective. In particular, we emphasize the fruitfulness of analyzing parameter spaces of systems, using this to draw beautiful conclusions—and beautiful images!
May 13th: Marius Huber
Dehn Surgery and the Berge and Cabling Constructions
In this pre-talk to Jacob Caudell's GTD Seminar talk, I will first cover what Dehn surgery is. After that, I will discuss Berge's doubly-primitive construction and the cabling construction.
Pre-talk for Jacob Caudell's talk in the GTD seminar