Slicing knots using topological Surgery Theory
A knot in the 3-sphere is called slice if it bounds a 2-disc in the 4-ball. Most techniques for constructing such slicing 2-discs are very hands-on, involving actually drawing or seeing the disc. I will discuss a famous exception to this, Freedman's theorem that Alexander polynomial 1 knots are (topologically) slice. I hope to mention some other similar results and explain why there are currently not many examples of such conditions.
From algebraic curves to ribbon disks and back
Here's a hot take from Moritz Epple's sizzling "Branch Points of Algebraic Functions and the Beginnings of Modern Knot Theory": knots were invented to study algebraic curves in complex Euclidean space. For background, I'll present a few triumphs of this perspective. Then we'll turn to a conjecture about the relationship between different knots associated to the same algebraic curve. We'll use a simple topological construction to show that the conjecture is false and that counterexamples are quite common.
An informal discussion of birational geometry and intersection theory
In this talk, I will introduce one of the basic tools from birational geometry: blowing up. First, I will introduce rational equivalence, the notion of intersection, and ample and nef divisors. The goal of this talk will be to consider the nodal curve embedded in the projective plane and describe the blow-up of the plane at the node, using the language of intersection theory. This example helps illustrate how birational geometry can be used to resolve singularities, and it also gives us examples of ample and nef divisors.
Introduction to Percolation Theory
How porous must a rock be so a fluid can permeate through it to get to the center? Modeling an answer to this question, Broadbent and Hammersley in 1957 gave a framework to what would become Percolation theory, a field that finds applications in physics, material science, and even public health!
I will discuss the simplest models of percolation, integer lattice graphs where each edge appears at random with probability p. We ask: For which values of p can we be certain that there is a path to infinity from 0? No background beyond some measure theory is assumed.
A discussion on the birational geometry of the Hilbert scheme of 2 points on P^2
Studying the ample, nef, and effective divisors of a scheme helps us understand it's geometry. It is interesting to study such divisors on moduli spaces - not only does this shed light on their geometry, but divisors on moduli spaces come with additional modular interpretation. In this talk, I will first give the necessary background information on the intersection product and nef, ample, and effective divisors. I will then give examples of such divisors on the Hilbert scheme of 2 points on P^2.
Bounds on Minimal Compression Bodies
In 1980, Casson and Long showed via disk surgery that given two simple closed curves on a surface, there are finitely many compression bodies in which the two curves compress. Taking a closer look at the surgery and using some nice facts and constructions from Biringer, Lecuire, and Vlamis, in fact on a surface S of genus g, there are at most 12(g-1)^2 minimal compression bodies! I will introduce compression bodies, we will look at examples of disk surgery, and discuss the bound. I'll mention how this relates to moving through disk sets in the curve complex.
An introduction to L-functions: Tate’s thesis
Tate’s thesis provided an elegant and unified treatment of the analytic continuation and functional equation of the L-functions attached to Hecke characters. It is now viewed as the theory of automorphic representations of GL(1), and so it remains a fundamental reference and starting point for anyone interested in the modern theory of automorphic representations. In this talk, I will start with some historical remarks on L-functions and explain both the local and the global theories presented in Tate’s work.
The 3-Sphere, Binary groups, and the McKay Correspondence
The 3-sphere is the primordial example of a compact Lie group, and one of my favourite miracles of nature. All of the theory of compact lie groups (conjugacy of maximal torii, surjectivity of the exponential map, etc.) can be seen and written down explicitly for the 3-sphere. In 1979 the miraculousness of 3-sphere grew further still when Mckay discovered a correspondence connecting the representation theory of the finite subgroups of the 3-sphere (the so-called binary groups) with the simply-laced (affine) Dynkin Diagrams (the so-called ADE diagrams). In this talk I’ll outline some of my favourite features of the 3-sphere, characterise the binary groups, and state an explicit version of the Mckay Correspondence, which has since grown far beyond McKay’s original vision.
Bing Topology: The amazing Theorem
We will go over the basics of Bing Topology and prove the amazing theorem, i.e. Bing’s shrinking criterion. We will then apply the criterion to prove the generalized Schonflies Theorem and apply it to prove funny facts about whitehead manifolds. Time permitting we might also say something about building non-trivial involution of the 3 sphere that are not conjugated to smooth involutions (this was Bing’s original application)
Tunnel number and knotted theta graphs in the 3-sphere
In 1957, Kinoshita presented the first example of a trivalent graph with two vertices in the 3-sphere, which has no knotted subgraph but is knotted itself. In this talk, I will introduce tunnel number of knots and some of its properties. Then I will show that the notion of tunnel number provides a geometric way to easily see that Kinoshita's example is indeed knotted. This proof is recent due to Scott Taylor. There will be lots of pictures.
Foliations in Characteristic p
I will define foliations on varieties defined over a field of characteristic p. I will explain two things they allow us to do:
1) study the “Galois theory” for purely inseparable field extensions.
2) connect the differential topology to the birational geometry of a variety.