On Fox's Conjecture of Quasi-Alternating Knots

We verify Fox’s trapezoidal conjecture for a family of alternating pretzel knots. We also extend the conjecture to quasi-alternating slice knots and verify the conjecture for a class of these knots. 

An Exploration of the Geometry of Hyperbolic Maps

We define an analog of polar coordinates for the hyperbolic plane and we propose a definition for the power function on the hyperbolic numbers. Calculus on the hyperbolic numbers allows the solution of the wave equation in much the same fashion as calculus on the complex numbers allows the solution to Laplace's Equation. In complex analysis, the technique of conformal mapping is used to create solutions of Laplace's equation with suitable boundary conditions. We study mappings of the hyperbolic plane with the hope of developing a similar technique to create wave equation solutions.

Lie Groups Fall 2023 Lecture Series

A Lie group is a group that is also a differentiable manifold. This fall at the University of Texas at Austin there will be a weekly seminar on Lie groups and algebras. The goal is to work through problems and examples in Lie Groups, Lie Algebras, and their representations. In week 2, I will give a talk on the matrix exponential,  logarithm, and polar decomposition.  

Junior Topology Seminar: 

On the topology of 4-manifolds 

In this talk we introduce the history of exotic structures and the motivation behind studying 4-manifolds. One of the field's central problems is the famous Poincaré conjecture, to this day the only solved Millennium Prize problem. Throughout the years, solutions to this question in specific dimensions have lead to multiple Fields Medals. In 1982, Freedman proved the conjecture in 4 dimensions, but could not show whether a 4-manifold that is homeomorphic to the 4-sphere is also diffeomorphic to it. Now known as the 4-D smooth Poincaré conjecture, this question is still open; in fact, 4 remains the only dimension for which it is unknown whether manifolds homeomorphic to the sphere are also diffeomorphic to it. One way to understand exotic structure is by studying surfaces embedded in 4-manifolds. 

Constructions of Exotic \R^4 and Stabilizations  

For \R^n where n is not equal to 4, smooth manifolds homeomorphic to \R^n are diffeomorphic. In this talk, we will go over constructions of 4-manifolds that are homeomorphic, but not diffeomorphic to \R^4. The first examples of exotic structure on \R^4 were first found around 1982 by using the theorems of Donaldson and Freedman. Earlier in 1964, Wall proved that any pair of homotopy-equivalent closed, oriented, simply-connected, smooth 4-manifolds become diffeomorphic after taking k-connected sums with (S^2 \times S^2). This technique is called stabilization. We will discuss some recent results involving stabilizations of exotic phenomena in four dimensions.