# Kyle Miller

**Office:** 1075 Evans

**E-mail:** kmill at math.berkeley.edu

I am a seventh-year Ph.D. student advised by Ian Agol in the Berkeley mathematics department.

**Research interests:** low-dimensional topology (especially knots and spatial graphs), representation theory, and computer-assisted proofs.
I am especially interested in using diagrammatic reasoning to bridge topology and algebra.

**CV** | **Research statement** | **Teaching statement**

**Mathblog** | **Notes** | **Student 3-Manifold Seminar**

**Math tools** | **KnotFolio**

**MathFacts** YouTube channel

**math.stackexchange.com**

**Current teaching:** Math 54, Spring 2020 with Prof. David Nadler.

## Papers

### Published

- (with Calvin McPhail-Snyder)
*Planar diagrams for local invariants of graphs in surfaces*, J. Knot Theory Ramifications**29**(2020), no.1, 1950093, 49, arXiv:1805.00575v2 [math.GT]. MR 4079619

### Preprints

- (with Alena Gusakov and Bhavik Mehta)
*Formalizing Hall’s Marriage Theorem in Lean*. 2101.00127 [math.CO]. - Chris Anderson, Kenneth L. Baker, Xinghua Gao, Marc Kegel, Khanh Le, Kyle A. Miller, Sinem Onaran, Geoffrey Sangston, Samuel Tripp, Adam Wood, and Ana Wright,
*Asymmetric L-space knots by experiment*. 1909.00790 [math.GT].

### Pre-preprints

*The homological arrow polynomial*. Draft 2020/11/15 (PDF).

### Exposition

*All the ways I know how to define the Alexander polynomial*(version 2020/11/14). A survey of the Alexander polynomial. Work in progress.

## Talks

Research:

- “A 2D TQFT approach to topological graph polynomials and graphs in thickened surfaces” for the Special Session on Developments in Spatial Graphs at the JMM, January 6, 2021.
- “A TQFT approach to topological graph polynomials” for the University of Virginia geometry seminar. December 3, 2019.
- “Invariants of graphs in thickened surfaces from topological graph polynomials” for the Rice topology seminar. November 18, 2019.
- “Invariants of virtual spatial graphs based on topological graph polynomials” (invited) for the Special Session on Invariants of Knots and Spatial Graphs, Fall Western Sectional Meeting of the AMS. November 10, 2019.
- “Diagrams on surfaces and an invariant of virtual spatial graphs” for the 3-manifold seminar. April 17, 2018. (About a paper joint with Calvin McPhail-Snyder.)

Student seminars:

- “Doing math the Lean way” for the Berkeley Lean Student Seminar. June 26, 2020.
- “The arithmeticity of figure eight knot orbifolds” for the 3-manifold seminar. February 12, 2019.
- “What is an alternating knot?” for the 3-manifold seminar. November 27, 2018.
- “The Lickorish-Wallace theorem” for the student low-dimensional topology seminar. September 27, 2018.
- “Introduction to the Jones polynomial and the Temperley-Lieb category” for GRASP seminar. September 4 & 12, 2018. Part 1, Part 2.
- Quandles for the topology topics course. November 2, 2017.
- Spatial graph invariants for the 3-manifold seminar. September 19 & 26, 2017.
- The Alexander Ideal for “Knot Another Seminar”. April 21, 2017.

## Toys or demonstrations

These require JavaScript to be enabled, and they are usually only tested with Chrome and Firefox.

### Calculus and differential equations

- Spring cart phase diagram. An interactive mass-damper-spring system showing the system’s vector field in phase space.
- Spring cart. A cart coupled by a spring to an oscillator.
- Heat equation & Wave equation. Click and drag an initial condition curve, then watch it evolve.

### Linear algebra

- Cat transformation. Experiment with $2\times 2$ matrix transformations by transforming the image of a cat.
- Cat transformation: eigenvector edition. Shows how the eigenbasis is just scaled by the transformation.
- Basis toy. Experiment with bases of ${\mathbb{R}}^{2}$.
- Basis change toy.
Experiment with
*two*bases of ${\mathbb{R}}^{2}$. - Row reducer. Computes the reduced row echelon form of a matrix mechanically, and displaying every step.
- Interactive cofactor expansion. See how all choices give the same result.

### Analysis

- zgraph is a complex function grapher that uses domain coloring. Documentation.
- Complex polynomials. Type in a
polynomial (for instance,
`x(x+2i)^2+1`), then drag the circle ($r{e}^{i\theta}$) on the left side around to see its image ($p(r{e}^{i\theta})$) on the right. One reason every polynomial has a root is that for large radii the image will wrap around zero $\mathrm{deg}p$ times, but for very small radii the image will wrap tightly around $p(0)$ — and somewhere in between the image must pass through zero. Arrow keys animate the domain coloring. - Complex polynomials (with derivatives). Similar, but has domain coloring for the derivative of the function as well. This is meant to illustrate the Gauss-Lucas theorem.
- Complex rational functions. Place zeros and poles of different orders on the plane, and see an animated domain coloring. The original idea was to let people have the rainbow go around points a certain number of times as a visualization of a first cohomology classes, which could make an interesting interactive exhibit at a museum. (More explanation.)
- Singularities of circle-valued functions. A variation on the previous one, but with a second type of singularity that arises for circle-valued harmonic functions.
- Julia set viewer. See this post for an explanation.

### Other

- Hopf. Fly through a 24-cell and the Hopf flow in ${S}^{3}$. WASD to move, click and drag to rotate perspective.
- Curves. Just a grid phasing through resolutions of the intersections, inspired by Vassiliev invariants. Tab, space, and ‘a’ do something.
- Trefoil. A rotating tubular neighborhood of a trefoil knot; nothing deep.