# Kyle Miller

**Office:** 1075 Evans

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

I am a sixth-year Ph.D. student in the
Berkeley mathematics department
advised by Ian Agol. My **CV**.

## Teaching

- Math 1B, Spring 2020
- Math 55, Spring 2017
- Math 54, Fall 2016
- Math 54, Summer 2016 (lecture notes)
- Math 54, Spring 2016
- Math 1B, Fall 2015
- Math 1A, Spring 2015
- Math 1A, Fall 2014

## Papers

- 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*, (submitted). 1909.00790 [math.GT]. - Calvin McPhail-Snyder and Kyle A. Miller,
*Planar diagrams for local invariants of graphs in surfaces*, to appear in Journal of Knot Theory and its Ramifications (2020), arXiv:1805.00575 [math.GT].

## Talks

Research:

- “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.

## Other artifacts

Math tools has programs for some possibly useful computations.

Notes has some notes.

## Toys or demonstrations

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

- Hopf. Fly through a 24-cell and the Hopf flow in ${S}^{3}$. WASD to move, click and drag to rotate perspective.
- 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.)
- Julia set viewer. See this post for an explanation.
- 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 ${\mathbb{R}}^{2}$. - Heat equation & Wave equation. Click and drag a curve, then watch it evolve.
- 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.
- Row reducer. Computes the reduced row echelon form of a matrix mechanically, and displaying every step.
- 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.