# 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.

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

## Papers

### 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).

## Talks

Research:

Student seminars:

## Toys or demonstrations

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

### Linear algebra

• Cat transformation. Experiment with $2×2$ 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 ($p\left(r{e}^{i\theta }\right)$) 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 $p\left(0\right)$ — 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.