# 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.

Mathblog

Student 3-Manifold Seminar

KnotFolio

## 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].

## 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 ($p\left(r{e}^{i\theta }\right)$) 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.)
• Julia set viewer. See this post for an explanation.
• 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 ${\mathbb{R}}^{2}$.
• Heat equation & Wave equation. Click and drag a curve, then watch it evolve.
• 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.