# Kyle Miller

**Office:** 747 Evans

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

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

## Teaching

- Math 55, Spring 2017
- Math 54, Fall 2016
- Math 54, Summer 2016
- Math 54, Spring 2016
- Math 1B, Fall 2015
- Math 1A, Spring 2015
- Math 1A, Fall 2014

## Talks

- 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. Someday it will have built-in documentation, but in the meantime I’m happy to explain it if you track me down!
- 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. - Curves. Just a grid phasing through resolutions of the intersections, inspired by Vassiliev invariants. Tab, space, and ’a’ do something.
- Julia set viewer. See this post for an explanation.
- Trefoil. A rotating tubular neighborhood of a trefoil knot; nothing deep.
- 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. 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.
- Interactive cofactor expansion.