# 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.

Mathblog

## 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 ($p\left(r{e}^{i\theta }\right)$) 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.
• 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 $2×2$ 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.