- Mathoverflow 40445 (Looking for Implementations of Domino Shuffling)
- Mathoverflow 81009 (Height Function from Max-Flow/Min-Cut)
- Mathoverflow #78302 (Rhombus Tilings with More than Three Directions

The original program was implemented by Sameera Iyengar who was a student at MIT in the mid 90’s and has since gone into theater. Programming has changed a bit in the past 20 years, and I have posted my own python implementation on the internet.

The height of the domino goes up and down around each tiling. What happens when we join some tiles together?

1-0-1-0-1-0-1-0 | | | | | | 2-3-2 I 2-3-2 I | | | | | | 1-0-1-0-1-0-1-0

Here `I`

means -1 (so as to only take up one space in the `ASCII`

art).

If you tile a rectangle with dominos, the height function defines a surface over the edges of the tiling.

Here are pictures of 3D surfaces you can get with the domino tiling projected to various angles.

3D partitions `------`

Lozenge tilings

The domino group is:

The lozenge group is:

These were calculated by William Thurston in his math monthly article. Cayley graphs are graphical ways of drawing the elements of a group presentation. Thurston takes it a step further and draws Cayley 2-complexes.

Tilings by T-tetraminos were analyzed by Korn and Pak.

_ _ _ _ |_ _| | | |_|_ | | _| |_| |_|_ _ _|

Are there any other cases with interesting height functions? It seems very difficult to solve the group equations outside the realm of domino’s or lozenges.

At one point I tried to work out the relation for the L-triomino in the plane. Any lattice polygons can be presented in a group with two generators.

_ _ _ | |_ | _| |_ _| y^(-2)x^2yx^(-1)yx^(-1) |_| xyxyx^(-2)y^(-2) _ _ _ _| | |_ | |_ _| x^2y^2x^(-1)y^(-1)x^(-1)y^(-1) |_| y^(-1)xy^(-1)xy^2x^(-2)

Manipulating the group words looks very difficult. Instead, we can visually merge two triominos to get

_ _ |_| _|_| |_| yxyx^{-1}y^{-1}xy^{-1}x^{-1} = 1 |_| xy^2xy^{-1}x^{-2}y^{-1} = 1

Then taking three of these pieces we can show a loop has order 3, `xyx^{-1}^{-1})^3=1`

. The resulting group is .

This non-commutative loop model looks very interesting… maybe a way of studying non-abelian cohomology of .

- Tiling with Polyominoes and Combinatorial Group Theory
- On Conway’s Tiling Groups
- Squaring Rectangles with Squares (for Dummies
- Domion Tilings and Planar Algebras Generated by a 3-Box
- Laying Train Tracks
- Rotation Numbers and the Jankins-Neumann Ziggurat
- Geodesics in Cat(0) Cubical Complexes
- Crystals, quivers and dessins d’enfants
- Cut and Project Tilings

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Meromorphic functions are allowed to blow up to infinity in a specific way. Obviously, is divergent at and it’s our prototype for a “pole”. Using the geometric series formula we can show it’s holomorphic everywhere else.

Infinite series tend to work in some regions and ‘break’ in others.

in the region |z| < 1 and otherwise.

So maybe the theory of analytic functions is the study of where 'long division' produces a legitimate infinite series expansion. If we do this in two variables, we are led to the theory of amoebas…

Then we'd like to say that is singular at 0 in the same way that is. Indeed if we zoom very closely around the origin we get where . This is kind of bizarre… but if we move continuously around a circle from 1 back to itself, moves from .

Integrals of complex-valued functions are better-behaved then real-valued functions. Cauchy's integral formula lets us deform integrals into small circles around the poles.

If we let

I remember once time in class we solved y’ = y(y-1)(y+1). We can separate the y and t into two sides and integrate.

The left side is a short partial fractions problem.

Residue calculus is like a great big partial fractions problem where we have to guess the coefficients on both sides. Let’s multiply both sides by y-1 and set y = 1

and get A = 1/2. If we multipled by y+1 instead and plugged in y = -1 we get C = 1/2. If we multiply by y and set y = 0 we get B = -1. We can do the integrals

and exponentiate

We don’t know whether inside the absolute value sign is + or -. It depends on where y lies on the real line

- y < -1
- -1 < y < 0
- 0 < y < 1
- 1 < y

In fact, y(t) can’t cross these lines since the log would be infinite, so this would take infinite time. So the equilibrium solutions y = -1,0,1 are **walls** so to speak.

The funciton is holomorphic everywhere and has no zeros, but if we chop off it’s Taylor series a polynomial of degree n must have exactly n roots in the complex plane so what happened?

This problem was solved by Szego and later reviewed by Dieudonne in the 1930’s (long before Mathematica could plot the roots!).

Let any circle around the origin.

We can estimate the contour integral using steepest descent. It can be approximated by a Gaussian around the critical point , **if** we can pick the right contour of integration.

This error term is uniform in . Where could the roots lie, inside the contour

and indeed if we take nth roots we get no solution the first case and for the second.

So what have we learned?

- contour integrals localize around a small circle
- residue theory is just partial fractions
- analytic expansions is just geometric series

I admit this is a very gimmicky (grossly oversimplified) way of learning things but maybe it can pave the way for a new approach.

In a way, this blog entry was a bust, since I was looking to for a simplification of the Kontesevich-Soibelman Wall-Crossing formula as proven by Gaiotto Moore and Neitzke. I’ll try again soon…

- Hitoshi Murayama’s Notes on Contour Integration
- arXiv:0709.1213 Locating the zeros of partial sums of exp(z) with Riemann-Hilbert methods
- arXiv:0807.4723 Four-dimensional wall-crossing via three-dimensional field theory
- Lecture Notes on BPS Wall-Crossing

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Technically, we should Wick rotate the time parameter , the Lorentzian worldsheet metric becomes Riemannian . We can pick normal coordinates and complexify and . Then the action looks like the Nambu-Goto action again (measuring the area of the worldsheet).

This action is invariant under conformal symmetries of the worldsheet

, where parameterizes the worldsheet. In 2 dimensions, the conformal group is especially large and there are infinitely many generators .

I couldn’t picture them so I drew them with mathematica. Technically, these are infinitesimal transformations but I drew them globally.

Tagged: conformal geometry, string theory ]]>