*Every -complex is homotopy equivalent* to (the realization of) a *simplicial complex.*

# Author: dantegrevino

# Too complex to be true but too symmetrical to be false

For , let be a complex domain and let . Let be a complex Brownian motion starting from . Set .

Suppose that there exists a conformal isomorphism such that . Set and define for

and .

Then and have the same distribution.

# What goes up must come down

Let such that . Prove that .

# The plane is too small 2

Let be the symmetric random walk in –, starting at the origin. So, and the transition probabilities are if and otherwise.

Let be the orthogonal projections of the random walk onto the diagonals and .

- Prove that are independent symmetric random walks in such that if and only if .
- Prove that , as , where and . Conclude that . This is equivalent to the recurrence of the random walk.

# The plane is too small

Prove that the planar Brownian motion is neighbourhood recurrent, but not point recurrent. Conclude that the path of planar Brownian motion is dense in the plane.

# It’s a matter of principles (of convergence)

Let be a sequence of entire functions such that, for every , there exists . Suppose that is continuous and . Let be an entire function such that . Prove that over compact sets if and only if , for every .