# Category Archives: Uncategorized

## MA5311. Take home exam

Exercise 1. Solve Exercise 44 in Chapter 1 of the book. Exercise 2.  Solve Exercise 3 in Chapter 1 of the book. Exercise 3.  Solve Exercise 39 in Chapter 1 of the book. Exercise 4. The heat kernel on is given by . By … Continue reading

## MA5161. Take home exam

Exercise 1. The Hermite polynomial of order is defined as Compute . Show that if is a Brownian motion, then the process is a martingale. Show that   Exercise 2. (Probabilistic proof of Liouville theorem) By using martingale methods, prove that if … Continue reading

## MA5311. Take home exam due 03/20

Solve Problems 1,2,8,9,10,11 in Milnor’s book. (Extra credit for problem 6)

## MA5161. Take home exam. Due 03/20

Exercise 1. Let . Let be a continuous Gaussian process such that for , Show that for every , there is a positive random variable such that , for every and such that for every , \textbf{Hint:} You may use … Continue reading

## MA5311. Non orientable manifolds

Here are some videos to visualize non orientability.