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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

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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

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MA5311. Take home exam due 03/20

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

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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

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MA5311. Non orientable manifolds

Here are some videos to visualize non orientability.  

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HW4 MA5161. Due February 24

Exercise. Let be a filtered probability space that satisfies the usual conditions. We denote and for , is the restriction of to . Let be a probability measure on such that for every , Show that there exists a right continuous … Continue reading

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HW4 MA5311. Due February 24

Exercise 1. Let be a smooth manifold and be a linear operator such that for every smooth functions , . Show that there exists a vector field on such that for every smooth function , . Exercise 2. Let be the open … Continue reading

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