Laura Shou
22 May 2018
10am - 12:40pm
Committee: Michael Aizenman, Alan Sly, Jonathan Hanselman
Special Topics: Probability, Statistical Physics
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Real Analysis (Sly)
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Prove the monotone convergence theorem.
Is there a nowhere dense set of positive measure? fat Cantor set
How would you define a density point?
State the Lebesgue density theorem. How would you prove it? Lebesgue differentiation
If E has positive measure, prove E-E has an interval. I gave the convolution proof, although I guess they were expecting the Lebesgue density theorem proof.
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Complex analysis (Aizenman)
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State the Riemann mapping theorem.
How would you map the upper half plane to the disk.
What is your favorite theorem in complex analysis?
If |f(z)| \le (1+|z|)^{3.5} what can you say about f? polynomial degree \le3.
How would you prove it. Cauchy integral formula, then take derivatives.
Can you write the resulting Cauchy estimate?
What are harmonic functions? How are they related to analytic functions?
Say you have a disk and you know the boundary values of a harmonic function. What can you say about the values of the function inside the disk.
What is the formula for the Poisson kernel? What about for a general domain? Explain the relation between Poisson kernel and Brownian motion.
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Algebra (Hanselman)
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State the Sylow theorems.
How would you prove it?
Classify groups of order 12.
What is the classification of f.g. abelian groups?
How would you prove it? Classification of f.g. modules over PID
Does the classification hold for more general rings? What if I just had an integral domain? Where do you use PID?
What are the invertible linear maps (Z/2)^3 -> (Z/2)^3? What is rational canonical form and how does it relate to the classification of f.g. modules over a PID?
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Probability (mostly asked by Sly, and some by Aizenman)
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Sn = sum X_i, X_i iid with mean 0, tau stopping time, is E[S_tau] = 0?
Define conditional expectation. Why does it exist? Radon-Nikodym
Let tau be the hitting time of 10 of a SRW in 1D. Then E[S_tau] = 10.
Can I make money betting this way? (play until gain $10)
What is E[tau]? infinity
What is the distribution of tau? they wanted me to write it in terms of the running maximum M_t and use the reflection principle
Do you know any theorems that would tell me a condition when E[S_tau] = 0? maybe for martingales? Optional stopping time theorem for martingales, defined uniformly integrable
Application of the optional stopping time theorem
Say S_n = sum X_i, X_i iid and mean 0.
When is E[S_tau] = 0?
Can you compute E[tau]?
When does S_{n\vee tau} converge in L^2?
What about when tau is bounded, or if just E[tau] < infinity?
Show that martingale increments are independent
How large is S_n? ~sqrt(n)
Can you generalize to stuff other than simple random walk? central limit theorem
A: Do you know about the Cauchy distribution? no...
He said he was going to ask about the sum of Cauchy distributions
A: Do you know about infinitely divisible random variables? no...
Then I forgot what they asked next.
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Statistical physics (Aizenman)
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We didn't have much time because Aizenman and Sly had another exam at 1:30pm, so this section was very short.
Define Gibbs measure (I wrote it in finite volume)
Define symmetry breaking. I gave the example of Z/2 symmetry in the Ising model.
What is the Mermin Wagner theorem? Can you say something about the proof or proof idea? Why does it work in d=2 but not d\ge3?
What do minimizers of the Dirichlet energy problem look like?
What are the Euler-Lagrange equations for this model? -\Delta u = 0, harmonic functions.
What do harmonic functions in d=2 look like? d=3? (Recall this is on Z^d)
State your favorite theorem/result in statistical physics.