Algebraic topology
[Casson] What is an H-space?
What special property does $\pi_1$ of an H-space have? Prove it.
[Stallings] Why can't $S^2$ be an H-space?
[Casson] What is the homology of $S^2\times S^2$? Cohomology?
How is the cohomology related to the homology?
What is the cup product structure?
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Combinatorics
[Sinclair] What is an Eulerian poset? What is graded? What is a rank function? What is the length of a chain? What is $\mu$ of an interval? Why is it called Eulerian?
[Sinclair] Consider monotonic paths from $(0,0)$ to $(n,n)$ consisting of unit steps either $+(1,0)$ or $+(0,1)$. $\alpha \ge \gamma$ if $\alpha$ is never below $\gamma$. Define a hill to be a $+(0,1)$ step followed by a $+(1,0)$ step. Define a valley to be a $+(1,0)$ step followed by a $+(0,1)$ step. Given $\alpha \ge \beta$, define hills of $\alpha$ and valleys of $\beta$ as good points. Define valleys of $\alpha$ and hils of $\beta$ as bad points. Show the number of good points is always greater than the number of bad points.
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Smooth Manifolds
[Casson] Define a Morse function. Define index. What is $h^{-1}(a,b)$ if $(a,b)$ does not contain any critical values? What if it contains exactly one critical value? If a morse function of a manifold $M$ has exactly two critical points, what can you say about $M$?
[Serganova] What is the Frobenius Integrability Theorem? What is a distribution? What is an integral submanifold? What is $[x,y]$? Can you give an example of a distribution which is not integrable?