Conan Wu's generals Commitee: David Gabai (Chair), Larry Guth, John Mather Topics: Metric Geometry, Dynamical Systems ----------------------------------------------------------------------- Real analysis: Mather: Construct a first category but full measure set. (I gave the intersection of decreasing balls around the rationals) Guth: $F:S^1 \rightarrow \mathbb{R}$ 1-Lipschitz, what can one say about its Fourier coefficients. (Decreasing faster than $c*1/n$ via intergration by parts) Mather: Does interegration by parts work for Lipschitz functions? (Lip imply absolutely continuous then Lebesgue's differentialation theorem) Mather: If one only has bounded variation, what can we say? ($f(x) \geq f(0) + \int_0^x f'(t) dt$) Mather: If $f:S^1 \rightarrow \mathbb{R}$ is smooth, what can you say about it's Fourier cooeficients? (Prove it's rapidly decreasing) Mather: Given a smooth $g: S^1 \rightarrow \mathbb{R}$, given a $\alpha \in S^1$, when can you find a $f: S^1 \rightarrow \mathbb{R}$ such that $g(\theta) = f(\theta+\alpha)-f(\theta)$ ? (A necessary condition is the intergal of $g$ needs to vanish, I had to expand everything in Fourier coefficients, show that if $\hat{g}(n)$ is rapidly decreasing, compute the diophantine set $\alpha$ should be in to garentee $\hat{f}(n)$ being rapidly decreasing. Gabai: Write down a smooth function from $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ with no critical points. (I wrote $f(x,y) = x+y$) Draw its level curves (straight lines parallel to x=-y) Gabai: Can you find a such function with the level curves form a different foliation from this one? (I think he meant that two foliations are different if there is no homeo on $\mathbb{R}^2$ carrying one to the other, After playing around with it for a while, I came up with an example where the level sets form a Reeb foliation, and that's not same as the lines!) We moved on to complex. ----------------------------------------------------------------------- Complex analysis: Guth: Given a holomorphic $f:\mathbb{D} \rightarrow \mathbb{D}$, if $f$ has $50$ $0$s inside the ball $B_{1/3}(\bar{0})$, what can you say about $f(0)$? (with a bunch of hints/suggestions, I finally got $f(0) \leq (1/2)^{50}$ -- construct polynomial vanishing at those roots, quotient and maximal modulus) Guth: State maximal modulus principal. Gabai: Define the Mobius group and how does it act on $\mathbb{H}$. Gabai: What do the Mobius group preserve? (Poincare metric) Mather: Write down the Poincare metric, what's the distance from $\bar{0}$ to $1$? (infinity) (I don't remember the exact distance form, so I tried to guess the denominator being $\sqrt{1-|z|}$, but then intergrating from $0$ to $1$ does not braely diverge''. Turns out it should be $(1-|z|^2)^2$.) Gabai: Suppose I have a finite subgroup with the group of Mobius transformations acting on $\mathbb{D}$, show it has a global fixed point. (I sketched an arguement based on each element having finite order must have a unique fixed point in the interior of $\mathbb{D}$, if two element has different fixed points, then one can construct a sequence of elements where the fixed point tends to the boundary, so the group can't be finite.) I think that's pretty much all for the complex. ----------------------------------------------------------------------- Algebra: Gabai: State Einsenstein's critieia (I stated it with rings and prime ideals, which leaded to a small discussion about for which rings it work) Gabai: State Sylow's theorem (It's strange that after stating Sylow, he didn't ask me do anything such as classify finite groups of order xx) Gabai: What's a Galois extension? State the fundemental theorem of Galois theory. (Again, no computing Galois group...) Gabai: Given a finite abelian group, if it has at most $n$ elements of order divisible by $n$, prove it's cyclic. (classification of abelian groups, induction, each Sylow is cyclic) Gabai: Prove multiplicative group of a finite field is cyclic. (It's embrassing that I was actually stuck on this for a little bit before being reminded of using the pervious question) Gabai: What's $SL_2(\mathbb{Z})$? What are all possible orders of elements of it? (I said linear automorphisms on the torus. I thought it can only be $1,2,4,\infty$, but turns out there is elements of order $6$. Then I had to draw the torus as a hexigon and so on...) Gabai: What's $\pi_3(S^2)$? ($\mathbb{Z}$, via Hopf fibration) Gabai: For any closed orientable $n$-manifold $M$, why is $H_{n-1}(M)$ torsion free? (Poincare duality + universal coefficient) ----------------------------------------------------------------------- We then moved on to special topics. Metric Geometry: Guth: What's the systolic inequality? (the term 'aspherical' comes up) Gabai: What's aspherical? What if the manifold is unbounded? (I guessed it still works if the manifold is unbounded, Guth 'seem to' agree) Guth: Sketch a proof of the systolic inequality for the n-torus. (I sketched Gromov's proof via filling radius) Guth: Give an isoperimetric inequality for filling loops in the 3-manifold $S^2 \times \mathbb{R}$ where $S^2$ has the round unit sphere metric. (My guess is for any 2-chain we should have $\mbox{vol}_1(\partial c) \geq C \mbox{vol}_2(c)$, then I tried to prove that using random cone arguement, but soon realized the method only prove $\mbox{vol}_1(\partial c) \geq C \sqrt{\mbox{vol}_2(c)}$.) Guth: Given two loops of length $L_1, L_2$, the distance between the closest points on two loops is $\geq 1$, what's the maximum linking number? (it can be as large as $c L_1 L_2$) ----------------------------------------------------------------------- Dynamical Systems: Mather: Define Anosov diffeomorphisms. Mather: Prove the definition is independent of the metric. (Then he asked what properties does Anosov have, I should have said stable/unstable manifolds, and ergodic if it's more than $C^{1+\varepsilon}$...or anything I'm formiliar with, for some reason the first word I pulled out was structurally stable...well then it leaded to and immediarte queation) Mather: Prove structural stability of Anosov diffeomorphisms. (This is quite long, so I proposed to prove Anosov that's Lipschitz close to the linear one in $\mathbb{R}^n$ is structurally stable. i.e. the Hartman-Grobman Theorem, using Moser's method, some details still missing) Mather: Define Anosov flow, what can you say about geodesic flow for neinatively curved manifold? (They are Anosov, I tried to draw a picture to showing the stable and unstable and finished with some help) Mather: Define rotation number, what can you say if rotation numbers are irrational? (They are semiconjugate to a rotation with a map that perhaps collapse some intervals to points.) Mather: When are they actually conjugate to the irrational rotation? (I said when $f$ is $C^2$, $C^1$ is not enough. Actually $C^1$ with derivative having bounded variation suffice) I do not know why, but at this point he wanted me to talk about the fixed point problem of non-separating plane continua (which I once mentioned in his class). Then they asked me to leave the room and shaked my hands after a couple minutes. The exam lasted 4.5 hours (New record? Infact it doesn't feel that long and was shocked when I looked at my watch at the end)