Conan Wu's generals
Commitee: David Gabai (Chair), Larry Guth, John Mather
Topics: Metric Geometry, Dynamical Systems
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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.
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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.
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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)
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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 $c L_1 L_2$)
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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)