General exam of Thomas Massoni
Thursday, January 23, 2020
1:30pm - 4:30pm
Committee:
John Pardon (chair), symplectic topology
Peter Ozsváth, algebraic topology
Adam Marcus
The committee decided to start with the standard topics. I said I didn't want to start with algebra and they decided to start with complex analysis.
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COMPLEX ANALYSIS
[AM] I just taught a multivariable calculus class and I was wondering: why is complex analysis an actual subfield of mathematics? Why is it different from doing analysis on R^2? For instance, what can you say about integrating a function along a path?
Talked about basic properties of holomorphic functions: complex analyticity, removal of singularities, Cauchy formula. Proved that the integral of a holomorphic function along a contractible loop is zero.
[AM] Write f = u+iv, what equations are satisfied by u and v?
Stated and proved the Cauchy-Riemann equations.
[JP] What can you say about u?
Harmonic. Mentioned that every harmonic function on a simply connected domain of C is the real part of holomorphic function.
[JP] What else can you say about harmonic functions?
Mentioned a few properties like mean value equality, maximum principle, Poisson formula.
[PO] When are two rectangles conformally equivalent?
Can assume that they have length 1 and widths r, R. Conformally equivalent iff r=R: using the Schwarz reflection principle, obtain biholomorphism between two cylinders/annuli. Then I proved the uniformization of annuli: using Schwarz reflection, the biholo extends to a biholo of the disk and the Schwarz lemma implies that r=R.
[PO] Prove Schwarz reflection for the upper half plane, write down explicitly the extension you use for the annulus.
I also mentioned the reflection principle for harmonic functions. The formula for the annulus involves a lot of bars!
[PO] What can you say about a biholomorphism of C fixing 0?
It’s linear: after multiplication by a complex number, can assume that the biholomorphism fixes 0 and 1, then it corresponds to a biholo of CP1 fixing 3 distinct points so it’s the identity.
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REAL ANALYSIS
[JP] What is the Fourier transform? What can you say about it?
Define Fourier transform of an L1 function, it is continuous and vanishes at infinity.
[JP] Prove, let say, that it is continuous. What is the Schwarz class? What can you say about it?
Schwarz class is preserved under Fourier transform. Fourier transform ‘‘trades regularity for decay and vice versa’’.
[JP] How do you define Fourier transform for distributions? What is the Fourier transform of a periodic function?
I didn't review distribution theory and didn't remember the precise definition of Fourier transform for tempered distribution. I nevertheless wrote down a fishy formula for a periodic function on the board and JP seemed satisfied.
[AM] What is the Fourier inversion theorem? How do you prove it?
Sketched a proof.
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ALGEBRA
[PO] I have a fake algebra question: what can you say about the subgroups of a free group?
They are free! Proved it using covering spaces of wedges of circles.
[PO] What can you say about index two subgroups of the free group in two generators? How many generators? Draw a degree two cover of the wedge of two circles. Can you draw another one? More generally, how many generators for an index d subgroup of a free group in n generators?
Used the Euler characteristic to compute the number of generators.
[JP] What is the Galois group of Q(sqrt(2), sqrt(3))/Q? Write down the elements of the group explicitly. What are the intermediate extensions? Which one is Q(sqrt(2)+sqrt(3))?
It's Q(sqrt(2), sqrt(3)) because the inverse of sqrt(2)+sqrt(3) is sqrt(3)-sqrt(2). They wanted me to give an argument involving the elements of the Galois group. At this point I said some inanities about Galois correspondence but I eventually corrected myself.
[JP] What is a Noetherian ring? Give several equivalent definitions. What is a Noetherian module? Prove that a module over a Noetherian ring is Noetherian iff it is finitely generated.
Three equivalent definitions: 1) ascending chain condition, 2) finitely generated ideals, 3) existence of maximal elements for every collection of ideals. Also showed that if 0->A->B->C->0 is a short exact sequence of modules, B is Noetherian iff A and C are.
[AM] I also have a fake algebra question: what are the eigenvalues of the Fourier transform (on L^2)?
Composing 4 times the Fourier transform gives the identity map (with suitable normalization), so the eigenvalues are in {-1, 1, -i, i}.
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ALGEBRAIC TOPOLOGY
[JP] Tell us about the Thom isomorphism. What is the Thom class?
[PO] What condition on the bundle?
Cohomology with coefficients in Z if orientable, Z/2Z if not.
[JP] How do you prove the uniqueness of the Thom class?
Mentioned the Mayer-Vietoris argument.
[JP] If the base of the fibration is a CW complex, prove existence and uniqueness of the Thom class inductively on the skeleton.
I went through some long exact sequence arguments, JP helped me along the way. It turns out that the only obstruction appears when extending on the 1-skeleton. Extending the Thom class from the k-skeleton to the (k+1)-skeleton is automatic for k>0!
[PO] What do you know about homotopy groups of spheres? Tell me an interesting one.
These groups are hard to compute! They are finitely generated, often torsion. I mentioned pi_3(S^2)=Z and proved it using the Hopf fibration. Also mentioned the quaternionic and octonionic Hopf fibrations.
[PO] Define fibration. How do you prove the homotopy long exact sequence of a fibration? What is the homotopy long exact sequence of a pair and what crucial property is used to prove it?
Homotopy lifting property!
[PO] Why is pi_3(S^3)=Z?
I mentioned it can be done by hand (cf. Milnor) but it's straightforward using Hurewicz theorem.
[PO] Write down the statement of Hurewicz theorem.
First forgot the simply connected assumption but then corrected myself.
[PO] Define torus knots Tp,q. When is Tp,q actually a knot? How could you tell when two torus knots are different?
I mentioned that (p-1)(q-1)/2 is a topological invariant (genus, unknotting number). I also said that one can use knot Floer homology which made PO laugh! Also mentioned the Alexander and Jones polynomials and the pi_1 of the knot complement.
[PO] How do you compute the pi_1 of the knot complement?
It is usually hard. Can get a presentation using a knot diagram.
[PO] What is your favorite tool for computing a pi_1?
Seifert-van Kampen theorem!
[PO] State it. What decomposition would you use for the torus knot?
If X is the complement of a neighborhood of Tp,q in S^3, write X=A∪B where A=small neighborhood of the solid torus, B=small neighborhood of complement of A in S^3. Then A∩B is a torus minus Tp,q which is homeomorphic to a band. In the end, get .
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SYMPLECTIC TOPOLOGY
[JP] What can you say about holomorphic spheres in CP^n in the class [CP^1] and passing through two generic points?
There is exactly one for the standard complex structure.
[JP] What can you say about the corresponding Gromov-Witten invariant?
The standard complex structure of CP^n is regular since U(n) is a compact lie group acting transitively by biholomorphisms on CP^n (JP didn't seem to know about this fact. This is Proposition 7.4.3. in McDuff-Salamon). Thus we can use this complex structure to compute GW invariants.
[JP] What can you say about holomorphic spheres in CP^1 x C^n in the class [CP^1 x pt] for the product complex structure?
They are of the form (f(z), m) where f : CP^1 -> CP^1 is a biholomorphism and m is a point in C^n.
[JP] What happens if we deform the complex structure on a compact subset?
There is a cobordism of moduli spaces induced by a choice of (generic) path between the two almost complex structures.
[JP] Why is the projection (u,J) -> J proper?
The curves can't escape to infinity because of monotonicity. Gromov compactness together with the indecomposability of the classe [CP^1 x pt] imply properness.
[JP] Let (M,w) be a symplectic manifold and H: M x R -> R a Hamiltonian. What is the characteristic foliation of the graph of H in (M x R^2, w + ds∧dt)?
I started to do some (wrong) computations. Fortunately time was running out so they stopped me there.
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They asked me to step out into the hall and deliberated for 1 or 2 minutes.
[JP] Congrats, you passed!
The exam was pretty long (3 hours) but my committee was very friendly. They helped me out when I was stuck or didn’t understand their questions. That happened quite a lot since some of the questions were very vague and general, like ‘‘what can you say about…’’. They almost never asked me about details of proofs and only wanted me to give the key ideas. Tip: remember to bring water! I was very thirsty and drank a lot during the exam. I also brought a stress icosahedron (thank you Leo!) with me to the exam hoping I would get asked about it but alas, as for Ben Lowe, they didn’t...