Transcript of the qualifying examination of Dmitri Pavlov. August 25, 2008, 14–17, room 959, Evans Hall. This is a rough transcript of my qual, which occurred on August 25, 2008. Since I wrote this transcript

using my memory, not the actual recording, the words ascribed to the participants do not coincide with the original phrases.

Participants: Dmitri Pavlov, Peter Teichner (advisor), Constantin Teleman (committee chair), Dan-Virgil Voiculescu, Raphael Bousso (Department of Physics).

Teichner: What do you want to start with? Me: Let’s start with topology. Teichner: How about the cohomology ring of S^{2} × S^{2}. [I compute group structure by Künneth formula.] Teichner: What about ring structure? [I use Poncaré duality. It leaves two variants for ring structure. I choose the wrong one.] Teichner: How did you obtain this? [I correct myself and point out the right ring structure.] Teichner: Yes, and Künneth formula is valid for ring structure. Can you write it down? [I write down how to obtain this ring structure via Künneth formula.] Teichner: But can give an example of manifold with the ring structure you originally wrote? Me: Maybe connected sum of two-dimensional projective space with itself will work. Teichner: Can you compute the cohomology of this manifold? Me: One can use Mayer-Vietoris sequence or de Rham cohomology. Teichner: You can use either. [I write down Mayer-Vietoris sequence and compute cohomology groups.] Teichner: But what about the ring structure? Me: We can use de Rham cohomology. We immediately obtain that xy = 0, where x and y are two

generators of second cohomology group.

Teichner: What about x^{2} and y^{2}? Me: By Poincaré duality x^{2} and y^{2} are generators of H^{4}. Whether we have x^{2} = y^{2} or x^{2} = −y^{2}

depends on orientation.

Teichner: Finally you came to the question of orientation of CP^{2}. First let’s decide whether CP^{2} and CP^{2} are different manifolds. In other words, can you tell us whether there is an orientation-reversing diffeomorphism of CP^{2}?

Me: Such a diffeomorphism should induce a negative identity on top cohomology. Teichner: You already computed ring structure on cohomology of CP^{2}. Me: Yes, from this structure one immediately obtains that there is no orientation-reversing diffeomor-

phism of CP^{2}.

Me: The manifolds CP^{2}#CP^{2} and CP^{2}#\overline{CP^{2}} have cohomology rings x^{2} = −y^{2} = z and x^{2} = y^{2} = z. Teichner: Therefore these manifolds are not homeomorphic. Teichner to Teleman: You want to ask some questions on topology? Teleman: Can you compute the cohomology of RP^{2} × RP^{2}. Me: Yes, we can again use Künneth formula. This time the cohomology has torsion and we have to use the version with Tor functors. [I write down the Künneth exact sequence. I make a mistake and write n − 1 instead of n + 1 in the

sum of Tor terms.] Teleman: Is it n − 1 or n + 1 there? Me: I think it is n − 1. Teleman: OK, go on and compute cohomology. Me: First let’s write down the left term of Künneth exact sequence. For this we need cohomology of

CP^{2}.

[I write down homology of RP^{2}.] Teleman: Are we computing homology or cohomology? Me: Oops, this is homology. Teleman: How do you compute cohomology from homology?

Me: Universal coefficient theorem. [I write down the universal coefficient theorem and apply it to RP^{2}. I say that Ext_{Z}(Z, Z) = Z but

then correct myself.]

[I substitute the cohomology of RP^{2} into Künneth exact sequence. The fifth cohomology group turns out to be nonzero.]

Me: Oops, we really should have n + 1 in the Künneth formula. Teleman: Signs should change when you pass from homology to cohomology. That’s how I remember

all these formulas.

Teichner: There are all these fancy topics in the syllabus. Let me choose one. What’s Dold-Thom theorem?

[I write down the Dold-Thom theorem. I explains what symmetric product is and say that it converts Moore spaces to Eilenberg-Mac Lane spaces.]

Teichner: Is symmetric product a monoid? Me: Yes, it is a commutative topological monoid. Teichner: Is it an abelian group? Me: No, the symmetric product is a free commutative topological monoid on a topological space,

therefore never a group.

Teichner: But Eilenberg-Mac Lane space is a group. Me: Yes. Teichner: Can you tell us what will happen if we replace free commutative monoid by free commutative

abelian group?

Me: We obtain zero singular chain group. Reduced zero singular chain group. Teichner: What I meant is connection with Eilenberg-Mac Lane spaces. Me: One can obtain Eilenberg-Mac Lane spaces by iterating classifying space construction. In this way

one immediately obtains that Eilenberg-Mac Lane space is an abelian group.

Teichner: Let’s make a five-minute break. [Break.] Teleman: Tell us about the relation between divisors and line bundles. [I define Weil divisor group as the free abelian group generated by hypersurfaces.] Teleman: Can you define hypersurfaces? [I write down the definition in local charts.] Teleman: These are smooth hypersurfaces. Me: Yes, generally in a local chart one should have a zero set of a holomorphic function. Teleman: Yes. OK, what about line bundles? Me: There is a canonical map from the group of divisors to the Picard group of line bundles. For each

hypersurface there is a unique line bundle and a global section of this bundle such that its zero set coincides with hypersurface.

Teichner: Is it well-defined? I think the map goes the other way. Me: Yes, it is well-defined. The map cannot go the other way because there is no canonical way to

attach a global holomorphic section to a line bundle.

Teichner: Yes, but is it well-defined? Me: There is another way to construct this map. Consider short exact sequence of sheaves 0 → O^{∗} →

K^{∗} → K^{∗}/O^{∗} → 0. The boundary map H^{0}(K^{∗}/O^{∗}) → H^{1}(O^{∗}) is exactly the map we need. Teleman: Sheaves of what? Me: Sheaves of abelian groups. Teleman: With what group structure? Me: Multiplicative. Teleman: Why H^{0}(K^{∗}/O^{∗}) and H^{1}(O^{∗}) are the groups we need? [I explain the isomorphism for the Picard group via local trivializations.] Teleman: What is K^{∗}? Me: Sheaf of meromorphic functions. Teleman: But you mentioned only holomorphic functions and their sets of zeroes. Me: For meromorphic functions one should subtract the divisor of poles from the divisor of zeroes.

Teleman: The map from divisors to line bundles, is it bijective, injective or surjective? Me: No. The long exact sequence I wrote down before tells us it isn’t. The kernel is the group of

principal divisors, H^{0}(K^{∗}). They are precisely the divisors defined by global meromorphic functions. Teleman: And the cokernel? Me: The cokernel is H^{1}(K^{∗}). It can also be nonzero. Teleman: Do you know when it is zero? Me: It is zero for projective manifolds. Teichner: What is Serre duality? [I write it down as an isomorphism.] Teleman: What about the pairing? [I write Serre duality as a nondegenerate pairing.] Teleman: What is the isomorphism from H^{n,n} to complex numbers? Me: Integration. Teleman: I would like to ask about the Hodge conjecture, but it isn’t on the syllabus. Teichner: It is. Teleman: Then tell us about the Hodge conjecture. [I write down the Hodge conjecture.] Teleman: This description suffices as the first approximation. But if we want to make sense of this

definition, we need to say something about the cohomology group in your statement.

Me: We can embed sheaf cohomology with complex coefficients into Dolbeault cohomology. Teleman: Yes, we can do this, but this doesn’t bring us any closer. We need something else. It is on

your syllabus.

Me: I don’t know. Teleman: Can you tell us what Hodge decomposition is? [I write down Hodge decomposition.] Teleman: So now we can identify de Rham cohomology with direct sum of Dolbeault cohomology. Teleman: Can you tell us what do you mean by the rational 1,1-cohomology? Me: It is the intersection of Dolbeault cohomology and rational singular cohomology. Teichner: Let’s have another five-minute break. [Break.] Voiculescu: What is a von Neumann algebra? Me: It is a weakly closed ∗-subalgebra of the algebra of bounded operators. Voiculescu: Can you give another definition? Me: It is a ∗-subalgebra of the algebra of bounded operators coinciding with its double commutant. Voiculescu: What I want to do is to ask some questions on commutative von Neumann algebras and

then go to type II_{1} factors. What can you say about commutative von Neumann algebras?

Me: Every von Neumann algebra is isomorphic to L^{∞} of some measurable space. Voiculescu: In your definition a von Neumann algebra acts on some Hilbert space. Me: L^{∞} acts on L^{2} by multiplication. Voiculescu: What is an isomorphism of von Neumann algebras? [I say that it is a norm-preserving bijection. By bijection I meant an isomorphism of ∗-algebras, but

forgot to say this. Voiculescu and Teichner were uncontent by such wording and eventually made write down all the properties of ∗-algebra isomorphism. Only then I noticed my terminological mistake.]

Voiculescu: So basically you want to say that an isomorphism of von Neumann algebras is a C*-algebra isomorphism.

Me: Yes. Voiculescu: This is correct, although usually another definition is used and then this statement is derived

as a corollary. Do you know another definition of isomorphism?

Me: No, I don’t. Voiculescu: You can replace norm-preservation by ultraweak continuity. Teichner: Can you say what ultraweak topology is? [I write down the definition.] Voiculescu: Can you define what a type II_{1} factor is?

Teichner: By the way, what is a factor? Me: A factor is a von Neumann algebra with a trivial center. Me: First we define an order on the set of all projections of a factor. [I write down the definition. I use non-standard notation for two orders on projections.] Voiculescu: You really want to use another notation for projections. [I rewrite the definitions using another notation.] Me: Now a type II_{1} factor is a factor whose ordering of projections is isomorphic to [0, 1] and the unit

projection is finite.

Voiculescu: What is a finite projection? Me: A finite projection is a projection that is not isomorphic to any of its subprojections. Voiculescu: Can you define type II_{1} factors in another way? Me: Yes. A type II_{1} factor is a factor with faithful ultraweak continuous normalized trace. Voiculescu: Yes, this definition is correct, even though you can omit some of the conditions. Teichner: So you can get rid of faithfulness? Voiculescu: Yes. You can get rid of faithfulness, you can get rid of ultraweak continuity, you can get

rid of almost anything.

Voiculescu: Can you say what is a general condition that guarantees that group algebra of a group is a type II_{1} factor?

Me: All conjugacy classes are infinite except for the unit. [Voiculescu then asked a question on a topic not on a syllabus. I did not know this topic. Voiculescu

did not insist, because this topic was not on my syllabus.]

[I leave the room and wait about five minutes. Then Teleman opens the door and congratulates me. Other committee members follow.]