Qual -- more details
This is the promised more detailed version of the qual write-up. It lasted almost three hours, but didn't feel like it at all. I should have brought more water with me, though (I just had a small bottle). The topology questions were good fun, even though half of them were off topic. They very rarely asked me to go through a proof I'd learnt (I think the only two they asked were Tychonoff's Theorem and Diamond in L. Unfortunatley, I didn't know the latter, but I did say that I could force to get diamond).
Bergman was very useful during the set theory, asking me to define terms the others had used in their questions where possible, which helped slow the pace down nicely (and allowed me to show I knew all the definitions!). Steel played a similar role during the topology. I don't think Leo asked any questions, although Steel did ask a question that no-one knew the answer to except for Leo!
After they asked me to leave at the end, they seemed to discuss for what seemed like an eternity, but I was pretty confident when I heard them laughing (I knew I hadn't done badly enough that they'd be laughing at my performance, so I assumed they were happy about passing me and someone told a joke).
Hidden below the cut are what I recall of the questions, which I'll put up on the MGSA website next week.
Topology:
- Give an example of a T_0 space which isn't T_1. What are the other separation axioms? Conjecture and prove classification theorems for finite T_2, T_1 and T_0 spaces.
- State and prove Tychonoff's theorem. What's it equivalent to over ZF? How does your answer change if you take the definition of compact to include T_2?
Basic Set Theory and Forcing:
- What can 2^{aleph_0} be? Sketch a proof.
- Is the product of ccc forcings necessarily ccc? What about the iteration? How are products and iterations related? Can you use this to prove that under MA+-CH the product of ccc forcings is cc?
- What's diamond? Is it equivalent if you change stationary to non-empty? How about unbounded? How about if rather than a Diamond-sequence being a sequence of subsets of omega_1, it's a sequence of countable sets of subsets of omega_1 and we just require A \cap \alpha \in \Sigma \alpha for stationarily many \alpha. Does Diamond imply CH? Does CH imply Diamond? Does GCH imply Diamond _{omega_2}?
- Show L models GCH. Are there any alpha less than omega_1 at which no new real appears in L_alpha? How long a gap can we get between ordinals where new reals occur?
Further Set Theory:
- What can you say about subsets of omega_1 under AD? How complex can such a subset be to code as a set of reals? This segued onto stuff about a^sharp under AD. Show subsets of omega_1 can be found in L[a] for some real a under AD.
Bergman was very useful during the set theory, asking me to define terms the others had used in their questions where possible, which helped slow the pace down nicely (and allowed me to show I knew all the definitions!). Steel played a similar role during the topology. I don't think Leo asked any questions, although Steel did ask a question that no-one knew the answer to except for Leo!
After they asked me to leave at the end, they seemed to discuss for what seemed like an eternity, but I was pretty confident when I heard them laughing (I knew I hadn't done badly enough that they'd be laughing at my performance, so I assumed they were happy about passing me and someone told a joke).
Hidden below the cut are what I recall of the questions, which I'll put up on the MGSA website next week.
Topology:
- Give an example of a T_0 space which isn't T_1. What are the other separation axioms? Conjecture and prove classification theorems for finite T_2, T_1 and T_0 spaces.
- State and prove Tychonoff's theorem. What's it equivalent to over ZF? How does your answer change if you take the definition of compact to include T_2?
Basic Set Theory and Forcing:
- What can 2^{aleph_0} be? Sketch a proof.
- Is the product of ccc forcings necessarily ccc? What about the iteration? How are products and iterations related? Can you use this to prove that under MA+-CH the product of ccc forcings is cc?
- What's diamond? Is it equivalent if you change stationary to non-empty? How about unbounded? How about if rather than a Diamond-sequence being a sequence of subsets of omega_1, it's a sequence of countable sets of subsets of omega_1 and we just require A \cap \alpha \in \Sigma \alpha for stationarily many \alpha. Does Diamond imply CH? Does CH imply Diamond? Does GCH imply Diamond _{omega_2}?
- Show L models GCH. Are there any alpha less than omega_1 at which no new real appears in L_alpha? How long a gap can we get between ordinals where new reals occur?
Further Set Theory:
- What can you say about subsets of omega_1 under AD? How complex can such a subset be to code as a set of reals? This segued onto stuff about a^sharp under AD. Show subsets of omega_1 can be found in L[a] for some real a under AD.