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age 26
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Graduate student (Cornell University).


May
20
accepted When is a filter generated by a (countable) chain?
Apr
20
accepted Existence of state on a C*-algebra satisfying $|\tau(ab)|=\|ab\|$
Apr
20
comment Classification of complex structures on $\mathbb{R}^{2n}$
This paper of Hjorth and Kechris may be of interest: projecteuclid.org/euclid.ijm/1255984956. It deals with using descriptive set theory to understand the classification problem for arbitrary Riemann surfaces (not just structures on $\mathbb{R}^{2n}$), and shows that even in complex dimension $1$, this is extremely complicated (the "moduli space" is "Borel equivalent" to the quotient of $\{0,1\}^{F_2}$ by the left shift action of the free group $F_2$). For higher dimensions, it is more complicated ("not classifiable by countable structure" in the subject's parlance).
Apr
20
asked Existence of state on a C*-algebra satisfying $|\tau(ab)|=\|ab\|$
Apr
7
awarded  Critic
Mar
19
comment Homotopy Type Theory: What is it?
@MikeShulman Yes, that answers my question, and also touches on the other questions I had in mind, namely can ZFC serve as a metatheory for HoTT (certainly, it would seem), and can HoTT serve as its own metatheory in a satisfactory way.
Mar
19
comment Homotopy Type Theory: What is it?
@MikeShulman I should say that I get that ZFC set theory sits inside this theory, and of course something like the independence of CH, has meaning "locally" there. But does it have meaning globally?
Mar
19
comment Homotopy Type Theory: What is it?
While not specifically about HoTT, Mike Shulman's blog post golem.ph.utexas.edu/category/2013/01/… is worth reading. It cleared some things up for me, in response to my comment on his answer below.
Mar
19
comment Homotopy Type Theory: What is it?
@MikeShulman Thank you for posting that; it was exactly what I was looking for. I may have some additional questions after I digest it, and the comments, a bit. I guess my main questions (as a set theorist) are about how we can recover independence phenomena in this framework. E.g., while the phrase "the Continuum Hypothesis is independent" has a very specific meaning to a set theorist, it's common meaning is simply "the usual methods of mathematics do not decide if the cardinality of $\mathbb{R}$ is $\omega_1$". Can this statement be recovered (and proved) from univalent foundations?
Mar
18
comment Homotopy Type Theory: What is it?
Could you say more about how "it more closely matches mathematical practice", in comparison to set theory? From a naive perspective, it seems like there would be many objects in mathematics that are just as painful to encode in this formalism as others are in set theory; they just might be different objects.
Mar
18
comment Decidabilty of the Hilbert lattice and quantum logic
I think, as in that paper, "quantum logic" is usually used to refer to the equational/quantifier-free theory. It just happens that I am interested in the question about the first-order theory (from which the corresponding fact about the equational theory would follow, of course). A negative result about the first-order theory would be interesting to me, for independent reasons (undecidability of lattices makes me think of the Turing degrees).
Mar
18
revised Decidabilty of the Hilbert lattice and quantum logic
edited tags
Mar
17
asked Decidabilty of the Hilbert lattice and quantum logic
Mar
15
comment What is the most useful non-existing object of your field?
For those not in the field, I think this is most dramatically stated in its equivalent form: a non-trivial elementary embedding $j:V\to V$. For those who live in $L$, a measurable cardinal would also be really nice if it existed, I suppose.
Mar
11
comment Metrization of weak convergence of signed measures
This is a very late comment, but how does this answer not contradict the fact that the dual space of a Banach space $X$ is weak*-metrizable if and only if $X$ is finite dimensional, as mentioned by Jochen below? It is true that the space of positive measures on $\Omega$ is metrizable, as in Theorem 3.1 of jstor.org/discover/10.2307/25048364.
Jan
26
comment Existence of a complementary closed subspace extending a given subspace
Got it. That is a useful lemma!
Jan
25
revised Existence of a complementary closed subspace extending a given subspace
edited title
Jan
25
comment Existence of a complementary closed subspace extending a given subspace
Does it follow from the existence of such a $Z$ that $X+Y$ has a complement? I might just be missing something simple.
Jan
25
asked Existence of a complementary closed subspace extending a given subspace
Oct
5
accepted Does every hypersurface in the projective plane contain a projective line?