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Aug
27 |
comment |
When do block sequences yield disjoint subspaces?
Bill: I'm a bit unfamiliar with the terminology (Banach space theory is a outside my usual area), what do you mean by "minimal sequence" and "M-basis"? |
Aug
24 |
asked | When do block sequences yield disjoint subspaces? |
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 |
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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. |