19,678 reputation
445152
bio website math.berkeley.edu/~theojf
location Berkeley
age 29
visits member for 5 years, 1 month
seen 2 days ago

I am a recent graduate from UC Berkeley. Starting Fall 2013, I am a Boas Assistant Professor at Northwestern University. My research is broadly centered on quantum field theory — my interests include category theory, representation theory, homological algebra, algebraic topology, Poisson geometry, and theoretical physics.


Nov
24
awarded  Nice Answer
Nov
20
awarded  Notable Question
Nov
10
comment Does the linear automorphism group determine the vector space?
... well formed. Indeed, the problem is that there is no functorial construction of $V$ from $\mathrm{GL}(V)$, so that, in particular, bundles of groups isomorphic to $\mathrm{GL}(V)$ do not lift to bundles of vector spaces isomorphic to $V$. (The first example I know of is a bundle whose base space is the suspension of $\mathbb R \mathbb P^2$.)
Nov
10
comment Does the linear automorphism group determine the vector space?
@LSpice Incidentally, I was led astray by your discussion of the underlying set $X$. Normally a "vector space structure" consists of particular maps --- it makes sense, then, to ask whether two given vector space structures on the same set are equal. The Erlangen program, at best, is about actually giving a manifold some structure in this strict sense. Your question, as I now understand, was whether the isomorphism type of the group $\mathrm{GL}(V)$ determines the isomorphism type of the vector space $V$. This is within a family of common questions, but I would argue is not really ...
Nov
9
awarded  Notable Question
Nov
8
comment Does the linear automorphism group determine the vector space?
@LSpice: Oh, I'm sorry --- I misunderstood the question. My apologies.
Nov
8
awarded  Good Answer
Nov
7
answered Does the linear automorphism group determine the vector space?
Nov
7
comment Does the linear automorphism group determine the vector space?
This does not answer the question posed, which is whether the vector space structure (in the sense that we teach in undergraduate linear algebra) on the underlying set of $V$ can be recovered from how the group $\mathrm{GL}(V)$ sits inside the group of all permutations of the underlying set.
Nov
6
comment Counterintuitive consequences of the Hahn-Banach theorem
@WillieWong I am reasonably well-educated in set theory and related questions, and am on record as being sick of AC questions on MO. But I was not aware of this "standard reference". I now am, and I am better for the knowledge.
Nov
3
accepted Does “simplicial” commute with “Bousfield localization”?
Nov
3
comment Does “simplicial” commute with “Bousfield localization”?
@KarolSzumiło: Great. Incidentally, I advocate the following notation: $X^Y$ means $X$-valued presheaves on $Y$, and not functors. I would write $^YX$ for the category of $X$-valued copresheaves on $Y$. The reason is that I think of right modules as contravariant functors.
Nov
3
comment Does “simplicial” commute with “Bousfield localization”?
As Tyler Lawson points out in the answers, I misunderstood (probably a lot about) injective versus Reedy structures; the "complicated" link doesn't illustrate just how complicated things are. Like, I think, many users of model categories, I in fact only case about Bousfield localizations of presheaf categories, and only for pretty well-behaved bases.
Nov
3
comment Does “simplicial” commute with “Bousfield localization”?
Or, no, I've misunderstood. For $M = \mathrm{sSet}$, they agree, not in general. My apologies.
Nov
3
comment Does “simplicial” commute with “Bousfield localization”?
Great. I agree that what I linked to does not define cofibrations levelwise. But I think that for $\Delta$, the Reedy and injective model structures agree? I would have to think a lot about how to prove that directly, but my impression was that it was reasonably known. The reference I know is to arxiv.org/abs/1110.1066, although the claim about $\Delta$ seems to be older.
Nov
3
asked Does “simplicial” commute with “Bousfield localization”?
Oct
31
accepted Which real Pin groups agree?
Oct
31
comment Chiral categories versus braided monoidal categories
I think it's not quite right to think of a chiral algebra as an $E_2$ algebra. The latter should give examples of the former (up to some framing, perhaps), but chiral algebras should not, in general, be "locally constant". Think about a much less categorical setting of functions. Locally constant functions are very different from holomorphic functions. Or am I misunderstanding the notion of "chiral algebra" --- is there a "de Rham" that I missed somewhere?
Oct
29
comment What are the “correct” conventions for defining Clifford algebras?
I second @KConrad, since you can walk down the hall and knock on his door --- my memory is that he is often in, and usually happy to talk to graduate students. And he certainly is sensitive to other conventions issues related to Clifford algebras in his Lie theory classes. If you get an answer, do be sure to post it here, of course.
Oct
24
comment Which real Pin groups agree?
Great, I'll take a look.