21,194 reputation
449163
bio website math.berkeley.edu/~theojf
location Berkeley
age 29
visits member for 5 years, 7 months
seen 3 hours 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.


May
19
comment Vectorisation of a category
Your $\mathrm{Cat}_p$ is, if I am reading correctly, equivalent to the category of $\mathrm{Vect}^{\mathrm{op}}$-enriched categories.
May
19
comment Vectorisation of a category
Oh, sorry, I think something funny must have happened with the computer. I wrote my comment having seen your first one but not the second, in spite of the time stamps, suggesting that for some reason the second one wasn't displaying on my end at the time.
May
19
answered Where is the exponential map a diffeomorphism?
May
19
comment Vectorisation of a category
Sorry, I don't know the notation "$\mathrm{Cat}_p$". Will you elaborate?
May
17
answered Vectorisation of a category
May
12
comment A model category of abelian categories?
@DavidRoberts No chance at updating the arXiv version to match the final refereed version? In any case, I hope you post an answer here recapping your comments and with some links to your upcoming paper.
May
10
comment Human Knot game
Here's a barely-related story. At Canada/USA Mathcamp (where I've been a counselor), there are a number of games played in groups like this, a popular one being "scream toes". N people stand in a circle, bow their heads, and each chooses a set of toes of some other person. At some signal, everyone looks up at the person whose toes they chose. If two people make eye contact, both scream, hence the name. That's the game; repeat until the neighbors complain about the noise. Anyway, a cute problem for high school math kids is the following: as $N \to \infty$, how many people scream each time?
May
7
comment Intuition behind the definition of quantum groups
In the case $x_ix_j = qx_jx_i$, you discover that quantum matrices have coordinates $a_{i,j}$ such that each $2\times 2$ submatrix satisfies the equations $(\star,\star\star)$ that I wrote. The determinant is something like $\sum_{\sigma \in S_n} (-q)^{\ell(\sigma)}a_{i,\sigma(i)}$.
May
7
comment Intuition behind the definition of quantum groups
Of course, actually for any matrix $(q_{ij})$ which is "log antisymmetric" in the sense that $q_{ji} = q_{ij}^{-1}$, you can write down a quantum $n$-space with $x_ix_j = q_{ij}x_jx_i$, and I think you can still get a monoid of quantum matrices. I don't remember if there's a central "quantum determinant".
May
7
comment Intuition behind the definition of quantum groups
@NoahSnyder You have more more expertise than I have --- certainly my impression was that SL(n) also arises in this way. I learned the story from Matt Tucker-Simmons during Kolya's 2009 quantum groups class. As far as I know, the story is due to Manin --- I don't know where to read about it. What I think you can do is to define quantum $n$-space $\mathbb A^n_q$ as having coordinates $x_1,\dots,x_n$ with $x_ix_j = qx_jx_i$ if $i<j$. Then quantum $n\times n$-matrices, quantum SL(n), and quantum GL(n) arise as various (semi)groups of transformations of these.
May
6
awarded  Nice Answer
May
5
asked When does Hochschild homology commute with infinite products?
May
5
comment Strategies for proving a category is Noetherian?
Thank you! I will take a look there. I realize that even for rings it's generally a very hard problem to decide if it is Noetherian, so a sharp practical criterion is definitely too much to ask for. This paper certainly provides some strategies I will attempt.
May
5
comment Is there a Lie II theorem for monoids?
The Krein part of Tannaka--Krein says that any topological monoid can be recovered from its category of all (including infinite-dimensional) continuous representations. So clearly I didn't mean that.
May
5
comment Is there a Lie II theorem for monoids?
Yes. I think I did say that "Rep^f" means "finite dimensional representations" in the first paragraph. I may have dropped those words in a comment.
May
5
revised Strategies for proving a category is Noetherian?
qiaochu's comment
May
5
comment Strategies for proving a category is Noetherian?
@QiaochuYuan Condition 1) is equivalent. I don't know about 2), but it is less likely to be the one that I want for my application. But I agree that the word "the" in the second sentence of the third paragraph is perhaps should be "a".
May
5
asked Strategies for proving a category is Noetherian?
May
5
revised Is there a Lie II theorem for monoids?
added 865 characters in body
May
5
comment Is there a Lie II theorem for monoids?
@RobertBryant Oh, I don't think they directly address my question, but certainly it's nice to be pointed to interesting mathematics. In any case, I'm asking a question the following question: of monoids with the given Lie algebra, which ones have the property that all Lie algebra representations extend to monoid representations? That class certainly includes the simply-connected groups, but perhaps there are non-groups in the class as well. (For example, it's not just the simply-connected groups, $\mathrm{SL}(2,\mathbb R)$ being the standard example.)