3,022 reputation
917
bio website ma.utexas.edu/users/…
location Austin, TX
age
visits member for 4 years, 5 months
seen Dec 23 at 0:14

Postdoc at UT Austin.


Nov
22
comment Is there a cotangent bundle of a stable $\infty$-category?
You may complain that this is more like the tangent bundle than the cotangent, but I think some kind of Koszul duality should relate sheaves on this odd tangent bundle with sheaves on the cotangent bundle (at least up to shifts by 2).
Nov
22
comment Is there a cotangent bundle of a stable $\infty$-category?
Not sure if this is relevant or not, but the categorical Hochschild homology (CHH) of a monoidal category seems to be a reasonable candidate for part 1 (no idea about part 2). By CHH I mean the category $\mathcal C \otimes{\mathcal C \times \mathcal C^{op}} \mathcal C$ associated to a monoidal category $(\mathcal C, \otimes)$. In the case $\mathcal C = QC(X)$, for X a scheme $CHH(\mathcal C) = QC(LX)$, where $LX = X\times_{X\times X} X$ is the derived loopspace (by a result of Ben-Zvi-Francis-Nadler). The derived loopspace is the same as the odd tangent complex by HKR.
Sep
24
awarded  Autobiographer
Sep
2
awarded  Revival
Jul
20
awarded  Yearling
Jul
6
awarded  Enlightened
Jul
2
awarded  Curious
Jun
15
comment in which sense is a mixed Hodge structure an extension of pure ones?
You are not missing anything. The term iterated extension means that (in your example): $W_1$ is an extension of $Gr_1$ by $W_0$, $W_2$ is an extension of $Gr_2$ by $W_1$ etc. So $H$ is "built up" from pure modules, in a similar way that a finite group is built as an iterated extension of finite simple groups.
May
27
comment The general Smith homomorphism in bordism
strung : string ?
May
25
comment Equivariant Sheaves, Local system
More naively, what you are asking for is the analogue of the statement that a 1-dimensional representation $\rho: G \to \mathbb C^\times$ of a group is a class function. The condition $m^\ast L = L \boxtimes L$ just says (roughly) that the stalk $L_{gh}$ at $gh$ is identified with $L_g \otimes L_h$. Thus we have $L_{ghg^{-1}} \simeq L_g \otimes L_h \otimes L_{g^{-1}} \simeq L_g \otimes L_{g^{-1}} \otimes L_h \simeq L_h$, which is (roughly) what it means to be $G$ equivariant. To get rid of the "roughly" you will need to express the above in terms of diagrams...
May
20
comment What's the relationship between these two isomorphisms involving G and T?
I don't know how to answer your question, but let me try to clarify one thing that I probably said hastily in a talk some time. Of course, there is no equivalence (or even a map) of stacks $T/W \to G/G$ - the stabilizers are all wrong, even if the orbit spaces agree. Let us write $N=N_G(T)$. You do have a map of stacks $BN \to BG$, which gives a map on loop spaces $N/N \to G/G$ ($BN$ is what you write as $BT/W$). Inside $N/N$ you have a copy of $T/N = (T/T)/W$ which also maps to $G/G$. Not sure what else to say yet.
Apr
16
answered Equivariant derived category and invariant divisor
Apr
8
comment Smooth mixed hodge modules - representations of fundamental group?
My expectation is that there will be no positive answer to your question. The data of a smooth (pure) Hodge module (aka variation of Hodge structure) involves a filtration on the holomorphic sections of the underlying vector bundle. This filtration is not flat, but rather satisfies the Griffiths transversality condition $\nabla \mathcal F^n \subseteq \mathcal F^{n-1}\otimes \Omega^1$. It doesn't look like it is possible to express these data and conditions in terms of the monodromy.
Mar
29
comment Why should noncommutative CYs be dgas?
I think of Calabi-Yau as being a property of a dga rather than differential graded being a property of a CY algebra...
Feb
14
answered regular singularities and comparison isomorphism
Jan
27
answered Most general “finiteness of de Rham cohomology” statement for holonomic $D$-modules in the algebraic case?
Dec
2
answered How is a descent datum the same as a comodule structure?
Nov
5
comment spectrum of an induced algebra
Ah, you got there first...
Oct
8
revised A question about flag variety of $SL(n,\mathbb{C})$
Changed SL(2,C) to SL(n,C) in the first line.
Oct
1
answered For G a Lie group, can I make sense of G/G as a derived manifold in a nice way?