12,958 reputation
12880
bio website front.math.ucdavis.edu/…
location Cornell
age 44
visits member for 4 years, 6 months
seen 8 hours ago
Math professor at Cornell, PhD 1996

20h
comment Does normalization of projective varieties preserve very ampleness
I don't understand the two comments above. Take $\mathcal O(1)$ on $\mathbb P^2$ and restrict to a nodal cubic $X$. Then $f^* L$ is degree $3$ on $\tilde X$, hence very ample, no?
Apr
15
revised The moduli space of special Lagrangian submanifolds
edited body
Apr
15
revised Error term in formula for products of necklaces
edited title
Apr
12
comment Is the big cell a principal open set?
This is missing perhaps the most concrete and interesting part, which is what the function is. The answer is $g \mapsto \prod_\omega \langle \vec v^\omega, g \cdot \vec v_\omega \rangle$ where $\omega$ ranges over the fundamental representations. The property you want of $G$ is that you have all these fundamental representations available. Note that this function generalizes the one you mentioned for $GL(n)$.
Apr
8
comment Cotangent bundle of coadjoint orbit is stein manifold?
To spell out the connection between the spaces more tightly: consider the map $(G \times \mathfrak{b})/B \to \mathfrak g$, $[g,X] \mapsto g\cdot X$, where $B$ acts on the right of $G$ and by conjugation on $\mathfrak b$. Then map further to $\mathfrak g/G = \mathfrak t/W$. The fibers of this flat map are $T^* G/B$ over $0$ and $G/T$ over a general point.
Mar
25
comment What is the combinatorial data classifying non-normal affine toric varieties?
There's even a game exploiting this difficulty: en.wikipedia.org/wiki/Sylver_coinage
Mar
25
comment Relation between intersection multiplicity and Hilbert-Samuel multiplicities
In $e_P(X) e_P(Y) e_P(X)$, should the third term be $e_P(Z)$? And if $K$ is infinite, what happens if you look instead at a point in $V \cap P$ where $V$ is a generic $3$-plane?
Mar
22
awarded  Announcer
Mar
22
comment What is the definition of “canonical”?
I'm surprised that noone has complained about the term "canonical bundle" in algebraic geometry, meaning the top exterior power of the cotangent bundle. Let me do so here.
Mar
22
comment Existence of parametrizations of rational orthogonal matrices
Once you've parametrized an open set on an algebraic group, cover the group with finitely many left translates, to get everything. Which is what's done in the reference above, where the translates are by diagonal matrices with entries $\pm 1$. Also: Kostant and Michor have a Cayley transform for other semisimple groups. (In the split case you can use the Bruhat decomposition.)
Mar
22
comment About structure of parabolic subgroups of finite classical algebraic groups
Let me spell out a framework I think is useful here. Given a set $S$ of simple roots (which will be the ones not in your parabolic), define the $S$-height of a root $\beta$ to be the sum of the coefficients on $S$, when expanding $\beta$ in simple roots. Let $m$ be the maximum $S$-height. Then for each $k\in [0,m]$ we can define a normal subgroup $J_k$ of $P$, using those roots of height $\geq k$, and $P = J_0 > J_1 > \ldots > J_m$. (Also $J_1 = Rad(P)$.) For your question, you want $m=1$. For $P$ maximal, $m$ is the coefficient of $P$'s missing simple root in the highest root.
Mar
22
comment Zariski closure of orbits of real groups on complex flag manifolds
Thanks! I will ponder this version of the question.
Mar
20
revised Weyl group action on complexified Iwasawa decomposition
added 115 characters in body
Mar
20
answered Weyl group action on complexified Iwasawa decomposition
Mar
20
revised The amplituhedron minus the physics
added 707 characters in body
Mar
20
revised The amplituhedron minus the physics
deleted 88 characters in body
Mar
20
comment The amplituhedron minus the physics
While quite readable, they do make the two points I'm complaining about above rather difficult! I will edit.
Mar
19
comment The amplituhedron minus the physics
Also, there's no orthogonality involved in defining the $G(2,m)$ bundles; rather, they live in the quotient space. (In ordinary algebraic geometry over a field this would be a distinction without a difference. But here it's very difficult to understand positivity when involving an inner product, and we don't have to.)
Mar
19
comment The amplituhedron minus the physics
I found the reference to the map $Y_{n,k,m}$, as taking two inputs, very confusing. $Z$ is fixed and one should take $Y_{n,k,m}$ as a function from $G_+(k,n)$. Also, the amplituhedron as a space is not what's interesting, but as a space plus meromorphic volume form. The positive-real structure is there to help nail down this form; it is supposed to have no poles on the interior of the amplituhedron.
Mar
18
comment Zariski closure of orbits of real groups on complex flag manifolds
I'm having trouble unpacking your description. What are the complex points of the space you suggest I find containing $G_{\mathbb C}/B_{\mathbb C}$ as the $\mathbb R$-points?