15,968 reputation
230100
bio website front.math.ucdavis.edu/…
location Cornell
age 45
visits member for 5 years, 8 months
seen 17 hours ago
Math professor at Cornell, PhD 1996

1d
comment Smooth morphism to homogeneous spaces and fibers
I usually use "$f$ is equivariant".
1d
comment Picard group of a quotient of a group by its maximal parabolic subgroup
You can't say "the" maximal parabolic; there's usually more than one, even up to conjugacy. But for every such $P$ this is true.
1d
answered Shared maximum eigenvector
Jun
28
revised What is deforming this non-complete intersection like?
made title more specific to the question
Jun
28
reviewed Approve What is deforming this non-complete intersection like?
Jun
26
awarded  Nice Question
Jun
26
comment Number of $\mathbb F_p$ points constant mod $p$?
Your #4 sounds like "what does it imply?" rather than "what does it suggest?" (what would imply it, that one should be on the lookout for?). I have a certain class of varieties; I don't know what properties they have, but now I'm going to try to prove rational connectivity directly for them (not actually imply it from the point-counting fact).
Jun
26
comment Number of $\mathbb F_p$ points constant mod $p$?
I'd certainly like to compute $\#X(\bf F_p)$ but don't expect to have much luck.
Jun
25
asked Number of $\mathbb F_p$ points constant mod $p$?
Jun
21
comment A homeomorphism between total spaces with same fiber and base spaces not homotopic
Or more generally, if $F\to E\to B$ is a bundle with $E$ not a product, then $F\times E \to E$, $F\times E \to F\times B$ works.
Jun
18
comment Combinatorial results by Poincaré duality
If you're willing to extend this from "Poincar\'e duality" to "properties of Betti numbers", then you could include Hard Lefschetz in your toolset, and get Stanley's proof of the Upper Bound Theorem for the number of faces of a simplicial polytope.
Jun
17
comment How to embed $U(1)$ into a bigger group, using Dynkin diagrams
The "remove the top root" step amounts to "take the centralizer in $E_7$ of a certain element of adjoint order $2$". Look around MO for Borel-de Siebenthal theory, which is about subgroups of the same rank as the big group.
Jun
16
comment q-Catalan numbers from Grassmannians
Hard Lefschetz tells you that the even Betti numbers have to increase to the middle, then decrease (they're "unimodal").
Jun
13
comment How to prove this polynomial always has integer values at all integers?
I really don't know how to connect $\mathbb Z$-valuedness and evenness. If $x^2=y$ so we can rewrite $P(x)$ as $P(y)$, then $P$ being $\mathbb Z$-valued at all $y\in\mathbb Z$ is (much?) stronger than its being $\mathbb Z$-valued at all $x\in\mathbb Z$. Do your computed polynomials have $\mathbb Z$-coefficients expanded in $\{ {y \choose k} \}$?
Jun
13
comment How to prove this polynomial always has integer values at all integers?
@DavidLoeffler : yes it has that effect, which means we've lost control of $a_0$ and $a_1$. I wrote "(assuming $Q(0)\in\mathbb Z$)" but meant $P(0)\in\mathbb Z$. How are you using evenness to control $a_1$, to establish this converse?
Jun
13
comment How to prove this polynomial always has integer values at all integers?
(1) The proper basis for the space of integer-valued polynomials is not monomials $\{x^k\}$ but binomial coefficients $\{ {x\choose k} \}$ (a polynomial is $\mathbb Z$-valued iff its unique expansion in this basis has $\mathbb Z$-coefficients). You should probably look into those expansions. (2) Let $Q(x) = P(x+1)+P(x-1)-2P(x)$, again even. If $P$ is $\mathbb Z$-valued, so is $Q$, and I suspect the converse is true for even functions (assuming $Q(0) \in \mathbb Z$). Perhaps you can use this for an inductive argument.
Jun
13
answered Why there is a relation among the second-order minors of a symmetric $4\times 4$ matrix?
Jun
12
comment sub-variety of (P^1)^4
It's the closure of a generic $T^4$-orbit on the Grassmannian $Gr(2,4)$, so there's something it lies inside.
Jun
8
revised Real and Quaternionic Representations according to Weights
added 23 characters in body
Jun
8
answered Real and Quaternionic Representations according to Weights