bio | website | math.stanford.edu/~church |
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location | Stanford | |
age | ||
visits | member for | 5 years, 1 month |
seen | yesterday | |
stats | profile views | 3,774 |
Nov 19 |
answered | Breaking up the free Lie algebra into Gl irreps |
Nov 14 |
comment |
Questions about the “universal elliptic curve” over the affine $j$-line punctured at 0 and 1728
What you had was right (if $\text{SL}_2(\mathbb{Z})$ acts only on $\mathcal{H}^\circ$, it won't descend). The problem is there's no ("fine") universal elliptic curve: Let $T$ be the randomly-chosen $T=\mathbb{C}/\langle m+n(1+i)\rangle$. Let $\mathbb{Z}$ act on $T\times \mathbb{R}$ by $(t,x)\mapsto (-t,x+1)$, and set $X=(T\times\mathbb{R})/\mathbb{Z}$. This is an elliptic curve bundle over $S^1$ with every fiber $\simeq T$, but it cannot be pulled back from your $E_2$ (or any family over the $j$-line). [$X$ is nontrivial because e.g. $H_1(X)=\mathbb{Z}\neq \mathbb{Z}^3=H_1(T\times S^1)$.] |
Nov 14 |
comment |
Questions about the “universal elliptic curve” over the affine $j$-line punctured at 0 and 1728
It looks to me like $E_2$ is a $\mathbb{P}^1$-bundle over $\text{SL}_2(\mathbb{Z})\backslash \mathcal{H}^\circ$; for $\gamma=-\text{Id}$ you have $\gamma\cdot(z,\tau)=(-z,\tau)$, so the fiber over $\tau\in \text{SL}_2(\mathbb{Z})\backslash \mathcal{H}^\circ$ is $E_\tau/\langle\pm 1\rangle\simeq \mathbb{P}^1$. (This commment may not make sense in the future if the question is revised.) |
Nov 9 |
awarded | Custodian |
Nov 9 |
reviewed | Reviewed CD - continuous development |
Nov 9 |
awarded | Custodian |
Nov 9 |
reviewed | Approve suggested edit on Questions related to a previous question about interpolation based on non-decreasing polynomials |
Oct 12 |
awarded | Yearling |
Aug 30 |
comment |
Corvallis 1979 proceedings
The AMS pages for these books are now ams.org/books/pspum/033.1 and ams.org/books/pspum/033.2; however they no longer seem to be freely available. |
Jun 26 |
reviewed | Reject suggested edit on nontrivial theorems with trivial proofs |
Jun 17 |
reviewed | Reject suggested edit on nontrivial theorems with trivial proofs |
Jun 15 |
revised |
$G_\mathbb{Z}$-homotopy type of rational Tits building $\Delta_{G, \mathbb{Q}}$
clarify result for orbits of chains, add remark on PID |
Jun 15 |
revised |
$G_\mathbb{Z}$-homotopy type of rational Tits building $\Delta_{G, \mathbb{Q}}$
correct ell = |K_0| to ell = |K_0| - 1 |
Jun 15 |
answered | $G_\mathbb{Z}$-homotopy type of rational Tits building $\Delta_{G, \mathbb{Q}}$ |
Jun 8 |
comment |
Integral of sin(x)/sqrt(x) from 0 to \pi
@NoamD.Elkies I agree that the downvotes may have been hasty; I didn't downvote, but I easily could have. However questions like this about evaluating definite integrals seem to be more at home at math.SE (even quite difficult integrals, which often receive extremely impressive answers there!). So the best outcome may be for this question to be moved to math.SE. |
Jun 4 |
awarded | Custodian |
Jun 4 |
reviewed | Reject suggested edit on nontrivial theorems with trivial proofs |
Jun 4 |
reviewed | Reject suggested edit on nontrivial theorems with trivial proofs |
May 14 |
comment |
Some question about polynomial representations of $GL(V)$
This is just about the difference between "rational" and "polynomial". Let's consider $\text{GL}(1)$. The irreducible rational representations of $\text{GL}(1)=\mathbb{C}^\times$ (or $\mathbb{G}_m$) are indexed by integers $k\in \mathbb{Z}$, with $k\in \mathbb{Z}$ corresponding to the 1-dimensional representation on which $z\in \mathbb{C}^\times$ acts by $z^k$. This is a rational function of $z$, matching the terminology. But $z^k$ is only a polynomial when $k\geq 0$, so the irreducible polynomial representations are indexed by $k\geq 0\in \mathbb{Z}$. |
May 12 |
comment |
How many proofs of the Polya's recurrent theorem are there?
For those who have access, this paper can be accessed online on the journal's website here: jstor.org/stable/10.4169/000298910X480072 For those who don't, it can also be accessed freely on the author's website here: pages.uoregon.edu/dlevin/polya.pdf |