User josh guffin - MathOverflow

Undergraduate Probability Topics

2010-09-01T18:10:21Z

I am teaching undergraduate probability this semester, and I am looking for some suggestions about inspiring applications that could be reasonably covered over the course of two one-hour lectures or less. For example, here are two very cool topics I covered last time I taught this course:

Search engines and Markov chains; AMS' description and The $25 billion eigenvector
Benford's Law and the Pareto Distribution

Thanks in advance!

Answer by Josh Guffin for The canonical line bundle of a normal variety

2010-09-27T22:26:07Z

Your formula is not quite right for toric varieties. In particular, the sum is not over "representatives of the class group", but over a set of minimal generators for the free group on torus-invariant divisors. Such a set is furnished by the 1-cones in the fan. More precisely,

Let $X_\Sigma$ be the toric variety associated to a fan $\Sigma$, and assume that $X_\Sigma$ has no torus factors. Then for each $\rho \in \Sigma(1)$, there is a torus invariant divisor $D_\rho$, and $$\mathcal O_{X_\Sigma}\Big(-\sum_{\rho} D_\rho\Big)\cong \omega_{X_\Sigma}$$

This is Proposition 8.2.7 in Cox Little Schenck.

An easy toric proof that no projective toric variety is Calabi-Yau is that, as you said, the minimal generators of the one-cones must lie in a hyperplane. The positive hull over such a set of generators is strongly convex, so that the support of the fan cannot be all of $N_{\mathbb R}$, and thus $X_\Sigma$ is not complete and thus not projective.

I believe that your question really concerns the existence of a natural set of generators for the "total coordinate ring" of an Calabi-Yau variety and if they obey a linear relation. The total coordinate ring is defined to be $$R=\bigoplus_{D\in Cl(X)} \Gamma(X,\mathcal O_X(D)).$$

Here $Cl(X)$ is the class group of $X$. See 0801.3995 for more details.

For toric varieties $X_\Sigma$, this is simply $\mathbb C[x_\rho | \rho \in \Sigma(1)]$. If this is indeed your question, it is probably too much to hope for, as $R$ is not known to be finitely generated! It is known to be so for toric varieties (Cox), and for varieties of Fano type (Birkar–Cascini–Hacon–McKernan). Some other specifically constructed examples (Prendergast-Smith) are known, but a general characterization is not.

Edit: Updated the links and fixed the unclear notation - thanks Artie!

Answer by Josh Guffin for Cox rings of toric varieties over arbitrary fields

2010-09-29T13:20:10Z

Parametrizations of toric varieties over any field: also available via google

Answer by Josh Guffin for practical applications of eigenvalues and eigenvectors

2010-09-29T12:46:51Z

I would comment on Peitro's answer, but I don't have enough reputation; for a marvelously-titled explanation of Google's Pagerank, see The $25,000,000,000 Eigenvector.

Answer by Josh Guffin for Jokes in the sense of Littlewood: examples?

2010-09-19T14:11:38Z

Another joke in the spirit of the chain rule; you solve separable differential equations by "multiplying by g(y)dx" $$g(y) dx \left(\frac{dy}{dx} = \frac{f(x)}{g(y)}\right) \Rightarrow g(y)dy = f(x) dx$$

Then, there's nothing to be done but integrating to get rid of the dx and dy.

I also like to point out to students who ask about cancellation in the chain rule that you can cancel there just like you can cancel the sixes and nines respectively in

$$\frac {16}{64} = \frac 1 4 \qquad \text{and} \qquad \frac{19}{95}=\frac 1 5;$$ that is, carefully, and when it makes sense to do so.

Comment by Josh Guffin

2010-10-31T21:10:59Z

Not exactly the same, but for high-energy physics there is spires: slac.stanford.edu/spires

Comment by Josh Guffin

2010-09-30T22:49:22Z

His condition that $S$ lies in a hyperplane is equivalent to the existence of a $\mathbb C^*$ action for which each monomial is of the same degree. The weights are given by the normal vector to $H$. For instance, in example 3, $(6,0,0).(1,2,3)=(1,1,1).(1,2,3)=\cdots=6$, so the exponents lie in the hyperplane $a + 2b + 3c = 6$.

Comment by Josh Guffin

2010-09-29T18:26:50Z

This also brings to mind the preface (books.google.com/…) from "Lectures on Partial Differential Equations" by Arnol'd. Unfortunately the google books version cuts out after the first page, and I can't find another English version online. You can find a Russian version by googling for "Лекции об уравнениях с частными производными".