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7 votes
1 answer
718 views

Example sought of an atomic domain R such that R[t] is not atomic

Recall that an integral domain $R$ is atomic if every nonzero nonunit admits at least one factorization into irreducible elements. (Indeed, hard-core factorization theorists have replaced the word "i …
Pete L. Clark's user avatar
25 votes
1 answer
3k views

What is a special parahoric subgroup?

Let me take this question again from the top. I would like to know what a special parahoric subgroup is. I think this is a "real" question, though not an especially good one -- it indicates my compl …
18 votes
2 answers
2k views

History of "no positive definite ternary integral quadratic form is universal"?

I am currently leading a graduate student research group on geometry of numbers (henceforth GoN) and its Diophantine applications, especially to quadratic forms. In an early lecture I gave a bit of a …
Pete L. Clark's user avatar
231 votes
13 answers
42k views

Is there an introduction to probability theory from a structuralist/categorical perspective?

The title really is the question, but allow me to explain. I am a pure mathematician working outside of probability theory, but the concepts and techniques of probability theory (in the sense of Kolmo …
Pete L. Clark's user avatar
9 votes
1 answer
2k views

Sums of two squares in (certain) integral domains

While giving the first of eight lectures on introductory model theory and its applications yesterday, I stated Hilbert's 17th problem (or rather, Artin's Theorem): if $f \in \mathbb{R}[t_1,\ldots,t_n] …
Pete L. Clark's user avatar
26 votes
3 answers
6k views

An unfamiliar (to me) form of Hensel's Lemma

In his very nice article Peter Roquette, History of valuation theory. I. (English summary) Valuation theory and its applications, Vol. I (Saskatoon, SK, 1999), 291--355, Fields Inst. Commun., …
Pete L. Clark's user avatar
47 votes
1 answer
3k views

Which small finite simple groups are not yet known to be Galois groups over Q?

The subject line pretty much says it all. To expand just a little bit: 1) What is the smallest simple group that is not yet known to occur as a Galois group over $\mathbb{Q}$? (Variants: not known …
Pete L. Clark's user avatar