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Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

108 votes
8 answers
15k views

What do heat kernels have to do with the Riemann-Roch theorem and the Gauss-Bonnet theorem?

I know the following facts. (Don't assume I know much more than the following facts.) The Atiyah-Singer index theorem generalizes both the Riemann-Roch theorem and the Gauss-Bonnet theorem. The Ati …
Qiaochu Yuan's user avatar
50 votes
Accepted

Given a polynomial f, can there be more than one constant c such that every root of f(x)-c i...

This is impossible by the Mason-Stothers theorem (which holds over any algebraically closed field of characteristic zero). We want to find $f, g, h$ such that $f + g = h$ where $g$ is a constant and …
Qiaochu Yuan's user avatar
47 votes
Accepted

Grothendieck says: points are not mere points, but carry Galois group actions

Suppose $k$ is a field, not necessarily algebraically closed. $\text{Spec } k$ fails to behave like a point in many respects. Most basically, its "finite covers" (Specs of finite etale $k$-algebras) c …
Qiaochu Yuan's user avatar
44 votes
2 answers
3k views

Why can't we take three loops?

Apologies for the vague title and soft question. According to Etingof, Igor Frenkel once suggested that there are three "levels" to Lie theory, which I guess could be given the following names: No l …
Qiaochu Yuan's user avatar
38 votes
6 answers
4k views

Are all polynomial inequalities deducible from the trivial inequality?

I remember learning some years ago of a theorem to the effect that if a polynomial $p(x_1, ... x_n)$ with real coefficients is non-negative on $\mathbb{R}^n$, then it is a sum of squares of polynomial …
Qiaochu Yuan's user avatar
37 votes

Does module Hom commute with tensor product in the second variable?

You can think about tensor products as a kind of colimit; you're asking the hom functor $\text{Hom}_A(L, -)$ to commute with this colimit in the second variable, but usually the hom functor only commu …
Qiaochu Yuan's user avatar
36 votes
Accepted

Why are polynomials so useful in mathematics?

Polynomials are, essentially by definition, precisely the operations one can write down starting from addition and multiplication. More formally, polynomials with coefficients in a commutative ring $R …
34 votes
2 answers
7k views

What is the geometric meaning of integral closure?

More precisely, how does one characterize integrally closed finitely generated domains (say, over C) based on geometric properties of their varieties? Given a finitely generated domain A and its inte …
Qiaochu Yuan's user avatar
32 votes
4 answers
4k views

Modular curves of genus zero and normal forms for elliptic curves

This is maybe the first question I actually need to know the answer to! Let $N$ be a positive integer such that $\mathbb{H}/\Gamma(N)$ has genus zero. Then the function field of $\mathbb{H}/\Gamma(N …
Qiaochu Yuan's user avatar
31 votes
1 answer
4k views

For which varieties is the natural map from the Chow ring to integral cohomology an isomorph...

My apologies if this question is too naive. Let $X$ be a smooth projective complex variety. There is a natural map $A^{\bullet}(X) \to H^{2\bullet}(X)$ of graded rings from the Chow ring of $X$ to th …
Qiaochu Yuan's user avatar
29 votes

There is a nice theory of quadratic forms. How about cubic forms, quartic forms, quintic for...

This is an extended comment on KConrad's discussion of symmetry groups. We can think of $k$-forms on a vector space $V$ (homogeneous polynomials of degree $k$) abstractly as elements of the symmetric …
Qiaochu Yuan's user avatar
28 votes

What is a field [Körper] really?

Fields are the simple (no nontrivial quotients) commutative rings. Grothendieck told us to work in nice categories with nasty objects rather than nasty categories with nice objects; fields are the nic …
Qiaochu Yuan's user avatar
28 votes

What is the field with one element?

Here's probably the simplest manifestation of the field-with-one-element phenomenon. Define a projective $n$-space of order $q$ to be a collection of points, lines, planes, etc. satisfying the usual …
Qiaochu Yuan's user avatar
26 votes

Why higher category theory?

Re: David Corfield's comment, this answer will mostly address "higher as in $(\infty, 1)$-categories" rather than "higher as in $n$-categories." I also do understand you need the notion of abelia …
24 votes

Interesting results in algebraic geometry accessible to 3rd year undergraduates

This isn't a result so much as a perspective, but it is one of the main reasons I first got interested in algebraic geometry. In basic algebraic number theory you learn that some extensions of the in …
Qiaochu Yuan's user avatar

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