All Questions

1
vote
4answers
2k views

Closest grid square to a point in spherical coordinates

I am programming an algorithm where I have broken up the surface of a sphere into grid points (for simplicity I have the grid lines are parallel and perpendicular to the meridians). Given a point A on ...
39
votes
1answer
16k views

What is inter-universal geometry?

I wonder what Mochizuki's inter-universal geometry and his generalisation of anabelian geometry is, e.g. why the ABC-conjecture involves nested inclusions of sets as hinted in the slides, or why such ...
3
votes
3answers
1k views

Conjugation in SU(2)

For any two matrices $P,Q \in SU(2)$, with $tr(P)=tr(Q)=0$, does there always exist some $G\in SU(2)$ such that $G P G^{-1} = -P$, and $G Q G^{-1} = -Q\ ?$
19
votes
3answers
2k views

Is any representation of a finite group defined over the algebraic integers?

Apologies in advance if this is obvious.
1
vote
8answers
2k views

The core question of topology

As I see it, the core question of topology is to figure out whether a homeomorphism exists between two topological spaces. To answer this question, one defines various properties of a space such as ...
5
votes
1answer
430 views

Do homotopy pullbacks commute with homotopy orbits (in spaces)?

Suppose we are given a diagram $X \to Z \gets Y$ of $G$-spaces ($G$ a discrete group). Let $(- \times^h -)$ denote homotopy pullback. Is $(X \times^h_Z Y)_{hG}$ weakly equivalent to $X_{hG} \times^h_{...
5
votes
2answers
386 views

Algorithms for semistable reduction of families of curves

This is a somewhat vague question which came up MSRI a few days ago: Suppose I have a family of curves over a one dimensional base, given in a computationally explicit way. For example, maybe I have a ...
15
votes
4answers
2k views

Arithmetic progressions without small primes

The following question came up in the discussion at How small can a group with an n-dimensional irreducible complex representation be? : Is it known that there are infinitely many primes p for which ...
7
votes
2answers
642 views

Is there a version of the valuative criteria for separateness/properness for varieties?

What I had in mind was something like the following: X is separated/proper iff for all curves C and all maps f : C \ c -> X, f extends to C in at most/exactly one way. Is there a good reason why ...
4
votes
2answers
413 views

Embedding abelian categories to have enough projectives

Is it true that the pro-objects of an abelian category form a category with enough projectives? In general, given an abelian category A, is there a canonical way to embed it a bigger abelian ...
-6
votes
3answers
3k views

Gaussian curvature and mean curvature. [closed]

Define Gaussian curvature for a nonorientable surface. Can you define mean curvature for a nonorientable surface?
25
votes
8answers
4k views

triangulated vs. dg/A-infinity

Someone recently said "derived/triangulated categories are an abomination that should struck from the earth and replaced with dg/A-infinity versions". I have a rough idea why this is true ("don't ...
66
votes
7answers
14k views

Teaching statements for math jobs?

What is the purpose of the "teaching statement" or "statement of teaching philosophy" when applying for jobs, specifically math postdocs? I am applying for jobs, and I need to write one of these ...
15
votes
5answers
4k views

Describing the universal covering map for the twice punctured complex plane

As is well known, the universal covering space of the punctured complex plane is the complex plane itself, and the cover is given by the exponential map. In a sense, this shows that the logarithm has ...
8
votes
4answers
1k views

cohomology of moduli spaces

Does anyone know if there's any reference on the $\ell$-adic cohomology of some simple moduli spaces/Shimura varieties, like Siegel moduli varieties $A_{g,N}$ of genus $g$ and level $N,$ for small $g$ ...
13
votes
2answers
1k views

Bad Categorical Quotients

Let $G$ be an algebraic group acting on a scheme $X$. Then $f: X \to Y$ is called a categorical quotient if it is constant on $G$-orbits and every $X \to Z$ constant on $G$-orbits factors through it ...
14
votes
7answers
2k views

Examples of rational families of abelian varieties.

I'd like to know examples of non-trivial families of abelian varieties over rational bases (e.g. open subschemes of the projective line P^1). One can generate many examples as Jacobians of rational ...
7
votes
7answers
1k views

Hochschild/Cyclic Homology of von Neumann Algebras: Useless?

Hochschild homology gives invariants of (unital) $k$-algebras for $k$ a unital, commutative ring. If we let our algebra $A$ be the group ring $k[G]$ for $G$ a finite group, we get group homology. ...
6
votes
2answers
1k views

What is an example of a ring in which the intersection of all maximal two-sided ideals is not equal to the Jacobson radical?

What is an example of a ring in which the intersection of all maximal two-sided ideals is not equal to the Jacobson radical? Wikipedia suggests that any simple ring with a nontrivial right ideal would ...
17
votes
4answers
2k views

Exhibit an explicit bijection between irreducible polynomials over finite fields and Lyndon words.

Let $q$ be a power of a prime. It's well-known that the function $B(n, q) = \frac{1}{n} \sum_{d | n} \mu \left( \frac{n}{d} \right) q^d$ counts both the number of irreducible polynomials of degree $n$...
2
votes
2answers
783 views

roots of analytic functions

Let $z$ be a complex variable and $f(z)$ be a formal power series with rational coefficients (an element in $\mathbb Q[[z]]$), with a finite radius of convergence, and assume $f(z)$ has a meromorphic ...
42
votes
95answers
66k views

Undergraduate Level Math Books [closed]

What are some good undergraduate level books, particularly good introductions to (Real and Complex) Analysis, Linear Algebra, Algebra or Differential/Integral Equations (but books in any undergraduate ...
2
votes
2answers
379 views

Does there exist a sequence of groups whose representation theory is described by plane partitions?

More precisely, does there exist a sequence G1 < G2 < ... of finite groups such that the irreducible representations of Gn are parameterized by the plane partitions of total size n?
10
votes
4answers
1k views

References for syntomic cohomology

Could anyone point to good readable references for learning about syntomic cohomology?
7
votes
10answers
2k views

What do models where the CH is false look like?

Additionally, is there any intuitive way to visualize the cardinalities that result?
13
votes
2answers
2k views

Co-induction understanding

Hi, I am studying coinduction(not induction) as part of a class on static analysis. Rummaging around the internet, I am simply not finding a clear, concise description of: How coinduction actually ...
4
votes
4answers
1k views

When is a map given by a word surjective?

Let $w(x,y)$ be a word in $x$ and $y$. Let $x$ and $y$ now vary in $SL_n(K)$, where $K$ is a field. (Assume, if you wish, that $K$ is an algebraically complete field of characteristic bigger than a ...
30
votes
6answers
4k views

“Points” in algebraic geometry: Why shift from m-Spec to Spec?

Why were algebraic geometers in the 19th Century thinking of m-Spec as the set of points of an affine variety associated to the ring whereas, sometime in the middle of the 20 Century, people started ...
9
votes
1answer
326 views

An “existence contra partition of unity” statement for integer matrices?

While reading a blog post on partitions of unity at the Secret Blogging Seminar the following question came into my mind. Let $n$ be a positive integer and let $B_1$ and $B_2$ be $n \times n$ ...
21
votes
4answers
2k views

algebraic group G vs. algebraic stack BG

I've gathered that it's "common knowledge" (at least among people who think about such things) that studying a (smooth) algebraic group G, as an algebraic group, is in some sense the same as studying ...
11
votes
9answers
2k views

What is the Tutte polynomial encoding?

Pretty much exactly what it says on the tin. Let G be a connected graph; then the Tutte polynomial T_G(x,y) carries a lot of information about G. However, it obviously doesn't encode everything about ...
7
votes
2answers
1k views

Is the Fourier transform of $\exp(-\|x\|)$ non-negative?

Is the $n$-dimensional Fourier transform of $\exp(-\|x\|)$ always non-negative, where $\|\cdot\|$ is the Euclidean norm on $\mathbb{R}^n$? What is its support?
21
votes
6answers
2k views

Formal consequences of Riemann-Roch (multiple answers welcome)

This question aims to pin down what Riemann-Roch can tell us about a divisor on a curve, without any "geometric thinking". It can be annoying to wonder if there is some clever trick you're missing ...
7
votes
1answer
1k views

Mirror symmetry for noncompact Calabi-Yau manifolds

In analogy with the Hodge diagram for ordinary de Rham cohomology, we should have some kind of diagram for Alexander-Spanier cohomology. Doing all the relevant duality stuff and assuming that now our ...
11
votes
1answer
1k views

Reference for the `standard' Tate curve argument.

I'd like a reference (e.g. something published somewhere that I can cite in a paper) for the proof of the following: Let $E$ be an elliptic curve over $\mathbb Q$ with minimal discriminant $\Delta$...
1
vote
1answer
280 views

Convergence of Affine Transformations

Hi, I was wondering if anyone could point me to any sources regarding the convergence of iterated affine transformation, i.e. sequences where {a_n} is a set of affine transforms and the sequence: ...
17
votes
2answers
3k views

Euler characteristic of a manifold and self-intersection

This is probably quite easy, but how do you show that the Euler characteristic of a manifold M (defined for example as the alternating sum of the dimensions of integral cohomology groups) is equal to ...
32
votes
7answers
3k views

Simplicial objects

How should one think about simplicial objects in a category versus actual objects in that category? For example, both for intuition and for practical purposes, what's the difference between a [...
10
votes
2answers
1k views

Finiteness conditions on simplicial sheaves/presheaves

Could someone give an overview, or just some examples, of "finiteness conditions" for simplicial sheaves/presheaves and/or simplicial schemes? Any answer or comment about this would be interesting, ...
27
votes
4answers
2k views

Handling arXiv feeds to avoid duplicates

I subscribe to feeds from the arXiv Front for a number of subject areas, using Google Reader. This is great, but there is one problem: when a new preprint is listed in several subject categories, it ...
19
votes
7answers
6k views

What is a cup-product in group cohomology, and how does it relate to other branches of mathematics?

I have a few elementary questions about cup-products. Can one develop them in an axiomatic approach as in group cohomology itself, and give an existence and uniqueness theorem that includes an ...
7
votes
8answers
2k views

Are good introductory/pedagogical problems in algebraic geometry rare?

I have just started reading Elementary Algebraic Geometry by Hulek. It is a nice book but I find that it doesn't give many problems (about 10 to 15 per chapter), and that the exercises present are a ...
1
vote
1answer
575 views

Are cyclotomic Khovanov-Lauda-Rouquier algebras symmetric?

Recall that for k a field, a finite dimensional k-algebra A is called symmetric if it is isomorphic to its dual as a bimodule of itself. Which is to say, there's a trace map t:A -> k such that t(ab)=...
12
votes
5answers
1k views

Can one make Erdős's Ramsey lower bound explicit?

Erdős's 1947 probabilistic trick provided a lower exponential bound for the Ramsey number $R(k)$. Is it possible to explicitly construct 2-colourings on exponentially sized graphs without large ...
21
votes
5answers
2k views

Homological algebra and calculus (as in Newton)

This question reminded me of a possibly stupid idea that I had a while back. On page 2 of this paper, while discussing Euclid's axioms of plane geometry and spatial geometry, Manin makes an extremely ...
8
votes
3answers
2k views

Stalks of sheaf-hom

Let $F$ and $G$ be sheaves on $X$. Under what conditions is the natural map from the stalk at $p$ of $SheafHom(F,G)$ to $Hom(F_p, G_p)$ an isomorphism?
62
votes
10answers
19k views

What is (co)homology, and how does a beginner gain intuition about it?

This question comes along with a lot of associated sub-questions, most of which would probably be answered by a sufficiently good introductory text. So a perfectly acceptable answer to this question ...
4
votes
3answers
921 views

Quotient of a category by a free group action

Let Cat denote the 1-category of small categories. The functor Mor : Cat -> Set which assigns to a category its set of morphisms (aka Hom([• -> •], -)) does not commute with most colimits. ...
11
votes
2answers
504 views

A complex manifold which is quasiprojective in two different ways

Does there exist a complex manifold M which is a quasiprojective variety in two "essentially" different ways? Let me be more specific. I'm looking for a complex manifold M together with two ...
3
votes
3answers
658 views

Is there a “universal LYM inequality?”

This question is based on a blog post of Qiaochu Yuan. Let P be a locally finite* graded poset with a minimal element, and w be a weight function on the elements of P. Suppose that the total weight ...

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