# All Questions

113,823
questions

**13**

votes

**4**answers

2k views

### Short Introduction to Planar Algebras

Are there any good short expositions of planar algebras out there? I am interested primarily in seeing the main definition and some explicit examples.

**5**

votes

**2**answers

1k views

### Higher vanishing cycles

The generalisation of the vanishing cycle formalism in SGA 7 is apparently since the 1970's an issue, Morava mentioned a connection with Bousfield localization. I find the Morava's remarks un-...

**25**

votes

**7**answers

4k views

### Etale covers of the affine line

In characteristic p there are nontrivial etale covers of the affine line, such as those obtained by adjoining solutions to x^2 + x + f(t) = 0 for f(t) in k[t]. Using an etale cohomology computation ...

**11**

votes

**0**answers

778 views

### Kato's log motives

What are they and what are their intended uses? Does anyone have notes/slides of this talk?
I am curious about "log motives" because there seems to exist a "log motivic yoga" among experts in ...

**1**

vote

**4**answers

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

**1**answer

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

**3**answers

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

**3**answers

2k views

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

Apologies in advance if this is obvious.

**1**

vote

**8**answers

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

**1**answer

466 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

**2**answers

406 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

**4**answers

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

**2**answers

711 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

**2**answers

444 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

**3**answers

3k views

### Gaussian curvature and mean curvature. [closed]

Define Gaussian curvature for a nonorientable surface. Can you define mean curvature for a nonorientable surface?

**27**

votes

**8**answers

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 ...

**70**

votes

**7**answers

15k 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 ...

**18**

votes

**5**answers

5k 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

**4**answers

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

**2**answers

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

**7**answers

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

**7**answers

2k 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

**2**answers

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

**4**answers

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

**2**answers

903 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 ...

**43**

votes

**95**answers

72k 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

**2**answers

386 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

**4**answers

2k views

### References for syntomic cohomology

Could anyone point to good readable references for learning about syntomic cohomology?

**7**

votes

**10**answers

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

**2**answers

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

**4**answers

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 ...

**33**

votes

**6**answers

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 ...

**10**

votes

**1**answer

349 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 $n \times n$ ...

**22**

votes

**4**answers

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 ...

**12**

votes

**9**answers

3k 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

**2**answers

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

**6**answers

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

**1**answer

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

**1**answer

2k 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

**1**answer

307 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

**3**answers

4k 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 ...

**35**

votes

**7**answers

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

**2**answers

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, ...

**29**

votes

**5**answers

3k 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 ...

**23**

votes

**7**answers

7k 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 ...

**11**

votes

**8**answers

3k 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

**1**answer

639 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)=...

**13**

votes

**5**answers

2k 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

**5**answers

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 ...

**13**

votes

**3**answers

3k 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?