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3
votes
3answers
413 views

What's the best reference for actual formulas for RT invariants?

If one really wants to understand the formulas for how to construct the Reshetikhin-Turaev 3-manifold invariants coming from quantum groups in terms of R-matrices and such, what's the best reference ...
20
votes
12answers
3k views

Homological Algebra for Commutative Monoids?

Homological algebra for abelian groups is a standard tool in many fields of mathematics. How much carries over to the setting of commutative monoids (with unit)? It seems like there is a notion of ...
4
votes
1answer
384 views

When does a transitive action of a profinite group have an infinite orbit?

That is: suppose G is a profinite group acting 1-transitively (but maybe not regularly) on a set X. Is there a reasonable criterion for when there is a g in G and a point a in X such that the g-orbit ...
7
votes
2answers
1k views

What is an example of a smooth variety over a finite field F_p which does not lift to Z_p?

Somebody answered this question instead of the question here, so I am asking this with the hope that they will cut and paste their solution.
1
vote
1answer
259 views

Request for info on the space of commuting matrices preserving a flag.

Fix a flag of subspaces V1 in V2 in V3, etc. all in Cn. Consider the space of pairs of commuting linear transformations A and B such that: A preserves the flag (i.e. A(Vi) is in Vi), and B strictly ...
12
votes
5answers
2k views

Existence of (smooth) models

Hi everyone, let X be a variety over a field k, S an integral scheme such that the function field K of S is contained in k. An S-scheme X is called model of X/k if X x_S k = X, i.e. if the generic ...
45
votes
5answers
5k views

Does homology have a coproduct?

Standard algebraic topology defines the cup product which defines a ring structure on the cohomology of a topological space. This ring structure arises because cohomology is a contravariant functor ...
8
votes
1answer
790 views

Generalized Teichmuller representatives

Fix a prime $p$. The Teichmuller representative associated to a $p$-adic integer $a$ is the unique root of $x^p - x$ in $Z_p$ congruent to $a$ mod $p$. One can identify this representative with the ...
12
votes
3answers
1k views

What is an example of a smooth variety over a finite field F_p which does not embed into a smooth scheme over Z_p?

Such an example of course could not be projective and would not itself lift to Z_p. The context is that one can compute p-adic cohomology of a variety X over a finite field F_p via the cohomology of ...
28
votes
4answers
1k views

How is tropicalization like taking the classical limit?

There is a folk — I can't call it a theorem — "fact" that the mathematical relationship between Complex and Tropical geometry is analogous to the physical relationship between Quantum and ...
33
votes
4answers
2k views

Does a scheme have a “separification”?

Background: (1) If C and D are categories and there is a forgetful functor U:C→D, then a C-ification functor F:D→C is an adjoint to U. For example, the (left) adjoint to the forgetful ...
40
votes
6answers
7k views

“A gentleman never chooses a basis.”

Around these parts, the aphorism "A gentleman never chooses a basis," has become popular. Question. Is there a gentlemanly way to prove that the natural map from $V$ to $V^{**}$ is surjective if $V$...
62
votes
18answers
69k views

Reading list for basic differential geometry?

I'd like to ask if people can point me towards good books or notes to learn some basic differential geometry. I work in representation theory mostly and have found that sometimes my background is ...
16
votes
4answers
2k views

What's the right way to think about “anomalies” in 3d TQFTs?

3d TQFTs constructed from modular tensor categories don't in general give an honest 3d TQFT, instead they have an "anomaly." My vague understanding from Kevin Walker's talks and from skimming Freed-...
21
votes
5answers
2k views

Do all 3D TQFTs come from Reshetikhin-Turaev?

The Reshetikhin-Turaev construction take as input a Modular Tensor Category (MTC) and spits out a 3D TQFT. I've been told that the other main construction of 3D TQFTs, the Turaev-Viro State sum ...
28
votes
6answers
7k views

Deformation theory and differential graded Lie algebras

There is supposed to be a philosophy that, at least over a field of characteristic zero, every "deformation problem" is somehow "governed" or "controlled" by a differential graded Lie algebra. See for ...
53
votes
9answers
19k views

Motivating the Laplace transform definition

In undergraduate differential equations it's usual to deal with the Laplace transform to reduce the differential equation problem to an algebraic problem. The Laplace transform of a function $f(t)$, ...
16
votes
4answers
2k views

Deligne's conjecture (the little discs operad one)

Deligne's conjecture states that the Hochschild cochain complex of an A-infinity algebra is an algebra over the operad of chains on the topological little discs operad. Of course the conjecture has ...
17
votes
5answers
2k views

Derived categories and homotopy categories

There are two constructions that look quite similar to me: the derived category of an abelian category, and the homotopy category of a model category. Is there any explicit relationship between these ...
2
votes
3answers
1k views

What is the base change in number theory?

I'm somewhat familiar with base change in scheme theory: sometimes a property of a morphism X \to Y survives a base change ...
81
votes
10answers
18k views

Motivation for algebraic K-theory?

I'm looking for a big-picture treatment of algebraic K-theory and why it's important. I've seen various abstract definitions (Quillen's plus and Q constructions, some spectral constructions like ...
54
votes
14answers
17k views

A reading list for topological quantum field theory?

Can you suggest a reading list, or at least a few papers that you think would be useful, for a beginner in topological quantum field theory? I know what the curvature of a connection is, know basic ...
162
votes
86answers
37k views

Examples of great mathematical writing

This question is basically from Ravi Vakil's web page, but modified for Math Overflow. How do I write mathematics well? Learning by example is more helpful than being told what to do, so let's try to ...
3
votes
3answers
548 views

What is a formula for the “group-like Drinfeld element”?

Any quantized universal enveloping algebra (in fact, any toplogically quasi-triangular Hopf algebra) has an (in its completion) an element u called the Drinfeld element which gives an isomorphism from ...
3
votes
3answers
863 views

When and how is a group of order n isomorphic to a regular subgroup of equal order?

In "Group Theory and Its Application to Physical Problems" by Morton Hamermesh, Morton states Cayley's theorem: Every group G of order n is isomorphic with a subgroup of the symmetric group Sn, which ...
7
votes
1answer
791 views

Why are torsion points dense in an abelian variety?

Hi everyone, let $A$ be an abelian variety of dimension $g$ over an algebraically closed field $k$ of characteristic $p\geqslant 0$. I'm trying to prove that the subgroup $A'$ which is the union of ...
23
votes
2answers
2k views

Is every functor a composition of adjoint functors?

My understanding of Ben's answer to this question is that even though associated graded is not an adjoint functor, it's not too bad because it is a composition of a right adjoint and a left adjoint. ...
6
votes
6answers
1k views

Does the space of $n \times n$, positive-definite, self-adjoint, real matrices have a better name?

This is also the space of real, symmetric bilinear forms in $\Bbb R^n$.
48
votes
3answers
12k views

What is Koszul duality?

Okay, let's make sure I'm on the same page with those who know homological algebra. What is Koszul duality in general? What does it mean that categories are Koszul dual (I guess representations of ...
0
votes
2answers
2k views

How to attack this diophantine equation in 3 variables?

From link: Find integers a, b and c such that: 987654321a + 123456789b + c = (a + b + c)³
29
votes
4answers
2k views

Spectrum of the Grothendieck ring of varieties

Here's a problem that may ultimately require just simple algebraic-geometry skills to be solved, or perhaps it's very deep and will never be solved at all. From the comments, some literature and my ...
7
votes
1answer
1k views

Dualizing sheaf on singular curves

I am trying to understand the stabilization map, which takes a prestable curve (a curve with some marked points, and at worst nodal singularities) and returns a stable curve (a curve with some marked ...
6
votes
2answers
2k views

Eigenvalues of Laplacian

What's the most natural way to establish the asymptotics of $\Delta$ on a compact Riemannian manifold $M$ of dimension $N$? The asymptotics should be $$ \#\{v < A^2\} = \mathrm{const}\ast\mathrm{...
4
votes
3answers
428 views

Functions on hyperbolic space and modular curves

The decomposition of $L^{2}\left(S^{2}\right)$ under $SO\left(3,\mathbb{R}\right)$ is well-known. Focus now on the hyperbolic plane $H$ presented as the quotient $SL\left(2,\mathbb{R}\right)/SO\left(...
5
votes
2answers
596 views

Are there interesting monoidal structures on representations of quantum affine algebras?

Is there a good monoidal structure on a category of integrable representations of a quantum affine algebra? In the ordinary affine Kac-Moody case, there is the usual tensor product (symmetric, adds ...
7
votes
3answers
3k views

Beilinson-Bernstein and Koszul duality

For geometric representation theorists down here. Consider the Beilinson-Bernstein theorem: Functor of global sections establishes the correspondence between twisted D-modules with fixed ...
10
votes
2answers
985 views

Elliptic curve over spectra?

Filling the gaps in my knowledge to understand the tmf question. So, what is the analogue of elliptic curve over the category of spectra?
10
votes
8answers
2k views

Which came first: the Fibonacci Numbers or the Golden Ratio?

I know that the Fibonacci numbers converge to a ratio of .618, and that this ratio is found all throughout nature, etc. I suppose the best way to ask my question is: where was this .618 value first ...
7
votes
3answers
2k views

What is the Theorem of the Cube?

What is the "theorem of the cube" for abelian varieties? What is the statement and how should I think about it?
2
votes
2answers
763 views

What is the difference between the Power Law and Zipf's Law?

I am new to statistics. Could somebody tell me what is the difference between a Power Law and Zipf's Law. The latter could be just for texts but I cant see any difference in their essence.
20
votes
3answers
2k views

What is a TMF in topology?

What is a topological modular form? How are they related to 'normal' (number-theoretic) modular forms?
10
votes
5answers
1k views

Questions about ordering of reals and irrationals

Three problems from G.Rosenstein "Linear orderings" (from the end of Chapter 2 and beginning of Chapter 4): 1) Is there a nondecreasing function from irrationals onto reals? 2) Is there a ...
4
votes
2answers
974 views

Non-zero sheaf cohomology

Let $\mathbb{R}$ denote the real line with its usual topology. Does there exist a sheaf $F$ of abelian groups on $\mathbb{R}$ whose second cohomology group $H^{2}\left(\mathbb{R},F\right)$ is non-zero?...
7
votes
2answers
758 views

Finding monochromatic rectangles in a countable coloring of $R^{2}$

Given a countable coloring of the plane, is it always possible to find a monochromatic set of points $\left\{ \left(x,y\right),\left(x+w,y\right),\left(x,y+h\right),\left(x+w,y+h\right)\right\} $ (the ...
2
votes
2answers
139 views

Are irregular points of an action necessarily in the closure of a larger orbit?

Suppose G is an affine algebraic group acting linearly on a vector space V. A point v∈V is stable if the orbit Gv is closed and v is regular (the dimension of the stabilizer of v is locally ...
25
votes
4answers
2k views

What is the universal property of associated graded?

Given a filtered vector space (or module over a ring) $0=V_{0}\subseteq V_{1}\subseteq\cdots\subseteq V$, you can construct the associated graded vector space $\mathrm{gr}\left(V\right)=\oplus_{i}V_{i+...
-2
votes
1answer
705 views

Fundamental: Division by Zero [closed]

All the articles I've read regarding "Division by Zero" the main argument for it being an undefined operation, because all proofs lead to contradictions. ...
17
votes
2answers
6k views

does a line bundle always have a degree

For curves there is a very simple notion of degree of a line bundle or equivalently of a Weil or Cartier divisor. Even in any projective space $\mathbb P(V)$ divisors are cut out by hypersurfaces ...
6
votes
4answers
960 views

Are abelian non-degenerate tensor categories semisimple?

A pivotal monoidal category is called non-degenerate if the inner product $\left(x,y\right) = Tr\left(xy^{*}\right)$ (where $y^{*}$ is the dual map) is non-degenerate. As a rule of thumb non-...
11
votes
5answers
2k views

When are Hilbert schemes smooth?

I know that Hilbert schemes can be very singular. But are there any interesting and nontrivial Hilbert schemes that are smooth? Are there any necessary conditions or sufficient conditions for a ...

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