All Questions

14
votes
5answers
2k views

How should I think about B-fields?

So, physicists like to attach a mysterious extra cohomology class in H^2(X;C^*) to a Kahler (or hyperkahler) manifold called a "B-field." The only concrete thing I've seen this B-field do is change ...
72
votes
54answers
21k views

Free, high quality mathematical writing online? [closed]

I often use the internet to find resources for learning new mathematics and due to an explosion in online activity, there is always plenty to find. Many of these turn out to be somewhat unreadable ...
15
votes
5answers
1k views

What do gerbes and complex powers of line bundles have to do with each other?

We all know how to take integer tensor powers of line bundles. I claim that one should be able to also take fractional or even complex powers of line bundles. These might not be line bundles, but ...
30
votes
3answers
1k views

Can algebraic varieties be rigidified by finite sets of points?

For an algebraic variety X over an algebraically closed field, does there always exist a finite set of (closed) points on X such that the only automorphism of X fixing each of the points is the ...
156
votes
76answers
56k views

Best online mathematics videos?

I know of two good mathematics videos available online, namely: Sphere inside out (part I and part II) Moebius transformation revealed Do you know of any other good math videos? Share.
-3
votes
3answers
6k views

Factorial of 3/2? [closed]

How do you compute the factorial of something like $3/2$ or $-2$? Wolfram Alpha gives an answer, but how does it arrive at that point?
0
votes
2answers
281 views

Mediated envy-free and efficient cake cutting with n=2?

Is there an algorithm in literature to compute an efficient (pareto optimal) and envy-free cake cutting when there are only $n=2$ players and a mediator?
2
votes
1answer
890 views

Range of a Certain Linear Operator

Consider the following hermitian form on the sobolev space H^1(I), of an interval I: g(u,v):= \int_I (du/dt dv/dt - \rho(t) u v)dt, where \rho is a nice bounded function on I. Riesz representation ...
53
votes
11answers
8k views

Why is the exterior algebra so ubiquitous?

The exterior algebra of a vector space V seems to appear all over the place, such as in the definition of the cross product and determinant, the description of the Grassmannian as a variety, the ...
10
votes
2answers
1k views

Graded or stacky Serre duality

I am considering the following situation. $A$ is a finitely generated ring over a field $K$ with non-negative grading and $A_0=K$ of Krull dimension n+1, but I don't necessarily assume A is generated ...
14
votes
5answers
698 views

Number of metric spaces on N points

Given $X = \{x_1, ..., x_n\}$, how many collections $C$ of subsets of $X$ are there such that $C$ is the listing of all open balls of some metric space? The first nontrivial example is $n=3$; let's ...
11
votes
4answers
2k views

How to do Computations Using the Decomposition Theorem for Perverse Sheaves

This is a follow-up to this post on the Decomposition Theorem. Hopefully, this will also invite some discussion about the theorem and perverse sheaves in general. My question is how does one use the ...
4
votes
1answer
6k views

What is the difference between a homogeneous stochastic process and a stationary one?

Hello. I am studying stochastic process. here, I don't know what is difference of "the process is homogeneous" and "the process is stationary" I feel confusing. It seems to similar to me.
4
votes
2answers
280 views

Can you do surgery on framed tangles?

Surgery on framed links give orientable compact closed connected 3-manifolds. Can you do surgery on framed tangles? Would Fenn-Rourke moves be invariant?
17
votes
1answer
1k views

When should I expect a quiver with potential to be rigid?

This question is pretty technical, but there are some very smart people here. Fix a quiver Q, WITH oriented cycles. Let k[[Q]] be the completed path algebra. (Like the path algebra, but we allow ...
6
votes
2answers
455 views

Can one calculate the (co)homology of the loopspace of a Lie group from its Lie algebra?

Compact connected simply-connected Lie groups have so much structure that you can calculate their cohomology from their Lie algebras using Lie algebra cohomology (certain Ext-groups) and similarly ...
4
votes
1answer
1k views

Can Walsh-Hadmard transform be used for convolution ?

The Walsh-Hadamard transform is very fast to compute. Can it be used to compute the convolution of two functions as it can be done with Fourier transform ?
3
votes
2answers
217 views

Properties of signomial Functions in one variable

I am interested in functions of the form: \sum_{j=1}^\infty a_j x^{p_j} where p_j can be any non-negative real number. Wikipedia has informed me that this is a subset of the signomial functions, but ...
17
votes
3answers
2k views

Simple example of a ring which is normal but not CM

I try to keep a list of standard ring examples in my head to test commutative algebra conjectures against. I would therefore like to have an example of a ring which is normal but not Cohen-Macaulay. I'...
0
votes
1answer
237 views

Correlation measure between signals of different dimensions?

I have several temporal signals of different dimensions, for example the motion of a point throughout time which would be of dimension 3, and the value of a temperature sensor, of dimension 1. I ...
19
votes
3answers
5k views

Finite type/finite morphism

I am not too certain what these two properties mean geometrically. It sounds very vaguely to me that finite type corresponds to some sort of "finite dimensionality", while finite corresponds to "...
5
votes
1answer
176 views

Homotopy type of stabilizers

Let X be a contractible metric space and G a topological group acting transitively on X (i.e. given any two points x,y \in X, there exists g \in G such that gx=y). My question is the following: is it ...
18
votes
3answers
2k views

K(F_1) = sphere spectrum?

I repeatedly heard that K(F_1) is the sphere spectrum. Does anyone know about the proof and what that means?
24
votes
3answers
8k views

Is 8 the largest cube in fibonacci sequence?

Can you prove that 8 is the largest cube in fibonacci sequence?
13
votes
2answers
815 views

Are generalized cohomology theories a homotopy category of some category of invariants?

I was taught to think of generalized cohomology theories as the homotopy category of (symmetric) spectra. But is there also a category of 'invariants', that is, some category of contravariant functors ...
7
votes
2answers
729 views

Is there a free digraph associated to a graph?

A little bit of background: A graph G is, of course, a set of vertices V(G) and a multiset of edges, which are unordered pairs of (not necessarily distinct) vertices. We say that two vertices v_1, v_2 ...
11
votes
4answers
1k views

When is a scheme a zero-set of a section of a vector bundle?

Are there any general results on when a closed subscheme X of a quasi-projective smooth scheme M can be written as the zero-set of a section of a vector bundle E on M? To put it in a diagram: When is ...
15
votes
5answers
1k views

Pairs of shortest paths

It is known that the binomial coefficient $2n \choose n$ is equal to number of shortest lattice paths from $(0,0)$ to $(n,n)$. The Catalan number $\frac{1}{n+1} {2n\choose n}$is equal to the number of ...
-2
votes
1answer
319 views

about Function of Random variables [closed]

Hello, I am studying random variables. Question is this: if rv X & a function g is known, what is the pdf of random variable Y = g(x)? in the textbook answer is explained as follows. P[y ≤ Y ≤...
2
votes
3answers
2k views

free homotopy groups — when do they exist?

Let (X,x) be a pointed space. There is an action of π1(X,x) on πn(X,x) -- determined by considering πn(X,x)=πn-1(ΩxX,x), where ΩxX denotes the space of loops in X based at x, ...
0
votes
2answers
129 views

Multiplicative stability for integration [closed]

I remember back in undergraduate to ask myself this question : In the general case, I is an interval, \int_I fg =! \int_I f \int_I g (*) But how to describe the egality case, i.e find all couples (f,g)...
7
votes
3answers
2k views

Special cases for efficient enumeration of Hamiltonian paths on grid graphs?

While the general problem of detecting a Hamiltonian path or cycle on an undirected grid graph is known to be NP-complete, are there interesting special cases where efficient polynomial time ...
45
votes
4answers
7k views

Is there a good way to think of vanishing cycles and nearby cycles?

Once in a while I run into literature that invokes vanishing cycle machinery with a cryptic sentence like, "this follows from a standard vanishing cycle argument." Is there a good way to look at ...
5
votes
1answer
299 views

Is there a canonical notion of principal divisor on a discrete dynamical system?

I hope this question is well-posed. Let (X, f) be a discrete dynamical system such that every x in X has finite period, i.e. there is some n such that f^n(x) = x. Let Div(X) be the free abelian ...
-1
votes
1answer
422 views

multidimensional multinomial density [closed]

I have data set X = {x_1, x_2, \ldots, x_N}, each x_i is a d-dimensional vector, where scalars are from some finite field (In practice they are categories, represented by integers from 1...C). If ...
12
votes
2answers
1k views

Can a singular Deligne-Mumford stack have a smooth coarse space?

Let XX be a Deligne-Mumford stack and let XX \to X be a coarse moduli space. Suppose that X is smooth. Is XX smooth? If not, what is an example? What if XX is of finite type over C (the complex ...
1
vote
5answers
929 views

Estimating the number of clusters

For a collection of points in $\mathbb{R}^n$, is there a statistic that I can compute which will estimate the number of clusters with some level of confidence?
6
votes
1answer
266 views

Can you construct a mapping space from local data? (looking for reference)

I'd to know if/where there is a reference for the following construction. Let C_*(maps(M, T)) denote the singular chains on the space of continuous maps from an n-...
23
votes
7answers
5k views

Quotients of Schemes by Free Group Actions

I've often seen people in seminars justify the existence of a quotient of a scheme by an algebraic group by remarking that the group action is free. However, I'm pretty sure they are also invoking ...
13
votes
3answers
874 views

Categorifying the Reals via von Neumann Algebras?

So one way to categorify the natural numbers is to replace them with vector spaces. Then the dimension of the vector space reproduces the natural number. More generally you can categorify integers to ...
19
votes
5answers
2k views

Point singularity of a Riemannian manifold with bounded curvature

Suppose you have an incomplete Riemannian manifold with bounded sectional curvature such that its completion as a metric space is the manifold plus one additional point. Does the Riemannian manifold ...
0
votes
7answers
1k views

What is the smallest integer whose primality status is not known?

Closely related: what is the smallest known composite which has not been factored? If these numbers cannot be specified, knowing their approximate size would be interesting. E.g. can current ...
4
votes
1answer
478 views

How do you rotate a matrix to maximum sparsity?

Given a matrix M, I want to find an orthogonal matrix U that maximizes the number of entries that are zero in the product MU. How do I go about doing this?
9
votes
3answers
2k views

Unstable Vector Bundles

As a follow up to me other question, what can be said about unstable vector bundles? I know this is rather open ended, but what sorts of horrible things does having a subbundle of strictly greater ...
0
votes
5answers
1k views

Is there a tool for finding probability distributions given some samples?

I'm looking for a tool that does "probability distribution fitting" given a set of data points. Sort of like curve fitting, but tries to fit to standard density distributions. For example if I input ...
17
votes
1answer
739 views

Is there a Murnaghan-Nakayama Rule for GL(n,q)?

The Murnaghan-Nakayama rule for S_n is a combinatorial rule to compute the irreducible characters of the symmetric group. Is there a q-analogue of this rule for GL(n,q) to compute the irreducible ...
34
votes
5answers
6k views

Induction and Coinduction of Representations

I'd like to understand the general framework of induction and coinduction of representations. If G is a finite group and H a subgroup, I know that there is a restriction functor from representations ...
4
votes
2answers
1k views

Pushforwards of Line Bundles and Stability

I recently finished reading this paper, and was wondering about a couple of things relating to theorem 1, which says that for any curve X there is a curve Y and f:Y->X such that pushforward is a ...
7
votes
1answer
474 views

Universal covers of domains in complex projective space

The Uniformization Theorem states that the universal cover of a Riemann surface is biholomorphic to the extended complex plane, the complex plane or the open unit disk. Each of these three is a domain ...
2
votes
2answers
339 views

Is the center of a free (as a module) algebra free?

A submodule of a free module need not be free (for instance, in the free Z[X]-module Z[X] the submodule generated by 2 and X is not free). But over a principal ideal domain, submodules of free modules ...

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