15
votes
2answers
218 views

Escape the zombie apocalypse

Consider zombies placed uniformly at random over $\mathbb{R}^2$ with asymptotic density $\mu$. You are placed at a random point and can move with speed $1$. Zombies move with speed $v\leq 1$ straight ...
9
votes
1answer
193 views
+50

Hecke-module structure implicit in definition of automorphic forms in Borel-Jacquet's Corvallis article

Let $G$ be a connected reductive group over a number field $F$, $G_\infty=\prod_{v\mid\infty} G(F_v)$, $\mathbf{A}$ the adèles of $F$, $\mathbf{A}_f$ the finite adèles of $F$. Fix a maximal compact ...
0
votes
0answers
3 views

laminations and branched surfaces

I am looking for a reference for this question: given a branched surface in a 3-manifold, how we can construct a lamination fully carried by that branched surface. any comments would be appreciated.
19
votes
2answers
932 views

Why do Lie algebras pop up, from a categorical point of view?

Groups pop up as automorphism groups in any category. Rings pop up as endomorphism rings in any additive category. Is there a similar way to attach a Lie algebra to an object in a category of a ...
0
votes
0answers
3 views

Sharp upper bounds on hypergeometric function ${}_2F_1[a,b,c;z]$ when $|z|\geq1$

Generally, hypergeometric function ${}_2F_1[a,b,c;z]$ is defined using Gauss series ${}_2F_1[a,b,c;z]=\sum_{n=0}^\infty\frac{(a)_n(b)_n}{(c)_nn!}z^n$ with $|r|<1$, and there seems to be a lot of ...
0
votes
1answer
66 views

Double Markovity

Suppose we have a double Markov relation for three random variables $X$, $Y$ and $W$ as follows $$X\to W\to Y,$$ and $$X\to Y\to W.$$ How to prove that there exist functions $f$ and $g$ such that ...
0
votes
1answer
67 views

Why is the Tate local duality pairing compatible with the Cartier duality pairing?

This question is a follow up to Why is the norm map dual to restriction under Tate local duality? Let $A$ and $B$ be dual abelian schemes over a base scheme $S$. For an integer $n \ge 1$, consider ...
0
votes
0answers
16 views

convergence interval of an infinite series without the general term [on hold]

I am trying to find the convergence of an infinite series of which I do not have the nth term. Instead of applying the ratio test for the nth term, I divided the first two terms, then the next two and ...
3
votes
1answer
577 views

“Aztec Diamond” analogue for Square-Octagon graph.

I have been reading David Speyer's Perfect Matchings and the Octahedron Recurrence, trying to carry out his "cross-wrenches" generalization of the Aztec diamond. In what follows, I'm asking for a ...
0
votes
0answers
10 views

Reference for “Newtonian capacity estimates probability that A is hit by a Brownian motion”

I am looking for the following statement "In fact, the Newtonian (logarithmic) capacity gives an estimate, up to a constant factor, the probability that A is hit by a Brownian motion started, say, ...
3
votes
0answers
88 views

Are inclusions from spaces of $C^\infty$ sections into spaces of $C^k$ sections homotopy equivalences?

[EDIT: The answer to my original question was obviously no, as user56365 pointed out. Here is what I should have asked.] For finite-dimensional smooth manifolds $E,M$, let $E\to M$ be a smooth fibre ...
2
votes
0answers
166 views

Multiplicative Identity for all elements in SU(n)

Let $\{P_i\}$ be a subset of $SU(n)$ such that for any $U$ in another subset (or perhaps subgroup) $H$ of $SU(n)$: $$P_1UP_2U\cdots P_mU=I$$ where $I$ is the identity element. Is there a sequence ...
12
votes
2answers
354 views

Why polarization of abelian varieties?

Maybe this question is not suitable for here, but I don't think I would receive a satisfactory answer in Math StackExchange. I could never understand the intuition behind polarization of abelian ...
3
votes
1answer
84 views

Prove that …, f(x-2), f(x-1), f(x), f(x+1), f(x+2),… is algebraically linearly independent without the Fourier transform

Suppose that $f$ is in $L^2(\mathbb{R})$ and consider the set of integer translates of this function, $V=\{f(x-k):k\in\mathbb{Z}\}$. This set is linearly independent: taking the Fourier transform of ...
3
votes
1answer
120 views

Positive definite quadratic forms on Banach spaces

This is a question about characterizing Hilbert spaces in terms of quadratic forms. Let $X$ be a real Banach space and $E$ a bounded quadratic form on it, it is called positive definite if ...
69
votes
10answers
7k views

Are there any very hard unknots?

Some years ago I took a long piece of string, tied it into a loop, and tried to twist it up into a tangle that I would find hard to untangle. No matter what I did, I could never cause the later me any ...
4
votes
1answer
88 views

Hypercovers of sheaves in classical and quasi-categories

I am interested in relating the definition of hypercovers in the $\infty$-topos of sheaves on an $\infty$-Grothendieck site to the classical definition of hypercovers of presheaves on a Grothendieck ...
8
votes
1answer
114 views

Do direct limits (filtered colimits) commute with pullbacks, in C*-algebras?

I asked this question already on math.stackexchange, but maybe it is also useful to ask this here, since it was not answered there. Suppose we have three directed sequences of $C^*$-algebras, say ...
1
vote
1answer
89 views

Conjugate surfaces: informations about the orbits

Consider a complex algebraic variety $X$ (namely a $\mathbb C$-scheme, of finite type, geometrically integral and separated); if $\sigma\in\textrm{aut}(\mathbb C)$, then is well defined the complex ...
4
votes
1answer
130 views

Which valuations of a field yield codimension $1$ subschemes of their 'models'

For a field $F$ (for example, a one generated by a finite number of its elements) there is a directed set of its 'models' (in this case those are 'arithmetic' schemes whose fraction field is $F$). It ...
1
vote
0answers
63 views

Default Orientation of Vectors [on hold]

When I started studying math in 1982 in Germany, there seemed to have been a change in the choice of the default orientation of vectors; while it was row-vectors till then, it changed to ...
4
votes
1answer
118 views

Horn's inequalities for n matrices

Where I can find necessary and sufficient conditions on eigenvalues of Hermitian matrices with the relation $$A_1 + A_2 + ... + A_n = A_0 ,$$ i.e. Horn's inequalities for n matrices? Can such ...
4
votes
1answer
173 views

Connections between Standard, Hodge and Tate conjectures on algebraic cycles?

What implications would a solution of the Standard Conjectures have on the Hodge and Tate Conjectures and reverse? What implications would a solution of the Tate Conjecture have on the Hodge ...
0
votes
1answer
161 views

Limits of functions with converging zeros

What can one say about the derivatives of a smooth function of several variables that is a limit of smooth functions with converging zeros? More precisely, suppose that $f_i: R^n \to R^m$ is a ...
3
votes
2answers
84 views

Heat kernel asymptotics for the sublaplacian on a contact Riemannian manifold

Let $\Delta$ denote a Laplace-type differential operator on a compact Riemannian manifold $(M,g)$. The asymptotics of the heat kernel and the heat operator trace of $\Delta$ are well-known (cf. ...
11
votes
2answers
256 views

Asymptotics for algebraic numbers of height less than one

Is an asymptotic equivalent known or conjectured for the number $N(d)$ of $\alpha \in \bar{\mathbb{Q}}$ with $h(\alpha) < 1$ and $[\mathbb{Q}(\alpha):\mathbb{Q}] \leq d$? The rather crude proof ...
5
votes
1answer
185 views

Motivic L-function vs motivic zeta function

Let $M$ be a pure motive over a field $k$. Roughly speaking, the L-function of $M$ is the product over all primes $p$ of $$L_p(M,s)=\det(I-Fr_p|_{M_\ell^I} N(p)^{-s})^{-1}$$ where $Fr_p$ is a ...
2
votes
3answers
194 views

Compact, densely ordered spaces

During my work with order preserving homeomorphisms, I got interested in the double arrow space and, subsequently, in the lexicographic square. I would really like to find examples of spaces like ...
1
vote
0answers
48 views

Reference request for instantons

I've been researching instantons lately and I'd like to learn more about them but would like some help finding what to read. I have read about the ADHM equations and their noncommutative analogues. ...
2
votes
1answer
153 views

Is the ordinary locus affine?

Let $p$ be a prime number and let $Y$ over $\mathbb F_p$ be a Siegel modular variety, with minimal compactification $X$. It is well known that $X^{\operatorname{ord}}$, the ordinary locus of $X$ is ...
0
votes
0answers
31 views

Multiplicativity of the Ideal Norm

I asked this question on StackExchange twice, and was never able to get an answer. This is not quite a research-level question, but it is sufficiently difficult and interesting from a pedagogical ...
2
votes
0answers
146 views
+50

Computing the chromatic polynomial of graph modulo $x-3$

The chromatic polynomial of graph $P(G,x)$ is univariate polynomial which counts the number of colorings of $G$ with $x$ colors for natural $x$. Graph is not $k$ colorable iff $P(G,k)=0$. The ...
17
votes
2answers
360 views

Adams Operations on $K$-theory and $R(G)$

One of the slickest things to happen to topology was the proof of the Hopf invariant one using Adams operations in $K$-theory. The general idea is that the ring $K(X)$ admits operations $\psi^k$ that ...
1
vote
1answer
52 views

$L^\infty_\mathrm{loc}$ assumption in global existence for Boltzmann equation

In short: In P. Gérard's paper on the existence of global solutions to the Boltzmann equation from 1988 (or equivalently Cercignani's book), why are the stated assumptions (especially $A_n \in ...
8
votes
1answer
223 views

Why does this antisymmetric product factor out a determinant?

Consider a generic $n \times n$ matrix $M$. Define the $(n-1) \times n$ matrix $M_q$ to be $M$ with the $q$th row omitted, and assume that $M_q$ possesses a right inverse, $R_q$: $$R_q = M_q^T (M_q ...
7
votes
3answers
701 views

Expectation of a simple function of multivariate gaussians iid rvs

I would like to compute analytically the following expected value: $E\left( \frac{X_i^2}{\sum_j \lambda_j^2 X_j^2}\right)$ where the $X_i \approx N(0,1)$ are iid. It seems an elementary integral, ...
1
vote
2answers
90 views

Why the intersection of a scott open (or \w the relatively compactness property) filter on a topology of a sober (and 2nd countable) space is compact?

Definitions and notations. Let $\mathcal{P}(X)$ the power set of $X$. Let $\tau_X\subseteq\mathcal{P}(X)$ a topology on X. We call $A$ irreducible if every time $A=B\cup C$ with $B,C$ closed set ...
0
votes
0answers
30 views

optimization problem, any solution?

The objective is as follows: $\min_{\mathbf{F}} a Tr(\mathbf{F} \mathbf{F}^H) - Re\{\mathbf{b}\mathbf{F}^H \mathbf{C} \mathbf{F} \mathbf{d}\}$ $s.t.\ \ \ Tr(\mathbf{F} \mathbf{F}^H)<p$ where $a$ ...
2
votes
1answer
99 views

Examples of cubic Julia sets

I'm looking for information on explicit cubic filled Julia sets with interesting properties such as: Having two different finite attractors (such as $f(z)=z^3-1.5z$) Being disconnected with ...
10
votes
2answers
391 views

Cut and Fold Polyhedron!

I have two convex polyhedra such that their sums of side areas are equal. It is true that I can cut one of them and flatten it on the plane, then fold the flattened polygon to reach the other ...
5
votes
1answer
374 views

Counterexample in cohomology for symmetric spectra?

Symmetric spectra are a particular model for spectra, introduced by Hovey, Shipley and Smith. They have the nice property that they have a well-behaved smash product. Our interest in spectra comes ...
11
votes
3answers
512 views

Is every paracompact, Hausdorff, locally contractible space homotopy equivalent to a CW complex?

Milnor proved that any paracompact Hausdorff space which is equi-locally convex (and hence in particular locally contractible) is homotopy equivalent to a CW complex. However, unlike being paracompact ...
6
votes
3answers
327 views

Ergodic theory and dynamical systems books references

I am arranging a weekly meeting of 2 hours with postgraduate students in ergodic theory (for a period of 3 weeks). I am asking here for an advice of a book (or maybe a set of papers) to look at ...
3
votes
0answers
64 views

Parametrized cancelations in stable Morse theory

Let $B$ be a closed manifold. Let $\pi : M\to B$ be a submersion such that each fiber is a manifold without boundary. Let $f : M \to \mathbb{R}$ be a function such that the restrictions $f_x$ to each ...
0
votes
0answers
32 views

A solution of a q-difference equation

Is it possible to find a solution of the $q$-difference equation $$f(q^{-1}x)-f(qx)=x(a-x)f(x),$$ with $f(0)=1$, (perhaps) in terms of basic hypergeometric series? Or in another rather explicit form? ...
-1
votes
0answers
56 views

Can first and second fundamental form be considered as differential 1-forms? [on hold]

Let $(M,g)$ be an abstract surface embedded into $\mathbb{R}^3$ via $f:M \to \mathbb{R}^3$. Denote the first and second fundamental form by $I$ and $II$, respectively and the smooth vector fields on ...
11
votes
4answers
312 views

Fibrations and Cofibrations of spectra are “the same”

My question refers to a folklore statement that I have now seen a couple of times, but never really precise. One avatar is: "For spectra every cofibration is equivalent to a fibration" (e.g. in the ...
2
votes
2answers
182 views

Extending vector bundles from subvarieties

Let $X$ be a smooth projective variety and let $Y\subset X$ be a smooth subvariety. Given a vector bundle $E$ on $Y$, when can $E$ be extended to a vector bundle $\tilde E$ on $X$? I.e., are there ...
0
votes
0answers
109 views

Basketball shots and stopping rule

Moved over from StackExchange. You are taken to play a basketball game where you can shoot basketballs at n slots using a machine that is equally likely to shoot the balls into those n slots. You can ...
6
votes
3answers
2k views

Au revoir, law of excluded middle?

In what way and with what utility is the law of excluded middle usually disposed of in intuitionistic type theory and its descendants? I am thinking here of topos theory and its ilk, namely synthetic ...

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