**75**

votes

**0**answers

6k views

### Dropping three bodies

Consider the usual three-body problem with Newtonian
$1/r^2$ force between masses. Let the three masses start off at rest,
and not collinear. Then they will become collinear a finite time ...

**21**

votes

**0**answers

1k views

### Correspondence between eigenvalue distributions of random unitary and random orthogonal matrices

In the course of a physics problem (arXiv:1206.6687), I stumbled on a curious correspondence between the eigenvalue distributions of the matrix product $U\bar{U}$, with $U$ a random unitary matrix and ...

**19**

votes

**0**answers

322 views

### Infinitely many planets on a line, with Newtonian gravity

(I previously asked essentially this on physics.stackexchange, but was actually
hoping for answers with something closer to a proof than what I got there.)
Suppose we have a unit mass planet at each ...

**13**

votes

**0**answers

418 views

### Catalan objects associated to a univariate polynomial

Given a monic degree $n$ polynomial $f(z)$ with no double roots, and a phase $0\leq \theta < \pi$, there are natural constructions which associate to this data:
a noncrossing matching on $2n$ ...

**12**

votes

**0**answers

343 views

### Which limit to take as a key applied math decision

The Borel-Kolmogorov paradox refers to situations where non-uniqueness in the notion of conditioning on a set of measure zero leads to apparent contradictions. As a formal matter, one requires ...

**11**

votes

**0**answers

926 views

### conformal blocks for beginners

I have given now a couple of talks that involve conformal blocks bundles on the moduli stack $\overline{\mathcal{M}}_{g,n}$, in front of a public of algebraic geometers but not specialists of the ...

**10**

votes

**0**answers

398 views

### Which journals publish applied mathematics with mostly pure mathematics content?

In the spirit of Which journals publish expository work? please advise:
What consistently high quality journals$^1$ today publish results that would otherwise go to a pure mathematics journal if ...

**10**

votes

**0**answers

302 views

### “extended TQFT” versus “TQFT with defects”

There are two ways in which higher categories appear in topological field theory: in extended TFTs and in TFTs with defects. How are these appearances related?
According to the Atiyah-Segal axioms, a ...

**9**

votes

**0**answers

291 views

### State of the art of BPS and Donaldson-Thomas invariants for toric Calabi-Yau threefolds

I am trying to understand what has been done with regards to computing BPS invariants and Donaldson-Thomas type invariants of Calabi-Yau threefolds. To make the question more focused, let's say that I ...

**9**

votes

**0**answers

474 views

### The open problem of nth quantization

In trying to explain a quote by E. Nelson, "First quantization is a mystery, but second quantization is a functor!" Baez points out what follows (full text available in this week find; I'm also ...

**9**

votes

**0**answers

389 views

### Parametrisations for Null Temperature Functions: nonuniqueness of solutions to the Heat Equation

Disclaimer I expect this is a highly open problem, but maybe I'm wrong and someone has come up with some answers besides those given here. In any case, all information appreciated, thanks!
Definition ...

**8**

votes

**0**answers

88 views

### What is the symmetry group fixing norms of elements of a unitary matrix?

Let $N\geq1$ be an integer and let us say that two matrices $U,V\in U(N)$ are related if $|U_{ij}|=|V_{ij}|$ for all indices $1\leq i,j\leq N$.
When exactly are two unitary matrices related in this ...

**8**

votes

**0**answers

381 views

### Equivariant quantum cohomology and K-theory of vector bundles over a Grassmannian

Consider a sum of m copies of the tautological bundle over the Grassmannian of n-planes in complex k-dimensional vector space. There is an obvious action of an (m+k)-dimensional torus T on the total ...

**8**

votes

**0**answers

740 views

### triangulated/derived categories in Physics and algebraic geometry

Why do physicists care about the triangulated/derived categories?
I mean what are the problems we want to approach using the machinery of triangulated/derived categories. e.g. in homological mirror ...

**7**

votes

**0**answers

154 views

### What's the appropriate notion of a Unitary representation of a Lie algebra?

Here Lie algebras/groups are real. The most straightforward definition might be:
Def: A representation $\rho:\mathfrak{g} \rightarrow \mathfrak{gl}(V)$ is unitary if $V$ is equipped with a Hermitian ...

**7**

votes

**0**answers

243 views

### Noncommutative geometry and line length

I would like to understand, in some formal sense, the relation between the Dirac operator and the line length introduced by Connes in noncommutative geometry. If $D$ is the Dirac operator, he sets $ds ...

**7**

votes

**0**answers

176 views

### Characterize spin cobordism invariants in dimer models

The paper by Cimasoni and Reshetikhin http://arxiv.org/abs/math-ph/0608070 shows that one can map problems about spin structures on a Riemann surface into problems about dimmer configurations on a ...

**7**

votes

**0**answers

537 views

### Curvature as infinitesimal holonomy

Let $P \to M$ be a principal $G$-bundle, assume as much regularity as you want (compact $G$ or compact base manifold, ect). Via parallel transport, a connection $A$ on $P$ gives rise to the holonomy ...

**7**

votes

**0**answers

239 views

### The space-time dimension of the N-superstring theory?

Let $\mathfrak{W}$ be the Lie algebra generated by $d_{n} = ie^{in\theta}\frac{d}{d\theta}$ and $\mathfrak{Vir} = \mathfrak{W} \oplus C \mathbb{C}$ its central extension:
$$
...

**7**

votes

**0**answers

70 views

### Explicit expression of WZ term for orthogonal groups

Consider the Wess Zumino term on the the space $W=I\times D$, where D is a two dimensional disk disk and $I$ is an interval, $[0,1]$, say, i.e.,
$$
\int_{I\times D} \langle(u^{-1} \, du)^3\rangle
$$
...

**7**

votes

**0**answers

145 views

### Deformation of Noether's first theorem

Noether's first variational theorem establishes a correspondence between symmetries and invariants. I would like to know what has been written on the following question: How do the invariants deform ...

**7**

votes

**0**answers

229 views

### What is known about the Yang-Mills stratification over 3-manifolds?

Rade proved in his thesis (Crelle's Journal, 1992, available here: digizeitschriften.de/dms/toc/?PPN=PPN243919689) that if $E\rightarrow M$ is a $U(n)$-bundle over a 3-manifold, then the gradient flow ...

**6**

votes

**0**answers

210 views

### Can one classify irreducible unitary representations of the Weyl algebra?

I saw in this MO post: Is there a machinery describing all the irreducible representations ? that classifying irreducible representations of the Weyl algebra is essentially intractable. My question is ...

**6**

votes

**0**answers

167 views

### Energy barriers between Hadamard matrices

Hadamard matrices may be characterized as $n\times n$ real orthogonal matrices $U$ that achieve the lowest possible "energy" as defined by the (scaled and shifted) entry-wise 1-norm:
$$
E(U)=n^2 ...

**6**

votes

**0**answers

163 views

### Density of odd and even eigenstates of an integral operator

Consider an integral operator $(Kf)(x)=\int_{-1}^{1}K(x-y)f(y)dy$, where the kernel $K(-x)=K(x)$ is an even function.
Let $\lambda_n$ be the ordered eigenvalues of $K$ and $f_n(x)$ the ...

**6**

votes

**0**answers

249 views

### Are there exactly solvable CFTs?

I am wondering if there are CFTs such that n-point correlation functions in them of the fields (may be the primaries or of some notion of twist fields) is exactly known.
Are there such?
Aren't ...

**6**

votes

**0**answers

171 views

### Spectral theory for Dirac Laplacian on a funnel

I would like to study the spectral theory of the Dirac Laplacian on a non-compact quotient of the hyperbolic plane by a discrete group (I am particularly interested in the simple case where the ...

**6**

votes

**0**answers

374 views

### Would a closed universe with special relativity violate causality? Does the universe have to be simply connected?

This question may be more appropriate for physics.stackexchange.com, but it would be helpful to get feedback from experts in Minkowski geometry.
The classic twin paradox is a false thought experiment ...

**6**

votes

**0**answers

287 views

### Birth-Death Process associated with Orthogonal Polynomials

I have read in various places the following objects are related:
orthogonal polynomials
birth-death processes
Lattice paths
continued fractions
After a lot of searching online, I found sketches ...

**6**

votes

**0**answers

821 views

### Harmonic maps into compact Lie groups

Consider locally minimizing harmonic maps from D-dimensional Euclidean space into a compact Lie group G. When $D=3$ the general regularity theory due to Schoen-Uhlenbeck,
Schoen, Richard; Uhlenbeck, ...

**5**

votes

**0**answers

108 views

### Can the compactification of a (co)tangent bundle equipped with Saski metric be viewed as a “Wick rotation”?

We can equip the (co)tangent bundle of a Riemannian manifold (B,g) with a Saski metric $\hat{g}$ (see, for example, "On the geometry of tangent bundles" by Gudmunssun & Kappos) that looks like
...

**5**

votes

**0**answers

74 views

### Fock Space Proof of $(g(x)\phi^4)_2$ Mass Gap?

Is there a proof that does not depend on Euclidean methods? Is this a proof? :
$V(g)$ can be written as $P+R$ where $P$ is non-negative and $R$ is $N$-bounded (and hence $(H_0+\lambda P)$-bounded). ...

**5**

votes

**0**answers

151 views

### TQFT characterization of braiding statistics

In the TQFT language, quasiparticles correspond to Wilson loop operators. It is well-known that quasiparticles can have non-trivial braiding statistics.
Take $2+1$ dimensional Abelian Chern-Simons ...

**5**

votes

**0**answers

109 views

### Stationary point processes with arbitrarily slow decorrelation

A point process $P$ (a probability measure on simple, locally finite point configurations $\mathcal{C}$ on $\mathbb{R}$ - I'm restricting to the one-dimensional setting) is stationary when ...

**5**

votes

**0**answers

103 views

### Implicit/Explicit Time Dependence for Melnikov Functions

My question concerns an article by Koiller and Carvalho found here: http://link.springer.com/article/10.1007/BF01260390
On page 645, they parameterize the time variable $t$ in terms of one of the ...

**5**

votes

**0**answers

75 views

### Short examples that are/are not quantum-ergodic

Are there any considerably short examples of manifolds that are/aren't quantum ergodic, or quantum unique ergodic?
Note that a (compact) Riemannian manifold is said to be quantum ergodic if ...

**5**

votes

**0**answers

257 views

### Elementary proof of lack of phase transition in Ising models with external fields

I have a question about the phase transitions in the Ising model in the presence of a (constant) external magnetic field. I will state the question on $\mathbb Z^2$ for simplicity. A definition of the ...

**5**

votes

**0**answers

366 views

### unitary equivalence

Let $U$ be the bilateral shift operator in $l^2(Z)$, and let $V$ stand for a rotation on an irrational angle $\alpha$ in $L^2(T, \mu)$, where $T$ is a circle with a rotation-invariant Lebesgue measure ...

**5**

votes

**0**answers

170 views

### Given that a conditional measure is Gaussian, how bad can the original measure be?

Let $X$ and $Y$ be Banach spaces, and let $\varphi : X \to Y$ be a continuous linear map. Suppose that $\mathbb P$ is a probability measure on $X$ which satisfies the continuous disintegration ...

**5**

votes

**0**answers

416 views

### Jones Polynomial and Quantum Field Theory

I am trying Witten's paper but unable to re-produce the computations presented in the paper.
I tried few things on internet but all these tutorials explicitly don't show the calculations and refer to ...

**5**

votes

**0**answers

191 views

### Generalization of the non-existence of a monostatic planar body

Domokos, Papadopulos, and Ruina showed that there does not exist a convex planar rigid body of uniform density which has only
one orientation of stable equilibrium and one orientation of unstable ...

**5**

votes

**0**answers

145 views

### BKS pairing in the SU(2) Chern-Simons theory

I know that usually, the way to compare the Hilbert spaces arising from $SU(2)$ Chern-Simons theory with different Kähler polarizations is via the Hitchin connection. However, it should be possible, I ...

**5**

votes

**0**answers

149 views

### what books to read to quickly understand adiabatic approximation

Hi group, I'm a theoretical ecologist with fairly adequate training in applied math (ODE, linear algebra, applied probability, some PDEs). In my current work, I've encountered the use of adiabatic ...

**5**

votes

**0**answers

268 views

### Quantum dynamics on varieties: asymptotic quantum trace distance on SHL varieties

The Question Asked
Definition: the Second-Hand Lion trace distance $D_k$
Let $\mathcal{M}^{(kk)}_r$ be the set of $k\times k$ complex matrices of rank $\le r$ having unit trace norm. Then the ...

**5**

votes

**0**answers

465 views

### Lebesgue measure on Frechet space?

It is well known that there are no Lebesgue measures on infinite-dimensional Banach spaces (see e.g. http://en.wikipedia.org/wiki/There_is_no_infinite-dimensional_Lebesgue_measure). However, I ...

**4**

votes

**0**answers

61 views

### Infinitesimal variation of spectrum of Schrödinger operator with changing domain

Suppose we have a Schrödinger operator
$$-\frac{d^2}{dx^2}+V(x)$$
defined on $[a,b]$ with Dirichlet boundary conditions. I am interested in whether there are any results for the variation of the ...

**4**

votes

**0**answers

106 views

### power laws emerging from the sandpile model

Is there a rigorous proof that the abelian sandpile model generates a power law distribution of avalanche lengths?

**4**

votes

**0**answers

97 views

### Ground State Degeneracy of 2+1D U(1) Chern Simons Theory?

I am a physics graduate student trying to understand more mathematical aspects of gauge theories.
How can I understand ground state degeneracy of a simple Chern Simons Theory: 2+1D U(1) $S= \int_M ...

**4**

votes

**0**answers

105 views

### First return time in an interval for N particles rotating on the circle at constant random speeds

Here is my problem: draw N velocities $v_1,v_2,\dots,v_n$ in $[-\pi,\pi]^N$ from some measure (Haar measure of uniform independent for simplicity) and make $N$ particles rotate around the circle with ...

**4**

votes

**0**answers

289 views

### The dogma of the natural numbers in physics

As is well known "God made the natural numbers; all else is the work of man" (Leopold Kronecker). However, "what would correspond more to the spirit of physics would be a mathematical theory of the ...