# All Questions

361 views

### Explicit Isomorphism between $Cl(8)$ and $\mathbb{R}(16)$

I am looking for a explicit isomorphism between $Cl(8)$ (Clifford algebra over $\mathbb{R}^8$ with standard Euclidean metric) and $\mathbb{R}(16)$ (algebra of $16\times 16$ matrices over ...
7k views

### Do we still need model categories?

One modern POV on model categories is that they are presentations of $(\infty, 1)$-categories (namely, given a model category, you obtain an $\infty$-category by localizing at the category of weak ...
1k views

### Analogues of Primitive Recursive Functions

Let $\mathbf{A}$ be an admissible set (possibly with urelements). I am wondering if there is some good notion of "primitive recursive arithmetic" relative to $\mathbf{A}$. More precisely, I would like ...
8k views

Some theorems give far more than you feel they ought to: a weak hypothesis is enough to prove a strong result. Of course, there's almost always a lot of machinery hidden below the waterline. Such ...
178 views

### Do the roots of this equation involving two Euler products all reside on the critical line?

This question loosely builds on the second part of this one. Take the Riemann $\xi$-function: $\xi(s) =\frac12 s\,(s-1) \,\pi^{-\frac{s}{2}}\, \Gamma\left(\frac{s}{2}\right)\, \zeta(s)$. Numerical ...
79 views

### Local factors of Tamagawa measure

This is a reference request to some computations which I hope can be found in the literature somewhere. Let $G\subset GL_n$ be a semisimple linear algebraic group over $\mathbb Q$. The Tamagawa ...
10k views

### Is the boundary $\partial S$ analogous to a derivative?

Without prethought, I mentioned in class once that the reason the symbol $\partial$ is used to represent the boundary operator in topology is that its behavior is akin to a derivative. But after ...
5k views

### Compelling evidence that two basepoints are better than one

This question is inspired by an answer of Tim Porter. Ronnie Brown pioneered a framework for homotopy theory in which one may consider multiple basepoints. These ideas are accessibly presented in his ...
315 views

### Q re Kaprekar's fixed mapping points

Jens Kruse Andersen in his comment in OEIS's A099009 noticed 3 families of numbers among Kaprekar's fixed mapping points (otherwise known as kernels of the Kaprekar's routine): "Let $d(n)$ denote ...
1k views

### Generalization of a theorem of Øystein Ore in group theory

Theorem (Øystein Ore, 1938): A finite group $G$ is cyclic iff its lattice of subgroups $\mathcal{L}(G)$ is distributive. Proof: see below. Let $(H \subset G)$ be an inclusion of finite groups and ...
312 views

### Applications of Lubotzky's linearity theorem?

Lubotzky's theorem is a necessary and sufficient set of conditions for a finitely generated discrete group to be linear, i.e. isomorphic to a subgroup of $GL_n(K)$, where $K$ is a field of ...
14k views

### “Philosophical” meaning of the Yoneda Lemma

The Yoneda Lemma is a simple result of category theory, and its proof is very straightforward. Yet I feel like I do not truly understand what it is about; I have seen a few comments here mentioning ...
426 views

### Convergence of random variables with hypergeometric distribution

This is a very interesting conjecture of large scale property of hypergeometric distribution. Let $a>1$ be a integer constant, $N\in\mathbb{N_+}$, for any $x<N-1$, consider $N+(a-1)x$ balls in ...
5k views

### equivalence of Grothendieck-style versus Cech-style sheaf cohomology

Given a topological space $X$, we can define the sheaf cohomology of $X$ in I. the Grothendieck style (as the right derived functor of the global sections functor $\Gamma(X,-)$) or II. the Čech ...
234 views

### Octonions product: inversion in the right and identity in the left

Once octonions product is studied, together with the relations with $Spin(8)$ and $SO(8)$ geometry (see for instance Robert Bryant's notes), one realises that the key fact bringing all the phenomena ...
449 views

### Local Inverse Galois Problem

It's a basic fact that a finite Galois extension $L/K$ of a local nonarchimedian field $K$ has solvable (in fact supersolvable) Galois group $G$. One sees this by using the ramification filtration ...
947 views

### What are the higher homotopy groups of a K3 suface?

All K3 surfaces have the same homotopy type. What are their higher homotopy groups? I know that $\pi_1$ is trivial, and $\pi_2$ is $\mathbb{Z}^{22}$. Even if the answer isn't known in all degrees, ...
64 views

316 views

### Can we construct a Baas-Sullivan presentation of TMF?

Quick Review: The Baas-Sullivan construction cones off generators $\alpha_1, ..., \alpha_n \in \pi_*(MU)$ from $MU$ to get a new spectrum $MU/(\alpha_1, ..., \alpha_n)$, which is isomorphic to some ...
10k views

### Why is it a good idea to study a ring by studying its modules?

This is related to another question of mine. Suppose you met someone who was well-acquainted with the basic properties of rings, but who had never heard of a module. You tell him that modules ...
499 views

### Geometric generic fibre

This is a pretty elementary question about schemes, but it came up in the course of research, so let's try it here rather than MSE. Question 1: Are the fibres of a family of complex varieties ...
4k views

### How should one think about sheafification and the difference between a sheaf and a presheaf

The first time I got in touch with the abstract notion of a sheaf on a topological space $X$, I thought of it as something which assigns to an open set $U$ of $X$ something like the ring of continuous ...
383 views

### Equidimensionality of stalks of $\operatorname{Proj} S$ when $S$ is equidimensional.

I would like to know a reference of the following statement (or counter example). Let $S$ be a (commutative) Noetherian standard graded ring over a local ring, i.e., $S = S_0[S_1]$, where $S_0$ is ...
12k views

### Slick proof?: A vector space has the same dimension as its dual if and only if it is finite dimensional

A very important theorem in linear algebra that is rarely taught is: A vector space has the same dimension as its dual if and only if it is finite dimensional. I have seen a total of one proof ...
4k views

### Is there a preferable convention for defining the wedge product?

There are different conventions for defininig the wedge product $\wedge$. In Kobayashi-Nomizu, there is $\alpha\wedge\beta:=Alt(\alpha\otimes\beta)$, in Spivak, we find ...
2k views

### Mysterious quotes (at least for me)

I heard two quotes, one from Alain Connes and an other one from Orlov. Alain Connes was talking about noncommutative geometry and he said the following: " a noncommutative algebra creates its own ...
5k views

### Riemannian surfaces with an explicit distance function?

I'm looking for explicit examples of Riemannian surfaces (two-dimensional Riemannian manifolds $(M,g)$) for which the distance function d(x,y) can be given explicitly in terms of local coordinates of ...
7k views

### Is there any formal foundation to ultrafinitism?

Ultrafinitism is (I believe) a philosophy of mathematics that is not only constructive, but does not admit the existence of arbitrarily large natural numbers. According to wikipedia, it has been ...
3k views

### Non-enumerative proof that there are many derangements?

Recall that a derangement is a permutation $\pi: \{1,\ldots,n\} \to \{1,\ldots,n\}$ with no fixed points: $\pi(j) \neq j$ for all $j$. A classical application of the inclusion-exclusion principle ...
302 views

### Foliations of Lorentzian manifolds by Spacelike Hypersurfaces

Suppose that $M$ is a Lorentzian manifold (not necessarily satisfying Einstein's equations). What conditions do we need in order to guarantee that $M$ admits a foliation by codimension-$1$ spacelike ...
2k views

### Operations via Morse Theory

I am interested in seeing if and how Morse Theory can "do everything". Some core things are handle decomposition, Bott periodicity, and Euler characteristic. But what do the normal (co)homology ...
4k views

### Basic software libraries for numerical analysis using modern programming languages?

I'm looking for a software library with a scope similar to "numerical recipes", but implemented in a modern programming language. "Modern" in this context means to me: object oriented (not C or ...
8k views

### Changing Careers: Becoming a Professional Mathematician

First, I apologize if this question is too soft or doesn't comport precisely with what is considered a good question, but I know of no other place to ask it. A little background: I obtained an ...
6k views

### japanese/chinese for mathematicians?

I'd like to learn to read math articles in Japanese or Chinese, but I am not interested in learning these languages from usual textbooks. Exist suitable texts, specialized for the needs for reading ...
6k views

### If $2^x$and $3^x$ are integers, must $x$ be as well?

I'm fascinated by this open problem (if it is indeed still that) and every few years I try to check up on its status. Some background: Let $x$ be a positive real number. If $n^x$ is an integer for ...
17k views

### Example of a good Zero Knowledge Proof.

I am working on my zero knowledge proofs and I am looking for a good example of a real world proof of this type. An even better answer would be a Zero Knowledge Proof that shows the statement isn't ...
8k views

### Computer Science for Mathematicians

This is a big-list community question, so I'm sorry in advance if it is deemed too soft but I haven't seen anything similar yet. I've seen computer scienctists post questions looking to learn things ...
16k views

### What is the difference between matrix theory and linear algebra? [closed]

Hi, Currently, I'm taking matrix theory, and our textbook is Strang's Linear Algebra. Besides matrix theory, which all engineers must take, there exists linear algebra I and II for math majors. What ...
19k views

### Good Algebraic Number Theory Books

I have just finished a master's degree in Mathematics and want to learn everything possible about algebraic number fields and especially applications to the generalized Pell equation (my thesis ...
186 views

### Self-adjointness of the components of the magnetic derivative

On $L^{2}(\mathbb{R}^{n})$ define the operator $\Pi_{j} u := (-i\partial/\partial x_{j} - A_{j})u$, where $A_{j} \in L^{2}_{loc}(\mathbb{R}^{n})$ represents the $j$-th component of the magnetic ...
7k views

### What is the symbol of a differential operator?

I find Wikipedia's discussion of symbols of differential operators a bit impenetrable, and Google doesn't seem to turn up useful links, so I'm hoping someone can point me to a more pedantic ...
8k views

### Irrationality of $\pi e, \pi^{\pi}$ and $e^{\pi^2}$

What is known about irrationality of $\pi e$, $\pi^\pi$ and $e^{\pi^2}$?
552 views

### Topological cobordisms between smooth manifolds

Wall has calculated enough about the cobordism ring of oriented smooth manifolds that we know that two oriented smooth manifolds are oriented cobordant if and only if they have the same ...
2k views

I often see in stochastic calculus books the terms 'adapted process' and 'progressively measurable process'. I know there is a small difference between them (every progressively measurable process is ...
1k views

### Function extensionality: does it make a difference? why would one keep it out of the axioms?

Yesterday I was shocked to discover that function extensionality (the statement that if two functions $f$ and $g$ on the same domain satisfy $f\left(x\right) = g\left(x\right)$ for all $x$ in the ...
4k views

### Volumes of n-balls: what is so special about n=5?

I am reposting this question from math.stackexchange where it has not yet generated an answer I had been looking for. The volume of an $n$-dimensional ball of radius $R$ is given by the classical ...
Define a growth function to be a monotone increasing function $F: {\bf N} \to {\bf N}$, thus for instance $n \mapsto n^2$, $n \mapsto 2^n$, $n \mapsto 2^{2^n}$ are examples of growth functions. Let's ...