**29**

votes

**0**answers

2k views

### The Work of Pierre Deligne

In this biography of Pierre Deligne, there is a quote of Jacques Tits which says that "quite a few of his best ideas have never been written!".
What are some of his best ideas that you have heard of ...

**22**

votes

**0**answers

2k views

### Why “open immersion” rather than “open embedding”?

When topologists speak of an "immersion", they are quite deliberately describing something that is not necessarily an "embedding." But I cannot think of any use of the word "embedding" in algebraic ...

**21**

votes

**0**answers

1k views

### Name of amateur who gave a new proof of the Ramanujan-Nagell theorem?

In an article by George Johnson in the New York Times back in 1999, it says that an amateur mathematician from India once sent Ian Stewart a proof of the Ramanujan-Nagell theorem that the Diophantine ...

**17**

votes

**0**answers

444 views

### Good ways to organize old personal mathematical resources

I am wondering how the other Mathematicians organize their old mathematical resources, like calculation drafts, class and seminar notes etc.
These old resources may be related to a wide range of ...

**16**

votes

**0**answers

960 views

### Origins of the Nerve Theorem

Recently, I've read two papers which have cited the Nerve Theorem, one crediting Borsuk with the result and another Leray. Here is the question:
Who was the first to prove the Nerve Theorem?

**14**

votes

**0**answers

479 views

### Motivation for Hall-Witt identity

I've wondered for a while about the (Hall-)Witt identity in group theory:
$[[a,b^{-1}],c]^b \cdot [[b,c^{-1}],a]^c \cdot [[c,a^{-1}]],b]^a = 1$.
(Here, $x^y$ means $y^{-1}xy$ and $[x,y]$ ...

**11**

votes

**0**answers

193 views

### Eilenberg's rational hiererchy of nonrational automata & languages — where is it now?

In the preface to his very influential books Automata, Languages and Machines (Volumes A, B), Samuel Eilenberg tantalizingly promised Volumes C and D dealing with "a hierarchy (called the rational ...

**9**

votes

**0**answers

333 views

### State of research in moduli space of flat connections

I am a recent PhD student trying to settle into a research topic. Even though I have a current project I am working on, I am not particularly enjoying it and would like to switch. Before braving the ...

**8**

votes

**0**answers

344 views

### How much algebraic geometry do I need to study complex geometry?

As one can deduce from the questions I have asked on MO, I'm interested in complex geometry. I am aware that there are many facets to the field, some of which I am more comfortable with than others. ...

**8**

votes

**0**answers

420 views

### Number Theory over $\mathbb{F}_q [t]$, why is it important/interesting?

This is something I was curious about, but not really sure why. I was hoping maybe someone could give me a good explanation. I understand that there are
similarities between $\mathbb{Z}$ and ...

**8**

votes

**0**answers

142 views

### Literature that helps explain what the theory of numerosities contributes with

Since 2003 a group of Italian mathematicians (Benci, Di Nasso and Forti) has developed a new measure for infinite sets that satisfies the Euclidian principle: The whole is greater than the part. The ...

**8**

votes

**0**answers

377 views

### Composition of two formal series

There are two formal semi-infinite Laurent series
$$
f_+(z)=z+\sum_{k=2}^{\infty} a_k z^k
$$
and
$$
f_-(z)=z+\sum_{k=0}^{\infty} b_k z^{-k}
$$
Their composition (we assume that this composition ...

**8**

votes

**0**answers

691 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

158 views

### Tangent space, metrics etc. on simplicial sets

Is there a way to attach some sensible notion of tangent space to a simplicial set? If yes, is it possible to transfer typical local data from differential geometry such as metrics to this setting?
...

**7**

votes

**0**answers

189 views

### Fixed marginals of joint distribution: status

One of the well-known problems of classical probability theory is the determination of the set of all extreme points in the convex set of all probability distributions in a product Borel space $\left ...

**7**

votes

**0**answers

225 views

### Is it possible to define a perverse $t$-structure for a certain triangulated category of sheaves of spectra?

The perverse t-structure for the derived category of complexes of sheaves is certainly a mighty tool for studying cohomology. My question is: does there exist any homotopy-theoretic analogue for it ...

**6**

votes

**0**answers

185 views

### Why is the Dynamical Mordell-Lang conjecture interesting?

The gist of the Dynamical Mordell-Lang conjecture is:
Let's say you have a set $S$. Inside $S$, you have a point $x$ where you start off at. Now, you also have a function $f: S \rightarrow S$ (is ...

**6**

votes

**0**answers

376 views

### Any reason to believe that $NP \neq P$ is unprovable in ZFC

We know $NP \neq P$ from a lot of point of view like empirical reason,or theoretical reasons such as finite model theory or descriptive complexity.Although we find so many reasons to believe $NP \neq ...

**6**

votes

**0**answers

98 views

### Duality between large and small scale structures

A rather immediate reaction to seeing the definition of a coarse structure, at least to me, is to be reminded of a uniform structure. The axioms for a coarse structure $\mathcal{C}$ (defined by a ...

**6**

votes

**0**answers

213 views

### History of the characterization of commutative Artin rings

When it comes to the world of "classical" (pre-homological) Noetherian commutative algebra, I tend to think of most of the results (Krull's intersection theorem, the principal ideal theorem, etc.) as ...

**6**

votes

**0**answers

277 views

### Where does the term “torsor” come from?

Is there a heuristic reason why pricipal homogeneus spaces of a group (object) $G$ (in some categories) are called $G$-torsors? Does it have anything to do with the idea of "torsion", somehow? When ...

**6**

votes

**0**answers

179 views

### what's the role of $H^{p}(\mathbb{R}^{n})$ in modern (harmonic) analysis?

The classical theory of $H^p$,due to it's heavy dependence on the complex function theory(such as Blaschke products), seemed to have an insurmountable obstacle barrying its extension to several ...

**6**

votes

**0**answers

519 views

### Mathematical etiquette: Rephrasing / restructuring a work, limited release (with attribution) acceptable?

Hello,
I am reading a mathematics textbook (which one is irrelevant, and I do not wish to insult the author if (s)he happens to be reading this). One section relies quite a bit on an appendix and ...

**5**

votes

**0**answers

376 views

### Refereeing a paper containing strong statements about other papers

The title says it all. Suppose you are refereeing a paper where the author A makes strong statements about other papers by a different author B, like: the proof of Theorem 1 in paper [B] is wrong and ...

**5**

votes

**0**answers

945 views

### “Must read ”papers on analytic number theory

Question: What would be some must-read
papers for an aspiring analytic number
theorist? In other words, what are the papers that any analytic number theorist would have read? (Background: ...

**4**

votes

**0**answers

190 views

### Etale Slice Theorem

I found the Luna's Slice Theorem very Technical. It will be helpful if someone illustrates the geometry involved in the theorem. Also why this theorem so useful?
This is Luna's Slice theorem from a ...

**4**

votes

**0**answers

166 views

### Why is the density function of a sum of many iid random variables (i.e., the Gaussian) self-dual?

In the usual proof of the central limit theorem via characteristic functions, we note that if $X_1, \dots, X_n$ are independent and identically distributed, with probability density function $f(x)$, ...

**4**

votes

**0**answers

154 views

### Stable homotopy of spheres non-locally

Are there any results/conjectures about the stable homotopy groups of spheres that relate the picture at different primes? Something like Gauss's reciprocity law in number theory?
I know about the ...

**4**

votes

**0**answers

94 views

### Categorical notions involving $\ell_p$ spaces.

First of all, apologies for a somewhat vague question but let me give a try. We know what the projective objects in the category of Banach spaces are: these are precisely $\ell_1(\Gamma)$-spaces. (One ...

**4**

votes

**0**answers

284 views

### Stability conditions for coherent sheaves and GIT

I am learning stability conditions for derived categories of coherent sheaves, following Bridgeland, and coming from a vector bundles background. $\mu$-stability for vector bundles has a clear GIT ...

**4**

votes

**0**answers

449 views

### How would you cite a result that it is not quite correct, but whose proof contains some useful ideas

There is a paper that was published 15 years ago; one of the theorems in it is wrong in general. A few years ago some people told the author that this theorem is wrong as stated, but yet a partial ...

**4**

votes

**0**answers

731 views

### What did Hilbert do on Hilbert spaces to deserve his name?

This question is just curiosity. When I had my first course in Functional Analysis, most of basic theorems about Banach spaces were presented to me as attributed to Banach (Hahn-Banach, ...

**4**

votes

**0**answers

527 views

### Two different theorems but only one fact?

Let me first state an example:
Let $X$ be the multiplication operator on the polynomials in $x$ defined by $Xf(x)=xf(x)$ and let $D$ be the differentiation operator defined by $Df(x) = f'(x).$
...

**3**

votes

**0**answers

203 views

### A paper by Elashvili (translation request)

I would like to know if there is an English version of a paper by Elashvili called "Centralizers of nilpotent elements in semisimple Lie algebras".
If not, is there atleast an online version of the ...

**3**

votes

**0**answers

187 views

### Is it difficult to prove that nature is chaotic?

If we have a Markov coding or another symbolic description of a dynamical system it is usually easy to prove that the system is chaotic (in the sense of of Li-York, Devaney, positive entropy of what ...

**3**

votes

**0**answers

255 views

### A Question Regarding Boolean-valued Models

What were the intuitions motivating the creation (or discovery, if you will) of Boolean-valued models? I have searched for the Scott-Solovay paper on the subject, but to no avail. There also seems to ...

**3**

votes

**0**answers

239 views

### Seminar Notes Repository

On Seminars, people actually talk a lot more about motivations than when they write in paper. It would be a good idea if there is an online repository where people can upload notes (handwritten, ...

**3**

votes

**0**answers

146 views

### Who defined the Inertia Group $I(M^n)\subset\Theta_n$ of a smooth manifold?

If you're unfamiliar with the definition, for an oriented smooth manifold $M^n$ we define the inertia group $I(M)$ to be the set of (h-coboridsm classes of) homotopy spheres $\Sigma^n$ such that ...

**3**

votes

**0**answers

1k views

### What would an ideal mathematical note-taking/organizer/PIM software look like?

I'm struggling with this problem for a long time, and I'm sure a lot of you out there are having similar problems too: when studying from a e-textbook I read and annotate in one app and make ...

**2**

votes

**0**answers

87 views

### Comprehensive survey on mathematical modelling of neural networks: from the basic ideas to contemporary research topics

I am looking for a comprehensive survey (paper(s) or book(s)) on mathematical modelling of neural networks (both artificial and biological).
It should start from the very basic concepts of modelling ...

**2**

votes

**0**answers

69 views

### Choice of MIP (mixed integer programming) solver

I would start using MIP solver for the research on the tiling.
I know (heard of) the open source solver jump:
https://github.com/JuliaOpt/JuMP.jl
and also the gold standard solver from IBM cplex.
...

**2**

votes

**0**answers

121 views

### characterization of all periodic tiling of a simple set of Wang Tile

Consider a set of Wang Tile such that all the edges are either 1 or 0.... there are 16 elements in such a set.
Now, I wish to characterize all the periodic tilings of this set (better if they are ...

**2**

votes

**0**answers

161 views

### Looking for author of calculus quote

When I was a lowly calculus student many many years ago, my calculus teacher quoted some famous mathemtician: "Calculus is the last course in arithmetic and the first course in mathematics that one ...

**2**

votes

**0**answers

256 views

### Descriptive set theory on $\mathbb{R}^\mathbb{N}$

The short version of my question is, What is a good source for learning about descriptive set theory on the space $\mathbb{R}^\mathbb{N}$, under the product topology coming from the discrete topology ...

**2**

votes

**0**answers

362 views

### From complexity to topology after a CS PhD

Let me start by apologizing for the soft and lengthy nature of the question.
I am a third year graduate student (in India) working in complexity theory. Early this year, I developed an interest for ...

**2**

votes

**0**answers

124 views

### Is there something interesting in the uniqueness condition for a sheaf?

After digesting the Presheaf definition by the very first time, one feels (at least I felt) a strange sensation noticing the existence and uniqueness conditions to graduate that Presheaf as a sheaf, ...

**2**

votes

**0**answers

304 views

### Different approaches to Shimura varieties

I have just started taking a look at some introductory papers about Shimura varieties, after a friend of mine suggested me to do so. They seem to be a sort of very interesting and many-sided topic, ...

**2**

votes

**0**answers

158 views

### Relationship between the notions of “excellent ring” and “universally catenary Nagata ring”

Every excellent ring is both universally catenary and Nagata. How "close" is a universally catenary Nagata ring to being excellent?
Context: I have not worked very much with the notions described ...

**1**

vote

**0**answers

49 views

### Unifying Geometry for Characteristic Classes

When working with characteristic classes (more concretely Chern classes), one finds at least four essentially distinct approaches:
Axiomatic Approach. See, for instance, Vector Bundles and K-Theory, ...

**1**

vote

**0**answers

124 views

### Algebraic properties of the semiring of open subsets.

Does anyone know of a useful general topological application of the algebraic properties of the semiring of open subsets of some topological space? Or examples of any such nontrivial properties at ...