Questions designed to get an overview of a specific subject or body of results or to understand the relations among similar definitions, techniques or concepts appearing in different sub-fields of mathematics. While such questions by their very nature sometimes cannot be made very narrow and ...

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29
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0answers
3k views

Grothendieck's manuscript on topology

Edit: Just in case anyone still thinks that Grothendieck's unpublished manuscripts are (by his letter) entirely out of sight: Declared as "national treasure", they seem to be in principle accessible ...
19
votes
0answers
1k views

more on “Transalgebraic Theories” (a 19th century yoga)?

Among the talks at occasion of the Galois Bicentennial, one is about "Transalgebraic Theories". Unfortunately I found only this article describing that fascinating idea as " an extremely powerful ...
13
votes
0answers
606 views

What is the appropriate setting for Cauchy's Integral Formula?

For a $C^1$ function $f:U\to\mathbb{C}$, where $D\subset U\subseteq \mathbb{C}$ with piecewise $C^1$ boundary $\partial D$, we have the following generalized Cauchy integral formula: $$ f(\zeta) = ...
12
votes
0answers
151 views

What do tangles teach us about braids?

A braid is a smooth level-preserving embedding $f\colon\, \{1,2,\ldots,n\}\times[0,1]\hookrightarrow \mathbb{R}^2 \times [0,1]$ such that $f(k,0)=(k,0)$ and $f(k,1) \in \{1,2,\ldots,n\} \times ...
12
votes
0answers
194 views

Is there a common framework for Tannaka and Gabriel-Ulmer reconstruction theorems?

Gabriel-Ulmer duality is a biequivalence between the 2-category of finite limit categories and the 2-category of locally finitely presentable categories. It allows for the reconstruction of a theory ...
11
votes
0answers
326 views

3D generalizations of permutations, RSK correspondence, contingency tables, etc.

I want to gather facts and questions related to 3D generalizations of permutations, RSK correspondence, contingency tables, etc. One reason I am interested in this is because it is potentially related ...
11
votes
0answers
1k views

Idea of presheaf cohomology vs. sheaf cohomology

Let $X$ be a topological space and $U$ an open cover of $X$. In this thread Angelo explained beautifully how presheaf cohomology (Cech cohomology) relates to sheaf cohomology: The zeroth Cech ...
10
votes
0answers
388 views

How come Cartan did not notice the close relationship between symmetric spaces and isoparametric hypersurfaces?

Elie Cartan made fundamental contributions to the theory of Lie groups and their geometrical applications. Among those, we can list the introduction of the remarkable family of Riemannian symmetric ...
8
votes
0answers
402 views

What is the origin of the formula for the Lie derivative along a Killing vector?

Background Let $(M,g)$ be an $n$-dimensioal riemannian manifold. A vector field $X$ on $M$ is said to be a Killing vector if the flow it generates is an isometry; that is, it preserves the metric ...
7
votes
0answers
272 views

Canonical Time Evolution for Type $II_{1}$-Factors?

This question was spurred by the answer of Steve Huntsman to the MO question here. The Tomita-Takesaki modular automorphism group gives rise to a canonical time evolution on a type $III$ factor ...
6
votes
0answers
373 views

Does mathematical fecundity ever deviate from its applicability?

We are all familiar with Wigner's "unreasonable effectiveness of mathematics" thesis (1), and of Hardy's opinion that "the great bulk of higher mathematics is useless" (2). I am wondering if there are ...
6
votes
0answers
466 views

Counting Lattice Points in Real Polytopes

Suppose one did have an exact formula for the number of $\mathbb{Z}^n$-lattice points intersecting an arbitrary dilate of a (not necessarily rational) finite, closed and convex $n$-polytope. As a ...
5
votes
0answers
67 views

Classifying two-faces of four-polytopes

Motivation: This question is related to my study of hyperbolic Coxeter polytopes. In general, if one put some restrictions on the type of their dihedral angles (say, all dihedral angles are equal to ...
4
votes
0answers
271 views

Q-construction and Gabriel-Zisman Localization

It might be a stupid question. When I took a look at the definition of Q-construction. It makes for an exact category $P$, one defines a new category $QP$ whose objects are the same as $P$ but ...
3
votes
0answers
465 views

Differential Galois number theory

Following http://math.stackexchange.com/questions/635659/can-the-error-term-involved-in-the-pnt-be-expressed-in-a-galois-theoretic-framew?noredirect=1#comment1341143_635659, I vainly tried to find ...
3
votes
0answers
194 views

Finite subgroups of the unimodular group

This is related to this MO question (and others as well). Hoping that this will not turn out to be too broad, I would like to know about the 'state of the art' of: 1) The problem of classifying ...
3
votes
0answers
204 views

ubiquitous modulicity?

On the one hand, as mentioned here, basically "everything" in algebraic geometry could be seen in the context of "moduli problems" - on the other hand, Grothendieck's few remarks on a possible "tame ...
3
votes
0answers
402 views

Quantized Calculus, the Hilbert Transform and the Upper Half-Plane

Given a *-algebra $\mathcal{A}$ over $\mathbb{C}$, a Fredholm module is a *-representation $\pi$ of $\mathcal{A}$ as operators on a Hilbert space $\mathcal{H}$ along with a self-adjoint operator $F$ ...
2
votes
0answers
239 views

Categories of sheaves and Kan Extensions

This is quite a broad question regarding constructions of categories of sheaves in geometry. Let $\textbf{Sch}$ denote the category of schemes. Let $\textbf{SchAff}$ denote the full subcategory of ...
1
vote
0answers
206 views

Classification of properties of structures

Is there a sensible classification of the properties of structures with a given signature $\sigma$, e.g. graphs with $\sigma = \lbrace R \rbrace$? For example like this: properties defined by ...