**7**

votes

**1**answer

1k views

### How much of modern algebraic geometry is there in modern complex(algebraic, analytic, differential) geometry?

Good day to you, people of mathoverflow. I'll get to the point. I wonder how much of modern abstract algebraic geometry is there in modern complex geometry?
What do I mean by complex geometry? ...

**38**

votes

**0**answers

4k views

### Grothendieck's manuscript on topology

Edit: Infos on the current state by Lieven Le Bruyn: http://www.neverendingbooks.org/grothendiecks-gribouillis
Edit: Just in case anyone still thinks that Grothendieck's unpublished manuscripts are (...

**28**

votes

**0**answers

545 views

### Atiyah-Singer theorem-a big picture

So far I made several attempts to really learn Atiyah-Singer theorem. In order
to really understand this result rather broad background is required: you need
to know analysis (pseudodifferential ...

**13**

votes

**0**answers

206 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 $f(k,1) \in \{1,2,\ldots,n\} \times \{1\}$....

**13**

votes

**0**answers

757 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

**0**answers

578 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 ...

**12**

votes

**0**answers

232 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 ...

**12**

votes

**0**answers

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 ...

**8**

votes

**0**answers

615 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 ...

**8**

votes

**0**answers

497 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 $g$...

**7**

votes

**0**answers

291 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

**0**answers

413 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 ...

**4**

votes

**0**answers

296 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

**0**answers

566 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

**0**answers

340 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 ...

**3**

votes

**0**answers

204 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

**0**answers

215 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

**0**answers

458 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

**0**answers

113 views

### Intuition behind the Duistermaat-Guillemin version of Weyl's law

The theorem in question (see this paper), after a modification by Ivrii (see this paper) states the following:
Let $(M, g)$ be a compact Riemannian manifold of dimension $n \geq 2$. Assume that the ...

**2**

votes

**0**answers

245 views

### Interpretation of Shannon Entropy Application

Consider a collection of increasing positive integers $\{a_i\}_{i=1}^m$ and the distribution $p_i=\frac{a_i}{\sum_{i=1}^ma_i}$. Let entropy of $\mathcal{A}=\{a_i\}_{i=1}^m$ be given
by $$H(\mathcal{A}...

**1**

vote

**0**answers

98 views

### Programmatically computing dual Hopf algebras: state of the art

Given a graded Hopf algebra of finite type, we know the (graded) linear dual is also a graded Hopf algebra. For instance the dual Hopf algebra to the polynomial algebra on an even degree generator, $R[...

**1**

vote

**0**answers

211 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 first-...