# Tagged Questions

This tag is used if a reference is needed in a paper or textbook on a specific result.

**37**

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2k views

### What does the theta divisor of a number field know about its arithmetic?

This question is about a remark made by van der Geer and Schoof in their beautiful article "Effectivity of Arakelov divisors and the theta divisor of a number field" (from '98) (link).
Let me first ...

**28**

votes

**0**answers

705 views

### Grothendieck's “List of classes of structures”

In Lawvere's article Comments on the Development of Topos Theory, the author writes:
Similarly, Grothendieck and others unerringly recognized which kinds of mathematical structures are 'preserved ...

**28**

votes

**0**answers

959 views

### Godel on recursion-theoretic hierarchies

At the end of his excellent article, "The Emergence of Descriptive Set Theory" (http://math.bu.edu/people/aki/2.pdf), Kanamori writes:
"Another mathematical eternal return: Toward the end of his ...

**25**

votes

**0**answers

440 views

### “Three great cocycles” in Complex Analysis as cohomology generators

In his lecture notes, C. McMullen discusses "the three great cocycles" in Complex Analysis: the derivative $$f\mapsto\log f',$$ the non-linearity $$f\mapsto (\log f')'dz$$
and the Schwarzian ...

**22**

votes

**0**answers

436 views

### Identities for power series like $\sum_n z^{n^3}$

Probably, one of the first power series that every mathematician encounter is the geometric series
$$\sum_{n=0}^\infty z^n = \frac1{1-z}, \quad z \in \mathbb{C},\; |z| < 1 .$$
Also, a particular ...

**22**

votes

**0**answers

920 views

### The most important facts, modern surveys, and readable introductions to p-adic cohomology theories (crystalline cohomology and the mysterious functor)

I would like to organize a seminar on crystalline cohomology; I dream of understanding the Beilinson's recent paper on the mysterious functor ...

**21**

votes

**0**answers

283 views

### Guess that group via product queries

Suppose someone (person B) knows a finite group $G$ of order $n$.
You (person A) know only the order $n$,
and that $1$ is the name of the identity element.
The group elements are named $1,2,\ldots,n$ ...

**21**

votes

**0**answers

722 views

### categorical characterization of large cardinals

Question 1. Is there a categorical representation of Kunen's inconsistency result?
Question 2. Is there a categorical characterization of very large cardinals (in particular for strong and ...

**21**

votes

**0**answers

2k views

### What are the possible singular fibers of an elliptic fibration over a higher dimensional base?

An elliptic fibration is a proper morphism $Y\rightarrow B$ between varieties such that the fiber over a general point of the base $B$ is a smooth curve of genus one.
It is often required for the ...

**20**

votes

**0**answers

567 views

### Real square roots of symmetric matrices

In joint work with Andreas Fischle (TU Dresden, Germany) and Patrizio Neff (U Essen, Germany) we needed to use the following statement: If $S$ is a real $n\times n$ matrix with $S^2$ symmetric, then ...

**20**

votes

**0**answers

528 views

### Are there “chain complexes” and “homology groups” taking values in pairs of topological spaces?

Throughout this question, notation of the form $(X,A)$ denotes a sufficiently nice pair of topological spaces. I think for most of what I'm saying here, it is enough to assume that the inclusion $A ...

**19**

votes

**0**answers

459 views

### The oriented homeomorphism problem for Haken 3-manifolds

Haken famously described an algorithm to solve the homeomorphism problem for the 3-manifolds that bear his name (fleshed out by many others, including Hemion and Matveev who fixed some gaps). But ...

**19**

votes

**0**answers

561 views

### Characteristic Classes for $E_8$ Bundles

Given a principal $E_8$ bundle $P\rightarrow X$ one can take the
adjoint representation $\rho :E_8\rightarrow SU(\mathbb C^{248})$
and form the associated vector bundle $V=P\times_{\rho}\mathbb
...

**19**

votes

**0**answers

640 views

### Is there a Mathieu groupoid M_31?

I have read something which said that the large amount of common structure between the simple groups $SL(3,3)$ and $M_{11}$ indicated to Conway the possibility that the Mathieu groupoid $M_{13}$ might ...

**18**

votes

**0**answers

555 views

### Reference request: Parallel processor theorem of William Thurston

Sometime in the 1980's or 1990's, Bill Thurston proved a theorem regarding the existence of a universal parallel processing machine, using a certain class for such machines having finite deterministic ...

**18**

votes

**0**answers

531 views

### Erdos-Kac for squarefree numbers

In its usual form, the Erdos-Kac Theorem states that if $f(n) : \mathbb{N} \rightarrow \mathbb{R}$ is a strongly additive function with $|f(p)| \le 1$ for all primes $p$, then
$$\frac{|\{n \le x : ...

**18**

votes

**0**answers

752 views

### local equivalence of loop group representations

Let $G$ be a compact, simple, connected, simply connected (cscsc) Lie group, and let its smooth loop group $LG:=C^\infty(S^1,G)$. Given an interval $I\subset S^1$, we have the local loop group
$$
L_IG ...

**16**

votes

**0**answers

254 views

+500

### Is this a model for $K$-theory of a triangulated category?

The recent question Complete the following sequence: point, triangle, octahedron, . . . in a dg-category reminded me of something I wanted to clarify long time ago; most likely this is now well known ...

**16**

votes

**0**answers

527 views

### How boundedly generated is $SL_3(\mathbb{Z})$?

The group $G = \mathrm{SL}_3(\mathbb{Z})$ is known to be boundedly generated, that is, there exists some $m \in \mathbb{N}$, and $g_1, \dots, g_m \in G$ such that we have the following equality of ...

**16**

votes

**0**answers

1k views

### Two conjectures by Gabber on Brauer and Picard groups

In a paper I need to make reference to 2 conjectures by Gabber
(see Conjectures 2 and 3, page 1975)
http://www.mfo.de/programme/schedule/2004/32/OWR_2004_37.pdf
1) Let $R$ be a strictly henselian ...

**15**

votes

**0**answers

730 views

### Derived functors - homotopical vs homological approach

This question is a crosspost of the second part of this MSE question.
In my first course in homological algebra, derived functors were defined in terms of universal $\delta$-functors. In the text ...

**15**

votes

**0**answers

280 views

### Partitions of ${\rm Sym}(\mathbb{N})$ induced by convergent, but not absolutely convergent series

Let $(a_n) \subset \mathbb{R}$ be a sequence such that the series
$\sum_{n=1}^\infty a_n$ converges, but does not converge absolutely.
Then there is a partition of the symmetric group ${\rm ...

**15**

votes

**0**answers

340 views

### Quasi-classical limit of representation theory

I am looking for a good reference on a general phenomenon of quasi-classical limit in representation theory, which relates "large" representations to measures on (co-adjoint orbits of) the associated ...

**14**

votes

**0**answers

375 views

### Chasing a 1950s thesis from the University of Dhaka on block designs

On behalf of a friend I am searching for a thesis on block designs from the 1950s. The details are below.
Author: Qazi Motahar Husein (Sometimes Husain or Hussein).
Title of the Thesis: Symmetrical ...

**14**

votes

**0**answers

457 views

### Can there be arbitrarily many cubic fields unramified outside $\{p,\infty\}$?

Observe, trivially, that since quadratic fields correspond to rational integers modulo squares (viz. discriminants), there are (roughly about, but certainly at most) $2^{|S|+1}$ quadratic fields ...

**14**

votes

**0**answers

334 views

### Connes' idea to use the hyperfinite $III_1$ factor for the archimedian place of Spec(Z)?

I recently gave a talk, where I talked about the tensor category
of (all, not just finite index) bimodules over the hyperfinite $III_1$ factor.
Vaughan Jones, who was in the audience, later told me ...

**14**

votes

**0**answers

1k views

### An open problem in convex geometry

Is it possible to find four norms $\| \cdot\|_k$ $( 1 \leq k \leq 4)$ on the plane such that a three-dimensional normed space containing four subspaces isometric to these normed planes does not exist? ...

**14**

votes

**0**answers

333 views

### Axiomatizations of the real exponential field

According to Marker's "Model Theory: An Introduction", the real exponential field has a $\forall\exists$ axiomatization (because it is model complete) but no-one has any idea what such an ...

**14**

votes

**0**answers

627 views

### Characterization of Fréchet-Urysohn spaces using sequential continuity at a point

A map $f \colon X \to Y$ is called sequentially continuous at the point $a$ if for every sequence $(x_n)$ such that $x_n\to a$, we also have $f(x_n)\to f(a)$.
$$x_n\to a \qquad \Rightarrow \qquad ...

**14**

votes

**0**answers

1k views

### Grothendieck 's question - any update?

This question is migrated from math.stackexchange. I ask because it is still unclear to me and I did not receive an answer.
I was reading Barry Mazur's biography and come across this part:
...

**14**

votes

**0**answers

403 views

### Any references on zeta-function like sums of inverse determinants over lattices of matrices?

I'm sorry for the title, it was little difficult to phrase..
Let us consider a matrix lattice $L\subset M_n(\mathbb{C})$. By this I mean a discrete additive group in $M_n(\mathbb{C})$. Let us ...

**13**

votes

**0**answers

492 views

### Arguments against Freiling's argument against Continuum Hypothesis

Freiling's axiom of symmetry ($\sf AS$) is known as a justification for falsity of Continuum Hypothesis. Freiling in his 1986 paper, Axioms of symmetry: throwing darts at the real number line, ...

**13**

votes

**0**answers

355 views

### Research situation in the field of Information Geometry

I am now doing an article survey on the field of information geometry started by S.Amari and Barndorff-Nielson. I want to know some research situation in this field.
I have read (4) and parts of (3). ...

**13**

votes

**0**answers

309 views

### Grothendieck on polyhedra over finite fields

In Grothendieck's Sketch of a Programme he spends a few pages discussing polyhedra over arbitrary rings and concludes with some intriguing remarks on specializing polyhedra over their "most singular ...

**13**

votes

**0**answers

380 views

### Is every projective space curve a set-theoretic intersection of two surfaces? What is known about this question?

I am sorry if this question has been already asked, couldn't find anything similar myself. I have recently recalled this long standing open problem of whether every irreducible curve in ...

**13**

votes

**0**answers

334 views

### Expanding disks lead to what packing of the plane?

Suppose one sprinkles points uniformly at random on the infinite Euclidean plane,
with some density $\rho$ per unit area.
View the points as disks of radius zero.
Now the radii $r$ of all disks grows ...

**13**

votes

**0**answers

309 views

### Relation Between Truncated Braid Groups and Regular Tilings of the Complex and Hyperbolic Plane

This is perhaps a vague question, but hopefully there exists literature on the subject. The question is motivated by an answer I gave to this question on math.SE.
There exists a rather remarkable ...

**13**

votes

**0**answers

342 views

### For how many primes does an elliptic curve over a totally imaginary field have supersingular reduction?

An elliptic curve over a finite field, $k$, of characteristic p is called supersingular if it has no $p$-torsion over $k^{\mathrm{alg}}$, or equivalently, if $\mathrm{End}(E)$ is an order in a ...

**13**

votes

**0**answers

648 views

### How much has been written down about Deligne's geometric approach to the order formula for a finite group of Lie type?

This is a follow-up to a recent mathoverflow question
34387
about computing the orders of finite unitary groups and the comments made there.
Between 1955 (Chevalley's Tohoku paper) and 1968 ...

**12**

votes

**0**answers

586 views

### “To operate the machine, it is not necessary to raise the bonnet.”

The quotation in the title is attributed to Frank Adams and appears in several places:
In the preface of [2002, Operads in algebra, topology and physics]: "to operate the machine, it is not ...

**12**

votes

**0**answers

195 views

### Entropy in elimination theory, or a brief remark by Gelfand-Kapranov-Zelevinsky

In the introduction to their book "Discriminants, resultants and multidimensional determinant", the authors state a very intriguing observation concerning the coefficients of monomials appearing in ...

**12**

votes

**0**answers

447 views

### Singular chains generated by manifolds with corners — does it really work?

Usually we define singular homology via the complex of singular chains:
$$C_\ast(X)=\bigoplus_{n\geq 0}\mathbb Z\langle\sigma:\Delta^n\to X\rangle$$
where the right hand side denotes the free abelian ...

**12**

votes

**0**answers

779 views

### Does this metric have an official name? Lévy metric? Ky Fan metric?

Let $X$ and $Y$ be random variables taking values in a separable metric space $(S,d)$. The metric I have in mind is
$$\rho(X,Y) = \mathbb{E}[\min\{d(X,Y),1\}]$$
if $X$ and $Y$ take values in the a ...

**12**

votes

**0**answers

316 views

### Has Cheeger's 'de Rham cohomology' of metric measure spaces been studied beyond its definition?

In J. Cheeger's 'Differentiability of Lipschitz Functions on Metric Measure Spaces' (Geometric and Functional Analysis, 1999, Vol. 9 pp 428-517, see here), a 'de Rham cohomology group' ...

**12**

votes

**0**answers

201 views

### Ising model - phase transition vs rapid mixing

Consider a graph $G=(V,E)$ and Ising model on that graph, i.e. configuration space is $\Omega=${$-1,+1$}$^V$ and energy of a configuration $s \in \Omega$ is given by:
$H(s) = -\beta \sum_{u \sim ...

**12**

votes

**0**answers

413 views

### The derived category of integral representations of a Dynkin quiver.

Let $Q$ be a Dynkin quiver. Let $\mathbb CQ$ be its complex path algebra. It is defined in a way such that modules over $\mathbb CQ$ are the same as representations of the quiver $Q$. Let's write ...

**12**

votes

**0**answers

331 views

### Complex manifold which is algebraic away from codimension \ge 2

If $X$ is a complex manifold and $Z$ is a closed subset of codimension $\ge 2$ such that $X-Z$ has an algebraic structure, then is $X$ algebraic; i.e. a scheme? If not is $X$ an algebraic space?
...

**12**

votes

**0**answers

549 views

### Representations of quantum groups at roots of unity

I'm interested in the semisimplified category of representations of a quantum group at a root of unity. I've heard that simple objects in this category correspond to certain "integral" conjugacy ...

**12**

votes

**0**answers

5k views

### Deligne's letter to Jean-Pierre Serre

I'm looking for another letter of Pierre Deligne, this time to Jean-Pierre Serre (from around 1974 I think), in which he proves that the Galois representation associated to a certain Hecke eigenform ...

**12**

votes

**0**answers

2k views

### Tanh version of a Fourier Transform?

I am trying to perform some computations in an environment where it is much easier to compute the hyperbolic tangent function (tanh) than cosines or sines. This prevents me from performing Fourier ...