# All Questions

100,960 questions

**26**

votes

**2**answers

898 views

### Are the inner automorphisms the only ones that extend to every overgroup?

Let H be a group. Can we find an automorphism $\phi :H\rightarrow H$ which is not an inner automorphism, so that given any inclusion of groups $i:H\rightarrow G$ there is an automorphism $\Phi: G\...

**7**

votes

**1**answer

283 views

### Universal property for collection of epimorphisms

Question Is there a nice universal property which captures the notion of "collection of all epimorphisms out of a given object". Of course I will have to consider two epimorphisms $X \rightarrow Y$ ...

**21**

votes

**1**answer

759 views

### Do DG-algebras have any sensible notion of integral closure?

Suppose R → S is a map of commutative differential graded algebras over a field of characteristic zero. Under what conditions can we say that there is a factorization R → R' → S ...

**10**

votes

**5**answers

6k views

### Applied mathematics Books (graduate level)

What are some good graduate level books on applied mathematics which explain in-depth the general modern problem-solving methods of the real-world typical hard problems?
There is a lot of books on ...

**8**

votes

**2**answers

429 views

### Are bicategories of lax functors also bicategories of of pseudofunctors?

Let A be a bicategory. Can I construct in some "natural" way a bicategory L(A) such that for every bicategory B, the bicategory of lax functors Lax(A, B) is (bi)equivalent to the bicategory of ...

**2**

votes

**3**answers

873 views

### How can I measure the Morse index in infinite dimensions?

Let $V$ be a vector space over $\mathbb R$, and $a: V\otimes V\to \mathbb R$ a symmetric bilinear pairing. Recall that the Morse index of $a$ is the maximal dimension of any subspace $V_- \subseteq V$...

**9**

votes

**5**answers

2k views

### Geodesics on a Grassmannian

Where can I find the most direct and simplest presentation of what geodesics on a (complex) Grassmannian look like? I know how to do it from scratch, but, if I want to provide a reference to, say, a ...

**2**

votes

**2**answers

842 views

### Periodic mapping classes of the genus two orientable surface

Please, any information on the periodic mapping classes of the genus two orientable surface, $O_2$, will be greatly thanked. We had been studying the topological structure of 3d surface bundles and ...

**7**

votes

**1**answer

247 views

### How to see meromorphicity of a function locally?

Given a germ of an analytic function on a (compact, for simplicity) Riemann surface, how can one see (locally) whether this is a "germ of meromorphic function"? I.e. if I do analytic continuation ...

**12**

votes

**2**answers

2k views

### The work of Thurston

I seem to remember written or said somewhere that at some point Thurston decided to stop writing down his theorems in order not to repel mathematicians from his field (maybe this is not correct?). I ...

**29**

votes

**3**answers

1k views

### Diameter of m-fold cover

Let $M$ be a closed Riemannian manifold.
Assume $\tilde M$ is a connected Riemannian $m$-fold cover of $M$.
Is it true that
$$\mathop{diam}\tilde M\le m\cdot \mathop{diam} M\ ?\ \ \ \ \ \ \ (*)$$
...

**21**

votes

**9**answers

2k views

### What methods exist to prove that a finitely presented group is finite?

Suppose I have a finitely presented group (or a family of finitely presented groups with some integer parameters), and I'd like to know if the group is finite. What methods exist to find this out? I ...

**14**

votes

**3**answers

3k views

### What is “restriction of scalars” for a torus?

I am starting on a Phd program and am supposed to read Colliot Thelene and Sansuc's article
on R-equivalence for tori. I find it very difficult and although I have some knowledge over schemes , I am ...

**21**

votes

**12**answers

3k views

### Statements in group theory which imply deep results in number theory

Can we name some examples of theorems in group theory which imply (in a relatively straight-forward way) interesting theorems or phenomena in number theory?
Here are two examples I thought of:
The ...

**2**

votes

**4**answers

2k views

### Splitting a space into positive and negative parts

Let $V$ be a vector space over $\mathbb R$. A symmetric bilinear pairing on $V$ is a linear map $a: V\otimes V \to \mathbb R$. Because $\mathbb R$ is characteristic not-two, I will freely confuse ...

**16**

votes

**7**answers

3k views

### Why is Riemann-Roch an Index Problem?

I was in a lecture not long ago given by C. Teleman and at some point he said "Well, since Riemann-Roch is an index problem we can do..."
Then right after that he argued in favour of such a sentence. ...

**10**

votes

**4**answers

1k views

### The category of finite locally-free commutative group schemes

I'm trying to understand the properties of the category $FL/S$ of finite locally-free commutative group schemes over an arbitrary base-scheme $S$. I know it is not in general an abelian category: Over ...

**35**

votes

**6**answers

2k views

### Clifford algebra as an adjunction?

Background
For definiteness (even though this is a categorical question!) let's agree that a vector space is a finite-dimensional real vector space and that an associative algebra is a finite-...

**7**

votes

**1**answer

395 views

### S-unit equation and small sets of places

Let $K$ be a number field, and let $S_x$ denote the set of primes of norm at most $x$. Is it possible to find a smaller set of places $T_x\subset S_x$ so that a lot of the solutions of the $S_x$-unit ...

**7**

votes

**3**answers

1k views

### Free subquotient of Galois representations coming from Hida theory

Let $\mathbf{T}$ be the reduced nearly ordinary Hecke algebra of level $N$ of Hida theory for $\operatorname{GL}_{2}$ over $\mathbb{Q}$ (or more generally over a totally real field $F$). Then $\mathbf{...

**2**

votes

**1**answer

346 views

### Parity, Balls and Boxes

Start with a distribution $\mu$ on [n], and drop m balls into these n+1 slots independently and according to the distribution &mu. That is, we have iid random variables x 1 through x m ...

**6**

votes

**2**answers

764 views

### What is the geometric meaning of reconstruction of quantum group via Ringel Hall algebra

If I remembered correctly. There are some work done by C.M.Ringel,he defined so called Ringel-Hall algebra on abelian category and then show that Ringel-hall algebra is isomorphic to positive part of ...

**17**

votes

**2**answers

2k views

### Lax Functors and Equivalence of Bicategories?

Lax functors of bicategories were introduced at the very inception of bicategories, and I'm trying to get a better feel for them. They are the same as ordinary 2-functors, but you only require the ...

**2**

votes

**6**answers

2k views

### How do we study the theory of reductive groups? [closed]

I am interested in the theory of reductive groups which is useful in the theory of automorphic forms. But the trouble boring me so long time is that I don't know the appropriate material for beginners ...

**61**

votes

**9**answers

12k views

### Why does the Gamma-function complete the Riemann Zeta function?

Defining $$\xi(s) := \pi^{-s/2}\ \Gamma\left(\frac{s}{2}\right)\ \zeta(s)$$ yields $\xi(s) = \xi(1 - s)$ (where $\zeta$ is the Riemann Zeta function).
Is there any conceptual explanation - or ...

**16**

votes

**4**answers

2k views

### Trace map attached to a finite homomorphism of noetherian rings

Let $f:A\rightarrow B$ be a homomorphism of noetherian rings which
makes $B$ into a finite $A$-module. Under what conditions on $f$, $A$,
$B$ can one associate to this map a canonical "trace map"
$$\...

**34**

votes

**21**answers

4k views

### Generalizations of Planar Graphs

This is a follow up to Harrison's question: why planar graphs are so exceptional. I would like to ask about (and collect answers to) various notions, in graph theory and beyond graph theory (topology; ...

**3**

votes

**2**answers

2k views

### Break Polyhedron into Tetrahedron

Given a polyhedron consists of a list of vertices (v), a list of edges (e), and a list of surfaces connecting those edges (...

**4**

votes

**2**answers

589 views

### Decomposition of Hölder continuous functions

Let $\alpha\in(0,1)$ and $\eta\in\Lambda_0^\alpha(\mathbb{R})$ be a compactly supported Hölder continuous function of order $\alpha$. I would like to show that, for any $n\in\mathbb{N}$, it is ...

**7**

votes

**3**answers

713 views

### Tropicalizing the learning

Could someone tell me if it is possible to do tropical geometry with NO knowledge(or with very few) of algebraic geometry (a la Hartshorne)?
By "do tropical geometry" I mean, to understand the ...

**19**

votes

**3**answers

3k views

### Twin Prime Conjecture Reference

I'm looking for a reference which has the first statement of the twin prime conjecture. According to wikipedia, nova, and several other quasi-reputable resources it is Euclid who first stated it, but ...

**14**

votes

**10**answers

5k views

### Set theory and alternative foundations

Every foundational system for mathematics I have ever read about has been a set theory, from ETCS to ZFC to NF. Are there any proposals for a foundational system which is not, in any sense, a set ...

**13**

votes

**2**answers

2k views

### Are the field norm and trace the unique “nice” maps between fields?

This seems like an obvious fact, but I'm not sure what the necessary meaning of "nice" is to get a result like this. I'm wondering if there is a theorem of the form:
For any <1> field extension $...

**8**

votes

**2**answers

853 views

### Hypercohomology of a dg-algebra

Can someone give me a reference (note I am looking for a reference and not a proof) for the following:
If a complex $C$ has a dg-algebra structure, then the hypercohomology
$H^0R\pi_*C$ has an ...

**0**

votes

**1**answer

918 views

### The Discrete Logarithm problem [closed]

I am puzzled with the following discrete logarithm problem:
Given positive integers b, c, m where (b < m) is True it is to ...

**12**

votes

**4**answers

957 views

### Mappings of mapping class groups

Let $X$ be a compact non-orientable surface, maybe with boundary, and let $\tilde X$ be the orienting cover of $X$. If I understand correctly, any smooth automorphism of $X$ lifts naturally to an ...

**29**

votes

**5**answers

1k views

### When are some products of gamma functions algebraic numbers?

I want to know when certain expressions of the form
$ {\Gamma(r_1/m) \Gamma(r_2/m) \ldots \Gamma(r_j/m) \over \Gamma(s_1/m) \Gamma(s_2/m) \ldots \Gamma(s_j/m)} $
are algebraic numbers. These ...

**7**

votes

**5**answers

2k views

### Indexing the line bundles over a Grassmannian.

As is well known, the line bundles over *CP*$^1$ are indexed by the integers. My question is how are the line bundles over *CP*$^n$, $n > 1$, and *Gr*$(n,k)$ indexed? Moreover, do there exist any ...

**16**

votes

**2**answers

2k views

### Topologically contractible algebraic varieties

From a post to The Jouanolou trick:
Are all topologically trivial (contractible) complex algebraic varieties necessarily affine? Are there examples of those not birationally equivalent to an affine ...

**4**

votes

**3**answers

536 views

### When is $A : C(X) \to C(Y)$ a composition operator?

A composition operator $C\_T : C(X) \to C(Y)$ with $T \in C(Y, X)$ is defined by $C\_T f := f \circ T, f \in C(X)$.
I read in the book about Composition Operators by Singh and others that a ...

**21**

votes

**9**answers

2k views

### When does the zeta function take on integer values?

Here $\zeta(s)$ is the usual Riemann zeta function, defined as $\sum_{n=1}^\infty n^{-s}$ for $\Re(s)>1$.
Let $A_n=${$s\;:\;\zeta(s)=n$}. The behaviour of $A_0$ is basically just the Riemann ...

**4**

votes

**1**answer

404 views

### Are any finitely generated reflexive module a 2nd syzygy?

Are any finitely generated reflexive module a second syzygy?
(I´m thinking especially in normal noetherian domains)
More general...
Are any divisorial lattice a second syzygy?
(I´m thinking ...

**118**

votes

**30**answers

54k views

### What are the most misleading alternate definitions in taught mathematics?

I suppose this question can be interpreted in two ways. It is often the case that two or more equivalent (but not necessarily semantically equivalent) definitions of the same idea/object are used in ...

**2**

votes

**1**answer

208 views

### Classifying strata for the adjoint representation of GL from first principles

How would one classify the strata for the standard nilpotent cone for $GL_{k}(\mathbb{C})$, using the definition from Hesselink's paper "Desingularizations of Varieties of Nullforms"? I know that they ...

**3**

votes

**1**answer

435 views

### When can one localize Ext?

Let $R\to S$ be a ring map such that $S$ is projective over $R$ (I am willing to assume $S=R[X_1,...,X_n]$). Let $M,N$ be finite $S$-modules. Let $P\in Spec R$ such that $M_P$ is $R_P$-flat. Under ...

**6**

votes

**2**answers

523 views

### Naive Z/2-spectrum structure on E smash E?

Let $E$ be a spectrum. Then $E \wedge E$ is a $\mathbb{Z}/2$-spectrum in the naivest possible sense, i.e., an object with $\mathbb{Z}/2$-action in the (∞,1)-category of spectra. Can I make it ...

**7**

votes

**1**answer

2k views

### finite surjective l.c.i morphism is flat

Let $X,Y$ be locally Noetherian schemes. Let $f:X\to Y$ be a finite, surjective, and locally complete intersection morphism, i.e., locally it can be decomposed as regular immersion followed by a ...

**14**

votes

**2**answers

536 views

### When is a commutative ring the limit of its local rings?

Let $A$ be a commutative ring. Then we get local rings $A_p$ by localizing at each prime ideal $p$. Moreover, we get $A_p \rightarrow A_q$ when $p$ contains $q$. So we get a big diagram indexed by the ...

**6**

votes

**1**answer

533 views

### Some examples of depth

This is related to the question I asked last time. This sounds a bit too specific, I hope this question is still acceptable on MO.
I am still not quite comfortable with the concept of depth, and ...

**15**

votes

**7**answers

846 views

### Extremal question on matrices

The following question was posed to me a while ago. No one I know has a given a satisfactory (or even a complete) proof:
Suppose that $M$ is an $n$ x $n$ matrix of non-negative integers. Additionally,...