All Questions

26
votes
2answers
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
1answer
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
1answer
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
5answers
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
2answers
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
3answers
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
5answers
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
2answers
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
1answer
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
2answers
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
3answers
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
9answers
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
3answers
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
12answers
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
4answers
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
7answers
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
4answers
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
6answers
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
1answer
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
3answers
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
1answer
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
2answers
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
2answers
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
6answers
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
9answers
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
4answers
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
21answers
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
2answers
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
2answers
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
3answers
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
3answers
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
10answers
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
2answers
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
2answers
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
1answer
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
4answers
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
5answers
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
5answers
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
2answers
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
3answers
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
9answers
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
1answer
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
30answers
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
1answer
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
1answer
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
2answers
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
1answer
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
2answers
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
1answer
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
7answers
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,...

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