15
votes
7answers
2k views

What are some Applications of Teichmüller Theory?

I'm trying to collect some specific examples of applications of Teichmüller Theory. Here are some things I have collected thus far: No-wandering-domain Theorem (Sullivan) Theorems of Thurston ...
3
votes
0answers
73 views

Poincaré-Birkhoff-Witt theorem for Leibniz algebras

Leibniz algebras can be seen as a non-skew-symmetric generalization of Lie algebras. I have already taken a look at some papers related to Leibniz algebras and extending main results of Lie algebras ...
2
votes
1answer
224 views

Proving the Eichler-Shimura Isomorphism defines a global section

Let $ f \in S_k(\Gamma)$ be a weight k modular cusp form of level $\Gamma$, with modular curve $Y_{\Gamma}$. Let $V^{k-2}$ be the homogenous polynomials in X and Y of degree k-2 with complex ...
10
votes
0answers
300 views

Between Being a Connection and Being an Elliptic Operator

Let $E$ be a smooth complex vector bundle over a smooth compact manifold $M$ and let $H$ be the $Z_{2}$-graded (with the $Z_{2}$-grading given by even/odd forms) Hilbert space of $L_{2}$ (with respect ...
8
votes
0answers
108 views

Is there an efficient algorithm for testing isomorphism of projective planes?

Isomorphism testing is a core problem in computational complexity. Recently, Babai has shown that Graph Isomorphism problem for general graphs can be solved in quasipolynomial time. Long time before ...
1
vote
0answers
62 views

Projective tensor product continuous?

For $V$, $W$ Banach spaces, is the canonical morphism $B(V) \times_\pi B(W) \to B(V \hat{\otimes}_\pi W)$ continuous, where $B(V)$ denotes the set of all linear bounded endomorphisms with operator ...
3
votes
2answers
336 views

The number of subgroups of ${\frak S}_n$

Because of my interest in this question, I listed the subgroups of ${\frak S}_n$ for $1\le n\le4$. I found that the number of subgroups are, respectively, $1,2,6,24$. It might be a coincidence, or it ...
116
votes
16answers
16k views

When should a supervisor be a co-author?

What are people's views on this? To be specific: suppose a PhD student has produced a piece of original mathematical research. Suppose that student's supervisor suggested the problem, and gave a few ...
22
votes
24answers
7k views

Is there an image for you that epitomizes mathematics? [closed]

Can you think of an image, whether technical or nontechnical, available for viewing online that says a lot about what you think mathematics or a particular field of mathematics is all about? For ...
2
votes
2answers
5k views

Count of full, binary trees with fixed number of leaves

How many ways is there to build an arithmetic expression with fixed number of terms and fixed order? Let’s assume we have only one distinct operation that is neither commutative nor associative. The ...
4
votes
3answers
498 views

Bounding the minimal maximum norm of a solution of a linear system.

I would be grateful for pointing me out a reference to some general bound on the $\ell_{\infty}$ norm of a solution of a linear system. To be specific, suppose that we have an underdetermined linear ...
108
votes
59answers
20k views

Jokes in the sense of Littlewood: examples? [closed]

First, let me make it clear that I do not mean jokes of the "abelian grape" variety. I take my cue from the following passage in A Mathematician's Miscellany by J.E. Littlewood (Methuen 1953, p. 79): ...
1
vote
0answers
203 views

Existence of Rational curve

Let $X$ be a compact Kähler manifold such that $K_X$ is nef. When there is no rational curve on $X$?
-3
votes
0answers
28 views

Probability problem [closed]

In an urn there are 8 blue and 5 red balls. Consequently, we are taking 1 ball without returning it, until all of one of the colors are taken out. Let X is the amount of balls. Find the type of ...
16
votes
1answer
405 views

What is the homomorphism between the third exterior and third symmetric power of the adjoint representation of a simple Lie algebra?

Let $\mathfrak{g}$ be the adjoint representation of a simple Lie algebra (which is not of type $A$). Then the space of intertwiners between the third exterior power of $\mathfrak{g}$ and the third ...
3
votes
1answer
213 views

Confusion surrounding the Koszul-Malgrange theorem

I recently had the need to appeal to some complex geometry in my research and have been trying to unravel the various relationships surrounding the Koszul-Malgrange theorem. According to nlab, the ...
27
votes
22answers
18k views

What are the worst notations, in your opinion ? [closed]

With which notation do you feel uncomfortable ?
3
votes
1answer
78 views

ideal generated from a truncated “real radical-like” set are still real radical?

Suppose $\mathbb{R}[X]$ is the normal multivariate polynomial ring where $X = x_1...x_n$. $\mathbb{R}[X]_t$ is the truncated ring such that $\mathbb{R}[X]_t =\left\{f: f \in \mathbb{R}[X], deg(f) \leq ...
5
votes
1answer
102 views

General properties of the Ruelle operator

Recently I have read Parry and Pollicott's book, Zeta functions and the periodic orbit structure of hyperbolic dynamics. I have been interested in some technical properties of the ...
73
votes
5answers
27k views

Consequences of the Riemann hypothesis

I assume a number of results have been proven conditionally on the Riemann hypothesis, of course in number theory and maybe in other fields. What are the most relevant you know? It would also be nice ...
19
votes
3answers
2k views

Half Cantor-Bernstein Without Choice

I had a discussion with one of my teachers the other day, which boiled to the following question: Assume ZF. Let $A,B$ be sets such that there exist $f\colon A\to B$ which is injective and ...
3
votes
1answer
156 views

Counting lattice points inside an ellipsoid subject to congruence conditions

Let $A,B,C$ be positive integers, and let $Z$ be a positive parameter. Let $M$ be a positive integer, and consider the set of points $$\displaystyle \{(x,y,z) \in \mathbb{Z}^3 : Ax^2 + By^2 + Cz^2 ...
11
votes
4answers
3k views

Pi1-sentence independent of ZF, ZF+Con(ZF), ZF+Con(ZF)+Con(ZF+Con(ZF)), etc.?

Let ZF1 = ZF, ZFk+1 = ZF + the assumption that ZF1,...,ZFk are consistent, ZFω = ZF + the assumption that ZFk is consistent for every positive integer k, ... and similarly define ZFα ...
114
votes
83answers
26k views

Examples of great mathematical writing

This question is basically from Ravi Vakil's web page, but modified for Math Overflow. How do I write mathematics well? Learning by example is more helpful than being told what to do, so let's try to ...
6
votes
2answers
847 views

On the pathwise uniqueness of solutions of SDEs(Stochastic Differential Equations)

Suppose that $(\Omega,\mathscr{F},P)$ is a complete probability space equipped a filtration $\{\mathscr{F}_t\}$ satisfying the usual conditions. $B_t$ is a 1-dimentional Brownian motion with respect ...
1
vote
0answers
31 views

Covariance of order statistics [closed]

I'm a researcher in social science and I have encountered the following math formulation of a problem in my field. Note that I have also posted on math.stackoverflow, but given that this seems to be ...
7
votes
0answers
409 views

Use of derivators to the theory of motives?

This is a rather imprecise question but i think this could become a interesting pool of ideas and comments. The theory of motives has evolved to a complex field of research the moment Voevodsky (and ...
71
votes
11answers
14k views

“Philosophical” meaning of the Yoneda Lemma

The Yoneda Lemma is a simple result of category theory, and its proof is very straightforward. Yet I feel like I do not truly understand what it is about; I have seen a few comments here mentioning ...
46
votes
1answer
2k views

A function whose fixed points are the primes

If $a(n) = (\text{largest proper divisor of } n)$, let $f:\mathbb{N} \setminus \{ 0,1\} \to \mathbb{N}$ be defined by $f(n) = n+a(n)-1$. For instance, $f(100)=100+50-1=149$. Clearly the fixed points ...
36
votes
9answers
4k views

What proportion of chess positions that one can set up on the board, using a legal collection of pieces, can actually arise in a legal chess game?

Many chess positions that one may easily set up on a chess board are impossible to achieve in a game of legal moves. For example, among the impossible situations would be: A position in which both ...
0
votes
0answers
65 views

Finding the number of integer points inside a sphere of radius R and dimension D centered at the Origin [on hold]

I am writing a computer program to count the number of integer points inside a sphere of radius R and Dimension D centered at the origin. In essence, if we have a sphere of dimension 2 (circle) and ...
97
votes
16answers
24k views

How do you keep your research notes organized? [closed]

One of the things I struggle with most in doing research is keeping my notes organized. Since research tends to do a lot of branching, keeping notes in a linear fashion seems useless to me. On the ...
8
votes
4answers
1k views

Grothendieck Topologies versus Pretopologies

The wikipedia article(s) as well as the nlab article(s) about Grothendieck topologies and Grothendieck pretopologies are careful to differentiate the two very emphatically and to point out that ...
22
votes
4answers
1k views

How is tropicalization like taking the classical limit?

There is a folk — I can't call it a theorem — "fact" that the mathematical relationship between Complex and Tropical geometry is analogous to the physical relationship between Quantum and ...
151
votes
3answers
7k views

A Game on Noetherian Rings

A friend suggested the following combinatorial game. At any time, the state of the game is a (commutative) Noetherian ring $\neq 0$. On a player's turn, that player chooses a nonzero non-unit element ...
4
votes
2answers
191 views

Reference for when a metric on a four-manifold is Kahler?

In a paper of Derdzinski (Proposition 5), he proved that if $\delta W^+=0$ and $W^+$ has at most two distinct eigenvalues, then the metric is (locally) conformally Kahler, and if in addition the ...
10
votes
2answers
348 views

Cohomology of $T^n/W$ for compact Lie groups

Let $G$ be a compact, connected and simply connected Lie group. Let $T\subset G$ be a maximal torus and let $W$ be the corresponding Weyl group. Then we have the diagonal action of $W$ on $T^{n}$ ...
15
votes
5answers
3k views

Factors of p-1 when p is prime.

Hi, I have no idea where to look for, so I'm hoping you can give me some pointers. I'm interested by numbers of form $p-1$ when $p$ is a prime number. Do they have a name, so that I can google them? ...
7
votes
3answers
698 views

Characterizing forcings that don't add any dominating reals

Regarding reals as functions from $\omega$ to $\omega$, let's say a real $f$ eventually dominates $g$ iff $(\exists n)(\forall m > n)[ f(m) > g(m)]$. Let's say that a (non-trivial separative) ...
9
votes
3answers
1k views

Why can projective varieties just have abelian group operations?

I just started to read Shimura - Automorphic forms and number theory (Lecture notes in mathematics, 54). On page 20 or so, he mentions that every projective variety which is an algebraic group, is ...
3
votes
1answer
465 views

Cohomology groups of quotient by finite group

I know there are already lots of questions about (co)homology groups of a quotient manifold, but please let me ask one more question. Let $G$ be a finite group acting on a manifold $M$ without fixed ...
-5
votes
0answers
71 views

Can any polynomial expressed as a chromatic polynomial? [on hold]

Is it possible any polynomial with integer coefficients to be (or converted to) a chromatic polynomial that corresponds to a graph?
4
votes
1answer
560 views

What do we mean by contractible for simplicial objects in a category?

EDIT: removed cruft from this question. Recall that extra degeneracies for an augmented simplicial set $X$ are maps $s_0\colon X_n \to X_{n+1}$ for $n=-1,0,1,2,\ldots$ which satisfy the usual ...
5
votes
2answers
866 views

intersection of finitely many maximal ideals

For what commutative rings with infinitely many maximal ideals we can say that the intersection of any combination of finitely many maximal ideals is not zero? Obviously it holds for Dedekind domains ...
11
votes
5answers
1k views

Alternative Arithmetics

Although, beyond any doubts, $ZFC$ is by and large the predominantly accepted theory of sets, there have been a few attempt to establish some serious competitors in town. I just quote two of them ...
14
votes
18answers
10k views

Good books on theory of distributions

Hi all. I'm looking for english books with a good coverage of distribution theory. I'm a fan of Folland's Real analysis, but it only gives elementary notions on distributions. Thanks in advance.
15
votes
4answers
2k views

Who thought that the Alexander polynomial was the only knot invariant of its kind?

I apologize that this is vague, but I'm trying to understand a little bit of the historical context in which the zoo of quantum invariants emerged. For some reason, I have in my head the folklore: ...
16
votes
9answers
10k views

What is the best algorithm to find the smallest nonzero Eigenvalue of a symmetric matrix?

see title. An algorithm is 'good' if it is able to distinguish between zero Eigenvalues and nonzero Eigenvalues.
3
votes
0answers
240 views

Beating Kadane's Algorithm

I am seeking some reference on already existing work for the following problem. Given an $n$-dimensional square matrix $A=DP$ where $D$ is a diagonal and $P$ is a permutation matrix (think of Gaussian ...
10
votes
2answers
475 views

Helix translates as geodesics

I believe one can fill $\mathbb{R}^3$ with horizontal translates of the helix $(\cos t, \sin t, t) \;,\; t \in \mathbb{R}$, so that every point of $\mathbb{R}^3$ lies in exactly one helix. I am ...

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