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https://mathoverflow.net/questions/19160 | 1 | First of all excuse my ignorance in number theory, the following question might have a well-known solution or it might be an open problem, I just don't know enough in that area of mathematics (and many others).
Let $P\in \mathbb{Z}[X]$ irreducible and of degree at least 1. For $k\in \mathbb{N}, k\geq 2$, denote by $S\_... | https://mathoverflow.net/users/3958 | Finite set of (perfect power) polynomial values? | As Qiaochu said in the comments, you must include Pell type equations as a special case, because they are the only counter example. At least for $k=2$, Siegel's theorem on integral points on algebraic curves implies that if your polynomial $P(x)$ has at least three distinct roots then $P(n)=m^2$ has only finitely many ... | 4 | https://mathoverflow.net/users/2384 | 19165 | 12,761 |
https://mathoverflow.net/questions/19148 | 21 | I always find the strong law of large numbers hard to motivate to students, especially non-mathematicians. The weak law (giving convergence in probability) is so much easier to prove; why is it worth so much trouble to upgrade the conclusion to almost sure convergence?
I think it comes down to not having a good sense... | https://mathoverflow.net/users/4832 | Motivation for strong law of large numbers | [Here](http://terrytao.wordpress.com/2008/06/18/the-strong-law-of-large-numbers/) is a nice post of T. Tao on SLLN. In the comments section he is asked a very similar question to which he answers the following: (I hope it's ok to reproduce it here, since it is buried down in the comments)
>
> Returning specifically... | 19 | https://mathoverflow.net/users/2384 | 19166 | 12,762 |
https://mathoverflow.net/questions/19149 | 6 | Let's say I have a linear regression model of the form $ y = B\_x x + I\_x + \epsilon $, where $B\_x$ is the beta coefficient of the $x$ term, $I\_x$ is the intercept term and $\epsilon$ is additive, normally distributed noise. If I have a dataset and perform linear regression, I get a value for $B\_x$, which indicates... | https://mathoverflow.net/users/4833 | Linear Regression Coefficients W/ X, Y swapped | Well, I think Mike McCoy's answer is "the right answer," but here's another way of thinking about it: the linear regression is looking for an approximation (up to the error $\epsilon$) for $y$ as a function of $x$. That is, we're given a non-noisy $x$ value, and from it we're computing a $y$ value, possibly with some n... | 3 | https://mathoverflow.net/users/4658 | 19171 | 12,764 |
https://mathoverflow.net/questions/19143 | 5 | Understanding adjoints has always been (and continues to be) a bit of a struggle for me.
Today I stumbled upon a property of adjoint functors which seemed extremely intuitive to me. I was wondering why this property isn't mentioned more often in introductory category theory literature, and whether or not it completel... | https://mathoverflow.net/users/2361 | "adjoint" =?= "inverse of composite endofunctor is uniform bi-composition" | (1) Yes. (2) Well, it doesn't give me any additional intuition. You didn't say why it helps you understand, so I can't judge what the advantage of it might be. I think this is really just a complicated way of giving the "bijection of hom-sets" condition.
(3) No, you need something more. For instance, let $r:B\to A$ b... | 4 | https://mathoverflow.net/users/49 | 19176 | 12,768 |
https://mathoverflow.net/questions/19186 | 2 | Dear all
While studying the overlap distribution for two random Cantor sets (long story made short), I came across the following problem.
$G(k)$ is a complex valued function, and satisfy the following condition:
$G(k\mu) = G(k)^2+ \beta$
with $\beta,\mu$ constant (in my case $\beta=\frac{2}{9}, \mu = \frac{4}{3... | https://mathoverflow.net/users/4626 | Finding Functional form for a given Scaling Condition | You do not give any smoothness requirement; I will look for an analytic $G$:
$$ G(k)=\sum\_{n=0}^\infty a\_nk^n.$$
In what follows, I assume also that $\mu=4/3$ and $\beta=2/9$. Expanding in a power series both sides of the equation and equating coefficients, we get that $a\_0=1/3$ or $a\_0=2/3$. In the first case we o... | 3 | https://mathoverflow.net/users/1168 | 19196 | 12,778 |
https://mathoverflow.net/questions/19213 | 4 | Let the ring R be a MU`*`-module via a ring homomorphism φ and suppose it satisfies the condition of the Landweber exact functor theorem such that we obtain a cohomology theory $R^\*(-) := R \otimes\_{MU\_\*} MU^\*(-)$. If ω denotes the complex orientation class in $\widetilde{MU}^2(\mathbb{C}P^\infty)$, then R`*` is o... | https://mathoverflow.net/users/4877 | Changing the orientation of a Landweber exact cohomology theory | The property of being Landweber exact is independent of the orientation. In terms of Landweber's criterion, this is generally phrased as saying that the element vn is invariant modulo the ideal (p,v1,...,vn-1), and so any change-of-orientation (which induces a strict isomorphism on the formal group law) does not change... | 6 | https://mathoverflow.net/users/360 | 19214 | 12,789 |
https://mathoverflow.net/questions/19215 | 7 | Let $(U\_i)\_{i\in I}$ be an open covering of a topological space $X$.
At <http://en.wikipedia.org/wiki/Nerve_of_an_open_covering>,
the nerve of the open covering is defined as follows:
>
> the nerve $N$ is the set of finite subsets of $I$ defined as follows:
>
>
> * the empty set belongs to $N$;
> * a finite s... | https://mathoverflow.net/users/1291 | Ambiguous definition of "nerve of an open covering" on wikipedia? | I think the second construction is not correct. If you replace the cover with the category whose objects are all **intersections of elements** of your original cover, then the two notions agree.
| 5 | https://mathoverflow.net/users/1231 | 19216 | 12,790 |
https://mathoverflow.net/questions/19218 | 9 | The following situation came up in my research:
Suppose two functions $f$ and $g$ map $[0,\infty)$ to (a subset of) itself. The function $f$ is linear and $g$ is quadratic, but $g$ is one-to-one on the interval $[0,\infty)$.
My conjecture/desired property: Any permutation of compositions of these two functions yiel... | https://mathoverflow.net/users/3400 | Uniqueness in Composition of Polynomials | Your special case is right. More generally:
Let $f\left(x\right)=x+b$ with $b\neq 0$.
Let $g\left(x\right)=cx^2+dx+e$ with $c>0$, $d\in\mathbb R$ and $e\in\mathbb R$.
In fact, it is clear that every composition of $f$'s and $g$'s is a polynomial of positive degree and with positive leading coefficient (since $c>0$)... | 7 | https://mathoverflow.net/users/2530 | 19221 | 12,792 |
https://mathoverflow.net/questions/19180 | 7 | Motivation: We have two examples:
(Abelian) Kummer theory (resp. Artin-Schreier theory) has a hidden cohomology theory given by Galois cohomology. The cocycle conditions become clear when you look at the multiplicative (resp. additive) form of Hlbert's theorem 90.
Descent theory for sheaves and stacks: In the case... | https://mathoverflow.net/users/1353 | Does the presence of cocycle conditions indicate the existence of an underlying cohomology theory? | I had lots of thoughts on that kind of question, and feel uneasy to speak as my answer can range from a tautology, through systematic and positive, but somewhat ignorant toward not-well understood cases, to mere impressions and (seeming?) "counterexample" oriented answer. The basic question is what you mean by a cocycl... | 5 | https://mathoverflow.net/users/35833 | 19230 | 12,799 |
https://mathoverflow.net/questions/19174 | 36 | Does the following exist, and if not, does anyone besides me wish it did? A web site where a mathematician (say) could find other mathematicians who want to study the same book or paper, and arrange to meet via videoconference, and run their own informal seminar around that topic, and then disband when they're done.
... | https://mathoverflow.net/users/4837 | Informal online seminars or reading groups via videoconferencing? | I do not see much of a point in seeing people's faces and with full video either resolution is low, or jittering or the badnwidth is huge. So the solution is to have a simultanous voice and shared white board, which should be controlled by the individual elctronic tablet devices (mouse is not good for drawing). The ele... | 13 | https://mathoverflow.net/users/35833 | 19231 | 12,800 |
https://mathoverflow.net/questions/19219 | 7 | When I first starting studying differential geometry, I asked my lecturer a question about smooth manifolds that didn't admit a partition of unity. He promptly told not to worry about such objects as they were only studied by the extremely eccentric. I would like to know if this is true, ie, does anyone study manifolds... | https://mathoverflow.net/users/1977 | Smooth manifolds that don't admit a partition of unity | The answer to your stated question ("Does anyone study non-paracompact manifolds?") is certainly yes. Here are a few papers which do just this:
>
> Gauld, David.
> Manifolds at and beyond the limit of metrisability. (English summary) Proceedings of the Kirbyfest (Berkeley, CA, 1998), 125--133 (electronic),
> Geom... | 13 | https://mathoverflow.net/users/1149 | 19237 | 12,803 |
https://mathoverflow.net/questions/19243 | 5 | My question is motivated by the previous discussion 'Why is a topology made of open sets?'. While the axioms for arbitrary unions and finite intersections are without doubt essential to the concept of a topological space, the 1st axiom (the set itself and the empty set are open) seems rather technical. So, do we really... | https://mathoverflow.net/users/1849 | Do the empty set AND the entire set really need to be open? | Here's a boring reason, and it may or may not convince you: any function $f : X \to Y$ between topological spaces has the property that the preimage of the entire space $Y$ is the entire space $X$, and the preimage of the empty subset of $Y$ is the empty subset of $X$. So if you allow topological spaces in which either... | 19 | https://mathoverflow.net/users/290 | 19248 | 12,811 |
https://mathoverflow.net/questions/19238 | 5 | For some reason my thinking is *very* fuzzy today, so I apologize for the following rather silly question below...
Let $T$ be an ergodic transformation of $(X,\Omega, \mathbb{P})$ and let $X$ be partitioned into $n < \infty$ disjoint sets $R\_j$ of positive measure. For $x \in R\_k$ define $\tau(x) := \inf \{\ell>0:T... | https://mathoverflow.net/users/1847 | Is the average first return time of a partitioned ergodic transformation just the number of elements in the partition? | I found a 2003 paper of Choe containing your sanity check, called "[A universal law of logarithm of the recurrence time](http://iopscience.iop.org/0951-7715/16/3/306?ejredirect=migration)". See the first few lines of section 3 on page 888. The "$K\_n$" used there is essentially your $\tau$, but corresponding to a parti... | 3 | https://mathoverflow.net/users/1119 | 19249 | 12,812 |
https://mathoverflow.net/questions/19264 | 12 | This is related to the previous question of how to define a conductor of an elliptic curve or a Galois representation.
>
> What motivated the use of the word "conductor" in the first place?
>
>
>
A friend of mine once pointed out the amusing idea that one can think of the conductor of an elliptic curves as "so... | https://mathoverflow.net/users/4872 | What is the etymology for the term conductor? | It is a translation from the German Führer (which also is the reason that
in older literature, as well as a fair bit of current literature, the conductor
is denoted as f in various fonts). Originally the term conductor appeared in
complex multiplication and class field theory: the conductor of an abelian extension is... | 24 | https://mathoverflow.net/users/3272 | 19265 | 12,822 |
https://mathoverflow.net/questions/19266 | 1 | Here by $P^n$ I mean $CP^n$, and what I want to do is to calculate the number of global sections of the holomorphic tangent bundle of $CP^n$.
If $n=1$, it is well known that $h^0(P^1, TP^1)=h^o(P^1,\mathcal{O}\_{P^1}(2))=3$.
If $n>1$, I did some calculation in local coordinates, and find out that
$h^0(P^n, TP^n) ... | https://mathoverflow.net/users/3569 | Computing the dimension of the module of global holomorphic vector fields for complex projective n-space | The dimension of $H^0(\mathbb P^n,T\mathbb P^n)$ is $(n+1)^2-1$ and
$h^1(\mathbb P^n, T \mathbb P^n)=0$. Using the Euler sequence (see for instance
Griffiths-Harris, Principles of Algebraic Geometry) you can reduce the computation of these guys to the computation of
the comology of $\mathcal O\_{\mathbb P^n}$ and $\ma... | 6 | https://mathoverflow.net/users/605 | 19268 | 12,823 |
https://mathoverflow.net/questions/19258 | 5 | Hi people,
I'm interested in results, such as the Gauß-Bonnet theorem, Fàry-Milnor theorem or classification theorems for manifolds, which give topological properties from geometric considerations. Can anyone recommend some good texts? In particular I'd like to see a nice proof of Fàry-Milnor and of the theorem of tu... | https://mathoverflow.net/users/4890 | Topological results from geometry | About the Fary–Milnor theorem. Milnor's original proof is already very nice (see [here](http://www.jstor.org/stable/1969467)). I also very much like [this proof](http://www.jstor.org/stable/119165) by Alexander & Bishop (see also a version of this proof in [my book](http://www.math.ucla.edu/~pak/book.htm)).
| 5 | https://mathoverflow.net/users/4040 | 19271 | 12,825 |
https://mathoverflow.net/questions/19240 | 48 | **4-colour Theorem.** Every planar graph is 4-colourable.
This theorem of course has a well-known history. It was first proven by Appel and Haken in 1976, but their proof was met with skepticism because it heavily relied on the use of computers. The situation was partially remedied 20 years later, when Robertson, San... | https://mathoverflow.net/users/2233 | Algebraic proof of 4-colour theorem? | There is a classical approach by Birkhoff and Lewis, which remained dormant for decades. It was recently revived by Cautis and Jackson (start [here](https://core.ac.uk/download/pdf/82080353.pdf) [“The matrix of chromatic joins and the Temperley-Lieb algebra”, *J. Combin. Theory* **89** (2003), 109–155] and proceed [her... | 16 | https://mathoverflow.net/users/4040 | 19274 | 12,827 |
https://mathoverflow.net/questions/19276 | 4 | Let $X$ be a finite CW complex. Is there an integer $N,$ such that $H^i(X,F)=0$ for all $i>N$ and all abelian sheaves $F$ on $X?$ The cohomology is defined to be the derived functor of the global section.
| https://mathoverflow.net/users/370 | singular cohomological dimension | You can take $N=\dim X$, according to Proposition 3.1.5 in Dimca, [Alexandru. Sheaves in topology. Universitext. Springer-Verlag, Berlin, 2004. xvi+236 pp. [MR2050072](http://www.ams.org/mathscinet-getitem?mr=MR2050072)]
| 4 | https://mathoverflow.net/users/1409 | 19278 | 12,830 |
https://mathoverflow.net/questions/19193 | 6 | I have been perusing Harthorne for some time, and I noticed something: it is well known that the class group on $\mathbb{P}^n\_k$ is $\mathbb{Z}$. But as I look at Harthorne's proof it seems to me that it works in much greater generality. Namely if I consider any projective scheme $X=\operatorname{Proj}(A)$, where $A$ ... | https://mathoverflow.net/users/4863 | Divisors on Proj(UFD) | Well, if you read on to Chapter 2, exercise 6.3, then it is stated that:
$$Cl(A) \cong Cl(X)/\mathbb Z[H]$$
here $[H]$ represents the hyperplane section. So the answer is yes.
There is a less well-known but very nice generalization. Suppose that $X$ is smooth. Let $R=A\_m$ be the local ring of A at the irrelevant ide... | 3 | https://mathoverflow.net/users/2083 | 19283 | 12,834 |
https://mathoverflow.net/questions/19285 | 30 | Let $X$ be a set and let ${\mathcal N}$ be a collection of nets on $X.$
I've been told by several different people that ${\mathcal N}$ is the collection of convergent nets on $X$ with respect to some topology if and only if it satisfies some axioms. I've also been told these axioms are not very pretty.
Once or twic... | https://mathoverflow.net/users/4783 | How do you axiomatize topology via nets? | Yes. This is given in Kelley's *General Topology*. (Kelley was one of the main mathematicians who developed the theory of nets so that it would be useful in topology generally rather than just certain applications in analysis.)
In the section "Convergence Classes" at the end of Chapter 2 of his book, Kelley lists the... | 30 | https://mathoverflow.net/users/1149 | 19288 | 12,837 |
https://mathoverflow.net/questions/19255 | 5 | I've been running into the following type of partition problem.
>
> Given positive integers *h*, *r*, *k*, and a real number ε ∈ (0,1), find *n* such that if every (unordered) *r*-tuple from an *n* element set *X* is assigned a set of at least ε*k* 'valid' colors out of a total of *k* possible colors, then you can ... | https://mathoverflow.net/users/2000 | Bounds on a partition theorem with ambivalent colors | I do not think that the lower bound could depend only on epsilon. Below is the sketch of my argument.
Fix h=3, r=2, eps=1/4, thus we color the edges of a graph, each with 25% of all the colors and we are looking for a "monochromatic" triangle. Let us take k random bipartitions of the vertices and color the correspond... | 4 | https://mathoverflow.net/users/955 | 19289 | 12,838 |
https://mathoverflow.net/questions/19303 | 7 | What's a good example to illustrate the fact that a function all of whose partial derivatives exist may not be continuous?
| https://mathoverflow.net/users/2612 | Good example of a non-continuous function all of whose partial derivatives exist | The standard example I have seen is: $f(x,y)=\frac{2xy}{x^2+y^2}$.
| 10 | https://mathoverflow.net/users/4500 | 19304 | 12,850 |
https://mathoverflow.net/questions/19309 | 10 | I am looking for a reference for the following fact:
The orthogonal group $O\_{2n}$ over an algebraically closed field of characteristic 2
has exactly two connected components.
To be more precise, let $O\_q$ denote the orthogonal group of the quadratic form $q(x)=x\_1 x\_2 +x\_3 x\_4+\cdots +x\_{2n-1}x\_{2n}$
over an... | https://mathoverflow.net/users/4149 | Connected components of the orthogonal group O(2n) in characteristic 2. | Presumably this is treated in detail in chapter 7 of the book
The Classical Groups and K-Theory, by A.J.Hahn and O.T.O'Meara.
On page 424 it says in theorem 7.2.23 that the elementary subgroup has index two.
And elementary matrices are in the connected component of 1.
Wilberd
| 8 | https://mathoverflow.net/users/4794 | 19327 | 12,862 |
https://mathoverflow.net/questions/19269 | 42 | What are the examples of some mathematicians coming very close to a very promising theory or a correct proof of a big conjecture but not making or missing the last step?
| https://mathoverflow.net/users/1851 | What are some examples of narrowly missed discoveries in the history of mathematics? | Freeman Dyson discusses a few examples of this in his article [*Missed Opportunities*](http://www.ams.org/bull/1972-78-05/S0002-9904-1972-12971-9/S0002-9904-1972-12971-9.pdf). One that I thought was particularly striking was that mathematicians could have discovered special relativity decades before Einstein just by st... | 35 | https://mathoverflow.net/users/290 | 19332 | 12,866 |
https://mathoverflow.net/questions/19348 | 15 | Space forms are complete (connected) Riemannian manifolds of constant sectional curvature.
These fall into three classes: Euclidean, with universal covering isometric to $\mathbb{R}^n$,
spherical, with universal covering isometric to $S^n$, and hyperbolic, with universal covering isometric to $\mathbb{H}^n$.
Does t... | https://mathoverflow.net/users/3304 | Fundamental group of a compact space form. | The fundamental group of a compact hyperbolic space form has exponential growth, according to a well-known theorem of Milnor [Milnor, J. A note on curvature and fundamental group. J. Differential Geometry 2 1968 1--7. [MR0232311](http://www.ams.org/mathscinet-getitem?mr=MR0232311)]. Bieberbach groups are, on the other ... | 15 | https://mathoverflow.net/users/1409 | 19352 | 12,879 |
https://mathoverflow.net/questions/19345 | 3 | Let $X$ be a scheme. Suppose you are given a sheaf of Abelian groups $\mathcal{A}$ over $X$. How can you determine if $\mathcal{A}$ is the underlying Abelian sheaf of a sheaf of $O\_X$-modules? In other words, is it possible to characterize in some (interesting) way the essential image of the forgetful functor from $Mo... | https://mathoverflow.net/users/4721 | How to characterize Abelian sheaves that are quasi-coherent? | 1) There is a very simple example that shows that it is impossible to answer the question of whether $\mathcal{A}$ comes from a quasi-coherent sheaf $\mathcal{F}$ on $X$ if all one is given is the underlying topological space $|X|$ and $\mathcal{A}$ as a sheaf on $|X|$. Namely, if $|X|$ is a point and $\mathcal{A}$ is ... | 13 | https://mathoverflow.net/users/2757 | 19361 | 12,885 |
https://mathoverflow.net/questions/19377 | 5 | It might be well-known (and sorry if it is), but a quick search did not return the answer.
Consider prime numbers $p \neq q$.
Are $\displaystyle \frac{p^q-1}{p-1}$ and $\displaystyle \frac{q^p-1}{q-1}$ relatively prime?
| https://mathoverflow.net/users/3958 | Prime numbers that lead to relatively prime | The answer is no. As the Wikipedia article in my comment states, the counterexample $p = 17, q = 3313$ was found by Stephens in 1971, but the stronger question of whether one can ever divide the other is a famous open problem because its solution would greatly simplify a step in the proof of the Feit-Thompson theorem.
... | 18 | https://mathoverflow.net/users/290 | 19387 | 12,903 |
https://mathoverflow.net/questions/19388 | 5 | I've always thought of the degree of a subvariety of projective space as the degree of the divisor that defines the (given) embedding into projective space. It's been pointed out to me that this works only for curves. Now I'm confused: is there a similar characterization of the degree of a general subvariety of some pr... | https://mathoverflow.net/users/3238 | Degrees of subvarieties of projective space | If $X\subset \mathbb P^n$ is a subvariety of dimension $m$ embedded by a linear system $V \subset H^0(X,\mathcal O\_X(D))$ then the degree of $X$ is equal to $D^m$.
| 7 | https://mathoverflow.net/users/605 | 19389 | 12,904 |
https://mathoverflow.net/questions/19390 | 41 | The modular curve $X\_0(N)$ has good reduction at all primes $p$ not dividing $N$. At such a prime, the Eichler-Shimura relation expresses the Hecke operator $T\_p$ (as an element of the ring of correspondences on the points of $X\_0(N)$ in $\overline{ \mathbb{F}\_p }$) in terms of the geometric Frobenius map. This is ... | https://mathoverflow.net/users/290 | Intuition behind the Eichler-Shimura relation? | (1) Short answer to first question: $T\_p$ is about $p$-isogenies, and in char. $p$ there is a canonical $p$-isogeny, namely Frobenius.
Details:
The Hecke correspondence $T\_p$ has the following definition, in modular terms:
Let $(E,C)$ be a point of $X\_0(N)$, i.e. a modular curve together with a cyclic subgroup
o... | 54 | https://mathoverflow.net/users/2874 | 19399 | 12,909 |
https://mathoverflow.net/questions/19414 | 19 | Let $S$ be an uncountable set. Does there exist a probability measure which is defined on *all* subsets of $S$, with $P({x}) = 0$ for any element $x$ of S ?
If I remove the condition $P({x}) = 0$, then I can trivially get a measure defined on all subsets as follows:
Fix some $a \in S$. For any subset $U \subset S$, d... | https://mathoverflow.net/users/4279 | Existence of probability measure defined on all subsets | The existence of such a measure is equiconsistent to the existence of a [measurable cardinal](https://en.wikipedia.org/wiki/Measurable_cardinal), one of the large cardinal notions, and if ZFC is consistent, cannot be proved in ZFC. (See the notion of real-valued measurable cardinal on the Wikipedia page.)
| 17 | https://mathoverflow.net/users/1946 | 19415 | 12,916 |
https://mathoverflow.net/questions/19363 | 8 | Background
----------
---
Let $X$ and $S$ be simplicial sets, i.e. presheaves on $\Delta$, the so-called *topologist's simplex category*, which is the category of finite nonempty ordinals with morphisms given by order preserving maps.
How can we derive the structure of the face and degeneracy maps of the join f... | https://mathoverflow.net/users/1353 | The Join of Simplicial Sets ~Finale~ | (A note: I am going to regard simplicial sets as also defined on the empty ordinal as well, with $X(\emptyset) = \*$, which is required for the join formula. This is implicit in your first definition and will remove the need for two extra cases for $d\_i$ at the end.)
Regarding the "minor" question. The short explana... | 12 | https://mathoverflow.net/users/360 | 19419 | 12,919 |
https://mathoverflow.net/questions/19420 | 133 | It's "well-known" that the 19th century [Italian school of algebraic geometry](https://en.wikipedia.org/wiki/Italian_school_of_algebraic_geometry) made great progress but also started to flounder due to lack of rigour, possibly in part due to the fact that foundations (comm alg etc) were only just being laid, and possi... | https://mathoverflow.net/users/1384 | what mistakes did the Italian algebraic geometers actually make? | As for a result that was not simply incorrectly proved, but actually false, there is the case of the [Severi bound](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=MR&pg8=ET&r=1&review_format=html&s4=severi&s5=massimo&s6=&s7=&s8=All&vfpref=html&y... | 74 | https://mathoverflow.net/users/4344 | 19428 | 12,923 |
https://mathoverflow.net/questions/19339 | 29 | I think I am confused about some terminology in algebraic geometry, specifically the meaning of the term "torsor". Suppose that I fix a scheme S. I want to work with torsors over S. Let $\mu$ be a sheaf of abelian groups over S. Then my understanding is that a $\mu$-torsor, what ever that is, should be classified by th... | https://mathoverflow.net/users/184 | Torsors in Algebraic Geometry? | As remarked by Brian Conrad above, there is an excellent explanation of all this in Milne's book *Étale cohomology*, Section III.4. There wouldn't be much point in reproducing the details here, but the main issues are:
* You need to decide whether a torsor is going to be a scheme over *S* which locally looks like a t... | 20 | https://mathoverflow.net/users/3753 | 19432 | 12,924 |
https://mathoverflow.net/questions/19431 | 5 | Take a cusp form $f$ and let $f(q) = q + a\_2q^2 + q\_3q^3 + \ldots$" denote its $q$-expansion (assume that the $a\_k$ are integers, and that $f$ comes from an elliptic curve $E$). Of course the series $f(1) = 1 + a\_2 + a\_3 + \ldots$ diverges, but I wonder whether there is any work on evaluating $f(1)$ via some regul... | https://mathoverflow.net/users/3503 | Values of cusp forms at q = 1 ? | I'll give a slightly uncertain answer, based somewhat on my recollection of conversations with Zagier a month ago about similar questions.
If we were to imitate Euler, we might consider $f(1)$ as
$$f(1) = \sum\_{n \geq 1} a\_n = \sum\_{n \geq 1} a\_n n^{-0} = L(f,0).$$
So the analytic continuation of the L-function s... | 2 | https://mathoverflow.net/users/3545 | 19433 | 12,925 |
https://mathoverflow.net/questions/19406 | 11 | For which $n \times n$ correlation matrix $C$ can one construct Bernoulli random variables $(B\_1, \ldots, B\_n)$ with correlation $C$ ?
Following the approach described in this MO [thread](https://mathoverflow.net/questions/18268/discrete-stochastic-process-exponentially-correlated-bernoulli), one can think of the f... | https://mathoverflow.net/users/1590 | Constructing Bernoulli random variables with prescribed correlation | Here's a pretty general construction. Take unit vectors $v\_1,\dots,v\_n$ in $\mathbb{R}^n$ and let $u$ be a random unit vector (chosen with the uniform probability measure on the unit sphere). Define $B\_i$ to be 1 if the inner product of $u$ and $v\_i$ is positive and -1 otherwise. Then the correlation between $B\_i$... | 10 | https://mathoverflow.net/users/1459 | 19436 | 12,926 |
https://mathoverflow.net/questions/19435 | 7 | Hello
Suppose given a polynomial $P=Q\_1\cdots Q\_k$ of degree $n$, where each $Q\_i$ is irreducible. Suppose also that I know the Galois group $G\_i$ (over the rationals) of each irreducible factor $Q\_i$.
Is there an easy correlation between the Galois group of $P$, and the $G\_i$?
| https://mathoverflow.net/users/416 | Galois group of a product of irreducible polynomials | The Galois group of $P$ will be a subdirect product of
the $G\_i$, that is a subgroup of $G\_1\times\cdots\times G\_k$
projecting surjectively onto each of the $G\_i$.
| 19 | https://mathoverflow.net/users/4213 | 19437 | 12,927 |
https://mathoverflow.net/questions/19413 | 3 | Searching for maths tutors online finds people willing to teach up to A-level. I'm looking for help at a more advanced level.
At the moment I'm trying to teach myself category theory from downloaded lecture notes, but I have my eye on other mathematical fields including having another go at algebraic geometry once my... | https://mathoverflow.net/users/4793 | How should I find a tutor for math-overflow level mathematics? | Some agencies do offer undergrad/postgrad-level tuition - in principle. (I know because I used to be a tutor [for one](http://bluetutors.co.uk/)). Your problem will be finding somebody with the specific knowledge you want. So other peoples' ideas about advertising directly to maths departments will probably be more hel... | 0 | https://mathoverflow.net/users/1256 | 19456 | 12,938 |
https://mathoverflow.net/questions/19453 | 4 | Suppose you have a scheme $X$ that is acted on by a torus $T$. Then the action induces a grading on the functions on $X$ by the character lattice of $T$. So for a fixed character $\lambda$, we can consider $\mathcal{O}\_{X,\lambda}$, the $\lambda$ graded part. Assuming the quotient $X/T$ exists, these graded parts shou... | https://mathoverflow.net/users/788 | Line Bundles on Torus Quotient | (**EDIT**: thought I had added this, guess I was wrong. As Brian points out, you definitely want all your tori to be split, or you won't have enough 1-d representations (for example, $S^1$ has no 1-d real representations); over an algebraically closed field, this is automatic.)
If $X/T$ is actually a nice scheme, and... | 2 | https://mathoverflow.net/users/66 | 19457 | 12,939 |
https://mathoverflow.net/questions/19458 | 10 | Question
--------
Let $G$ be a group, and let $X$ be a $G$-biset that is (weakly) invertible with respect to the contracted product. Is $X$ necessarily a bitorsor?
Background
----------
By $G$-biset, I mean a set equipped with commuting left and right $G$-actions. There is a standard tensor product on the categor... | https://mathoverflow.net/users/396 | Is an invertible biset necessarily a bitorsor? | A torsor is a faithful transitive $G$-set. If the left $G$-action on $X$ is not faithful, the left $G$-action on $X\times\_G Y$ will not be faithful. If the left $G$-action on $Y$ is not transitive, the left $G$-action on $X\times\_G Y$ will not be transitive. By symmetry, it follows that a $G$-biset with a left and ri... | 10 | https://mathoverflow.net/users/250 | 19462 | 12,941 |
https://mathoverflow.net/questions/19392 | 4 | I ran into an expression calculating the expected value of $\exp(i t \sigma)$ where $\sigma$ is the total number of cycles in a uniformly chosen $S\_n$ element. The expression is
$$E\_n (\exp(i t \sigma)) = \Gamma(n + \exp(it)) / (\Gamma(\exp(it)) n!)$$
where $E\_n$ denotes the expectation under the uniform distributio... | https://mathoverflow.net/users/4923 | Generalized binomial coefficients and Gaussian density | Note that $exp(it) = 1 + it - t^2/2 + O(t^3)$ uniformly in $t \in \mathbb{R}$. Thus $n^{exp(it)-1} = exp(it \cdot \log n - \log n \cdot t^{2}/2 + O(t^3 \cdot \log n))$ and also by Taylor's theorem $1/\Gamma(exp(it)) = 1 + O(t)$ when $t$ is small (but in fact also for all real $t \in \mathbb{R}$ by periodicity). Thus $$... | 6 | https://mathoverflow.net/users/3882 | 19464 | 12,942 |
https://mathoverflow.net/questions/19466 | 6 | Let $A$ be your favorite finite dimensional algebra, and $P\_i$ be a sets of representatives for the indecomposible projectives (or [PIMs](http://en.wikipedia.org/wiki/Principal_indecomposable_module), if you like). Then we have the Cartan matrix $C$ of the algebra, whose entries are $\dim Hom(P\_i, P\_j)$. You can thi... | https://mathoverflow.net/users/66 | Does a finite dimensional algebra having a Cartan matrix with determinant 1 imply finite global dimension (possibly with more hypotheses)? | In general, no. See [Burgess, W. D.; Fuller, K. R.; Voss, E. R.; Zimmermann-Huisgen, B. The Cartan matrix as an indicator of finite global dimension for Artinian rings. Proc. Amer. Math. Soc. 95 (1985), no. 2, 157--165. [MR0801315](http://www.ams.org/mathscinet-getitem?mr=MR0801315)]
It does work for artin algebras o... | 7 | https://mathoverflow.net/users/1409 | 19467 | 12,944 |
https://mathoverflow.net/questions/19459 | 32 | This is motivated by pure curiosity (triggered by [this question](https://mathoverflow.net/questions/19402)). A map $f:\mathbb R^n\to\mathbb R^m$ is said to be *Lebesgue-Lebesgue measurable* if the pre-image of any Lebesgue-measurable subset of $\mathbb R^m$ is Lebesgue-measurable in $\mathbb R^n$. This class of maps i... | https://mathoverflow.net/users/4354 | Is every smooth function Lebesgue-Lebesgue measurable? | It seems that your example of bijection that sends one Cantor set with positive measure to another Cantor set with zero measure can be made $C^\infty$.
Am I missing something?
| 20 | https://mathoverflow.net/users/1441 | 19468 | 12,945 |
https://mathoverflow.net/questions/19471 | 22 | Is the sum of two measurable set measurable? I think it is not...
| https://mathoverflow.net/users/4928 | Is the sum of 2 Lebesgue measurable sets measurable? | Evidently, there are [measure zero sets with a non measurable sum](http://www.math.wvu.edu/~kcies/prepF/89A+A/89A+A.pdf). The article begins as follows:
>
>
> >
> > Krzysztof Ciesielski,
> > Hajrudin Fejzi´c, Chris Freiling,
> >
> >
> > **Measure zero sets with non-measurable sum**
> >
> >
> >
> > >
> > ... | 28 | https://mathoverflow.net/users/1946 | 19472 | 12,948 |
https://mathoverflow.net/questions/19116 | 20 | In the category of smooth real manifolds, do all small colimits exist? In other words, is this category small-cocomplete? I can see that computing push-outs in the category of topological spaces of smooth manifolds need not be manifolds, but this is not a proof.
| https://mathoverflow.net/users/4517 | Colimits in the category of smooth manifolds | I'd like to recast Reid's (excellent) answer slightly. The essence of it is the following principle:
>
> To show that a limit or colimit doesn't exist in some category, embed your category in one where limits or colimits do exist and find some diagram in the original category whose colimit in the larger category do... | 17 | https://mathoverflow.net/users/45 | 19473 | 12,949 |
https://mathoverflow.net/questions/19398 | 1 | It is provable that $f\_\lambda\to f\Rightarrow f\_\lambda\*g\to f\*g$ if $g$ has a compact support (shown in my textbook). In my particular case, $g=u(t+\triangle t)-u(t-\triangle t)$. Does for that particular case, $f\_\lambda\*g\to f\*g\Rightarrow f\_\lambda\to f$?
In other words: It is easy to prove that the exis... | https://mathoverflow.net/users/4925 | On the convolution of generalized functions | If I understand correctly what you are asking then the answer is: "No".
Here's where I may be misunderstanding: I assume that $\Delta t$ is fixed. If this is correct, we can argue as follows.
Let me write $r = \Delta t$ since it is fixed and I want to disassociate it from $t$. We consider the operator $A\_r \colon ... | 3 | https://mathoverflow.net/users/45 | 19484 | 12,956 |
https://mathoverflow.net/questions/19475 | 6 | I'm going to postpone the motivation for this question because the question itself involves no complicated maths and may well have a very simple solution so I don't want to put anyone off with high falutin' symbols.
Here's the question. I have a smooth curve $c \colon (0,1) \to \mathbb{R}^2$ which does not intersect ... | https://mathoverflow.net/users/45 | Can I detect the point of impact without looking at it? | Andrew's comments showed me that in my first answer I was misunderstanding several aspects of his question. Since I am still not entirely sure that I am capturing the spirit of the problem, let be begin this answer by stating in my own words (in very dry mathematical terms) what I interpret the question(s) to be, so th... | 4 | https://mathoverflow.net/users/2757 | 19492 | 12,958 |
https://mathoverflow.net/questions/19478 | 23 | Let $K$ and $L$ be two subfields of some field. If a variety is defined over both $K$ and $L$, does it follow that the variety can be defined over their intersection?
| https://mathoverflow.net/users/4948 | Fields of definition of a variety | Yes, if varieties are interpreted as subvarieties closed subschemes of base extensions of a fixed ambient variety scheme (e.g., affine space or projective space).
More precisely, suppose that $k \subseteq F$ are fields and the variety $X$ is an $F$-subvariety a closed subscheme of $\mathbf{P}^n\_F$. Say for a field $... | 20 | https://mathoverflow.net/users/2757 | 19494 | 12,960 |
https://mathoverflow.net/questions/19490 | 37 | I have often heard in the folk-lore that Feynman Path Integral can be used to compute geometric invariants of a space.
Coming from a background of studying Quantum Field Theory from the books like that of Weinberg, I have myself used Feynman Path Integrals to compute scattering of particles.
Earlier I had done co... | https://mathoverflow.net/users/2678 | Doing geometry using Feynman Path Integral? | Try:
Witten, Quantum field theory and the Jones polynomial
Witten, The index of the Dirac operator in loop space
I have found both of these papers quite difficult to understand. I don't know any easier references, and would greatly appreciate it if anybody could suggest some.
Anyway, I guess the basic idea is v... | 18 | https://mathoverflow.net/users/83 | 19498 | 12,962 |
https://mathoverflow.net/questions/19496 | 3 | Hello all, it is well-known by transcendence results or Galois theory that famous geometric problems such as trisecting an angle or "squaring the circle" (i.e. given a disk of radius 1 construct a square of equal area) are not solvable by compasses-and-ruler only
constructions.
On the other side, it is equally well-kn... | https://mathoverflow.net/users/2389 | Approximate solutions for trisecting the angle or squaring the circle | Well, for trisection it's very simple. You could divide angle into $2^n$ parts, then just take $\lfloor\frac{2^n}{3}\rfloor$ parts. Of course it could be made as close to one third as you want, but might be hard to do.
For circling the square - draw the $2^n$-gon, then a rectangle with sides $a\_n \cdot 2^n$ and $R/... | 8 | https://mathoverflow.net/users/1888 | 19500 | 12,964 |
https://mathoverflow.net/questions/19505 | 39 | I have studied differential geometry, and am looking for basic introductory texts on Riemannian geometry. My target is eventually Kähler geometry, but certain topics like geodesics, curvature, connections and transport belong more firmly in Riemmanian geometry.
I am aware of earlier questions that ask for basic texts... | https://mathoverflow.net/users/nan | Introductory text on Riemannian geometry | Personally, for the basics, I can't recommend John M. Lee's "Riemannian Manifolds: An Introduction to Curvature" highly enough. If you already know a lot though, then it might be too basic, because it is a genuine 'introduction' (as opposed to some textbooks which just seem to almost randomly put the word on the cover)... | 33 | https://mathoverflow.net/users/4281 | 19508 | 12,970 |
https://mathoverflow.net/questions/19454 | 6 | For $i, j \in \{ 1, \ldots, n \}$, let $X\_{i,j}$ be a real-valued random variable uniformly distributed on the interval $[0,1]$. The $X\_{i,j}$ are independent.
Let $A\_{i,j}$ be the indicator random variable of the event that $X\_{i,j}$ is a local maximum, i. e. it is the largest of the five random variables $X\_{i... | https://mathoverflow.net/users/143 | Limit law for the number of local maxima in a square lattice of IID random variables | There are quite a few extensions of the Central Limit Theorem to dependent random variables whose dependence is controlled. This includes the case of a sequence of sums of identically distributed random variables whose dependency graphs have uniformly bounded degrees. ["On Normal Approximations of Distributions in Term... | 5 | https://mathoverflow.net/users/2954 | 19516 | 12,977 |
https://mathoverflow.net/questions/18797 | 13 | I'm sure this is standard but I don't know where to look. Let $M$ be a contractible compact smooth $n$-manifold with boundary. Does it have to be homeomorphic to $D^n$? What about diffeomorphic?
[UPDATE: the answer is well-known to be negative as many people kindly pointed out. But actually I assume more about the ma... | https://mathoverflow.net/users/4354 | Contractible manifold with boundary - is it a disc? | Given a function $\psi:\mathbb R\to \mathbb R$,
set
$$\Psi=\psi\circ\mathrm{dist}\_ {\partial M},\ \ \ \ \ f=\Psi\cdot(R-\mathrm{dist}\_ p)$$
for some fixed $R>\mathrm{diam}\ M$.
Further,
$$d\,f =
(R-\mathrm{dist}\_ p)\cdot d\,\Psi-\Psi\cdot d\,\mathrm{dist}\_ p$$
Thus, we may choose smooth increasing $\psi$,
suc... | 3 | https://mathoverflow.net/users/1441 | 19522 | 12,981 |
https://mathoverflow.net/questions/19521 | 11 | This question was inspired by
[How to prove that the subrings of the rational numbers are noetherian?](https://mathoverflow.net/questions/19480/how-to-prove-that-the-subrings-of-the-rational-numbers-are-noetherian/19481#19481)
which some people found too routine to be of interest. So I have decided to liven things ... | https://mathoverflow.net/users/1149 | For which fields K is every subring of K…? | Regarding question (c), I can tell you exactly which integral domains have only Noetherian subrings by quoting the aptly titled [*Integral domains with Noetherian subrings*](https://doi.org/10.1007/BF02567320 "Commentarii Mathematici Helvetici 45, 129–134 (1970)") by Robert Gilmer:
If $K$ is the field of fractions an... | 14 | https://mathoverflow.net/users/3143 | 19523 | 12,982 |
https://mathoverflow.net/questions/19527 | 3 | Let *m* the n-dimensional Lebesgue measure on $R^n$.By definition of product measure, on each borelian set E
$m(E)=\inf \left(\sum\_{j=1}^\infty m(R\_j),\:\: E\subseteq \bigcup R\_j , \:\:R\_j \text{ rectangles}\right)$
It is also true that lebesgue measures are regular, so
$m(E)=\inf \left(m(U), E\subseteq U, \: ... | https://mathoverflow.net/users/4928 | Lebesgue measure of a set | It follows from Vitali's covering theorem but not in an entirely trivial
fashion. We can reduce to the case where $E$ is open of finite measure.
The set of all open balls contained in $E$ is then a Vitali cover. By Vitali's
covering theorem there is a sequence of disjoint balls $(B\_n)$
whose union is a subset $U$ of $... | 3 | https://mathoverflow.net/users/4213 | 19532 | 12,985 |
https://mathoverflow.net/questions/17306 | 4 | To say that I am a novice at deformation theory is to grossly overestimate my abilities in this area. I've come across the following theorem in a paper, and I'd like to know how far one is able to generalize it:
### Theorem
Let $R$ be a complete DVR, $X$ a proper smooth curve over $R$, and $D$ a simple divisor on $... | https://mathoverflow.net/users/3238 | Deformations of Tame Coverings | A paper I'm reading now is a PERFECT reference for this: "Deformation of tame admissible covers of curves" by Stefan Wewers is written in an expository style. (corollary 3.1.3 is exactly the theorem stated in the question.)
| 3 | https://mathoverflow.net/users/2665 | 19537 | 12,989 |
https://mathoverflow.net/questions/19530 | 30 | There seems to be some confusion over what the tangent space to a singular point of an orbifold is.
On the one hand there is the obvious notion that smooth structures on orbifolds lift to smooth $G$-invariant structures on $\mathbb R^n$ ($G$ being the finite group so that the orbifold is locally (about some specific ... | https://mathoverflow.net/users/2031 | What is meant by smooth orbifold? | Disclaimer: I don't talk to people about orbifolds. This answer may not represent the opinions of orbifolders.
As I understand it, the orbifold $\mathbb R^n/G$ is characterized by how manifolds map to it,† not by how it maps to manifolds. In particular, the orbifold *is not* determined by the ring of smooth functions... | 27 | https://mathoverflow.net/users/1 | 19542 | 12,994 |
https://mathoverflow.net/questions/19397 | 6 | I often hear mention of two theorems, [Mostow's rigidity theorem](http://en.wikipedia.org/wiki/Mostow_rigidity) and [Liouville's theorem on conformal mappings](http://en.wikipedia.org/wiki/Liouville%2527s_theorem_%2528conformal_mappings%2529), which superficially sound similar: they say that a set of geometric structur... | https://mathoverflow.net/users/2819 | Analogy of Liouville conformal mapping theorem with Mostow rigidity? | There is a connection in some way: if I remember right, you usually prove Mostow rigidity by looking at the hyperbolic space, which is the universal cover of your hyperbolic manifold, then you consider its boundary, which is the flat conformal sphere. Any isometry of the hyperbolic space induces a conformal transformat... | 1 | https://mathoverflow.net/users/4961 | 19544 | 12,996 |
https://mathoverflow.net/questions/19526 | 3 | I ran into a "well-known identity" on page 345 of Shepp and Lloyd's [On ordered cycle lengths in a random permutation](http://www.jstor.org/pss/1994483):
$$\int\_x^{\infty} \frac{\exp(-y)}y dy = \int\_0^x \frac{1-\exp(-y)}y dy - \log x - \gamma, $$
where $\gamma$ is the Euler constant. I am clueless as to how it is de... | https://mathoverflow.net/users/4923 | Reference request for a "well-known identity" in a paper of Shepp and Lloyd | You can apply WZ theory to such identities. In particular, both sides satisfy
$$x\*z''(x) + (x+1)z'(x)$$
Picking $x=1$ as the initial condition (since the DE is regular there, that helps), we see that both sides evaluate to $Ei(1,1)$ and their derivatives both evaluate to $-1/e$, so they are equal.
I got that differe... | 6 | https://mathoverflow.net/users/3993 | 19546 | 12,998 |
https://mathoverflow.net/questions/19547 | 7 | Say we have three infinite sequences $\{a\_i\},\{b\_i\},\{c\_i\}$ of natural numbers, satisfying the equations $$a\_1+b\_1=c\_1,\dots, a\_n+b\_n=c\_n,\dots $$
Assume further that $gcd(a\_i,b\_i,c\_i)=1$ for each $i$ and that $(a\_i,b\_i,c\_i)\neq (a\_j,b\_j,c\_j)$ for all $i,j$. Now let's define $S$ as the set of prime... | https://mathoverflow.net/users/2384 | An S-unit equation, with S an infinite and sparse set of primes. | Yes, in fact, you can make $S$ grow as slowly as you like. This follows, for example, from the fact that there exist 3-term arithmetic progressions of primes $(p,q,r)$ with $\min(p,q,r)$ arbitrarily large. For each such arithmetic progression, you can take $(a\_i,b\_i,c\_i)=(p,r,2q)$. Now just choose these arithmetic p... | 10 | https://mathoverflow.net/users/2757 | 19554 | 13,001 |
https://mathoverflow.net/questions/19485 | 2 | The famous hopf theorem says that a smooth map from a oriented closed dimension $p$ manifold to $S^{p}$ is homotopic if and only if $f$ and $g$ have the same brower degree. To prove the theorem Milnor suggested us three theorems in the book 'topology from the differential view point':
* Theorem A: any two such homoto... | https://mathoverflow.net/users/4437 | a small questions about hopf theorem | I realize that this doesn't answer your question, but there is also an approach using the methods of homotopy theory and CW complexes. If $M$ is a closed smooth orientable $p$-manifold, then $M$ is homeomorphic to a finite CW complex with cells of dimension $\leq p$, and $H^p(M)=\mathbb{Z}$.
We may construct a $K(\m... | 3 | https://mathoverflow.net/users/1345 | 19562 | 13,008 |
https://mathoverflow.net/questions/16128 | 4 | Bayesian probabilities are usually justified by the Cox theorems, that can be written this way:
*Under some technical assumptions (continuity, etc, etc...), given a set $P$ of objects $A, B, C, \ldots$, with a boolean algebra defined over it with operations $A \wedge B$ (and) and $A | B$ (or) such that*:
1) $A \wed... | https://mathoverflow.net/users/757 | Generalized Cox Theorems, valuations on boolean sets, bayesian probabilities and posets | Thanks for noting our work in this area.
This has been worked out in even more detail here:
Lattice duality: The origin of probability and entropy. Neurocomputing. 67C: 245-274. DOI: 10.1016/j.neucom.2004.11.039
<http://knuthlab.rit.albany.edu/papers/knuth-neurocomp-05-published.pdf>
Its fundamental application to ... | 5 | https://mathoverflow.net/users/4963 | 19563 | 13,009 |
https://mathoverflow.net/questions/19568 | 7 | If you get your PhD in math , and then work for 1 or 2 years in a non-academic institution and then turn to apply for postdoc or tenure-track position in math like usual, is there any disadvantage (I mean for your application for postdoc or tenure-track position)?
An appendix: I just want to make sure whether or not ... | https://mathoverflow.net/users/2391 | Is there any disadvantage from non-academic job turn to academic job in math | My personal opinion is that such a career path can contribute a lot to mathematics, because such candidates can often be informed by a more practical or utilitarian focus in their mathematical research, providing an important and invigorating perspective. For example, for someone to arrive at hard-core mathematical res... | 17 | https://mathoverflow.net/users/1946 | 19569 | 13,013 |
https://mathoverflow.net/questions/19548 | 1 | It is common that you have some interesting object (set, group, algebra or something, whatever) which has certain properties, structure etc. You may try describe it in pure algebraic way. Sometimes You cannot find representation different from this you are working on.
How to distinguish its algebraic properties form ... | https://mathoverflow.net/users/3811 | How to distinguish property of particular representation from property of algebraic structure? | I'm not sure I completely understand what you're asking, but here is some information that appears to be relevant.
In the context you're describing, you have two languages: the pure language L0 of groups and the augmented language L1 of groups together with a linear representation over some field (see note). You seem... | 4 | https://mathoverflow.net/users/2000 | 19578 | 13,018 |
https://mathoverflow.net/questions/19574 | 3 | Most people define a function, f(n) on N recursively. I think I can calculate f(n) without dealing with f(n-r) for any 0 < r < n. How do I know that my method isn't still going through the same calculations needed to find f(n-1) (or whatever previous terms are required to find f(n) recursively) -- ?
1. If my method t... | https://mathoverflow.net/users/2907 | Can you tell if you have escaped from a recursive definition? | You inquire about comparing your algorithm to a given recursive algorithm, but the more fundamental question would seem to be how good is your algorithm just by itself?
There are numerous ways to measure the efficacy of a computational algorithm using the ideas of [computational complexity](http://en.wikipedia.org/w... | 8 | https://mathoverflow.net/users/1946 | 19580 | 13,020 |
https://mathoverflow.net/questions/19529 | 7 | Has anybody done computations of such a theory? Is there a place I could look up and see what the answers are for low crossing knots?
| https://mathoverflow.net/users/4304 | Computations of the Link homology categorifying the second colored Jones polynomial | Slava Krushkal and I have an alternative approach set inside of Dror Bar-Natan's universal construction. It should agree with results obtained by Webster and Frenkel, Stroppel Sussan. Computations are reasonable in our setting. We hope to place the paper on the arxiv shortly.
| 3 | https://mathoverflow.net/users/4960 | 19582 | 13,022 |
https://mathoverflow.net/questions/19552 | 6 | Let $A$ be a Noetherian commutative ring and $I$ an ideal in $A$. It is pretty much trivial to see that every free $A/I$-Module is obtained from a free $A$-module by tensoring over $A$ with $A/I$: Just choose preimages for every generator and take the free module generated by them. I'm wondering whether the same remain... | https://mathoverflow.net/users/473 | Does every projective A/I-module come from A? | Thomas already [gave](https://mathoverflow.net/a/19564) an example, but let me make a general point I wish he had said: lots of rings with complicated sets of projectives are quotients of rings with simple sets of projectives. For example, any finitely generated projective module over a polynomial ring in any field is ... | 17 | https://mathoverflow.net/users/66 | 19583 | 13,023 |
https://mathoverflow.net/questions/19571 | 3 | Let $M$ and $N$ be two topological manifolds. Denote by $[M,N]^{\text{diff}}$ and $[M,N]^{\text{cont}}$ the set of homotopy classes of differential and continuous maps respectively. Is it true that $[M,N]^{\text{diff}}=[M,N]^{\text{cont}}$ ? Any reference? Thank you.
**Edit:**
Thanks to the comments below, I should a... | https://mathoverflow.net/users/2348 | Homotopy classes of differential maps VS those of continuous maps | There's no way this can be literally true:
$$[M,N]^{diff} = [M,N]^{cont}$$
Most of the continuous functions from $M$ to $N$ are not differentiable. So there's no way the above equality can be an equality of sets. I think what you want to ask is if the inclusion:
$$[M,N]^{diff} \to [M,N]^{cont}$$
a bijection? Th... | 10 | https://mathoverflow.net/users/1465 | 19592 | 13,027 |
https://mathoverflow.net/questions/19589 | 12 | Background
----------
Inside the Temperley-Lieb algebra $TL\_n$ (with loop value $\delta=-[2]$ and standard generators $e\_1,\ldots,e\_{n-1}$), the Jones-Wenzl idempotent is the unique non-zero element $f^{(n)}$ satisfying
$$ f^{(n)}f^{(n)} = f^{(n)} \quad \textrm{and} \quad e\_i\;f^{(n)} = 0 = f^{(n)}e\_i \quad \tex... | https://mathoverflow.net/users/813 | Do Jones-Wenzl idempotents lift to anything interesting in the Hecke algebra? | Yes. No problem. This is the $q$-analogue of the symmetriser. In terms of $T\_i$ we have
$$ \frac{1}{[n]!}\sum\_{\pi\in S\_n} q^{\ell(\pi)} T\_\pi$$
where for $T\_\pi$ we take a reduced word for $\pi$ and $\ell(\pi)$ is the length of a reduced word.
There is another presentation for the Hecke algebra which I am used ... | 8 | https://mathoverflow.net/users/3992 | 19594 | 13,029 |
https://mathoverflow.net/questions/7089 | 8 | There's an old problem in 4-manifold theory that, as far as I know, doesn't have a name associated with it and really deserves a name.
Let $M$ be a smooth 4-manifold with boundary. Let $S$ be a smoothly embedded 2-dimensional sphere in $\partial M$. Assume $S$ does not bound a ball in $\partial M$, but $S$ is null-h... | https://mathoverflow.net/users/1465 | A problem/conjecture related to 4-manifolds that deserves a name. What name does it deserve? | I think that the conjecture is wrong. The following leads to counterexamples in the topological category and probably also smoothly: Take a closed oriented 4-manifold N with infinite cyclic fundamental group and remove an open neighborhood of a generating circle. Then you get a 4-manifold M with boundary $S^1 \times S^... | 10 | https://mathoverflow.net/users/4625 | 19596 | 13,031 |
https://mathoverflow.net/questions/19584 | 70 | One of the more misleadingly difficult theorems in mathematics is that all finitely generated projective modules over a polynomial ring are free. It involves some of the most basic notions in commutative algebra, and really sounds as though it should be easy (the graded case, for example, is easy), but it's not. The qu... | https://mathoverflow.net/users/66 | What is the insight of Quillen's proof that all projective modules over a polynomial ring are free? | Here is a summary of what I learned from a nice expository account by Eisenbud (written in French), can be found as number 27 [here](https://web.archive.org/web/20120710030443/http://www.msri.org/%7Ede/papers/index.html).1
First, one studies a more general problem: Let $A$ be a Noetherian ring, $M$ a finite presented... | 60 | https://mathoverflow.net/users/2083 | 19603 | 13,037 |
https://mathoverflow.net/questions/19599 | 7 | Regarding the sphere as complex projective line (take $(0,0,1)$ as the infinite point), the Gauss map of a smooth surface in the 3 dimensional space pulls a complex line bundle back on the surface.
My question is, what the bundle is?
(In the trivial case, if the surface is sphere itself, the bundle is just the taut... | https://mathoverflow.net/users/2913 | About the Gauss map of a surface in euclidean 3 space | I assume that your surface is closed. Suppose you have a fixed vector bundle $\xi$ over $S^2$ (no matter which one). You have an oriented surface $M$ embedded in $\mathbb R^3$, which defines the Gauss map $\nu:M\to S^2$, which defines the vector bundle $\nu^\*\xi$ on $M$. You want to know whether $\nu^\*\xi$ depends on... | 10 | https://mathoverflow.net/users/4354 | 19606 | 13,039 |
https://mathoverflow.net/questions/19607 | 12 | What I said is Lusztig's conjecture about representation of quantum group at root of unity and representation of Lie algebra at positive characters.
It seems that Andersen-Jantzen-Soergel ever wrote a book on this conjecture.
Is it solved? Any recent development? I am looking for reference talking about it.
Tha... | https://mathoverflow.net/users/1851 | Is Lusztig's conjecture solved? | The result of that book is that the conjecture is true for sufficiently large, but unspecified characteristic. (First fix a Dynkin type.)
More recently Peter Fiebig has given actual bounds. See
An upper bound on the exceptional characteristics for Lusztig's character formula
by Peter Fiebig arXiv:0811.1674v2 at <http... | 14 | https://mathoverflow.net/users/4794 | 19609 | 13,041 |
https://mathoverflow.net/questions/19635 | 7 | Given a smooth projective complex variety $X$, instead of using Mumford's GIT to construct the moduli of rank $n$ topologically trivial vector bundles, we can also take the gauge theory approach.
To classify all topologically trivial vector bundles is same as to classify all possible complex structures on the topolo... | https://mathoverflow.net/users/4975 | Gauge theory construction of moduli of vector bundles | In the case of a nonsingular algebraic curve, I guess this is the point of Atiyah and Bott's Yang-Mills on Riemann surfaces, or the work of Narasimhan and Seshadri. Choose a Hermitian pairing on your complex bundle and then search for a unitary connection with central curvature. There is a really great book by Kobayash... | 8 | https://mathoverflow.net/users/4304 | 19637 | 13,053 |
https://mathoverflow.net/questions/19652 | 9 | I am currently writing up some notes on the max-plus algebra $\mathbb{R}\_{\max}$ (for those that have never seen the term "max-plus algebra", it is just $\mathbb{R}$ with addition replaced by $\max$ and multiplication replaced by addition. For some reason, authors whose main interest is control-theoretic applications ... | https://mathoverflow.net/users/4977 | Is there an "adjacency matrix" for weighted directed graphs that captures the weights? | It looks like in your definition the weight of a path is the **sum** of the weights of its edges. In many combinatorial applications a natural definition of the weight of a path is the **product** of the weights of the edges, and there one uses precisely the weighted adjacency matrix $A\_{ij} = w(i, j)$ (as an element ... | 10 | https://mathoverflow.net/users/290 | 19653 | 13,063 |
https://mathoverflow.net/questions/19649 | 18 | Miranda's book on Riemann surfaces ignores the analytical details of proving that compact Riemann surfaces admit nonconstant meromorphic functions, preferring instead to work out the algebraic consequences of (a stronger version of) that assumption. Shafarevich's book on algebraic geometry has this to say:
> A harmo... | https://mathoverflow.net/users/290 | "Physical" construction of nonconstant meromorphic functions on compact Riemann surfaces? | For this and most other things about Riemann surfaces, I recommend Donaldson's [Notes on Riemann surfaces](http://www2.imperial.ac.uk/~skdona/RSPREF.PDF), which are based on a graduate course I was once lucky enough to see, and which may eventually make it into book format.
In his account, the "main theorem for compa... | 14 | https://mathoverflow.net/users/2356 | 19658 | 13,067 |
https://mathoverflow.net/questions/19661 | 4 | Let P(X,Y,Z) be a homogeneous polynomial in ℂ[X,Y,Z] whose locus M in ℂℙ2 is a nonsingular curve of genus ≥ 2.
Define M to be *maximally symmetric* if the following is **not** true:
---
There exists a continuous family { Pt | t ∊ [0,1] } of homogeneous polynomials in ℂ[X,Y,Z] such that 1), 2), and 3) hold:
... | https://mathoverflow.net/users/5484 | Maximally symmetric smooth projective varieties in CP^2 | By the same averaging trick that shows that finite-dimensional complex representations of a finite group are unitary with respect to some inner product, your question is equivalent to the one obtained by replacing ambient isotropy groups with linear automorphism groups in the sense of algebraic geometry. Here *linear* ... | 5 | https://mathoverflow.net/users/2757 | 19672 | 13,079 |
https://mathoverflow.net/questions/19633 | 10 | For each $d$, I have a matrix $M$ with values
$$
M\_{ij} = \begin{cases}
\frac{4ij}{d} - \binom{2d}{d} & i \neq j & \\\\
\frac{4i^2}{d} - \binom{2d}{d} -
\frac{\binom{2d}{d}}{\binom{d}{i}^{2}} & i = j
\end{cases}
$$
I want to show that, for every $d=2,3,\ldots$, the matrix is negative-definite.
**An elegant ... | https://mathoverflow.net/users/4974 | Showing a matrix is negative definite [formerly Showing a sum is always positive] | It's easier to prove the result about the matrix without resorting to determinants. What we need is the inequality
$$
\left(\sum\_{i=0}^d 2ix\_i\right)^2\le d{2d\choose d}\left(\sum\_{i=0}^d x\_i\right)^2
+d\sum\_{i=0}^d \frac{{2d\choose d}}{{d\choose i}^2}x\_i^2
$$
Now recall that $\sum\_{i=0}^d (2i-d)^2{d\choose i}^2... | 18 | https://mathoverflow.net/users/1131 | 19677 | 13,083 |
https://mathoverflow.net/questions/19393 | 3 | This question is rather vague. Are there any natural situations which involve Laurent polynomials of the form
$$\sum q^{a\_i}\in\mathbb{Z}[q,q^{-1}]$$
where the $a\_i$'s are either Euler characteristics of some spaces (possibly all subspaces of one fixed space), or more generally, indices of some elliptic operators? I'... | https://mathoverflow.net/users/492 | Euler characteristics and operator indices as exponents for Laurent polynomials | 1. Knot polynomials like the Jones polynomial
2. Perturbative expansions of Feynman Integrals
3. Heat kernel asymptotics, and other universal polynomials in characteristic classes.
4. Generating functions associated to combinatorial problems.
5. Poincare Polynomials of Topological Spaces.
6. Hilbert Polynomials.
7. Cer... | 6 | https://mathoverflow.net/users/4304 | 19681 | 13,085 |
https://mathoverflow.net/questions/19692 | 15 | I once heard from a graduate student that the ABC conjecture implies the Riemann hypothesis. I can't find a reference for this, but given the department the student is from I tend to believe he might know about these things.
I looked through [Goldfeld's paper](http://www.math.columbia.edu/~goldfeld/ABC-Conjecture.pdf... | https://mathoverflow.net/users/4872 | Is the ABC conjecture known to imply the Riemann hypothesis? | I am pretty sure that the answer to the question is no: no two of those big conjectures are known to imply the third. But I feel somewhat sheepish giving this as an answer: what evidence can I bring forth to support this, and if nothing, why should you believe me?
The only thing I can think of is that in the function... | 17 | https://mathoverflow.net/users/1149 | 19694 | 13,093 |
https://mathoverflow.net/questions/19697 | 1 | If $P, Q$ are prime, and $P > Q$, then let $K$ be the set of all numbers $(P-Q)$. Is there a way to determine $\frac{|K|}{|\mathbb{Z}^+|}$? Is this even a converging value? What kind of numbers are in set $K$?
So far:
if $P-Q = d$ is odd, then $P, Q$ are of different parity and $Q = 2$, so $d = P-2$.
But, if $d$ i... | https://mathoverflow.net/users/4990 | Possible values for differences of primes | Since there are infinitely many primes, the set $K$ is certainly infinite, so in the expression $\frac{|K|}{|\mathbb{Z}^+|}$, you are attempting to divide two infinite cardinalities. This is not a meaningfully defined operation.
Not so much is known about the set $K$ unconditionally. However, an old conjecture of [Al... | 2 | https://mathoverflow.net/users/1149 | 19699 | 13,097 |
https://mathoverflow.net/questions/18884 | 18 | Not long back I asked [a question](https://mathoverflow.net/questions/17599/existence-of-multi-variable-p-adic-l-functions) about the existence of p-adic L-functions for number fields that are not totally real; and I was told that when the number field concerned has a nontrivial totally real or CM subfield, then there ... | https://mathoverflow.net/users/2481 | Stark's conjecture and p-adic L-functions | Conjecturally, the answer is yes, but the amount of work required is not trivial at all. The general set-up is roughly as follows: the special values of $L$-functions (in your case, for Tate motives) are predicted by the Tamagawa Number Conjecture, and by the Equivariant Tamagawa Number Conjecture (ETNC) when one wishe... | 7 | https://mathoverflow.net/users/2284 | 19715 | 13,110 |
https://mathoverflow.net/questions/19716 | 1 | Hello,
where can I read about some basic properties of twisted D-Modules? I would like to know, a reference, that describes how to glue these modules together/pull them back/push them forward.
| https://mathoverflow.net/users/2837 | TDO basic facts reference request | [Twisted Differential operators](http://www.math.harvard.edu/~gaitsgde/grad_2009/BB%2520-%2520Jantzen.pdf)
| 2 | https://mathoverflow.net/users/1851 | 19718 | 13,111 |
https://mathoverflow.net/questions/19705 | 6 | So from the $\overline{\partial}$-Poincare lemma, there is a short exact sequence of sheaves on $X = \mathbb{P}^1$
$$0 \to \Omega \to A^{1,0} \to Z^{1,1} \to 0$$
where $\Omega$ is the sheaf of holomophic 1-forms, $A^{1,0}$ is the sheaf of (1,0)-forms, $Z^{1,1}$ is sheaf of closed (1,1)-forms and the surjection is a... | https://mathoverflow.net/users/7 | Dolbeault Cohomology of $\mathbb{P}^1$ | I wrote a [blog post](http://sbseminar.wordpress.com/2010/01/12/residues-and-integrals/) about almost exactly this question. I'll give a summary here:
Since $H^{1,1}(X)$ is one dimensional, I could answer your question by giving anythng with the correct integral. However, I'll try to give you the kind of cocycle whic... | 7 | https://mathoverflow.net/users/297 | 19720 | 13,112 |
https://mathoverflow.net/questions/19312 | 33 | I want to know exactly how derived functor cohomology and Cech cohomology can fail to be the same.
I started worrying about this from Dinakar Muthiah's [answer](https://mathoverflow.net/questions/4214/equivalence-of-grothendieck-style-versus-cech-style-sheaf-cohomology/4217#4217) to [an MO question](https://mathoverf... | https://mathoverflow.net/users/184 | Example Wanted: When Does Čech Cohomology Fail to be the same as Derived Functor Cohomology? | Q1: A very simple example is given in Grothendieck's Tohoku paper "Sur quelques points d'algebre homologique", sec. 3.8. *Edit:* The space is the plane, and the sheaf is constructed by using a union of two irreducible curves intersecting at two points.
Q2: Cech cohomology and derived functor cohomology coincide on a ... | 36 | https://mathoverflow.net/users/1784 | 19723 | 13,114 |
https://mathoverflow.net/questions/19651 | 4 | The relevant paper is ["An estimate of the remainder in a combinatorial central limit theorem" by Erwin Bolthausen](https://doi.org/10.1007/BF00533704 "Z. Wahrscheinlichkeitstheorie verw Gebiete 66, 379–386 (1984)"). I would like to understand the estimate on page three right before the sentence "where we used independ... | https://mathoverflow.net/users/4923 | Stein's method proof of the Berry–Esséen theorem | If you take expectation first with respect to $S\_{n-1}$, then by Fubini's theorem the last term gives
$$
E \left[\frac{|X\_n|}{\sqrt{n}}\frac{1}{\lambda} \int\_0^1 P\left(z-t\frac{X\_n}{\sqrt{n}}
\le S\_{n-1} \le z-t\frac{X\_n}{\sqrt{n}} + \lambda\right) dt\right].
$$
Now if $Y$ is a standard Gaussian random variable ... | 4 | https://mathoverflow.net/users/1044 | 19727 | 13,117 |
https://mathoverflow.net/questions/19729 | 1 | I would like to preface by saying that I have no significant experience working with set theory, so I'm probably making an intuitive mistake. I have figured out where the mistake probably is, but I can't figure out why it IS a mistake. I figured that this was the best outlet to ask my question.
I was reading about th... | https://mathoverflow.net/users/1982 | Explicitly constructing an infinite set with particular size | The problem lies with your interpretation of the notation $2^{\aleph\_0}$ and $\aleph\_0^n.$ These do not mean that your set is constructed from finite sets with a specified rate of growth.
Let $A$ be the cardinality of a set. e.g. $A = 2$ or $A=\aleph\_0.$
$A^n$ is the cardinality of the n-fold cartesian product o... | 8 | https://mathoverflow.net/users/4872 | 19731 | 13,119 |
https://mathoverflow.net/questions/19648 | 2 | Again from the Shepp and Lloyd paper "ordered cycle lengths in a random permutation", I found this puzzling equality. This one might require access to the paper itself since it's quite a mouthful:
In equation (15), they claimed it is straightforward that if there is an $F\_r$ such that
$$\int\_0^1 \exp(-y/\xi) dF\... | https://mathoverflow.net/users/4923 | method of moments and Laplace transform from Shepp and Lloyd | This is a general fact: assume that $X$ is a positive random variable and that, for a given nonnegative function $g$, $\displaystyle E(\mathrm{e}^{-y/X})=\int\_y^{\infty}g(x)\mathrm{d}x$ for every positive $y$. Then $\Gamma(s+1)E(X^s)=\displaystyle\int\_0^{\infty}x^sg(x)\mathrm{d}x$ for every positive $s$.
To prove t... | 7 | https://mathoverflow.net/users/4661 | 19737 | 13,124 |
https://mathoverflow.net/questions/19732 | 12 | While doing some research, I came up with a problem of proving that
$ f(a,b,c)=\begin{cases}1 &\text{ if }A(a,b)=c\\ \\\\ 0 &\text{ otherwise }\end{cases} $
is primitively recursive ($A$ is the Ackermann's function).
Any references, ideas or proofs?
(This may not be a good MO question, but since the participan... | https://mathoverflow.net/users/4925 | Ackermann-related function | Here's a sketch of an argument which I expect could be made into a proof.
The key fact is that the Ackermann function fails to be primitive recursive only because it grows so quickly. More formally:
**Claim**. There exists a Turing machine T and a primitive recursive function f(a, b, c) (which is an increasing func... | 8 | https://mathoverflow.net/users/126667 | 19742 | 13,128 |
https://mathoverflow.net/questions/19747 | 14 | Let $P$ be a polyhedron which satisfies the following three conditions:
1. $P$ is built out of regular hexagons and regular pentagons.
2. Three faces meet at each vertex.
3. $P$ is topologically a sphere.
An easy Euler characteristic argument tells you that $P$ has exactly twelve pentagonal faces.
An example of a... | https://mathoverflow.net/users/5000 | The Symmetry of a Soccer Ball | Only soccer ball or dodecahedron.
---
Clearly 3 hexagons can not meet at one vertex.
Thus we have only 3 choices for one vertex:
* 3 pentagons
* 2 pentagons + 1 hexagon
* 1 pentagons + 2 hexagon
Note that if $[pq]$ is an edge then $p$ has the same type as $q$ (the type is determined by angle at $[pq]$).
Thus... | 13 | https://mathoverflow.net/users/1441 | 19753 | 13,134 |
https://mathoverflow.net/questions/19684 | 21 | In the study of number theory (and in other branches of mathematics) presence of Hecke Algebra and Hecke Operator is very prominent.
One of the many ways to define the Hecke Operator $T(p)$ is in terms of double coset operator corresponding to the matrix $ \begin{bmatrix} 1 & 0 \\ 0 & p \end{bmatrix}$ .
On the oth... | https://mathoverflow.net/users/4291 | Relation between Hecke Operator and Hecke Algebra | The fact that Hecke operators (double coset stuff coming from $SL\_2(\mathbf{Z})$ acting on modular forms) and Hecke algebras (locally constant functions on $GL\_2(\mathbf{Q}\_p)$) are related has nothing really to do with the Satake isomorphism. The crucial observation is that instead of thinking of modular forms as f... | 24 | https://mathoverflow.net/users/1384 | 19757 | 13,137 |
https://mathoverflow.net/questions/19745 | 2 | How to obtain an upperbound for knots up to k crossings?
I think I've found something which involves the genus but I'm not sure.
| https://mathoverflow.net/users/5001 | Counting knots with fixed number of crossings | There are some known exponential bounds on the number. For example, if kn is the number of prime knots with n crossings, then Welsh proved in "On the number of knots and links" (MR1218230) that
>
> 2.68 ≤ lim inf (kn)1/n ≤ lim sup (kn)1/n ≤ 13.5.
>
>
>
The upper bound holds if you replace kn by the much large... | 10 | https://mathoverflow.net/users/428 | 19759 | 13,139 |
https://mathoverflow.net/questions/19638 | 1 | Hello all, let $n$ be an integer $\geq 2$ and let $\alpha$ be an algebraic number
of degree $n$. Let $R$ be the ring of algebraic integers in ${\mathbb Q}(\alpha)$, and
let $B$ be the subset of $R$ containing the elements whose degree is exactly
$n$. Any $\beta \in B$ has a minimal polynomial
$X^n+b\_{n-1}X^{n-1}+ \ld... | https://mathoverflow.net/users/2389 | Infinite collection of elements of a number field with very similar annihilating polynomials | For $n>4$, almost all fields of degree $n$ will have $r>1$:
Fix a field $K$ with discriminant $D\_0$. Fix the $n-1$ coefficients $b\_{n-1},...,b\_{i+1}, b\_{i-1},..., b\_0$. The discriminant of the polynomial $x^n+b\_{n-1}x^{n-1}+...$ is a polynomial $D(b\_i)$ in the single variable $b\_i$, and is of degree at least ... | 3 | https://mathoverflow.net/users/2024 | 19774 | 13,148 |
https://mathoverflow.net/questions/19766 | 17 | A homotopy (limits and) colimit of a diagram $D$ topological spaces can be explicitly described as a geometric realization of simplicial replacement for $D$.
However, a homotopy colimit can also be described as a derived functor of limit. A model category structure can be placed on the category $\mathrm{Top}^I$, whe... | https://mathoverflow.net/users/2532 | Homotopy colimits/limits using model categories | In the context of Goodwillie's paper, he's got an explicit natural transformation $f:holim\_I(X)\to holim\_J(X|\_J)$, where $X:I\to Top$ is a functor to spaces, and $J\subset I$ is a subcategory of $I$. With the construction of holim he's using, this map is always a fibration.
What if you tried to use a different con... | 14 | https://mathoverflow.net/users/437 | 19780 | 13,152 |
https://mathoverflow.net/questions/19721 | 2 | This is sort of a follow-up to:
[Gauge theory construction of moduli of vector bundles](https://mathoverflow.net/questions/19635/gauge-theory-construction-of-moduli-of-vector-bundles)
If I have a complex compact algebraic curve with at worst nodal singularities, is there an analytic description of holomorphic structu... | https://mathoverflow.net/users/3500 | gauge theory construction of vector bundles on singular varieties | There are subtleties even in the simplest case - $C$ a compact, irreducible complex curve with one node, $Pic\_0(C)$ the Picard variety of line bundles of degree $0$ - so why not start there?
Pulling back line bundles via the normalisation map $\nu\colon \tilde{C}\to C$ defines a map $Pic\_0(C)\to Pic\_0(\tilde{C})$... | 4 | https://mathoverflow.net/users/2356 | 19781 | 13,153 |
https://mathoverflow.net/questions/19775 | 32 | There are a couple of ways to define an action of $\pi\_1(X)$ on $\pi\_n(X)$. When $n = 1$, there is the natural action via conjugation of loops. However, the picture seems to blur a bit when looking at the action on higher $\pi\_n$. All of them have the flavor of the conjugation map, but are more geometric than algebr... | https://mathoverflow.net/users/2532 | Different way to view action of fundamental group on higher homotopy groups | If G is a topological group, then the group acts on itself by conjugation, and this action is base-point-preserving. In particular, for an element $g \in \pi\_0(G)$ and a higher homotopy element $\alpha \in \pi\_{n-1} G = [S^n, G]$, one can check that the conjugate $g \alpha g^{-1}$ is well-defined and defines an actio... | 20 | https://mathoverflow.net/users/360 | 19787 | 13,157 |
https://mathoverflow.net/questions/19791 | 14 | Is infinite (say complex) projective space a scheme? More generally, can schemes have infinite cardinal dimension? It seems that infinite dimensional projective space is not a manifold, since it is not locally Euclidean for any R^n.
Related question. If inifinite projective space is a scheme, then take a nonclosed po... | https://mathoverflow.net/users/nan | Infinite projective space | Starting with the affine case, if you try to define infinite dimensional affine space as Spec of k{x1,x2,...], then you realise that this is not a vector space of countable dimension, but something much larger. If you want a vector space over k of countable dimension, then this will not be a scheme, but instead will be... | 13 | https://mathoverflow.net/users/425 | 19795 | 13,162 |
https://mathoverflow.net/questions/19740 | 11 | This is maybe a dumb question. $SL\_2(\mathbb{R})$ has a natural action on the upper half plane $\mathbb{H}$ which is transitive with stabilizer isomorphic to $SO\_2(\mathbb{R})$. For this reason, people sometimes write $\mathbb{H}$ as the coset space $SL\_2(\mathbb{R})/SO\_2(\mathbb{R})$.
Now, it's clear how this d... | https://mathoverflow.net/users/290 | How do you recover the structure of the upper half plane from its description as a coset space? | **Edit:** I should have put a short version of the answer in the beginning, so here is how the various structures are recovered. To get a smooth manifold structure on the quotient, you use the fact that $SL\_2(\mathbb{R})$ is a real Lie group and $SO\_2(\mathbb{R})$ is a closed subgroup. To get a hyperbolic structure, ... | 4 | https://mathoverflow.net/users/121 | 19796 | 13,163 |
https://mathoverflow.net/questions/19783 | 6 | (repost from the topology Q&A board)
I have a (T\_1), Normal, countably paracompact space X. I would like to know if every locally countable open cover of X (i.e. an open cover such that every x in X has a neighbourhood which intersects only countably many members of the cover) has a locally finite refinement.
My s... | https://mathoverflow.net/users/4959 | Countable paracompactness, normality and locally countable open covers | In Caryn Navy's thesis under Mary Ellen Rudin she constructed several spaces that are normal, countably paracompact and paralindelöf (every cover has a locally countable refinement) but not paracompact.
All such spaces provide counterexamples (we can refine a cover without a locally finite refinement to a locally count... | 4 | https://mathoverflow.net/users/2060 | 19798 | 13,165 |
https://mathoverflow.net/questions/19768 | 4 | I was reading Mehta and Seshadri's paper "Moduli of vector bundles on curves with parabolic structures".
In the second paragraph, they wrote:
"Suppose that $H$ mod $\Gamma$ has finite measure ($H$ is the complex upper half plane, and $\Gamma$ is a discrete group). Let $X$ be the smooth projective curve containing $... | https://mathoverflow.net/users/4975 | Upper half plane quotient by a discrete group | I like to think about this geometrically. $H/\Gamma$ is a topological metric space. At most points of $H$ (that are not fixed by any element of $\Gamma$, the quotient looks just like the hyperbolic plane $H$ itself. The singularities come from elliptic elements of $\Gamma$, i.e., (locally) rotations, where you get a co... | 7 | https://mathoverflow.net/users/5010 | 19799 | 13,166 |
https://mathoverflow.net/questions/19760 | 12 | Suppose $E/\mathbb{Q}$ is an elliptic curve with rank zero. According to the conjecture of Birch and Swinnerton-Dyer, the special value $L(1,E\_{/\mathbb{Q}})$ should be equal (up to some harmless factors) to the order of the Tate-Shafarevich group Sha$(E/\mathbb{Q})$. Now, suppose $\chi$ is a Dirichlet character, and ... | https://mathoverflow.net/users/1464 | Decomposition of Tate-Shafarevich groups in field extensions | Fix a prime p which doesn't divide the degree of K over ${\mathbb Q}$, and let ${\mathcal O}$ denote the ring of integers of ${\mathbb Q}\_p(\chi)$ i.e. an extension of ${\mathbb Q}\_p$ containing the values of $\chi$. Then the group algebra ${\mathcal O}[G]$ decomposes into a direct sum of 1-dimensional pieces over ${... | 8 | https://mathoverflow.net/users/86179 | 19821 | 13,180 |
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