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https://mathoverflow.net/questions/68432 | 4 | Consider a triangulation of some bounded region of $R^3$ with a (finite) set of tetrahedra (like in Regge calculus). It can be thought of as a simplicial 3-complex with specified lengths of edges. The other way around - a 3-complex with specified lengths of edges can sometimes be isometrically embedded in $R^3$. Now le... | https://mathoverflow.net/users/11329 | When can a 3-dimensional triangulation be isometricaly embedded in R^n? | The ratio $E/V$ has no upper bound. Moreover, for every $n\ge 4$ there exists a triangulation of a tetrahedron in $\mathbb R^3$ with $V=n$ and $E=n(n-1)/2$ (so that the edges form a complete graph).
To construct such a triangulation, consider a curve $\gamma:\mathbb R\to\mathbb R^4$ defined by
$$
\gamma(t) = (t,t^2,... | 6 | https://mathoverflow.net/users/4354 | 68552 | 42,123 |
https://mathoverflow.net/questions/68544 | 4 | I was searching through the small groups database in GAP to find counterexamples to a certain conjecture (which is not important here). I was checking non-nilpotent solvable groups and noticed that all the permutation modules ($1\_H^G$ for $H\leq G$) in characteristic 2 have self-dual socles. Obviously this is true in ... | https://mathoverflow.net/users/15904 | Do permutation modules of solvable groups have self-dual socle in characteristic 2? | The closest work I can think of of this type is a 2010 paper in Journal of Algebra by Natalie Naehrig
"On the Endomorphism Rings of Permutation Modules". This was not confined to characteristic 2,
nor to solvable groups, and usually considered the permutation module (in characteristic p) on the
cosets of a Sylow p-subg... | 2 | https://mathoverflow.net/users/14450 | 68553 | 42,124 |
https://mathoverflow.net/questions/68550 | 5 | I think that my question is easily answerable. The question is: What is a nice subcategory of topological spaces where the subobjects are subspaces. I would like the category of compactly generated Haussdorf spaces to be such a category, since this category is convienent in many other ways.
Some backround definitions... | https://mathoverflow.net/users/14167 | A subcategory of top where subspaces and subobjects coincide? | Well, I'd say that compact Hausdorff spaces are at least a *nice* category of spaces where subspaces are equivalent to subobjects. One reason it is nice is that it is a category of algebras for a monad on $Set$, with all the nice properties that entails, for example Barr exactness. It is of course *not* "convenient" in... | 6 | https://mathoverflow.net/users/2926 | 68557 | 42,128 |
https://mathoverflow.net/questions/68551 | 11 | The $g$-fold symmetric product of a Riemann surface of genus $g$ naturally has both a symplectic structure as well as a complex structure. Is it in fact Calabi-Yau? If so, is anything known about a mirror for it (in the sense of mirror symmetry)?
My motivation for this comes from Mirror Symmetry and Heegaard-Floer Ho... | https://mathoverflow.net/users/5323 | Is $Sym^g$ of a Riemann Surface of genus $g$ Calabi-Yau? | This is not a Calabi-Yau if $g\ne 1$ (for any definition of Calabi-Yau). Indeed, there is a degree one map from the symmetric power of the curve to a torus of dimension $g$. Pull-back of the volume form on the torus to $Sym^gS$ will have zeros at the set where the differential of the map is degenerate, this set is non-... | 17 | https://mathoverflow.net/users/943 | 68559 | 42,130 |
https://mathoverflow.net/questions/68424 | 5 | According to the nlab, the categorical trace of a 1-endomorphism $F:C\to C$ in a 2-category is the set hom$(1\_C, F)$ of global elements of $F$. If $F$ is a functor in the 2-category Cat, the categorical trace is a set of natural transformations that assign to each object of $C$ a coalgebra of $F$ such that the obvious... | https://mathoverflow.net/users/756 | In what sense do the categorical trace and coend count fixed points? | Simon Willerton explains it all very well here: <http://www.simonwillerton.staff.shef.ac.uk/ftp/TwoTracesBeamerTalk.pdf>
| 3 | https://mathoverflow.net/users/756 | 68564 | 42,132 |
https://mathoverflow.net/questions/68570 | 4 | In reference to 1961 paper "On Non Computable Functions" by T. Rado.
Motivation - Scott Aaronson's [Who Can Name the Bigger Number?](http://www.scottaaronson.com/writings/bignumbers.html).
M is an n-state binary Turing machine. A valid BB-n entry is a set $(M,s)$ where M halts in exactly s steps. $E\_n$ is the se... | https://mathoverflow.net/users/15951 | Existence of a set of valid Busy-Beaver entries. | The question of whether a given fixed Turing machine $M$
halts or not is something that can be independent of our
fundamental axioms of mathematics.
For example, let $M$ be the Turing machine that searches
for a proof of a contradiction from ZFC, say, halting only upon
finding one. One could in principle write down... | 10 | https://mathoverflow.net/users/1946 | 68574 | 42,137 |
https://mathoverflow.net/questions/68506 | 7 | Here is a simply described but fiendishly diophanterrorizing problem
I asked on AMM eons ago. Maybe you can shed some light upon it.
0.2 (base 4) = 0.2 (continued fraction)
0.24 (base 6) = 0.24 (continued fraction)
Find all examples of
0.$xyz$... (base B) = 0.$xyz$... (continued fraction).
First of all, b... | https://mathoverflow.net/users/11504 | Unsolved Problem from AmMathMonthly | Here is a solution for the case you ask. But first let me say that given the nature of the question it would probably get better answers at artofproblemsolving. What follows is a lot of very elementary number theory, and an appeal to a result of Ljunggren from 1942.
So we have $B\geq 2$ and $x,y\in \{0,1,\dots,B-1\}$... | 7 | https://mathoverflow.net/users/2384 | 68575 | 42,138 |
https://mathoverflow.net/questions/68465 | 18 | Early this year, I started to learn about p-adic modular forms. Very recently, a mathematician tells me Emerton constructed an object called completed cohomology group with very rich structure, and the author could use it to prove fantastic results about Galois representations. (see Emerton's paper "Local-Global Compat... | https://mathoverflow.net/users/15783 | Katz Modular Functions and Emerton's Completed Cohomology | You already have two helpful answers related to general aspects of Eichler--Shimura isomorphisms in a $p$-adic context. Here is an answer that more directly addresses your original question.
---
I will begin by recalling/stating some facts on the $p$-adic modular form side:
Fix a tame (i.e. prime-to-$p$) level ... | 30 | https://mathoverflow.net/users/2874 | 68593 | 42,150 |
https://mathoverflow.net/questions/68600 | 0 | Let $Q$ be an open interval of ${\mathtt R}$ and $E$ be the space of continuous and bounded functions in $Q\to \mathtt{R}$.
I call $E^\*$ the set of linear functionals over $E$ and $E\_+^\*$ the subset of positive linear functionals.
My question is whether, for $x\in E$, the condition $\forall s\in Q,\; x(s) > 0$ i... | https://mathoverflow.net/users/15956 | Linear functionals and continuous functions on open intervals | No. Let w.l.o.g $Q:= (0,1)$. There is a bounded linear functional $f$ on $E$ such that for any $x\in E$ one has: $\liminf \_ {s\to 0} x(s)\le f(x) \le \limsup \_ {s\to 0} x(s) $. This functional is positive, still vanishes on some functions which are strictly positive on $Q$.
**rmk.** For the construction of $f$, you... | 1 | https://mathoverflow.net/users/6101 | 68602 | 42,154 |
https://mathoverflow.net/questions/68601 | 2 | I am looking for a reference or short explanation of a proof by E. Brieskorn.
In his famous work "Singularities of complex hypersurfaces" Milnor proves that the (nowadays called) Milnor Number (in the sense of the local degree) of a polynomial $f\in\mathbb{C}{[X\_1,X\_2,...,X\_{n+1}]}$ with an isolated singular point... | https://mathoverflow.net/users/15782 | Brieskorn's proof of a theorem by Milnor about the Milnor number | I guess that in his book (1968) Milnor refers to the proof later published in Brieskorn's paper
"Die Monodromie der Isolierten Singularitäten von Hyperflächen", Manuscripta Mathematica 2, 103-161 (1970).
See in particular Satz 1.
In fact, at the beginning of the Appendix Brieskorn writes
>
> "Wir haben in Sat... | 3 | https://mathoverflow.net/users/7460 | 68603 | 42,155 |
https://mathoverflow.net/questions/68576 | -1 | The differential equations are :
$$ ( n\_{j,k,0} )'(x) = - \frac {jn\_{j,k,0}(x)} {a-x}, $$
$$ ( n\_{j,k,b} )'(x) = \frac { (j-b+1)n\_{j,k,b-1}(x) - (j-b)n\_{j,k,b}(x) } {a-x}, $$
for $ 0\lt b\lt c $.
$$ (f\_{j,k})'(x) = \frac{ (j-c+1)n\_{j,k,c-1}(x) }{a-x}.$$
Here, the second equation holds with $0\lt b\lt c$.... | https://mathoverflow.net/users/15955 | Solve the following system of differential equations | If b is an integer, you can simply solve the equations successively, starting with b=0. If b is supposed to be real, you need to explain what $n\_{j,k,b-1}$ is supposed to mean for b between 0 and 1. Also, what is the point of the index k? Nothing in the equations depends on k!
| 0 | https://mathoverflow.net/users/12120 | 68605 | 42,157 |
https://mathoverflow.net/questions/68567 | 3 | Suppose we have a
[combinatorial bracelet](http://en.wikipedia.org/wiki/Bracelet_(combinatorics)) composed of natural numbers.
(Two bracelets are equivalent if you can get from one to the other via rotation or reflection.)
What is the number of different bracelets whose elements sum up to a previously fixed natura... | https://mathoverflow.net/users/4102 | Number of partitions of a number on a combinatorial bracelet | There is almost a bijection between your partition bracelets adding to $n$ and bracelets of length $n$ with $2$ colors. Let the colors be pluses "+" and commas "," and put a $1$ between each two beads. Then the bracelet $++,$ corresponds to the partition bracelet $1+1+1,$ or $(3)$. The bracelet
$+,+,$ corresponds to $... | 7 | https://mathoverflow.net/users/2954 | 68621 | 42,165 |
https://mathoverflow.net/questions/56388 | 8 | What maps of simplicial sets exist between
* the image under the Dold-Kan correspondence of a chain complex shifted up in degree
* and the image under the right adjoint to simplicial looping of the DK-image of the unshifted complex
?
Here is the same question in detail:
Write
$$
(G \dashv \bar W) : sGrp \st... | https://mathoverflow.net/users/381 | delooping under Dold-Kan and simplicial delooping | There's an explicit natural isomorphism between the two functors.
Rick Jardine says as much, but for the image of the functors in the category of chain complexes (i.e. after applying the normalization). You can find this in Goerss, Jardine Remark III.5.6, or in greater depth in section 4.6 of Jardine's book on Gener... | 3 | https://mathoverflow.net/users/9581 | 68624 | 42,166 |
https://mathoverflow.net/questions/68615 | 5 | Let $K$ be a local field (of characteristic 0) with (finite) residue field of characteristic $l$ and let $p$ be a prime.
Considering the cases, whether the $p$-th roots of unity are in $K$ and whether $l$ equals $p$ (and maybe whether $p=2$) or not, my question is:
How many Galois extensions of $K$ of degree $p$ ex... | https://mathoverflow.net/users/12668 | number of galois extensions of local fields of fixed degree | If $K$ contains the $p$-th roots of unity, then Kummer theory tells us that the degree $p$ Galois extensions of $K$ are in bijective correspondence with the subgroups of $K^{\times}/(K^{\times})^p$ of order $p$. The structure of $K^{\times}$ is well-known; see <http://en.wikipedia.org/wiki/Local_fields> or any decent b... | 9 | https://mathoverflow.net/users/7443 | 68631 | 42,168 |
https://mathoverflow.net/questions/52915 | 7 | There is a map $BG \to A(\ast)$ where $BG$ classifies stable spherical fibrations and $A(\ast)$ is
Waldhausen's algebraic $K$-theory of a point. The map is induced by applying Quillen's plus construction to the inclusion
$$
BGL\_1(S^0) \to BGL\_\infty(S^0)
$$
where $BGL\_1(S^0)$ is $BG$. Here $BGL\_\infty(S^0)$ can b... | https://mathoverflow.net/users/8032 | Why does the map $BG\to A(*)$ fail to split? | Question 1: There are several arguments.
In degree 2 there is a reference: the proof of corollary 3.7 of Waldhausen's "Algebraic K-theory of spaces, a manifold approach". See <http://www.math.uni-bielefeld.de/~fw/> for a copy. Consider the maps $BG \to A(\ast) \to K(Z)$ and apply $\pi\_2$. Here $\pi\_2 BG = Z/2$, the... | 5 | https://mathoverflow.net/users/9684 | 68645 | 42,175 |
https://mathoverflow.net/questions/68633 | 4 | Hi,
I have been interested in foundations for a while, especially categories as foundations. I am of the opinion that, as long as we present the theory of categories in SET, we will not be able to give a reasonable justification for categories as a foundation. (that could be a question: does the persistent presentati... | https://mathoverflow.net/users/10007 | linear logic, diagrammatic calculus and foundations | Just to address one of the issues raised by the questioner: the formal theory of categories does not depend on $Set$, any more than the formal theory ZFC depends on $Set$ (what would the latter even mean?). It's just a first-order theory. One formal syntactic presentation of it can be found [here](http://ncatlab.org/nl... | 8 | https://mathoverflow.net/users/2926 | 68646 | 42,176 |
https://mathoverflow.net/questions/68612 | 2 | Let $S\subset\mathbb{P}^g$ be a smooth polarized K3 surface of genus $g$. I am interested in the existence of certain cuspidal curves in the linear system. We know a general hyperplane section $H\cap S$ is smooth, and to have a nodal singularity is a codimension 1 condition (Let's say the dual variety of $S$ is a divis... | https://mathoverflow.net/users/10646 | cuspidal curves in K3 surfaces | Xi Chen has a theorem that says that rational curves on K3's in a linear system of dimension $>3$ are nodal. I suppose you don't need this curve to be rational, but his techniques might help you in your quest. At least his theorem tells you that you cannot expect too many cusps.
I couldn't find the paper online, I in... | 5 | https://mathoverflow.net/users/10076 | 68670 | 42,191 |
https://mathoverflow.net/questions/68672 | 1 | A few days ago I asked a similar [question](https://mathoverflow.net/questions/68386/can-we-say-anything-about-the-krull-dimension-of-a-localization) about Krull dimension and got fantastic answers. Unfortunately, for the application I have in mind (a question on ring spectra), Krull dimension doesn't generalize correc... | https://mathoverflow.net/users/11540 | Is there a relationship between the right global dimensions of R and R[1/v]? | In general, if $R$ is any ring and $S$ is any right denominator set, the right globaldimension of the localization $R\_S$ does not exceed that of $R$.
If $R$ is of finite global dimension and Noetherian, and $S$ is left and right denominator set, then $R\_S$ and $R$ have the same dimension iff there is a simple $R$-m... | 1 | https://mathoverflow.net/users/1409 | 68675 | 42,193 |
https://mathoverflow.net/questions/68674 | 7 | I am looking for a reference for the following fact which must be classical (which makes it harder, for me, to track a reference down). I am interested because there are similar (more complicated) statements about the cohomology of symmetric groups.
If $P$ is a partition, namely $p\_{1} + \cdots + p\_{k} = n$, we let... | https://mathoverflow.net/users/4991 | Tensor products of permutation representations of symmetric groups. | Hi Dev,
It looks to me like a proof of this fact is given in the answer to Exercise 7.84(b) of Richard Stanley's Enumerative Combinatorics, volume 2, along with a reference to Example I.7.23(e), page 131, of I. G. Macdonald's Symmetric Functions and Hall Polynomials (2nd edition).
| 8 | https://mathoverflow.net/users/36466 | 68684 | 42,198 |
https://mathoverflow.net/questions/68678 | 7 | If $A$ is copositive, what about $A^3$? Is it also copositive? More generally,
my question is whether the odd power of a copositive matrix is still copositive.
Any reference is appreciated
| https://mathoverflow.net/users/3818 | The odd power of copositive matrix | A counter-example is given by
$$\pmatrix{.6,0,0,.6,1}\pmatrix{
1 & -1 & 1 & 1 & -1 \\\
-1 & 1 & -1 & 1 & 1 \\\
1 & -1 & 1 & -1 & 1\\\
1 & 1 & -1 & 1 & -1\\\
-1 & 1 & 1 & -1 & 1} ^3 \pmatrix{.6 \\\ 0 \\\ 0\\\ .6 \\ 1}=-0.44.$$
while the matrix in the centre is actually copositive. It is easy to check that there are no... | 11 | https://mathoverflow.net/users/2384 | 68686 | 42,199 |
https://mathoverflow.net/questions/68687 | 10 | I am interested in this claim:
>
> The $n$th symmetric power $C^{(n)}$ of a genus $g$ curve $C$ is isomorphic to the projectivization $\mathbb{P}(E\_n)$ of the sheaf $E\_n := \pi\_\ast(P\_n)$ over the Jacobian $J(C)$, where $P\_n$ is a degree $n$ Poincare bundle over $C \times J(C)$ and $\pi$ is the projection $C \... | https://mathoverflow.net/users/83 | Symmetric powers of a curve = projective bundle over Jacobian, and the relative version thereof | This is worked out in excruciating detail in the article [Jacobians and Symmetric products](https://projecteuclid.org/journals/illinois-journal-of-mathematics/volume-7/issue-2/Jacobians-and-symmetric-products/10.1215/ijm/1255644637.full) by Schwarzenberger. I think the arguments there are perfectly good in the families... | 8 | https://mathoverflow.net/users/4707 | 68691 | 42,202 |
https://mathoverflow.net/questions/68596 | 7 |
>
> How do I make the following sort of argument work in characteristic p?
>
>
>
Let $f:X \to Y$ be a proper equidimensional map of smooth algebraic varieties, assume all fibres are reduced. Say at some point $y \in Y$, I have computed the differential and know that $df(T\_x X) \supset V$ for all $x \in X\_y$ an... | https://mathoverflow.net/users/4707 | What is the replacement for a "sufficiently small disc" in characteristic p? | I think it would be difficult to give a general result that covers everything you want and can do but there is a collection of techniques (maybe better described as a dictionary) that works in many cases.
* A polydisc should be replaced by the strict Henselisation of some (smooth) subvariety $T$ passing through $y$. ... | 10 | https://mathoverflow.net/users/4008 | 68704 | 42,211 |
https://mathoverflow.net/questions/68560 | 5 | Let $G$ be a primitive permutation group of degree $n$, that is $G$ acts transitively and faithfully on a set consisting of $n$ elements and $G$ preserves no nontrivial partition of $X$. In a sense primitive groups are the 'simple' permutation groups.
For example one can show that a primitive group has at most 2 mini... | https://mathoverflow.net/users/2042 | How many normal subgroups a primitive group can have? | I have been trying to find results in the literature on this topic, but finding it frustrating. The difficulty is that estimates on composition and chief series lengths of various types of groups are typically studied not as end in themselves, but as a means to proving other results, such as bounding the minimal genera... | 6 | https://mathoverflow.net/users/35840 | 68712 | 42,215 |
https://mathoverflow.net/questions/68710 | 1 | If we let $R=\mathbb{Z}[x]$ and $D=\mathbb{Z}[[x]]$. We say that $z\in D$ is rational if there is $g\in R$, $g\ne 0$ such that $zg\in R$. Let $S$ be the set of all rational elements in $D$. Then $S$ is a subring of $D$.
So, in this case, we may assume that $zg=f\in R$ with $f,g$ have no common factors. By some comput... | https://mathoverflow.net/users/11228 | Rational power series | For any domain $R$ an element of $R[[X]]$ is invertible if and only if the constant term is invertible in $R$.
Applying this repeatedly, one gets that an element of $\mathbb{Z}[[X\_1,\dots,X\_n]]$ is invertible (in this domain) if and only if its constant term is invertible in $\mathbb{Z}$ thar is it is in $\pm 1 + ... | 4 | https://mathoverflow.net/users/nan | 68716 | 42,218 |
https://mathoverflow.net/questions/68715 | 3 | Assume $M$ is a $2n-$dimensional differentiable manifold. Let $(U\_{i})$ be a open covering of $M$. With respect to this covering let $\rho\_{i}$ be a partition of unity. Assume that on each $U\_{i}$ we have a symplectic form $\omega\_{i}$. Is then $\omega := \sum\_{i} \rho\_{i} \omega\_{i}$ a symplectic form ? If, not... | https://mathoverflow.net/users/15975 | symplectic form with partition on unity | Suppose you are in $\mathbb{R}^{2n}$, and endow it with a symplectic form $\omega$. Let $U\_1 = \{x\_1>-\varepsilon\}$ and $U\_2=\{x\_1<\varepsilon\}$, with two symplectic forms $\omega\_1 = \omega|\_{U\_1}$, $\omega\_2=-\omega|\_{U\_2}$. Notice that if $n$ is even, the $\omega\_i$ induce the same orientation on the ov... | 9 | https://mathoverflow.net/users/13119 | 68717 | 42,219 |
https://mathoverflow.net/questions/68719 | 9 | Let me first recall some basic well-known definitions: Let $R$ be a ring (as always commutative). A (cocomplete) *abelian tensor category* is defined to be a symmetric monoidal category, whose underlying category is also a $R$-linear (cocomplete) abelian category, such that the tensor product is right exact (cocontinuo... | https://mathoverflow.net/users/2841 | Dualizable objects are flat? | A dual pair as you describe above induces an adjunction $(X^\*\otimes-)\dashv(X\otimes-)$, and since right adjoints preserve limits, $X$ is flat.
| 16 | https://mathoverflow.net/users/8482 | 68723 | 42,221 |
https://mathoverflow.net/questions/58059 | 4 | I hope that my question is appropriate for MO, since it might turn out te be mainly a question about GAP or other group theory software.
>
> Is there an algorithm to produce all non-nilpotent groups of odd order (up to some given upper bound)?
>
>
>
All groups of odd order are solvable by the famous Feit-Thomp... | https://mathoverflow.net/users/12858 | Finding groups of odd order without non-cyclic nilpotent quotients | Hi Tom,
the answer (at least to your second, refined question) is "Yes! or at least "Yes, soon!" :). I first wanted to post this as a comment, but since it is rather lengthy, I figured it made more sense to give this as an answer, even though it might not be completely satisfying.
There are algorithms that can gen... | 4 | https://mathoverflow.net/users/8338 | 68727 | 42,224 |
https://mathoverflow.net/questions/68549 | 1 | Hello,
for any given function $F$ of the Selberg class $\mathcal{S}$, let $A\_{F}$ be the set of coefficients $a\_{n}$ of the Dirichlet series defining $F(s)$ for $\Re(s)>1$, and let $A=\bigcup\_{F\in\mathcal{S}}A\_{F}$. Is it true that $\mathbb{Q}(A)=\mathbb{C}$? Same question for $F$ running through $\mathbb{P\_{\m... | https://mathoverflow.net/users/13625 | The "maximal" field associated to the Selberg class | Take any Dirichlet series $L(s)=\sum\_{n \geq 1} \frac{a\_n}{n^s}$ in the [Selberg class](http://en.wikipedia.org/wiki/Selberg_class). If $L(s)$ has no pole at $s=1$, then for any $\theta \in \mathbf{R}$, the additive twist $L\_{\theta}(s)=L(s+i\theta) = \sum\_{n \geq 1} \frac{a\_n n^{-i\theta}}{n^s}$ still belongs to ... | 3 | https://mathoverflow.net/users/6506 | 68730 | 42,226 |
https://mathoverflow.net/questions/68708 | 3 | Hello!
I have a few questions on Reshetikhin Turaev invariants.
By RT any ribbon category ${\mathcal C}$ yields an invariant of oriented, framed links labelled with objects of ${\mathcal C}$.
Is there a general way to build from this an invariant of *unframed*, oriented links? At least in the case where one consi... | https://mathoverflow.net/users/3108 | Invariants of unframed, oriented links from Reshetikhin Turaev construction | The ribbon element (i.e. the move that changes framing) acts on a simple object $V$ by a scalar $\theta\_V$. The writhe changes by 1 when you change framing. Hence $\theta\_V^{-w(K)}RT\_V(K)$ is an invariant of unframed oriented links (depending on your conventions I may have a sign wrong here). Note that this only wor... | 2 | https://mathoverflow.net/users/22 | 68741 | 42,234 |
https://mathoverflow.net/questions/68667 | 11 | A classical theorem of Clifford states that if G is a finite group and K a field, then every irreducible right KG-module is a completely reducible right KN-module, where N is any normal subgroup of G.
Is there a Lie theoretic analog of this result? That is, if L is a finite-dimensional Lie algebra, I an ideal of L, an... | https://mathoverflow.net/users/14653 | Is there an analog of Clifford Theorem in the setting of Lie algebras? | The positive results seem at most to be tied to finite dimensional representations in positive characteristic:
Let $\frak H$ be the Heisenberg algebra with basis $x,y,$ where $c$ is central and $[y,x]=c$. It acts on polynomials $k[x]$ with $x$ acting by multiplication by $x$ and $y$ acting as $d/dx$ and $c$ acting as... | 3 | https://mathoverflow.net/users/4008 | 68753 | 42,237 |
https://mathoverflow.net/questions/68754 | 2 | I wonder if the following holds in an arbitrary Riemannian manifold $M$:
assume $x\in M$, $h\in T\_x M$, do we have for $u\in T\_x M$ exponentiable (if necessary of small enough norm) that:
$$\lim\_{t\to 0} \frac{d(\exp\_{\exp\_x u}t\tau(h), \exp\_x (u+th))}{t}=0$$
where $\tau(h)$ is the parallel transport of $h$ a... | https://mathoverflow.net/users/9152 | Tangential behavior of Riemannian exponential | The second identity is always true because both arguments of $d$ are smooth functions of $u$ and $t$ and they coincide when $u=0$.
The first one holds true for all $u$ and $h$ only if the metric is flat. Indeed, the l.h.s. is the length of the difference of the initial velocity vectors of two curves $t\mapsto \exp\_{... | 4 | https://mathoverflow.net/users/4354 | 68756 | 42,239 |
https://mathoverflow.net/questions/68742 | 2 | I would like to have a better understanding of a notion I've met in the beautiful book of Nikishin and Sorokin "Rational approximation and Orthogonality", since they do not provide examples.
As it is classical to do in potential theory, denote for $\mu$ in $M\_1(K)$, the set of probability measures on a compact set $... | https://mathoverflow.net/users/15517 | Compact sets of the complex plane having the K-property ? | I do not have a satisfactory answer to your question, just a pointer. In the paper:
Białas-Cież, Leokadia
Markov sets in ${\bf C}$ are not polar.
Bull. Polish Acad. Sci. Math. 46 (1998), no. 1, 83–89
for a compact subset $E$ of $\mathbb{C}$ which satisfies Markov inequality (i.e., certain estimate for derivatives o... | 5 | https://mathoverflow.net/users/14493 | 68762 | 42,244 |
https://mathoverflow.net/questions/67903 | 29 | Let $X$ be a complex manifold and $g$ a hermitian metric on $X$. Consider the Riemannian exponential $\exp\_p: T\_p X \to X$.
If $\exp\_p$ is holomorphic for every $p \in X$, then $(\exp\_p)^{-1}$, suitably restricted, provide holomorphic normal coordinates near $p$, with respect to which the metric osculates to ord... | https://mathoverflow.net/users/35428 | Complex manifolds in which the exponential map is holomorphic | NB: I've had a little time to think about this and can now improve my answer, in particular, removing the real-analytic assumption, which, as I suspected, was not necessary. Here is the improved answer:
If the metric $g$ is Kähler, then having the exponential map from a point $p\in M$ be holomorphic makes it flat in ... | 24 | https://mathoverflow.net/users/13972 | 68766 | 42,246 |
https://mathoverflow.net/questions/68707 | 13 | Does there exist a group $G$ (finite or infinite) with three subgroups $A, B, C \leq G$ satisfying the following three conditions?
>
> 1. $A = N\_G(A)$, $B = N\_G(B)$, $C = N\_G(C)$;
> 2. $AB = BC = CA = G$;
> 3. $A \cap B = B \cap C = C \cap A = 1$.
>
>
>
(This question turned up in a more specific setting, b... | https://mathoverflow.net/users/12858 | Groups with triple system of self-normalizing subgroups | I revise earlier edits to give a coherent account of the construction which shows that such subgroups can exist.
The underlying idea of the strategy is as follows: Let $X$ be a non-trivial finite group with trivial center which admits an automorphism $\alpha$ fixing only the identity (informally, and by a slight abu... | 16 | https://mathoverflow.net/users/14450 | 68769 | 42,249 |
https://mathoverflow.net/questions/68657 | 1 | Q1.Given a quasi-projective variety $X$ over $\mathbf{C}$, is it always possible to find a $\Delta$-complex structure on $X$?
Q2. What is a good reference which gives a survey about what we know of $CW$-complex structures of quasi-projective varieties over $\mathbf{C}$?
| https://mathoverflow.net/users/11765 | On delta complex structures of complex quasi-projective varieties | There are very general triangulability results for real (semi)algebraic sets (sets cut out by inequalities of real polynomials), and even for semianalytic and subanalytic sets. Lojasiewicz has some papers from the 60s on semianalytic sets; Hironaka and Hardt also have papers on the subject; and the book Real Algebraic ... | 3 | https://mathoverflow.net/users/4042 | 68792 | 42,259 |
https://mathoverflow.net/questions/68201 | 7 | L is a holomorphic line bundle on a compact complex manifold X. The Kodaira dimension of L is defined as the maximal dimension of the image of the map associated to the powers $ mL(m \in N)$. I want to prove the asymptotic estimate
$$ h^0 (X,mL) \leq O(m^{k(L)})$$
I heard that it is an easy consequence of the Schwa... | https://mathoverflow.net/users/15882 | The asymptotic growth of global sections of powers of a complex line bundle | Hi,
An enlightening and very elementary proof of this fact can be found in the very complete book of X. Ma and G. Marinescu "Holomorphic Morse inequalities and Bergman kernels".
You will find this in Chapter 2. Their approach is exactly what you are looking for (only elementary complex analysis in several variables... | 3 | https://mathoverflow.net/users/9871 | 68796 | 42,261 |
https://mathoverflow.net/questions/43377 | 12 | We know that the classical Maass forms on GL(3) are depicted, for instance, in D.Goldfeld's book. I wonder that if there exists "holomorphic" automorphic forms on GL(3) as an analogue of GL(2) case. If those forms exist, where can I find the materials which concretely tell the story of them?
Thanks in advance.
| https://mathoverflow.net/users/1930 | Automorphic forms on GL(3) | As in the comments and earlier answer: in short, there is nothing comparably elementary or accessible for GL(3), as holomorphic things for GL(2).
Even the explication of this apparent fact is not, and perhaps could not be, as immediate as the direct exhibiting of holomorphic things for GL(2): to demonstrate the *abse... | 17 | https://mathoverflow.net/users/15629 | 68805 | 42,268 |
https://mathoverflow.net/questions/67957 | 12 | Background
----------
Recall (from Cisinski's Astérisque volume 308) that given a small category $A$, we define an $A$-localizer to be a class $W$ of morphisms of $\mathrm{Psh}(A)$ satisfying the following axioms:
* The class $W$ satisfies 2-for-3
* The class $W$ contains $\mathrm{rlp}(\mathrm{Mono}(A))$, where $\m... | https://mathoverflow.net/users/1353 | Is the simplicial completion of a localizer always a bousfield localization of the injective model structure? | Let $A$ be a small category, and $W$ an $A$-localizer. Then we say that $W$ is *regular* if any presheaf $X$ over $A$ is canonically the homotopy colimit of the representable presheaves above $X$; see Definition 3.4.13 (all references are in Astérisque 308). Except stated otherwise, all the assertions below about regul... | 19 | https://mathoverflow.net/users/1017 | 68809 | 42,270 |
https://mathoverflow.net/questions/68810 | 3 | Let $f: \Bbb{R}^n \rightarrow \Bbb{R}$ be a non-negative function that vanishes on a set $\Omega$ that is compact and has positive measure. What is the minimial amount of regularity required of $f$ to guarantee that $\Omega$ contains an open set? I'm interested in classes of the form $C^k$ or $C^{k,\alpha}$ ($k$-times ... | https://mathoverflow.net/users/15856 | Vanishing on Bad Sets | By the Whitney extension theorem any closed set of $\mathbb{R}^n$ can be the zero-set of a non-negative $C^\infty$ function. And, of course, there are closed sets with positive measure and empty interior.
| 4 | https://mathoverflow.net/users/6101 | 68812 | 42,272 |
https://mathoverflow.net/questions/68795 | 4 | Given an explicit polynomial, is there any kind of trick/algorithm to check whether it is a pfaffian of a matrix with linear entries?
The pfaffian can be defined as $\sqrt{{\rm det}(A) } $ when $A$ is skew symmetric, or explicitly $${\rm pf}(A) = \frac{1}{2^n n!}\sum\limits\_{\sigma \in S\_{2n}}{\rm sgn} (\sigma)\pr... | https://mathoverflow.net/users/4096 | Detecting if a polynomial is a Pfaffian | As Bruce Westubury noticed, the answer to this question is trivial as it is stated.
Surprisingly enough, however, the situation becomes very interesting when one considers representations of *homogeneous* polynomials as pfaffians of matrices with *linear* entries.
More precisely, let us consider the following versi... | 18 | https://mathoverflow.net/users/7460 | 68814 | 42,273 |
https://mathoverflow.net/questions/68813 | 6 | I have one issue with the Jacquet Langlands correspondence. The Weyl law for $H$ modulo a congruence subgroup and the Weyl law for cocompact groups are different. So why does this not contradict this functoriality? What am I missing?
I have not yet studied the Jacquet Langlands correspondence explicitly yet. How expl... | https://mathoverflow.net/users/10400 | Jacquet Langlands correspondence | In what sense is the Weyl law different for congruence subgroups and cocompact groups?
At any rate, the Jacquet-Langlands correspondence is not a bijection between the two cuspidal spectra. More precisely, let $D$ be a quaternion algebra over a number field $F$, and consider the groups $G=PD^\times$ and $G'=PGL\_2$.... | 4 | https://mathoverflow.net/users/11919 | 68815 | 42,274 |
https://mathoverflow.net/questions/68838 | 6 | For boolean algebra, let's take Roman Sikorski's *Boolean Algebras* as our reference. After giving a set of axioms, he proves (p.9) that the join of A and B is the least element of the algebra such that A and B are its subelements. He also asserts that since that's so, the join of A and B can be defined in terms of the... | https://mathoverflow.net/users/8224 | Is it possible to define a closure operator in terms of partial ordering? | Oh sure, this is quite well-known. The closure of an element is the smallest *closed* element which is greater than or equal to the given element. Dually for the interior operator.
In general, a closure operator $\phi: P \to P$ on a poset $P$ is an order-preserving, inflationary ($x \leq \phi(x)$), idempotent ($\phi... | 8 | https://mathoverflow.net/users/2926 | 68849 | 42,297 |
https://mathoverflow.net/questions/68098 | 4 | I want to show that conformally immersed Riemann surfaces in $\mathbb{R}^4$ are leaves of a 2-foliation $\mathcal{F}$. I start with the generalized Weierstrass representation of the surfaces: take 4 holomorphic functions $ \{\phi(z)\_\alpha, \psi(z)\_\alpha\},~\alpha=1,2$ that satisfy a Dirac equation
$\partial\_z \p... | https://mathoverflow.net/users/15387 | Conformally immersed Riemann surfaces and foliations | Um, $V,W$ are vector fields on the *surface* but not on $\mathbb{R}^4$. You need a distribution of $2$-planes on (an open subset of) $\mathbb{R}^4$ to use Frobenius to cook up a foliation.
| 2 | https://mathoverflow.net/users/1143 | 68864 | 42,305 |
https://mathoverflow.net/questions/68841 | 1 | Suppose that $(A,m)$ is a Noetherian local ring, $M$ is an $A$-finite module. Assume that $x\_1, ..., x\_n$ are elements in $m$. Is the following equality true:
$$
\mbox{ann}(M/(x\_1, ..., x\_n)M) = (x\_1, ..., x\_n) + \mbox{ann}(M).
$$
| https://mathoverflow.net/users/16012 | The annihilator of the quotient module | By modding out ${ann} (M)$ one can assume that $ann(M)=0$. Then the following is true:
$$I \subseteq ann(M/IM) \subseteq \bar I $$
Here $\bar I$ denotes the integral closure of $I$. You can prove it using the determinantal trick (the one used in the proof of Nakayama's Lemma). In particular equality happens if $I$ ... | 5 | https://mathoverflow.net/users/2083 | 68871 | 42,310 |
https://mathoverflow.net/questions/68772 | 7 | Let's consider $K\_t(M)$, the Kauffman bracket skein module (see [this](http://arxiv.org/abs/q-alg/9604013) and [this](http://www.ams.org/journals/tran/2000-352-10/S0002-9947-00-02512-5/) papers) of a three-manifold $M$. When $t=-1$, $K\_t(M)$ is easily seen to be isomorphic to the ring of functions on the character va... | https://mathoverflow.net/users/35353 | Trace identities and the Kauffman Bracket skein module | The old answer is that the trace identity you give in 2) is not quite right,
Let $R=\sum\_i a\_i\otimes b\_i$ be the $R$ matrix for $U\_q(sl\_2)$ and let $t$ be the
$4$th root of $q$, then
$$t tr(XY)+ t^{-1}tr(S(X)Y)=\sum\_itr(a\_iX)tr(b\_iY),$$
where $S$ is the antipode and $tr$ is the ordinary trace in the fundam... | 10 | https://mathoverflow.net/users/4304 | 68876 | 42,314 |
https://mathoverflow.net/questions/68875 | 13 | In a certain model of a stat-physics type, one encounters a matrix
$$
A\_n:=\left[\binom{n}{2j-i}\right]\_{i,j=1}^{n-1}.
$$
The determinant of this matrix (equal to $2^{\binom n2}$) counts the number of all possible configurations, and our understanding of the model would greatly increase if we would know the inverse ... | https://mathoverflow.net/users/979 | How to invert the matrix [n choose 2j - i] ? | This is more an idea to explore than a complete answer.
You may interpret the binomial coefficient $\binom{n}{k}$ as the elementary symmetric function $e\_k$ of $1,1,\ldots,1$ ($n$ variables evaluated at $1$). The coefficients of the adjoint matrix of $A\_n$ become skew Schur functions of $1,1,\ldots,1$. Then there ... | 11 | https://mathoverflow.net/users/6768 | 68880 | 42,316 |
https://mathoverflow.net/questions/65107 | 8 | I am reading an [outstanding paper](http://arxiv.org/pdf/1101.5851) by Bateman and Katz, improving the best known bounds on the cap set problem (Roth's theorem over $\mathbb{F}\_3^N$).
The paper contains some technical lemmas for which I believe there must be an excellent geometric intuition -- which I am afraid I am... | https://mathoverflow.net/users/1050 | Fourier analysis, orthogonality, and Plancherel for finite abelian groups | I don't know whether it would be perceived as "geometric", but an intuition that "works for me" on this and related matters is that "Fourier analysis" on finite abelian groups is "abelian" Fourier analysis (e.g., on products of circles or lines, in the classical analytic scenarios) without the need to "do analysis".
... | 5 | https://mathoverflow.net/users/15629 | 68886 | 42,319 |
https://mathoverflow.net/questions/68888 | 4 | According to the Kneser-Milnor prime decomposition theorem for 3-manifolds, any compact, connected, orientable 3-manifold $M$ is diffeomorphic to $S^3 / \Gamma\_1$ # $\cdots$ # $S^3/ \Gamma\_n$ # $(S^2 \times S^1)\_1$ # $\cdots$ # $(S^2 \times S^1)\_r$ # $K( \pi\_1,1)$ # $\cdots$ # $K( \pi\_m,1)$, where # is the connec... | https://mathoverflow.net/users/12782 | Is a compact, connected, orientable 3-manifold with $\mathbb{Z}^K$ fundemental group uniquely determined? | NO, since the three-torus $T^3$ does not have this form.
**EDIT** if the OP really means a free product of $\mathbb{Z}$s, so the free group $F\_k,$ then the answer is YES. It is a fact (see Hempel's book, chapter 7) that every splitting of the fundamental group of $M^3$ as a free product comes from a connected sum de... | 10 | https://mathoverflow.net/users/11142 | 68889 | 42,321 |
https://mathoverflow.net/questions/68890 | 8 | Let $S$ be a finite set. Now $\mathop{End}(S)$ is a monoid, and we may build a ring $R$ by allowing formal sums of functions.
Preliminary questions, since $R$ is surely well-known: What is it called? In a general category, what is the name of the construction that builds a ring out of the endomorphisms of an object?
... | https://mathoverflow.net/users/9068 | The ring generated by all functions from a set to itself | The monoid of all maps on $n$ letters is denoted $T\_n$ and called the full transformation monoid. Your intuition is both right and wrong. The irreducible representations of $T\_n$ are in bijection with irreducible representations of all symmetric groups of degree at most $n$. The character table is block upper triangu... | 13 | https://mathoverflow.net/users/15934 | 68894 | 42,323 |
https://mathoverflow.net/questions/68891 | 4 | It is curious to know whether the following assertion is ture or not?
If $A-B$ and $B$ are copositive matrices (implying $A$ is copositive) of the same size, then $\rho(A)\ge \rho(B)$, where $\rho$ means the spectral radius.
For positive definite matrices class and nonnegative (entrywise) matrices class, this is ob... | https://mathoverflow.net/users/3818 | Spectral order of copositive matrices | The assertion is false. Here is how to construct a counterexample.
1. Let $A = XX^T + Y + Y^T$ where $Y \ge 0$ (elementwise)
2. Let $B = XX^T$
Then, by construction $A$ is a copositive matrix (sum of semidefinite plus symmetric nonnegative matrix), and $B$ is copositive too (because it is semidefinite). Moreover, $... | 4 | https://mathoverflow.net/users/8430 | 68900 | 42,326 |
https://mathoverflow.net/questions/68899 | 6 | I was rereading basic results on de Rham cohomology, and this led me inevitably to the fact that $H^q(X,\Omega^p)$ converges to $H^\*(X)$ for any smooth proper variety (over any field). How does one view this spectral sequence "maturely" as a Grothendieck spectral sequence?
| https://mathoverflow.net/users/5756 | How does one view the De Rham spectral sequence as a Grothendieck spectral sequence? | If by "Grothendieck spectral sequence" you mean the spectral sequence associated to the composite of functors (fulfilling the Grothendieck condition) then I am skeptical as to whether this is possible. Also I do not see that there would be any particular point in being able to view it in that light (unless the functors... | 6 | https://mathoverflow.net/users/4008 | 68903 | 42,328 |
https://mathoverflow.net/questions/68906 | 3 | Dear Sir/friends,
How to give manifold structure to set of all $C^2$ path over any manifold.
| https://mathoverflow.net/users/16031 | Is it possible to see Path Spaces as manifold | If by "path" you mean a map with domain $[0,1]$ then this is a standard construction and is independent of the class of maps (providing it is contained in $C^0$). You can find it in many places, search MathSciNet for "manifold" and "mapping space", or you can *almost* find it in my paper [Constructing Smooth Manifolds ... | 5 | https://mathoverflow.net/users/45 | 68907 | 42,330 |
https://mathoverflow.net/questions/68883 | 2 | Consider a symmetric algebra $H$ over a field $k$. By definition, this is a $k$-algebra $H$ with a *symmetrizing trace* $\tau$, which is a $k$-linear map $\tau:H\to k$ such that $\tau(hh')=\tau(h'h)$ for all $h,h'\in H$ and the corresponding bilinear form is non-degenerate. I have been using chapter 7 of Characters of ... | https://mathoverflow.net/users/3318 | When does a symmetric algebra over a field of characteristic 0 fail to be semisimple? | The answer is no. The reason is that every algebra can be embedded into a symmetric algebra, the so called trivial extension:
If $A$ is a $K$-algebra, then define $D(A):=A\oplus Hom\_K(A,K)$. $I:=Hom\_K(A,K)$ is a $A$-$A$-bimodule via
$a\cdot \phi \cdot b:=x\mapsto \phi(bxa)$
Hence you can define an $K$-algebra s... | 6 | https://mathoverflow.net/users/3041 | 68918 | 42,335 |
https://mathoverflow.net/questions/66272 | 1 | Let's suppose that a language $L \in \operatorname{NSPACE}(f(n))$ where $f(n) = \Omega(\log(n))$. And now let's suppose that i have a probabilistic turing machine. Can this machine run in $O(f(n))$ space and answer yes for a $x \in L$ with Pr(yes)>1/2 and for a x that doesn't belong,answer no with Pr(no)=1? Le's suppos... | https://mathoverflow.net/users/15379 | Can i achieve something better with the probabilistic turing machine in matter of space? | Yes, if you do not care about running time, then you can simulate nondeterminism by a randomized algorithm with only a linear increase in space.
Assume that $f(n)\ge\log n$ is space-constructible, and let $L\in\mathrm{NSPACE}(f)$. By definition, there exists a nondeterministic Turing machine $M\_0$ working in space $... | 2 | https://mathoverflow.net/users/12705 | 68933 | 42,342 |
https://mathoverflow.net/questions/68920 | 9 | The group $GL\_n(\mathbb{Z})$ acts properly and isometrically on the space of homothety classes of scalar products on $\mathbb{R}^n$. This is a Riemannian manifold of nonpositive sectional curvature.
Is there a similar space for the case of $GL\_n(F\_p[x])$. Maybe one can construct a building or something like this. ... | https://mathoverflow.net/users/3969 | On which space does $GL_n(F_p[X])$ act nicely? | The group $GL\_n(\mathbb{F}\_p[x])$ acts on the Bruhat-Tits building for $GL\_n$. The vertex set is $GL\_n(\mathbb{F}\_p((x^{-1})))/GL\_n(\mathbb{F}\_p[[x^{-1}]])$, and the higher simplices form sets of the form $GL\_n(\mathbb{F}\_p((x^{-1})))/I$ for various parahoric groups $I$. The action of $GL\_n(\mathbb{F}\_p((x^{... | 10 | https://mathoverflow.net/users/121 | 68935 | 42,343 |
https://mathoverflow.net/questions/68919 | 2 | Is there an integer $m\geq 1$ such that $2^m+3^m$ is a perfect power?
| https://mathoverflow.net/users/75935 | ${2}^{p}+{3}^{p}={a}^{n}$ , then n=1 for any p ? | If you really wanted to prove this (and I'm afraid that I'm not sure why you would), you could invoke a Theorem of Darmon and Merel for $n=2$ and $3$, check that there are no solutions with $p \leq 5$, say, and then write down the usual $(n,n,n)$ Frey curve, assuming $n \geq 5$ is prime (which leads to a weight $2$, le... | 6 | https://mathoverflow.net/users/7302 | 68940 | 42,345 |
https://mathoverflow.net/questions/68902 | 1 | Hello?
I have a simple question.
Is $\mathbb{Z}\_p$ flat $\mathbb{Z}\_pG$-module for a finite $p$-group $G$?
Here, $p$ is prime and $\mathbb{Z}\_p$ means the integers localized at $(p)$.
If not, is it false even for a finite abelian $p$-group $G$?
Please let me know.
| https://mathoverflow.net/users/15728 | Is $\mathbb{Z}_p$ flat $\mathbb{Z}_pG$-module for a finite $p$-group $G$? | *(This is answering a comment to the main question)*
$\newcommand\ZZ{\mathbb Z}$
If $G$ is cyclic of order $p$, then there is a resolution of $\ZZ$ looking like $$\cdots\to\ZZ G\xrightarrow{d\_{\mathrm{odd}}} \ZZ G\xrightarrow{d\_{\mathrm{even}}}\cdots\to\ZZ G\xrightarrow{d\_{\mathrm{even}}} \ZZ G\xrightarrow{d\_{\ma... | 3 | https://mathoverflow.net/users/1409 | 68943 | 42,347 |
https://mathoverflow.net/questions/68921 | 1 | In every lecture on Riemannian geometry it is standard to prove that geodesic curves are locally length minimizing.
The only thing I find confusing about this is, that here length minimizing means: compared to all piecewise smooth curves
in contrast to, say, all continuous curves. So my question is:
Are geodesics loc... | https://mathoverflow.net/users/1272 | Are geodesics locally minimizing in continuous curves? | Your question will be trivial once you give a definition of the length of curve in a Riemannian manifold.
For example, you may define distance as infimum of lengths piecewise smooth curves connecting given points.
Then you define length of general curve as you do it in a metric space...
| 6 | https://mathoverflow.net/users/1441 | 68945 | 42,349 |
https://mathoverflow.net/questions/68953 | 5 | The (right) big finitistic dimension of a ring is Findim$(R) =$ sup{proj.dim(M) | $M$ a right $R$-module of finite projective dimension}. The (right) little finitistic dimension findim$(R)$ is the sup over f.g. right modules of finite projective dimension.
The right global dimension of a ring is r.gl.dim$(R) =$ sup{p... | https://mathoverflow.net/users/11540 | An example where finitistic dimension does not equal right global dimension? | A non-semisimple self-injective algebra like $k[t]/(t^2)$ has the property that its modules are either of infinite projective dimension or projective. So its Findim is actually zero, while its gldim is infinite.
For your second question, pick a ring $R$ with global dimension $n$ and consider the direct product ring $... | 8 | https://mathoverflow.net/users/1409 | 68957 | 42,354 |
https://mathoverflow.net/questions/68936 | 23 | I want to understand algebraic geometry from the functorial viewpoint. I've found a set of notes (linked below) that develop algebraic geometry from the elementary beginnings in this framework. They go under the name "Introduction to Functorial Algebraic Geometry" (following a summer course held by Grothendieck), and a... | https://mathoverflow.net/users/2857 | Source on functorial algebraic geometry | Since your question might interest other readers, allow me to expand it.
Given a scheme $T$, you can associate to it the contravariant functor $h\_T: \mathcal{ Schemes}^\text{opp} \to \mathcal{Sets}$. In a nutshell, Eivind's request is for documents showing how you can study the scheme $T$ by studying the functor $h... | 30 | https://mathoverflow.net/users/450 | 68958 | 42,355 |
https://mathoverflow.net/questions/68938 | 2 | Hi all, here's my question which I have no idea how to approach.
Fix a complex number q such that |q| < 1. Describe all entire functions f such that f(z)/f(qz) is a linear function of z.
| https://mathoverflow.net/users/16040 | Entire function with special conditions | Consider zeros of $f$. If $f(0)=0$, we can, for some n, write $f(z)=z^n g(z)$, $f(qz)=z^n q^n g(qz)$, and we find that $g(z)/g(qz)$ is also a linear function. Now let us say $g(z)/g(qz)=az+b$. By plugging in $z=0$, we find $b=1$. Moreover, $g(-1/a)=0$. Since $az+1$ has no poles, we recursively find $g(-1/(qa))=g(-1/(q^... | 1 | https://mathoverflow.net/users/12120 | 68959 | 42,356 |
https://mathoverflow.net/questions/68952 | 24 | Is there an elliptic curve in CP^2 whose induced Remannian metric ( induced from the Fubini-Sudy metric on CP^2) is Euclidian flat?
| https://mathoverflow.net/users/1643 | Geometry of complex elliptic curves | According to this [paper](http://www.numdam.org/numdam-bin/item?id=CM_1977__35_1_57_0) by Linda Ness the Gaussian curvature of a curve $C\subset \mathbb P^2$ defined by the
zeros of a degree $d>1$ homogeneous polynomial $F \in \mathbb C[x,y,z]$ at a smooth
point $p$ is given by
$$
K(p) = 2- \frac{\|p\|^6 \cdot | \rm{H... | 28 | https://mathoverflow.net/users/605 | 68963 | 42,358 |
https://mathoverflow.net/questions/68956 | 2 | Let $R$ be a noetherian local ring and let $M$ be a finite $R$-module. Assume that the annihilator of $M$ is zero. Consider a minimal presentation of M as follows: $R^n\stackrel{\varphi}{\longrightarrow}R^m\longrightarrow M\longrightarrow0$. Can we conclude that $m>n$, or is it also possible to have $m\leq n$ with all ... | https://mathoverflow.net/users/16046 | Presentation of finite modules with null annihilator | Graham's comment gave some simple counterexamples. I will show that even if $R$ is nice, say a Gorenstein domain, there will always be a lot of counter-examples.
Let $M$ be a non-free maximal CM module over $R$. Consider a minimal presentation:
$$ 0 \to N \to R^n \to R^m \to M \to 0 $$
If $m\leq n$ we found our coun... | 2 | https://mathoverflow.net/users/2083 | 68964 | 42,359 |
https://mathoverflow.net/questions/68961 | 1 | let G be an algebraic group. which subgroups of G are codimension one subgroups.
| https://mathoverflow.net/users/16049 | what are the subgroups of an algebraic group with codimension one | Perhaps it is better to phrase the question in terms of Lie algebras. For instance, if you want to know which are the possible codimension one Lie subalgebras of a given finite dimensional Lie algebra then there is a result of [Tits](http://www.ams.org/mathscinet-getitem?mr=120308) which address exactly this.
>
> ... | 8 | https://mathoverflow.net/users/605 | 68965 | 42,360 |
https://mathoverflow.net/questions/68950 | 20 | Let $k$ be a field. In 1984 Andreas Blass [proved](http://www.math.lsa.umich.edu/~ablass/bases-AC.pdf) that the axiom "for every extension $K|k$, every vector space over $K$ has a basis" implies the axiom of choice. He also raised the question
>
> Does the axiom "every vector space over $k$ has a basis" imply the ... | https://mathoverflow.net/users/10194 | Axiom of choice and bases of vector spaces over a fixed field | It has been shown for $K=\mathbb F\_2$ (the field with two elements) by Keremedis ([Available here](http://www.ams.org/journals/proc/1996-124-08/S0002-9939-96-03305-9/S0002-9939-96-03305-9.pdf))
In [the dictionary of AC equivalences](http://consequences.emich.edu/file-source/htdocs/conseq.htm) it shows that not a lot... | 15 | https://mathoverflow.net/users/7206 | 68966 | 42,361 |
https://mathoverflow.net/questions/68960 | 6 | Let $M$ be a pseudo-Riemannian manifold. Assume that $M$ comes with a zero-torsion affine connexion $\nabla.$ There is no need for $\nabla$ to be the Levi-Civita connexion. Recall that the curvature tensor of $\nabla$ is given by $R(X,Y)Z = \nabla\_X\nabla\_YZ - \nabla\_Y\nabla\_XZ - \nabla\_{[X,Y]}Z,$ while the Ricci ... | https://mathoverflow.net/users/16048 | Symmetric Ricci Tensor | They are both correct. `:-)`
I gave a somewhat detailed write-up last year on my [blog](http://williewong.wordpress.com/2009/12/09/parallel-volume-forms/#more-231), but the gist of the argument is that if $\tau = 0$, then $d\tau = 0$. On the flip side, if $d\tau = 0$, locally you can lift $\tau = du$ for some functio... | 8 | https://mathoverflow.net/users/3948 | 68972 | 42,365 |
https://mathoverflow.net/questions/68973 | 7 | Let $\beta:\widetilde{X}\mathrel{\mathop:}=\mathop{\mathrm{Bl}}\_Z(X)\to X$ be the blow-up of a nonsingular algebraic variety $X$ along a nonsingular subvariety $Z$. Let $E\mathrel{\mathop:}=\beta^{-1}(Z)$ be the exceptional divisor. Now, let us assume I have a divisor $D$ on $X$. Then I was told that $\beta^\ast D \si... | https://mathoverflow.net/users/9947 | Strict Transform under Blow-Up along nonsingular Subvariety | First of all, let $X$ be a smooth variety and $D$ an effective divisor on $X$. Denote $\operatorname{mult}\_x(D)$ the multiplicity of $D$ at a point $x\in X$. The function $x\mapsto\operatorname{mult}\_x(D)$ is known to be upper-semicontinuous on $X$. Therefore, if $Z\subset X$ is any irreducible subvariety, one can de... | 7 | https://mathoverflow.net/users/9871 | 68977 | 42,369 |
https://mathoverflow.net/questions/68825 | 15 | Constructing quantum field theories is a well-known problem. In Euclidean space, you want to define a certain measure on the space of distributions on R^n. The trickiness is that the detailed properties of the distributions that you get is sensitive to the dimension of the theory and the precise form of the action.
I... | https://mathoverflow.net/users/14689 | Quantum field theory in Solovay-land | I don't know anything about the space of all distributions dual to smooth test functions, but do know a fair bit about computable measure theory (from a certain perspective).
First, you mention that you have a computable algorithm which generates a probability distribution. I believe you are saying that you have a c... | 5 | https://mathoverflow.net/users/12978 | 68981 | 42,371 |
https://mathoverflow.net/questions/68980 | 9 | I like to think in terms of commutative diagrams rather than referring to elements. So to me a group is really a [group object](http://eom.springer.de/g/g045250.htm), i.e. an object with some maps satisfying certain commutative diagrams. You can define rings and modules similarly. You can define a field object as a com... | https://mathoverflow.net/users/11540 | Is there a way to define a prime ideal object via diagrams in the category of rings? | The lattice of ideals of $R$ is isomorphic to the lattice of regular quotients of $R$. Here, a regular quotient is an equivalence class of regular epimorphisms $R \to S$ in the category of rings (which are precisely the surjective ring homomorphisms). So this also serves as a categorical definition of an ideal. Also th... | 13 | https://mathoverflow.net/users/2841 | 68982 | 42,372 |
https://mathoverflow.net/questions/68947 | 3 | This is a follow-up question to
[When does a symmetric algebra over a field of characteristic 0 fail to be semisimple?](https://mathoverflow.net/questions/68883/when-does-a-symmetric-algebra-over-a-field-of-characteristic-0-fail-to-be-semisim)
Let $H$ be a symmetric algebra over $\mathbb{R}$ with symmetrizing trace ... | https://mathoverflow.net/users/3318 | Algebra with positive definite symmetrizing trace is semisimple. | There is a lemma due to Dieudonné that is similar and may be the generalization you are seeking. (My reference is Nathan Jacobson's book "Structure and Representations of Jordan algebras", in Chapter VI on pages 239, 240.) We replace the real numbers with a field $F$---this can be any field.
Let $A$ be a finite-dimen... | 5 | https://mathoverflow.net/users/6486 | 68993 | 42,377 |
https://mathoverflow.net/questions/68998 | 4 | I'm writing for a broad audience about a collection of topics of math, and including the dates of birth and death of all people mentioned, just to help the reader to keep a general idea of when different things happened. I still could not manage to find the dates for Meigu Guan (who contributed with the formulation of ... | https://mathoverflow.net/users/16050 | Basic biographical data of contemporary mathematicians | I tried to find out DOB and DOD for Meigu Guan myself while preparing a biographical addendum for a book, but haven't succeeded - I hope someone will help both you and me. H. Peyton Young was born in 1945 (source: Library of Congress cataloging-in Publication Data).Michel Balinski was born in 1933 (source: A mathematic... | 2 | https://mathoverflow.net/users/4925 | 69003 | 42,383 |
https://mathoverflow.net/questions/69005 | 4 | Let $(\Omega, \mathcal{F}, P)$ be a probability space, on which
$\mathcal{F}\_t$ is filtration satisfying general conditions. $W\_t$ is
a standard Brownian motion. By law of iterated logarithm, one has
$$ P\Big(\lim\_{t\to \infty} \frac{W\_t}{t} = 0\Big) = 1 $$
This implies
$$ P\Big(\lim\_{t\to \infty} \frac{t + W\_t}{... | https://mathoverflow.net/users/5656 | law of iterated logrithm | Girsanov's theorem tells you that on any *finite* interval $[0,T]$ you can find an equivalent probability that makes $t+B\_t$ a Brownian motion. You just gave the proof that it cannot be done on the whole real line.
| 10 | https://mathoverflow.net/users/1590 | 69010 | 42,387 |
https://mathoverflow.net/questions/69007 | 8 | Can someone explain (or give a reference) why the Galois representation attached to a p-divisible group over the ring of integers of a p-adic ring is Crystalline?
| https://mathoverflow.net/users/9516 | The Galois representation of a p-divisible group is crystalline | This is shown in §6 of Fontaine's paper "Sur certains types de représentations p-adiques du groupe de Galois d'un corps local; construction d'un anneau de Barsotti-Tate", see the point i) after Theorem 6.2.
The paper is available on jstor: <http://www.jstor.org/pss/2007012>
| 8 | https://mathoverflow.net/users/2308 | 69011 | 42,388 |
https://mathoverflow.net/questions/68954 | 7 | Let $X$ be a (singular, reducible) affine plane real algebraic curve of degree $d$.
How we can estimate maximal number of connected components of it's complement in $R^2$ in terms of degree?
| https://mathoverflow.net/users/16044 | Maximal number of connected components of complement to an affine plane real algebraic curve | We can also prove that the maximal number of components for reducible nonsingular curve is $\frac{n^2+n+2}{2}$ by the following computation.
Let $m$ be a partition of $n$, where $m\_i$ is the degree of $i$-th irreducible component of our curve.
Denote $s=\{i|m\_i=1\}$, $t=\{i|m\_i>1\}$.
Then, using Harnack inequality... | 2 | https://mathoverflow.net/users/16044 | 69013 | 42,390 |
https://mathoverflow.net/questions/68994 | 3 | Say we have a PID $R$, integers $1 \leq a \leq b$, and $R$-homomorphisms $R^a \stackrel f\to R^b \stackrel g\to R^a$ with $g \circ f$ of full rank.
For $h = f, g, g \circ f$, let $c(h)$ be the characteristic ideal of $\mathrm{coker}(h)\_\mathrm{tors}$, i.e. the index of the image of $h$ in its saturation. (In particu... | https://mathoverflow.net/users/367 | spurious torsion under compositions of linear maps | Here is an obvious way to construct examples with $a=1,b=2$. By elementary divisor theory, with these numerics one always has $f=uf\_0,g=vg\_0$ with $c(f\_0),c(g\_0)=1$ for some $u,v \in R$, so it suffices to study the situation with $f,g$ replaced by $f\_0,g\_0$, i.e. to assume that $c(f),c(g)=1$.
Take $w \in R$ arb... | 0 | https://mathoverflow.net/users/367 | 69017 | 42,393 |
https://mathoverflow.net/questions/69002 | 3 | Let $X\subset \mathbb{P}\_{\mathbb{C}}^N$ be irreducible generically smooth closed subscheme and
let $\mathrm{Hilb}\_{lines}^{x}(X)$ denote
the Hilbert scheme of lines contained in $X$
and passing through the point $x\in X$.
Is it true that the set
$$
\{ x\in X : \mathrm{Hilb}\_{lines}^{x}(X) \mbox{ is smooth } \}
$$... | https://mathoverflow.net/users/15606 | On the place where $\mathrm{Hilb}_{lines}^{x}(X)$ is smooth. | Yes. This can be seen as follows:
Let $Hilb\_{lines}(X)$ denote the Hilbert scheme of lines in $X$ and let $\Gamma \subset X \times Hilb\_{lines}(X)$ be the correponding universal family. Then $Hilb\_{lines}^x(X) = p^{-1}(x)$ where $p:\Gamma \to X$ is induced by the first projection. So we are reduced to the folowing... | 5 | https://mathoverflow.net/users/519 | 69023 | 42,398 |
https://mathoverflow.net/questions/69024 | 1 | Given a fiber bundle $p:E\to B$ and a point $x\in B$, is the evaluation map $\varepsilon:\Gamma^0(E)\to p^{-1}(x)$ defined by $\varepsilon(\sigma):=\sigma(x)$ a weak homotopy equivalence when $\Gamma^0(E)$ is endowed with the compact-open topology? I'm having trouble coming up with a good counter example but also faile... | https://mathoverflow.net/users/16062 | Are evaluation maps for sections of a fiber bundle weak homotopy equivalences? | In your situation, the base space (a disk) is contractible, so your smooth fiber bundle is actually diffeomorphic to a product bundle, i.e. $E = B\times p^{-1} (x)$. (This follows, for example, by passing to the associated principal bundles - with structure group the diffeomorphism group of the fiber, say.) Now the spa... | 1 | https://mathoverflow.net/users/4042 | 69030 | 42,402 |
https://mathoverflow.net/questions/69031 | 1 | 1)Why embedding of ( not necessarily finite-dimensional) vector spaces $V\rightarrow W$ produces embedding of tensor algebras $T(V)\rightarrow T(W)$.
I can prove it using Hamel basis in $W$ but is there a nicer ( more functorial ) argument?
2) How to prove the same statement for modules over an algebra instead of vect... | https://mathoverflow.net/users/15805 | Tensor algebra question | If $V$ is a subspace of $W$, consider the inclusion $f:V\to W$ and any map $g:W\to V$ such that $g\circ f=1\_V$; to construct $g$, you need to use bases or something equivalent, for it does not exist over, say, a general ring...
Now $T(-)$ is a functor, so $T(g)\circ T(f)=T(1\_V)=1\_{T(V)}$. It follows that the map $... | 2 | https://mathoverflow.net/users/1409 | 69033 | 42,404 |
https://mathoverflow.net/questions/69044 | 5 | I just came across Charles Weibel's [Development of Algebraic K-Theory until 1980](http://www.math.rutgers.edu/~weibel/papers-dir/khistory.pdf), and found it really helpful. Is there been anything analogous which surveys the developments in the last 30 years? I'd be particularly interested in understanding links (if th... | https://mathoverflow.net/users/9581 | Survey of Algebraic K-Theory Since 1980? | I recommend the [Handbook of K-theory](http://www.math.uiuc.edu/K-theory/handbook/). It was published in 2005 and Part II seems to contain what you're looking for.
| 6 | https://mathoverflow.net/users/11540 | 69045 | 42,407 |
https://mathoverflow.net/questions/69019 | 1 | I am an undergrad student who wants to know about the representation theory over
arbitrary finite fields or finite rings of characteristic p (p a prime). (called modular
representation theory.)
In particular, I would like to study the representation theory of modular Heisenberg groups
(entries in a finite fiel... | https://mathoverflow.net/users/16060 | Does it make sense that "Representations of groups over finite ring" ? | EDIT: Everything below the line is from before the OP changed his question. It was attempting to study linear maps $\rho: R \rightarrow GL(V)$ for $R$ a finite ring and $V$ a vector space. Now that the question has been cleared up, we see that the OP desired to study $\rho: G \rightarrow GL(R)$ for $G$ a group and $R$ ... | 2 | https://mathoverflow.net/users/11540 | 69049 | 42,411 |
https://mathoverflow.net/questions/69059 | 10 | I am wondering if there is a nice presentation of the Hochschild cohomology of $A\_n$ the Weyl algebra. It is known that $H^m(A\_n,A\_n)=0$ for $m>0$, and thus it is rigid. A proof can be found [in Sridharan](http://www.ams.org/journals/tran/1961-100-03/S0002-9947-1961-0130900-1/S0002-9947-1961-0130900-1.pdf), but this... | https://mathoverflow.net/users/348 | A simple proof of the Weyl algebra's rigidity. | (I write $HH^\bullet(\Lambda)$ what you write $H^\bullet(\Lambda,\Lambda)$. When I become Emperor of Notation, everyone will!)
Since $A\_n\cong A\_1\otimes\cdots\otimes A\_1$, the Künneth formula for Hochschild cohomology (which is proved in Cartan-Eilenberg, Theorem XI.3.1, for example) tells you that $HH^\bullet(A\... | 19 | https://mathoverflow.net/users/1409 | 69065 | 42,416 |
https://mathoverflow.net/questions/68111 | 32 | Let G be a finite directed acyclic graph, with sources $A=\{a\_1,\ldots,a\_n\}$ and sinks $B=\{b\_1,\ldots,b\_n\}$, with edge weights $w\_{ij}$. The *weight* of a directed path P is the product of weights of edges in P. Set
$e(a,b)= \sum\limits\_{P\colon\, a\to b}w(P).$
Then we can form a matrix $M=\left(e(a\_i,b... | https://mathoverflow.net/users/2051 | How much linear algebra can be done with graphs? | I will try to contribute a partial answer. First I want to comment on the Lindstrom-Gessel-Viennot determinant coming from quantum mechanics stuff, in physics this is known as the [Slater determinant](http://en.wikipedia.org/wiki/Slater_determinant), giving the formula for the wavefunction of a multi-fermionic system. ... | 14 | https://mathoverflow.net/users/2384 | 69067 | 42,418 |
https://mathoverflow.net/questions/69060 | 5 | I am trying to show that for an elliptic curve $E/K$ with complex multiplication the action of $G\_{\overline{K}/ K}$ on the $T\_{l}(E)$, the Tate module is abelian.
An approach: Let $\rho$ denote the Galois representation on the (rational) Tate module $(T\_{l} \otimes \mathbb{Q}\_{l})$.
Complex Multiplication imp... | https://mathoverflow.net/users/14812 | Basic Question on action of Galois group on Tate module | Since $E$ has CM over $K$, the ring $F:= \operatorname{End}\_K(E) \otimes \mathbb{Q}$ is an imaginary quadratic field. Suppose $\ell$ is a prime integer unramified in $F$. Now $F\_\ell:= F \otimes\_\mathbb{Q} \mathbb{Q}\_{\ell}$ is either two copies of $\mathbb{Q}\_{\ell}$ or a quadratic extension of $\mathbb{Q}\_{\ell... | 6 | https://mathoverflow.net/users/11786 | 69070 | 42,421 |
https://mathoverflow.net/questions/69064 | 10 | To explain, I will use the following concrete example: Let $\mathcal{M}\_g$ be the functor for the moduli problem of classifying genus $g$ smooth projective curves (taking a scheme $S$ to the set of ways that $S$ parametrizes genus $g$ curves). This, as is well known, has a coarse moduli space: $M\_g$.
Let $\sigma \i... | https://mathoverflow.net/users/5309 | Do coarse moduli spaces respect Galois actions? | If $X$ is the coarse moduli scheme associated to a functor $F$ on schemes, then in particular, there is a natural transformation $F\to h\_X$, where $h\_X$ is the functor of points of $X$.
Unless I am missing something, if you apply the naturality of this transformation to the map $\mathrm{Spec}(\mathbb{C}) \to \math... | 4 | https://mathoverflow.net/users/12107 | 69071 | 42,422 |
https://mathoverflow.net/questions/29961 | 9 | **My question is:**
How do I find **sharp** upper bounds on $P(|q|\leq \epsilon)$ uniformly over a set of gaussian polynomes $q$ of degree two.
**Notations and definitions** (to make the question rigorous)
* Let me define $\mathcal{X}\_{2}^\*$ as the set of real random variables $q$ that can be written $$q=c+\sum\... | https://mathoverflow.net/users/6531 | Small crown probabilities (and infinite dimensional margin assumption) | Here is a solution for problem 2, with power $1/2$, using your idea 1. First some computations. Let A, B real numbers, let z ~ N(0,1), and X=$B(z^2-1) + Az$. The Fourier transform of the distribution of X (i.e.: the characteristic function of X with some $\pi$) is
$\; \; \; \;\; \; \; \;E(exp(-2\pi i\xi X) = \frac{e^... | 1 | https://mathoverflow.net/users/15828 | 69082 | 42,426 |
https://mathoverflow.net/questions/69077 | 5 | So let $f(x)\in\mathbf{Z}[x]$ be a monic polynomial of degree $d$ and let $K$ be the splitting field of $f$. Let us define
the "heigt of $f$" $:=||f||$ to be the maximum of the abolute values of
the coefficients of $f$. (Instead of the height it might be better to work with the abolute value of the discriminant of $K... | https://mathoverflow.net/users/11765 | Effective Chebotarev density results for arbitrary number fields | You are looking for the very useful paper
Effective versions of the Chebotarev density theorem, J. C. Lagarias and A. M. Odlyzko, pp. 409-464 in Algebraic Number Fields, A. Frohlich (ed.), Academic Press, 1977.
The bounds there are quite large, as I recall, especially if you don't want to assume GRH. There is very... | 10 | https://mathoverflow.net/users/431 | 69087 | 42,429 |
https://mathoverflow.net/questions/69085 | 18 | In Chern-Weil theory, we choose an arbitrary connection $\nabla$ on a complex vector bundle $E\rightarrow X$, obtain its curvature $F\_\nabla$, and then we get Chern classes of $E$ from the curvature form. A priori it looks like these live in $H^\*(X;\mathbb{C})$, but by an argument that I don't really understand they'... | https://mathoverflow.net/users/303 | Can one use Atiyah-Singer to prove that the Chern-Weil definition of Chern classes are $\mathbb{Z}$-cohomology classes? | It's true that in some sense the Atiyah-Singer Index Theorem has led to some integrality results. The theorem states that for an elliptic operator on a compact manifold two numbers are equal. One of them, the "analytic index", is obviously an integer. The other one, the "topological index", which depends on the symbol ... | 15 | https://mathoverflow.net/users/6666 | 69090 | 42,431 |
https://mathoverflow.net/questions/69089 | 4 | I am looking for a paper by Irena Swanson on a result on comparison of ordinary and symbolic powers of prime ideals in complete local rings.
The paper is referenced in problem 0.9 here
<https://aimath.org/WWN/integralclosure/Huneke.pdf>
I don't know the name of the paper, and so far my searches have been to no avail. I... | https://mathoverflow.net/users/16078 | Paper by I. Swanson on symbolic powers | That would be number 11 on her paper site "Linear equivalence of topologies".
As for Problem 0.9, it is known for regular local rings over fields by Ein-Lazarsfeld-Smith and Hochster-Huneke. The most recent result is for isolated singularities, see the paper Craig Huneke. Daniel Katz. Javid Validashti. "Uniform equiv... | 3 | https://mathoverflow.net/users/2083 | 69092 | 42,433 |
https://mathoverflow.net/questions/69080 | 5 | I'm thinking about properties of "limits" of p-adic representations, in the following sense.
Notations: $p$ denotes a prime. For a field $F$, let $G\_F$ be the absolute Galois group of $F$. Representations are always continuous.
Definition: Let $\rho:\ G\_{\Bbb{Q}\_p}\rightarrow GL\_d(\Bbb{Q}\_p)$ be a representati... | https://mathoverflow.net/users/15783 | Limits of p-adic Representations | Both $2$ and $3$ are immediately false by considering limits of ($p$-adic) powers of the cyclotomic character.
| 5 | https://mathoverflow.net/users/nan | 69095 | 42,436 |
https://mathoverflow.net/questions/69086 | 28 | A Lawvere theory is a small category with finite products such that every object is isomorphic to a finite product of copies of a distinguished object x. A model of the theory in a category with finite products is a product preserving functor from the theory to that category.
This notion is supposed to be the right c... | https://mathoverflow.net/users/15934 | Lawvere theories versus classical universal algebra | My own experience is that Lawvere theories help one "think outside the box" in ways that I really don't think are too likely with classical universal algebra. Qiaochu has already pointed to what is the key idea: that they enable one to consider models other than in $Set$. Actually, you could put it more strongly. Namel... | 32 | https://mathoverflow.net/users/2926 | 69097 | 42,437 |
https://mathoverflow.net/questions/69116 | 3 | I am working on the chamber homology for $SL(2,F)$, and stuck at some basic stuff on D.S. reps of $SL(2,F)$.
Let $ I=\left(
\begin{array}{cc}
\mathcal{O}\_{F} & \mathcal{O}\_{F} \\
\varpi\_{\mathbb{F}}\mathcal{O}\_{F} & \mathcal{O}\_{F}\\
\end{array}
\right)\cap SL(2, F)$. Now, let $ w\_{0}= \left(
\begin{array... | https://mathoverflow.net/users/9842 | Discrete Series representations for $SL_{2}$ over $p$-adic field. | Inducing a "cuspidal" repn from SL(2,o) produces a finite sum of supercuspidals of SL(2,F). The easiest "cuspidal" repns of SL(2,o) are the ones that factor through SL(2,k), where k is the residue field. The "cuspidal" repns of SL(2,k) can be quasi-explicitly produced via the finite-field version of the Weil/theta pair... | 8 | https://mathoverflow.net/users/15629 | 69121 | 42,450 |
https://mathoverflow.net/questions/69120 | 9 | On wikipedia, the normal crossing divisor is defined to be (by my understanding):
(Assume $X/k$ be a smooth geometrically integral scheme of finite type over a field $k$).
Let $D = \sum\_{i=1}^n C\_i$ be a Weil Divisor, here $C\_i$ are irreducible closed subsets of codimension 1 of $X$. Endow $C\_i$ with the reduce... | https://mathoverflow.net/users/5482 | normal crossing divisor v.s. strict normal crossing divisor | Your definition of normal crossings divisor is, as you say, often called a strict normal crossings divisor. People who use this terminology allow normal crossings divisors to have components which are not necessarily smooth; they are only required to looks like a smooth components meeting transversally locally in the a... | 14 | https://mathoverflow.net/users/519 | 69126 | 42,454 |
https://mathoverflow.net/questions/69115 | 1 | Let $\mathbb{F}\_2$ denotes the free group generated by a,b, denote this group by $G$. Then consider the von Neumann algebra $L(G)$ generated by the family
$\{L\_{x\_g} : g \in G\}$, here, with $g \in G$, we denote by $x\_g$ the function on $G$ that takes the value 1 at g and 0 at other elements of $G$. Then, note tha... | https://mathoverflow.net/users/9305 | Restriction on the coefficients for an operator in the free group factor $ L(\mathbb{F}_2) $ | It is more common to just write $L\_g$ for $L\_{x\_g}$. As $L(G)$ admits a finite trace, there is a natural injective map $L(G)$ into $\ell^2(G)$-- this is your map $A \mapsto (\mu\_g)$. It is absolutely not true that this map surjects (Open Mapping Theorem). It is obviously sufficient that $(\mu\_g)\in\ell^1(G)$ for t... | 1 | https://mathoverflow.net/users/406 | 69130 | 42,455 |
https://mathoverflow.net/questions/69129 | 2 | Let $X$ be a topological space and $Y\subseteq X$, the sequential closure of $Y$ is the set of elements in $X$ that are limit of **sequences** belonging to $Y$.
Let $\mathcal M\_{\text{fin}}(\mathbb Z)$ be the set of finitely supported probability measures on $\mathbb Z$. The general question is: who is the weak\* se... | https://mathoverflow.net/users/13809 | Who is the weak* sequential closure of the set of finitely supported measures on the integers? | The finitely support probability measures on $\mathbb Z$ are all members of $\ell^1(\mathbb Z)$. So we could ask a slightly more general question:
>
> What is the sequential closure of (the probability measures in) $\ell^1(\mathbb Z)$ in $\ell^\infty(\mathbb Z)^\*$?
>
>
>
If $(a\_n)$ is a sequence in $\ell^1(\... | 4 | https://mathoverflow.net/users/406 | 69136 | 42,458 |
https://mathoverflow.net/questions/68897 | 3 | This is related to the earlier [question here](https://mathoverflow.net/questions/39934/when-does-lusztigs-canonical-basis-have-non-positive-structure-coefficients)
In Conjecture 25.4.2 in his "Introduction to Quantum Groups," Lusztig conjectures that "If the Cartan datum is symmetric, then the structure constants $m... | https://mathoverflow.net/users/3545 | What is the current status for Lusztig's positivity conjecture for symmetric Cartan datum? | You should take this with a grain of salt, but I would guess that this is stated in the literature in type A and no other types. It follows in type A from the Beilinson-Lusztig-MacPherson construction, I believe. This is discussed a bit in [this paper](http://arxiv.org/abs/1007.5384) of Yiqiang Li.
For ADE type, a cl... | 2 | https://mathoverflow.net/users/66 | 69158 | 42,469 |
https://mathoverflow.net/questions/69146 | 5 | Let $C$ be a compact Riemann surface, let $C^2$ be the cartesian square of $C$, let $J(C)$ be the degree zero Jacobian of $C$, and let $\delta : C^2 \to J(C)$ be the map $(x,y) \mapsto [\mathcal{O}(x-y)]$.
In this paper <http://arxiv.org/abs/math/9810054> of Hain and Reed, page 9, they say that it is an elementary ex... | https://mathoverflow.net/users/83 | A very basic question about Abel-Jacobi map | Let me write $x\_i$ and $y\_i$ for a symplectic basis of cohomology of $C$, and $a\_i$, $b\_i$ for the linear dual basis of the first homology of $C$. It is enough to find $\delta^\*(dx\_i)$ and $\delta^\*(dy\_i)$ in terms of $x\_i$ and $y\_i$.
But to do this we just evaluate $\langle \delta^\*(dx\_i), a\_i \otimes 1... | 5 | https://mathoverflow.net/users/318 | 69166 | 42,471 |
https://mathoverflow.net/questions/69167 | 4 | Hi!
Novikov's additivity theorem states that if you glue together two compact oriented 4n-manifolds along a connected component of their boundaries, the [signature](http://en.wikipedia.org/wiki/Signature_of_a_manifold) of the resulting manifold is simply the sum of the signatures of the pieces. It is proved for exam... | https://mathoverflow.net/users/9114 | First appearance of Novikov's additivity theorem | Here is as Novikov [himself](http://www.mccme.ru/edu/index.php?ikey=n-rohlin) describes it (in russian):
>
> Rokhlin in 1965 drew my attention repeatedly to the fact that for prime p (large enough for a given dimension), the definition of combinatorial Pontryagin-Hirzebruch classes modulo p is unknown, and this iss... | 8 | https://mathoverflow.net/users/14551 | 69170 | 42,473 |
https://mathoverflow.net/questions/69180 | 9 | I would like to automate a huge amount of computation that involves basic arithmetic operations with $p$-adic numbers. I have found a Mathematica package for it, but it is old and acts quite erratically. Do you know of any computational software that does it reliably?
| https://mathoverflow.net/users/3635 | On $p$-adic arithmetic softwares | SAGE has p-adic arithmetic (for example, see <http://www.math.utah.edu/~carlson/cimat/python-sage.pdf>), and has the added benefit of being completely free and open-source!
| 14 | https://mathoverflow.net/users/15331 | 69181 | 42,476 |
https://mathoverflow.net/questions/69171 | 4 | This comes as a question in Beauville's Algebraic surfaces book (III.24 (2)). We work over $\mathbb{C}$.
All geometrically ruled surfaces (grs) $p:S\longrightarrow C$ over a curve $C$ can be seen as $S=\mathbb{P}(E)$ where $E$ is a vector bundle of rank $2$ over $C$, i.e. a locally free sheaf of rank $2$ over $C$. Th... | https://mathoverflow.net/users/1887 | Elementary transformations of ruled surfaces as maps of vector bundles | The kernel $E'$ of $u\_s \colon E \to F$ is a *torsion free* sheaf over the curve $C$, hence it is necessarily a vector bundle (torsion free sheaves over curves are locally free).
Since the generic rank of $E'$ is $2$, it follows that $E'$ is a rank $2$ vector bundle.
The point you are missing is that the inclusion... | 5 | https://mathoverflow.net/users/7460 | 69183 | 42,478 |
https://mathoverflow.net/questions/69188 | 5 | This is a follow-up to question [Completeness vs Compactness in logic](https://mathoverflow.net/questions/68788/completeness-vs-compactness-in-logic) 68788. One common theme was that compactness in logic is a purely semantic notion, so should have no need of completeness.
The definition of compactness seems to depend... | https://mathoverflow.net/users/9896 | Semantic definition of sentence | One way to completely avoid syntax is to use [Ehrenfeucht–Fraïssé games](http://en.wikipedia.org/wiki/Ehrenfeucht%E2%80%93Fra%C3%AFss%C3%A9_game). If $\mathfrak{A}$ and $\mathfrak{B}$ are two structures with the same signature, then two $k$-tuples $\bar{a}$ and $\bar{b}$ from these respective structures satisfy the sam... | 7 | https://mathoverflow.net/users/2000 | 69195 | 42,483 |
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