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https://mathoverflow.net/questions/513 | 7 | Suppose that $M$ is a finitely generated module over $A=k[X\_1,\ldots,X\_n]$ of Krull dimension $m$ with $k$ an infinite field. Then one version of Noether normalisation says there is an $m$-dimensional $k$-subspace $W$ of the $k$-vector space spanned by $X\_1,\ldots,X\_n$ such that $M$ is finitely generated over $\ope... | https://mathoverflow.net/users/345 | Generic Noether normalisation | Ok, I might be missing something (I often am) but I believe that [this](http://books.google.com/books?id=7ua4WsmpDbMC&pg=PA453&lpg=PA453&dq=noether+normalization+works+generically&source=bl&ots=Rb24oAphIe&sig=fXu2RhCrSf77Dyi0IYHg-mmXMMM&hl=en&ei=ODveSo_2KsbV8AbsiJFn&sa=X&oi=book_result&ct=result&resnum=4&ved=0CBgQ6AEwA... | 4 | https://mathoverflow.net/users/622 | 1515 | 952 |
https://mathoverflow.net/questions/1510 | 3 | It's hard to prove a number is transcendental (non-algebraic) yet there are some wonderful examples amongst them like π,e and Liouville's number. What's so special about them?
Are most numbers transcendental?
| https://mathoverflow.net/users/836 | What's so special about transcendental numbers? | The set of real numbers is uncountable, but the set of algebraic numbers is countable, so "most" real numbers are transcendental in a very strong sense of "most". This is actually a capsule description of Cantor's proof of the existence of transcendental numbers; just note that an uncountable set cannot be empty. Looki... | 16 | https://mathoverflow.net/users/3304 | 1517 | 954 |
https://mathoverflow.net/questions/1522 | 2 | A submodule of a free module need not be free (for instance, in the free Z[X]-module Z[X] the submodule generated by 2 and X is not free). But over a principal ideal domain, submodules of free modules are free.
I was wondering about the center of a free (as a module) algebra. Is it always free? or are there weaker co... | https://mathoverflow.net/users/336 | Is the center of a free (as a module) algebra free? | Choose a ring R of characteristic not 2 which does not satisfy the condition "every projective is free" (e.g. R is not local). Pick a nonfree projective R-module M and make M into a commutative R-algebra in some way. Pick a complement N such that M + N (direct sum) is free and make N into an anticommutative R-algebra v... | 4 | https://mathoverflow.net/users/344 | 1531 | 963 |
https://mathoverflow.net/questions/1380 | 21 | Suppose $X$ and $Y$ are Banach spaces whose dual spaces are isometrically isomorphic. It is certainly true that $X$ and $Y$ need not be isometrically isomorphic, but must it be true that there is a continuous (not necessarily isometric) isomorphism of $X$ onto $Y$?
| https://mathoverflow.net/users/792 | Isomorphisms of Banach Spaces | Indeed, $\ell\_1$ provides a strong counterexample. As noted by Matt, the spaces C(X), where X is countable and compact, provide nonisomorphic Banach spaces whose duals are isomorphic to $\ell\_1$. If X is countable and compact (and Hausdorff, of course!) then X is homeomorphic to a closed ordinal interval [0, *a*] (eq... | 17 | https://mathoverflow.net/users/848 | 1538 | 966 |
https://mathoverflow.net/questions/1493 | 9 | I am looking for software that can find a global minimum of a polynomial function over a polyhedral domain (given by, say, some linear inequalities) in $\mathbb R^n$. The number of variables, $n$, is not more than a dozen. I know it can be done in theory (by Tarski's elimination of quantifiers in real closed fields), a... | https://mathoverflow.net/users/806 | Software for rigorous optimization of real polynomials | I used [QEPCAD](https://www.usna.edu/CS/qepcadweb/B/QEPCAD.html) once for this sort of problem, with reasonable success, although I think my problem was a bit smaller than yours.
| 6 | https://mathoverflow.net/users/126667 | 1539 | 967 |
https://mathoverflow.net/questions/1523 | 8 | The Uniformization Theorem states that the universal cover of a Riemann surface is biholomorphic to the extended complex plane, the complex plane or the open unit disk. Each of these three is a domain in the extended complex plane. In particular, then, the universal cover of a domain in the extended complex plane is bi... | https://mathoverflow.net/users/3304 | Universal covers of domains in complex projective space | Consider a tubular neighborhood of three generic lines on P^2.
The fundamental group is Z. The universal covering will contain
an infinite chain of P^1's, and in particular two
disjoint P^1's. Thus it cannot be a domain in P^2.
| 10 | https://mathoverflow.net/users/605 | 1544 | 970 |
https://mathoverflow.net/questions/1546 | 10 | As a follow up to me other [question](https://mathoverflow.net/questions/1527/pushforwards-of-line-bundles-and-stability), what can be said about unstable vector bundles? I know this is rather open ended, but what sorts of horrible things does having a subbundle of strictly greater slope imply?
| https://mathoverflow.net/users/622 | Unstable Vector Bundles | Well, I don't know about horrible. There's a lot you can say that's good! I'll start rambling and see where I end up.
I'm going to pretend you said principal GL(n)-bundle instead of rank n vector bundle. Same thing, really, since we have the standard representation.
The collection Bun(n,C) of all principal GL(n) bu... | 7 | https://mathoverflow.net/users/35508 | 1561 | 976 |
https://mathoverflow.net/questions/1564 | 2 | For a collection of points in $\mathbb{R}^n$, is there a statistic that I can compute which will estimate the number of clusters with some level of confidence?
| https://mathoverflow.net/users/812 | Estimating the number of clusters | This is an age-old question, which actually does not have (I think even cannot have) a definite answer, because first you need to define what you mean by a cluster and so on. A famous saying in this regard is that "cluster is in the eye of a beholder". It is easy to construct examples where somebody could see one clust... | 4 | https://mathoverflow.net/users/861 | 1571 | 984 |
https://mathoverflow.net/questions/1470 | 4 | Poisson brackets play the very important roles in Symplectic geometry and Dynamical system. I'm interested in some conserved quantities of Poisson brackets.
Let's say we are working on T^n x R^n (T^n is the torus in R^n, T^n = R^n/Z^n). Assume that I have the Hamiltonian H: T^n x R^n \mapsto R, where H=H(x,p) and H i... | https://mathoverflow.net/users/823 | What are some conserved quantities of Poisson brackets? | My impression is that for most choices of H this is a hard question. In general, the largest possible cardinality of a set of independent Poisson commuting functions on a 2n-dimensional symplectic manifold (where by independent I mean that their differentials are linearly independent at each point in their domain--in p... | 4 | https://mathoverflow.net/users/424 | 1575 | 988 |
https://mathoverflow.net/questions/1565 | 13 | Let XX be a Deligne-Mumford stack and let XX \to X be a coarse moduli space. Suppose that X is smooth. Is XX smooth? If not, what is an example? What if XX is of finite type over C (the complex numbers)? What are conditions we can put on XX to make this true?
| https://mathoverflow.net/users/2 | Can a singular Deligne-Mumford stack have a smooth coarse space? | The answer is yes, a singular DM stack can have a smooth coarse space. Let U=Spec(k[x,y]/(xy)) be the union of the axes in **A**2, and consider the action of G=**Z**/2 given by switching the axes: x→y and y→x. Then take XX to be the stack quotient [U/G]. This is a singular Deligne-Mumford stack (since it has an etale c... | 17 | https://mathoverflow.net/users/1 | 1584 | 994 |
https://mathoverflow.net/questions/1480 | 2 | I recall being told -- at tea, once upon a time -- that there exist models of the real numbers which have no unmeasurable sets. This seems a bit bizarre; since any two models of the reals are isomorphic, you'd expect any two models to have the same collection of subsets. Can anyone tell me exactly what the story here i... | https://mathoverflow.net/users/35508 | Models of the reals which have no unmeasurable sets | As John Goodrick is asking in a few places, you have to be careful in stating what you mean by "a model of the reals". If you're going to talk about sets of reals, then you need to have variables ranging over reals, and also variables ranging over sets of reals. You also of course want symbols in your language for the ... | 22 | https://mathoverflow.net/users/445 | 1587 | 996 |
https://mathoverflow.net/questions/1501 | 6 | **Background:** Properness is a much more robust notion than projectiveness. For example, properness descends along arbitrary fpqc covers (see, for example, Vistoli's [Notes on Grothendieck topologies, fibered categories and descent theory](http://arxiv.org/abs/math/0412512), Proposition 2.36). This is far from true fo... | https://mathoverflow.net/users/1 | Does projectiveness descend along field extensions? | For finite extensions, this is stated explicitly as Corollary 6.6.5 in EGA II and it is also stated there that the result is true for arbitrary extensions. One may reduce the general case to the finite case as follows:
First, we may asume that K is finitely generated since any projective scheme is defined by finitely... | 4 | https://mathoverflow.net/users/519 | 1591 | 997 |
https://mathoverflow.net/questions/1047 | 11 | The size of a finite skeletal category C [in the sense of Leinster](http://arxiv.org/abs/math.CT/0610260) is defined as follows: Label the objects of C by integers 1,2,...,n and let aij be the number of morphisms from i to j (for i and j between 1 and n). The **size (or Euler characteristic) of C** is defined as the su... | https://mathoverflow.net/users/296 | What is the size of the category of finite dimensional F_q vector spaces? | Following the observations made in the comments one can compute the sum of the entries of Qn-1. It turns out that Kevin Costello's formula is true for every n.
Let (a1, a2, ..., an) be the the transpose of the kth column vector of Qn-1. (Of course, this vector depends on k, but we omit the index k.) Qiaochu Yuan sugg... | 5 | https://mathoverflow.net/users/296 | 1609 | 1,008 |
https://mathoverflow.net/questions/1614 | 15 | Are there any general results on when a closed subscheme X of a quasi-projective smooth scheme M can be written as the zero-set of a section of a vector bundle E on M?
To put it in a diagram: When is X the fiber product of M -> E <- M , where one arrow is the zero section and the other arrow is the section I'm looking ... | https://mathoverflow.net/users/473 | When is a scheme a zero-set of a section of a vector bundle? | As for the first question, the class of X has to be the product of the Chern roots of the bundle, so in the Chow ring, it is the class of a complete intersection.
As for the second question, you would have to find classes that will solve the class of X in the Thom-Porteus formula, see Fulton's intersection theory 14.... | 14 | https://mathoverflow.net/users/404 | 1615 | 1,010 |
https://mathoverflow.net/questions/1607 | -1 | Hello,
I am studying random variables.
Question is this:
if rv X & a function g is known, what is the pdf of random variable Y = g(x)?
in the textbook answer is explained as follows.
P[y ≤ Y ≤ y + dy] = P[x ≤ X ≤ x + dx]
F\_y(y + dy) - F\_y(y) / dy dx = F\_x(x + dx) - F\_x(x) / dx dy
why is left side of dx... | https://mathoverflow.net/users/877 | about Function of Random variables | What you're looking at is known as "the transformation theorem" and is just an integral change of variables written in probability notation.
Suppose g is an increasing function and Y = g(X). Then
```
F_Y(y) = P( g(X) < y ) = P( X < g^{-1}(y) ) = F_X( g^{-1}(y) )
```
To obtain the PDF, differentiate both sides o... | 4 | https://mathoverflow.net/users/136 | 1616 | 1,011 |
https://mathoverflow.net/questions/1624 | 28 | Can you prove that 8 is the largest cube in the Fibonacci sequence?
| https://mathoverflow.net/users/887 | Is 8 the largest cube in the Fibonacci sequence? | For a much more accessible treatment, and history of the result, see [Andrejic (2006) "On Fibonacci powers"](http://www.doiserbia.nb.rs/img/doi/0353-8893/2006/0353-88930617038A.pdf)
| 21 | https://mathoverflow.net/users/261 | 1629 | 1,019 |
https://mathoverflow.net/questions/1628 | 21 | I repeatedly heard that K(F\_1) is the sphere spectrum. Does anyone know about the proof and what that means?
| https://mathoverflow.net/users/451 | K(F_1) = sphere spectrum? | I understand that this is because GLn(F1) is supposed to be Sigman, the symmetric group on n letters. Thus K(F1) = K(finite sets) which is the sphere spectrum by the Barratt-Priddy-Quillen-Segal theorem.
But I have no idea why GLn(F1) should be Sigman...
| 16 | https://mathoverflow.net/users/318 | 1630 | 1,020 |
https://mathoverflow.net/questions/1256 | 9 | I've been reading the [wonderful slides by Terry Tao](http://terrytao.files.wordpress.com/2009/07/primes1.pdf) and thought about this question.
Primes appear to be quite random, and the formal statement should be that there are some characteristics of primes that are indistinguishable by any algorithm from the a sequ... | https://mathoverflow.net/users/65 | Primes are pseudorandom? | There is no general statement, but there is a general philosophy.
The general idea in mathematics is that things do not happen for no reason. For example, almost every mathematician would be willing to bet that alpha=e^e+pi^sqrt(2) is irrational, as the 'generic' number is irrational, and a genuine reason is needed f... | 6 | https://mathoverflow.net/users/806 | 1637 | 1,023 |
https://mathoverflow.net/questions/1621 | 15 | I was taught to think of generalized cohomology theories as the homotopy category of (symmetric) spectra. But is there also a category of 'invariants', that is, some category of contravariant functors from a suitable category of topological spaces to a suitable category of algebraic objects, which has a model category ... | https://mathoverflow.net/users/798 | Are generalized cohomology theories a homotopy category of some category of invariants? | Here is a short argument why we don't expect generalized cohomology theories to behave so well.
In the stable homotopy category, there is a generalized homology/cohomology theory represented by the sphere spectrum $S$, so that $S\_\*(X)$ are the stable homotopy groups of $X$. It has a multiplication-by-2 self-map $f$... | 13 | https://mathoverflow.net/users/360 | 1642 | 1,026 |
https://mathoverflow.net/questions/1610 | 16 | It is known that the binomial coefficient $2n \choose n$ is equal to number of shortest lattice paths from $(0,0)$ to $(n,n)$. The Catalan number $\frac{1}{n+1} {2n\choose n}$is equal to the number of shortest lattice paths that never go above the diagonal. Here, the diagonal may be viewed as a path from $(0,0)$ to $(n... | https://mathoverflow.net/users/296 | Pairs of shortest paths | The answer is (2n)! (2n+1)! / (n)!^2 (n+1)!^2 .
You can get this by the Gessel-Viennot method suggested above. One difficulty is that GV wants to count paths which don't touch at all, even at vertices, while you just want to count paths that don't cross. To solve this, take your lower path and slide it south-east. Yo... | 16 | https://mathoverflow.net/users/297 | 1650 | 1,032 |
https://mathoverflow.net/questions/1036 | 20 | Do I remember a remark in "Sketch of a program" or "Letter to Faltings" correctly, that acc. to Grothendieck [anabelian geometry](http://www.math.okayama-u.ac.jp%2F~h-naka%2Fzoo%2Fpeacock%2FNTM.ps "anabelian survey") should not only enable finiteness proofs, but a proof of FLT too? If yes, how?
Edit: [In this](http:/... | https://mathoverflow.net/users/451 | "Fermat's last theorem" and anabelian geometry? | See the papers of Minhyong Kim. For example, begin by looking at the MR review 2181717 of his paper Invent. Math. 161 (2005), no. 3, 629--656.
| 13 | https://mathoverflow.net/users/930 | 1659 | 1,039 |
https://mathoverflow.net/questions/1600 | 4 | Let (X,x) be a pointed space. There is an action of π1(X,x) on πn(X,x) -- determined by considering πn(X,x)=πn-1(ΩxX,**x**), where ΩxX denotes the space of loops in X based at x, and **x** denotes the constant loop -- given simply by conjugation. We can speak unambiguously of πn(X), the *free* (i.e., not necessarily ba... | https://mathoverflow.net/users/303 | free homotopy groups -- when do they exist? | For the last part of your question: given a group π1 which acts on an abelian group πn, there is always as space X with these homotopy groups with this action, and you can manufacture one using Eilenberg-MacLane spaces. You can make the group π1 act on the Eilenberg-MacLane space K(πn,n) in such a way that realizes the... | 5 | https://mathoverflow.net/users/437 | 1660 | 1,040 |
https://mathoverflow.net/questions/1647 | 1 | I have several temporal signals of different dimensions, for example the motion of a point throughout time which would be of dimension 3, and the value of a temperature sensor, of dimension 1.
I would like to find out if these signals are correlated. Is there a measure of correlation that works for signals that do no... | https://mathoverflow.net/users/180 | Correlation measure between signals of different dimensions? | Yes, you can use multiple regression, with temperature as the dependent variable and the three space dimensions as the independent variables.
| 2 | https://mathoverflow.net/users/619 | 1663 | 1,042 |
https://mathoverflow.net/questions/1662 | 5 | The Walsh-Hadamard transform is very fast to compute.
Can it be used to compute the convolution of two functions as it can be done with Fourier transform ?
| https://mathoverflow.net/users/903 | Can Walsh-Hadmard transform be used for convolution ? | Not in the sense I think you mean it. First of all, the Walsh-Hadamard transform **is** a Fourier transform - but on the group (Z/2Z)^n instead of on the group Z/NZ. That means you can use it to compute convolutions with respect to the space of functions (Z/2Z)^n -> C. Unfortunately, unlike the case with Z/NZ you can't... | 5 | https://mathoverflow.net/users/290 | 1678 | 1,051 |
https://mathoverflow.net/questions/1592 | 8 | While the general problem of detecting a Hamiltonian path or cycle on an undirected grid graph is known to be NP-complete, are there interesting special cases where efficient polynomial time algorithms exist for enumerating all such paths/cycles? Perhaps for certain kinds of k-ary n-cube graphs? I hope this question is... | https://mathoverflow.net/users/774 | Special cases for efficient enumeration of Hamiltonian paths on grid graphs? | There are certainly special graphs that are always Hamiltonian (if every vertex of a graph of n vertices has degree at least n/2, say) and these have efficient algorithms associated with them.
For instance, [this paper](http://www.math.cmu.edu/~af1p/Texfiles/5out.pdf) proves the graph of a random 5-outregular digraph... | 5 | https://mathoverflow.net/users/441 | 1687 | 1,057 |
https://mathoverflow.net/questions/1590 | 62 | Once in a while I run into literature that invokes vanishing cycle machinery with a cryptic sentence like, "this follows from a standard vanishing cycle argument." Is there a good way to look at vanishing cycles, nearby cycles, and specialization? I have a decent idea how some of it works for studying the cohomology of... | https://mathoverflow.net/users/121 | Is there a good way to think of vanishing cycles and nearby cycles? | In general I don't think there's anything easy about nearby and vanishing cycles. However, I tend to find it enlightening to just consider their topology. Namely, if $f:X \to \Bbb C$ is a function on a complex algebraic (or analytic) variety, then the stalk cohomology of the nearby cycles functor applied to some comple... | 22 | https://mathoverflow.net/users/916 | 1691 | 1,061 |
https://mathoverflow.net/questions/1675 | 13 | This is a follow-up to [this post](https://mathoverflow.net/questions/1039/explicit-direct-summands-in-the-decomposition-theorem) on the Decomposition Theorem. Hopefully, this will also invite some discussion about the theorem and perverse sheaves in general.
My question is how does one use the Decomposition Theorem ... | https://mathoverflow.net/users/788 | How to do Computations Using the Decomposition Theorem for Perverse Sheaves | To supplement Ben's answer, basically every aspect of the decomposition theorem is hard.
To give you a simple example of something which is implied by the decomposition theorem but is far from trivial is the following statement: given a proper smooth map of smooth varieties f : X -> Y the direct image of the constant... | 13 | https://mathoverflow.net/users/919 | 1694 | 1,064 |
https://mathoverflow.net/questions/1664 | 7 | Compact connected simply-connected Lie groups have so much structure that you can calculate their cohomology from their Lie algebras using Lie algebra cohomology (certain Ext-groups) and similarly their homology from their Lie algebras using Lie algebra homology (certain Tor-groups).
Is there similar theorem that gi... | https://mathoverflow.net/users/798 | Can one calculate the (co)homology of the loopspace of a Lie group from its Lie algebra? | Yes.
At least, rationally.
The result that you want starts on p68 of "Loop Groups" by Pressley and Segal. There, they prove surjectivity of H\*(L𝔤;ℝ) → H\*(LG;ℝ). The basic idea of the argument is as follows: for reasonably simple reasons, the cohomology of LG is easily obtainable from that of G. This yields speci... | 2 | https://mathoverflow.net/users/45 | 1697 | 1,066 |
https://mathoverflow.net/questions/1620 | 9 | A little bit of background: A graph G is, of course, a set of vertices V(G) and a multiset of edges, which are unordered pairs of (not necessarily distinct) vertices. We say that two vertices v\_1, v\_2 are adjacent if {v\_1, v\_2} is an edge. A directed graph, or digraph is essentially the same thing, except that the ... | https://mathoverflow.net/users/382 | Is there a free digraph associated to a graph? | I like to use the following definitions, which give a nonstandard definition of undirected graph but produce particularly nice categories.
>
> A directed graph is a pair of sets V and E together with two maps s, t : E -> V. Call the category of these DirGraph.
>
>
> An undirected graph is a pair of sets V and E t... | 8 | https://mathoverflow.net/users/126667 | 1703 | 1,068 |
https://mathoverflow.net/questions/1684 | 62 | The exterior algebra of a vector space V seems to appear all over the place, such as in
* the definition of the cross product and determinant,
* the description of the Grassmannian as a variety,
* the description of irreducible representations of GL(V),
* the definition of differential forms in differential geometry,... | https://mathoverflow.net/users/290 | Why is the exterior algebra so ubiquitous? | Just to use a buzzword that Greg didn't, the exterior algebra is the symmetric algebra of a purely odd supervector space. So, it isn't "better than a symmetric algebra," it is a symmetric algebra.
The reason this happens is that super vector spaces aren't just Z/2 graded vector spaces, they also have a slightly diffe... | 38 | https://mathoverflow.net/users/66 | 1705 | 1,069 |
https://mathoverflow.net/questions/1467 | 17 | One makes precise the vague notion of "curve with a fractional point removed" (see for instance [these slides](http://www-math.mit.edu/~poonen/slides/campana_s.pdf)) using stacks -- one should really consider Deligne-Mumford stacks whose coarse spaces are curves, and the "fractional points" correspond to the residual g... | https://mathoverflow.net/users/2 | Are curves with `fractional points' uniquely determined by their residual gerbes? | Here's an example of two non-isomorphic Deligne-Mumford stacks whose coarse spaces are **A**1, and the only non-trivial residual gerbe in each case is B(**Z**/2) at the origin.
First, take the **Z**/2 action on **A**1 given by reflection around 0, given by x→-x. The stack quotient [**A**1/(**Z**/2)] has coarse space ... | 23 | https://mathoverflow.net/users/1 | 1715 | 1,074 |
https://mathoverflow.net/questions/1722 | 76 | I often use the internet to find resources for learning new mathematics and due to an explosion in online activity, there is always plenty to find. Many of these turn out to be somewhat unreadable because of writing quality, organization or presentation.
I recently found out that "The Elements of Statistical Learning... | https://mathoverflow.net/users/812 | Free, high quality mathematical writing online? | [John Baez's stuff](http://www.math.ucr.edu/home/baez/) is a fantastic resource for learning about - well, whatever John Baez is interested in, but fortunately that's a lot of interesting stuff. Scroll down for a link to TWF as well as his expository articles.
| 30 | https://mathoverflow.net/users/290 | 1723 | 1,079 |
https://mathoverflow.net/questions/1721 | 16 | We all know how to take integer tensor powers of line bundles. I claim that one should be able to also take fractional or even complex powers of line bundles. These might not be line bundles, but they have some geometric life. They have Chern classes, and one can twist differential operators by them. How should I think... | https://mathoverflow.net/users/66 | What do gerbes and complex powers of line bundles have to do with each other? | Complex powers of line bundles are classes in $H^{1,1}$, or equivalently sheaves of twisted differential operators (TDO) (let's work in the complex topology). This maps to $H^2$ with $\mathbb{C}$ coefficients, or modding out by $\mathbb{Z}$-cohomology, to $H^2$ with $\mathbb{C}^\times$ coefficients. The latter classifi... | 7 | https://mathoverflow.net/users/582 | 1737 | 1,091 |
https://mathoverflow.net/questions/1743 | 14 | Or at least it's order of magnitude.
I've only ever heard it described as "huge", and a google search turned up nothing.
Also, given that the Strassen algorithm has a significantly greater constant than Gaussian Elimination, and that Coppersmith-Winograd is greater still, are there any indications of what constant ... | https://mathoverflow.net/users/942 | What is the constant of the Coppersmith-Winograd matrix multiplication algorithm | In your second question, I think you mean "naive matrix multiplication", not "Gaussian elimination".
Henry Cohn et al had [a cute paper](http://arxiv.org/pdf/math.GR/0307321.pdf) that relates fast matrix multiply algorithms to certain groups. It doesn't do much for answering your question (unless you want to go and p... | 8 | https://mathoverflow.net/users/598 | 1749 | 1,101 |
https://mathoverflow.net/questions/430 | 26 | Homological algebra for abelian groups is a standard tool in many fields of mathematics. How much carries over to the setting of commutative monoids (with unit)? It seems like there is a notion of short exact sequence. Can we use this to define ext groups which classify extensions? What works and what doesn't work and ... | https://mathoverflow.net/users/184 | Homological algebra for commutative monoids? | Your question can be understood as how to do Homological Algebra over the Field with one Element.
Deitmar, in <http://arxiv.org/abs/math/0608179> , section 6, gives an example of what can go wrong if you try to do sheaf cohomology directly via resolutions...
You might also want to look at his <http://arxiv.org/abs/... | 7 | https://mathoverflow.net/users/733 | 1752 | 1,102 |
https://mathoverflow.net/questions/1672 | 9 | Hello.
I am studying stochastic process.
here,
I don't know what is difference of
"the process is homogeneous"
and
"the process is stationary"
I feel confusing. It seems to similar to me.
| https://mathoverflow.net/users/877 | What is the difference between a homogeneous stochastic process and a stationary one? | A process is (strictly) stationary if any sequence of n consecutive points has the same distribution as any other sequence of n consecutive points. There are weaker definitions, for example weak stationarity is based only on the first two moments.
A (discrete valued) process is homogeneous if the transition probabili... | 6 | https://mathoverflow.net/users/261 | 1761 | 1,107 |
https://mathoverflow.net/questions/1720 | 32 | For an algebraic variety X over an algebraically closed field, does there always exist a finite set of (closed) points on X such that the only automorphism of X fixing each of the points is the identity map? If Aut(X) is finite, the answer is obviously yes (so yes for varieties of logarithmic general type in characteri... | https://mathoverflow.net/users/930 | Can algebraic varieties be rigidified by finite sets of points? | I get that the answer is "no" for an abelian variety over the algebraic closure of Fp with complex multiplication by a ring with a unit of infinite order. Since you say you have already thought through the abelian variety case, I wonder whether I am missing something.
More generally, let X be any variety over the alg... | 25 | https://mathoverflow.net/users/297 | 1768 | 1,112 |
https://mathoverflow.net/questions/1726 | 15 | So, physicists like to attach a mysterious extra cohomology class in H^2(X;C^\*) to a Kahler (or hyperkahler) manifold called a "B-field." The only concrete thing I've seen this B-field do is change the Fukaya category/A-branes: when you have a B-field, you shouldn't take flat vector bundles on a Lagrangian subvariety,... | https://mathoverflow.net/users/66 | How should I think about B-fields? | Let me add a few words of explanation to Aaron's comment. Perturbative string theory is (at least at the level of caricature) concerned with describing small corrections to classical gravitational physics on the spacetime X. So, to do perturbative string theory on X, you need to choose a "background" metric on X. You m... | 5 | https://mathoverflow.net/users/35508 | 1772 | 1,115 |
https://mathoverflow.net/questions/1765 | 9 | I was looking at a [paper](http://www.google.com/url?sa=t&source=web&ct=res&cd=1&ved=0CA4QFjAA&url=http%3A%2F%2Fwww.mathematik.hu-berlin.de%2F~farkas%2Fsdg-2.pdf&ei=3KTfSoXmFo-0sgPoydHhCA&usg=AFQjCNHJWSHKXWGpaxtDsY6CxCJEDGYIBg&sig2=0Yq2ygZ4ikWs5tBgAakIeA) of Farkas and the following confusing point came up.
Let $\mat... | https://mathoverflow.net/users/2 | Kodaira-Spencer Theory and moduli of curves | By standard deformation theory (see e.g., Hartshorne III Ex 4.10, but there are probably better references), the tangent sheaf of $\mathscr{M}\_g$ is $R^1\pi\_{\ast}(\mathscr{C}, T\_{\mathscr{C}/\mathscr{M}\_g})$, which is Serre dual to $\pi\_{\ast}\mathscr{F}$. The tangent sheaf is dual to what you wanted.
| 7 | https://mathoverflow.net/users/121 | 1779 | 1,120 |
https://mathoverflow.net/questions/1781 | 8 | The following question came up in my research. I suspect that it has a slick answer,
but I can't seem to find it.
Fix an integer n>=2 and a prime p. Define X(n) to be the set of primitive
vectors in the Z-module Z^n and Y(n,p) to be the set of "lines" in the vector space (Z/pZ)^n (ie the spans of non-zero vectors). T... | https://mathoverflow.net/users/317 | Lifting bases for (Z/pZ)^n to Z^n | Here's a unified argument based on my comments to Scott's post that doesn't use quadratic reciprocity in any form. Suppose n=2 and p >= 5, and lift each line of slope i in Y(2,p) to a point (ai+pbi, iai+pci).
Since each pair of lifts should give a basis of Z2 and thus a matrix with determinant \pm 1, taking each pair... | 4 | https://mathoverflow.net/users/428 | 1787 | 1,125 |
https://mathoverflow.net/questions/769 | 23 | Let $q$ be a power of a prime. It's well-known that the function $B(n, q) = \frac{1}{n} \sum\_{d | n} \mu \left( \frac{n}{d} \right) q^d$ counts both the number of irreducible polynomials of degree $n$ over $\mathbb{F}\_q$ and the number of [Lyndon words](http://en.wikipedia.org/wiki/Lyndon_word) of length $n$ over an ... | https://mathoverflow.net/users/290 | Exhibit an explicit bijection between irreducible polynomials over finite fields and Lyndon words. | In Reutenauer's "Free Lie Algebras", section 7.6.2:
A direct bijection between primitive necklaces of length $n$ over $F$ and the set of irreducible polynomials of degree $n$ in $F[x]$ may be described as follows: let $K$ be the field with $q^n$ elements; it is a vector space of dimension $n$ over $F$, so there exist... | 31 | https://mathoverflow.net/users/961 | 1800 | 1,134 |
https://mathoverflow.net/questions/1809 | 14 | Given a homomorphism f:G→H between smooth algebraic groups, we get an induced homomorphism of algebraic stacks Bf:BG→BH, given by sending a G-torsor P over a scheme X to the H-torsor PxGH, whose (scheme-theoric) points are {(p,h)|p∈P,h∈H}/∼, where (pg,h)∼(p,f(g)h).
Is every morphism of algebraic stacks BG→BH of the f... | https://mathoverflow.net/users/1 | Does every morphism BG-->BH come from a homomorphism G-->H? | Depends on the base scheme and the topology being used. For example if you're working over a field k in the etale or the flat topology, and take the group G to be trivial, you're asking if H^1(k,H) is trivial, which is obviously false in general. This is, in a sense, the only obstruction: for any base scheme S, giving ... | 8 | https://mathoverflow.net/users/986 | 1819 | 1,147 |
https://mathoverflow.net/questions/1788 | 62 |
>
> Precisely, if an R-module M *has* a finite presentation, and Rk → M is some unrelated **surjection** (k finite), is the kernel necessarily also finitely generated?
>
>
>
Basically I want to believe I can choose generators for M however I please, and still get a finite presentation. I have reasons from algeb... | https://mathoverflow.net/users/84526 | Does "finitely presented" mean "always finitely presented"? (Answered: Yes!) | $\require{begingroup} \begingroup$
$\def\coker{\operatorname{Coker}}$
$\def\im{\operatorname{Im}}$
Suppose that we have a short exact sequence $0 \to K \to R^m \to M \to 0$ with $K$ finitely generated over $R$ and that $0 \to K' \to R^n \to M \to 0$ is another short exact sequence. Your question is: is $K'$ necessari... | 71 | https://mathoverflow.net/users/493 | 1824 | 1,150 |
https://mathoverflow.net/questions/1818 | 4 | As someone whose knowledge of cohomology is patchy and picked up on a need-to-know basis, and whose algebraic geometry is even worse, I wondered if someone could help with this question. (I ran into it a while back while trying to answer some questions about the *Fourier* algebras of compact Lie groups.) The solution c... | https://mathoverflow.net/users/763 | What is the homology of the real coordinate ring of SO(n,R)? Other compact matrix groups? | I am very far from an expert on the subject, but I think the Hochschild homology of the coordinate ring should be the algebraic de Rham complex of your variety SO(n, R)--*not* the cohomology of the complex, just the groups in the complex (with 0 differential if you like). This is the Hochschild-Kostant-Rosenberg theore... | 4 | https://mathoverflow.net/users/126667 | 1834 | 1,156 |
https://mathoverflow.net/questions/452 | 7 | $f:\mathbf{R}^d\to \mathbf{R}\_{\ge 0}$ is *log-concave* if $\log(f)$ is concave (and the domain of $\log(f)$ is convex).
Theorem: For all $\sigma$ on the sphere $\Bbb S^{d-1}$ and $r\in \mathbf{R}$,
$$
g\_\sigma(r) := \int\limits\_{\sigma\cdot x=r}f(x)\,\mathrm{d}S(x)
$$
is a log-concave function of $r$. (Note: $g$,... | https://mathoverflow.net/users/302 | Characterizing the Radon transforms of log-concave functions | If I understand the question correctly, I think the answer is no.
Start with the following : if $f$ is the indicator function of the unit ball, then the function $g\_\sigma(r)$ is strictly log-concave close to 0 (this function does not depend on $\theta$).
Now, let $h$ be the indicator function of the ball of radiu... | 5 | https://mathoverflow.net/users/908 | 1838 | 1,160 |
https://mathoverflow.net/questions/1634 | 22 | I am not too certain what these two properties mean geometrically. It sounds very vaguely to me that finite type corresponds to some sort of "finite dimensionality", while finite corresponds to "ramified cover". Is there any way to make this precise? Or can anyone elaborate on the geometric meaning of it?
| https://mathoverflow.net/users/nan | Finite type/finite morphism | I definitely agree with Peter's general intuitive description.
In response to some of the subsequent comments, here are some implications to keep in mind:
**Finite** ==> **finite fibres** (1971 EGA I 6.11.1) and **projective** (EGA II 6.1.11), hence **proper** (EGA II 5.5.3), but *not conversely*, contrary to popu... | 22 | https://mathoverflow.net/users/84526 | 1839 | 1,161 |
https://mathoverflow.net/questions/1827 | 18 | So here's my problem: I have no intuition for how a "generic" module over a commutative ring should behave. (I think I should never have been told "modules are like vector spaces.") The only examples I'm really comfortable with are
* vector spaces,
* finitely generated modules over a PID, and
* modules over a group a... | https://mathoverflow.net/users/290 | What representative examples of modules should I keep in mind? | Yes, there is a big class of modules that have an intuition different from the abstract algebra, namely the ones that come from an **algebraic geometry**. If $R$ is a (say, Noetherian) commutative ring, then you consider a scheme $\mathrm{Spec}\, R$ and (finitely generated) modules over $R$ correspond to coherent sheav... | 14 | https://mathoverflow.net/users/65 | 1845 | 1,167 |
https://mathoverflow.net/questions/1805 | 6 | Let g be a finite dimensional Lie algebra over k, and let U be its universal enveloping Lie algebra. Is there a left module M of U which is projective but not free? That is, is the Quillen-Suslin theorem still true for enveloping algebras?
Quillen-Suslin says this there are no non-free projectives for S(g), the assoc... | https://mathoverflow.net/users/750 | Non-free projective modules for a Universal Enveloping Algebra? | In [this paper](http://www.numdam.org/numdam-bin/item?id=CM_1985__54_1_63_0) Stafford shows that whenever g is a finite-dimensional non-abelian Lie algebra the enveloping algebra has non-free but stably free (and therefore projective) right ideals. He also shows how to construct them.
| 5 | https://mathoverflow.net/users/345 | 1860 | 1,179 |
https://mathoverflow.net/questions/1828 | 3 | Suppose w^(2n)=1 (w is a complex number).
For which n (if any) \sqrt(w) \in Q(w) ?
| https://mathoverflow.net/users/966 | Is an nth root of unity a square? | The key point is to understand the field Q(w) for w a primitive kth root of unity. Call this field Qk. In particular, you want to know that Q4n \neq Q2n.
The key fact here is that the field extension Qk/Q has degree phi(k), where phi(k) is the Euler phi function, and phi(4k) \neq phi(2k). For a proof that Qk/Q has de... | 3 | https://mathoverflow.net/users/297 | 1862 | 1,180 |
https://mathoverflow.net/questions/1812 | 20 | While many of us have had the experience of learning mathematics informally by osmosis or more formally in classes, there are times when we have to sit down and systematically learn, without the benefit of a class, large amounts of mathematics. For instance, there might be a technique that we need from a field we are n... | https://mathoverflow.net/users/812 | Learning new mathematics | Many have remarked that they first understood a subject only when they first taught it.
| 14 | https://mathoverflow.net/users/454 | 1871 | 1,185 |
https://mathoverflow.net/questions/1878 | 5 | Functions on an algebraic subvariety X of A^n are the same as functions on A^n restricted to X. So the statement that functions on X extend to all of A^n follows by the definition. My question is: does the analogous statement hold for C^n and closed complex submanifolds (maybe even closed analytic subvarieties), and if... | https://mathoverflow.net/users/788 | Extending Functions on Closed Submanifolds of C^n | Yes, this is true. It follows from "Cartan's Theorem B" which says that H^1 of any coherent analytic sheaf on a closed submanifold of C^n is 0; the same result is also true for analytic subspaces. Look up any book on several complex variables for a proof. (It is quite possible that there is a more elementary proof.)
... | 4 | https://mathoverflow.net/users/519 | 1883 | 1,192 |
https://mathoverflow.net/questions/1822 | 9 | The *norm* of a graph is the largest eigenvalue of the adjacency matrix. I'll write ||`G`|| for the norm of `G`.
Now, fix some graph `G` with a chosen vertex `*`, and consider the family of graphs `G_k` obtained by adding a chain of `k` edges to `*`.
For many such examples, the sequence `{`||`G_k`||`^2}_k` appears ... | https://mathoverflow.net/users/3 | How can I prove that a sequence of squares of graph norms is never cyclotomic? | This is a vague thought: is there some simple recurrence for the characteristic polynomials of the charctertistic polynomials of the corresponding matrices. For example, if you look at the A\_n chains, the polynomials are the Chebyshev polynomials, whose roots are cyclotomic, and which obey a simple resursion.
Even i... | 3 | https://mathoverflow.net/users/297 | 1885 | 1,194 |
https://mathoverflow.net/questions/1876 | 9 | There are a lot of results in textbooks concerned with canonical forms of matrices under certain complex groups of transformations, e.g. GL(n|C), O(n|C),...
Could anybody give me references where the canonical forms of real matrices under the action of SO(p,q|R) were found. Of most interest is the canonical form of an... | https://mathoverflow.net/users/985 | Orbits of real groups, canonical forms of matrices | In Chapter 2 of [the PhD thesis of Charles Boubel](http://cat.inist.fr/?aModele=afficheN&cpsidt=204208) (in French, though), you can find the normal forms of a symmetric or antisymmetric bilinear form under the automorphism group of a nondegenerate bilinear form, again either symmetric or skewsymmetric. In particular y... | 10 | https://mathoverflow.net/users/394 | 1891 | 1,197 |
https://mathoverflow.net/questions/1858 | 9 | Suppose that X and Y are finite sets and that f : X → Y is an arbitrary map. Let PB denote the pullback of f with itself (in the category of sets) as displayed by the commutative diagram
PB → X
↓ ↓
X → Y
[Terence Tao](http://terrytao.wordpress.com/2007/04/01/open-question-triangle-and-diamond-densiti... | https://mathoverflow.net/users/296 | Determinant of a pullback diagram | Write Xn = {x1, ..., xk}. For each 1 ≤ i ≤ k let wi be the vector whose jth component is the cardinality of the inverse image of xj in Xi. Then your matrix is the sum w1w1T + ... + wkwkT, a sum of positive semidefinite matrices, so it is positive semidefinite and in particular has nonnegative determininant.
| 10 | https://mathoverflow.net/users/126667 | 1892 | 1,198 |
https://mathoverflow.net/questions/1894 | 5 | Given groups $G\_1, G\_2, G\_3$ and injections $A\_1 \to G\_1$ and $A\_1
\to G\_2$ , from $A\_2 \to G\_2$ and $A\_2 \to G\_3$, let $G\_1 \*\_{A\_1} \*G\_2 \*\_{A\_2} G\_3$ be the amalgam formed these groups and maps.
Then is it true that $G\_1 \*\_{A\_1} \*G\_2 \*\_{A\_2} G\_3$ is the same as (G\_1 \*\_{A\_1} G\_2 ) ... | https://mathoverflow.net/users/996 | is amalgamation of groups associative | Amalgamation of groups is a categorical construction known as a "pushout": <http://en.wikipedia.org/wiki/Pushout_(category_theory)>
By general category theory, pushouts are associative up to unique isomorphism, i.e., the two things you wrote are isomorphic in a unique way (subject to commuting with the inclusions fro... | 7 | https://mathoverflow.net/users/321 | 1898 | 1,202 |
https://mathoverflow.net/questions/1887 | 0 | I have a graph with Edge `E` and Vertex `V`, I can find the spanning tree using [Kruskal algorithm](http://www-b2.is.tokushima-u.ac.jp/~ikeda/suuri/kruskal/Kruskal.shtml), now I want to find all the cycle bases that are created by utilitizing that spanning tree and the edges that are not on the tree, any algorithm that... | https://mathoverflow.net/users/807 | Given a Spanning Tree and an Edge Not on the Spanning Tree, How to Form a Cycle Base? | I think you're likely to get better answers if you post this question on Stack Overflow rather than here.
But anyways, if your graph doesn't have weights on edges, you don't need Kruskal's algorithm to find a spanning tree; you can just use DFS. As you compute the tree, store for each vertex its parent and the distan... | 3 | https://mathoverflow.net/users/126667 | 1901 | 1,205 |
https://mathoverflow.net/questions/1893 | 6 | I've been reading some "introduction to categories" type materials and have been impressed with the all-encompassing nature, but the skeptic in me wonders: is there any mathematical object that categories *can't* describe?
To be quite specific, I'd be interested any of these:
a.) Objects that can be described by c... | https://mathoverflow.net/users/441 | What can't be described by categories? | I don't quite understand your question, but if you're asking whether category theorists should worry about set-theoretic problems the answer seems to be "sometimes". I'm not an expert in this area, but it seems that people tend to avoid universal constructions like limits over large diagrams, and in other cases, people... | 6 | https://mathoverflow.net/users/121 | 1903 | 1,207 |
https://mathoverflow.net/questions/1908 | 2 | Recall that a ring homomorphism A->B is geometrically regular if for all primes p of A, the fiber of B over p is geometrically regular over k(p). A Grothendieck ring (or, G-ring) is one for which A\_p->A\_p\* is regular for all primes p. These are the maps from the local rings of A to their completions.
If A is an or... | https://mathoverflow.net/users/100 | Are non-maximal orders in number fields Grothendieck rings? | Yes: if R is excellent, so is any finite type R-algebra (apply this to Z and A).
| 5 | https://mathoverflow.net/users/986 | 1911 | 1,213 |
https://mathoverflow.net/questions/1931 | 27 | Grothendieck's approach to algebraic geometry in particular tells us to treat all rings as rings of functions
on some sort of space. This can also be applied outside of scheme theory (e.g., Gelfand-Neumark theorem
says that the category of measurable spaces is contravariantly equivalent to the category
of commutative v... | https://mathoverflow.net/users/402 | Bimodules in geometry | In "commutative geometry," I think bimodules tend to be a little concealed. People are more likely to talk about "correspondences" which are the space version of bimodules: A correspondence between spaces X and Y is a space Z with maps to X and Y.
When you think in this language, there are lots of examples you're mis... | 20 | https://mathoverflow.net/users/66 | 1943 | 1,235 |
https://mathoverflow.net/questions/1949 | 1 | This is not a homework question.
Just wondering if there is a general formula for the gaussian curvature at point (x,y,f(x,y)) in terms of x, y, and f(x,y).
I didn't see any thing like that on the [wikipedia article](http://en.wikipedia.org/wiki/Gaussian_curvature), but maybe it's hidden behind all these letters/sy... | https://mathoverflow.net/users/1007 | Gaussian curvature of a z=f(x,y) function | There are a few explicit formulas at <http://mathworld.wolfram.com/GaussianCurvature.html>.
| 4 | https://mathoverflow.net/users/303 | 1952 | 1,239 |
https://mathoverflow.net/questions/1916 | 6 | In a complex analysis course I have been given the following definition:
Let $X$ be a Riemann surface, denote by $H(1,0)$ the space of all $(1,0)$-holomorphic forms on $X$ and consider the quotient vector space (over $\mathbb{C}$) of $H(1,0)$ by $$\{f \in H(1,0) \mid f = d(\phi) \text{ for some } \phi \in C(X)\}.$$ T... | https://mathoverflow.net/users/997 | Space of $(1,0)$-holomorphic forms on a Riemann surface | There is the introductory graduate-level text Riemann Surfaces by Otto Forster which approaches the subject from just the angle suggested by the definition you were given. If you read French there is the book Quelques Aspects des Surfaces de Riemann by Eric Reyssat, a gentle introduction with a broad outlook. Rather mo... | 9 | https://mathoverflow.net/users/3304 | 1953 | 1,240 |
https://mathoverflow.net/questions/1832 | 16 | My understanding is that an analogy along the following lines is (roughly) true:
"The Alexander polynomial is to knot Floer homology is to gl(1|1)
as the Jones polynomial is to Khovanov homology is to sl(2)
as a-lot-of-other-specializations-of-HOMFLY are to Khovanov-Rozansky homology are to sl(n)."
1) To what ... | https://mathoverflow.net/users/361 | HOMFLY and homology; also superalgebras | In terms of just the knot polynomials, there's a simple explanation for what's going on that makes $\mathfrak{gl}(1|1)$ seem totally in place:
The knot polynomial attached to the defining representation of $\mathfrak{gl}(m|n)$ only depends on m-n (the dimension of that representation in the categorical sense); you ju... | 10 | https://mathoverflow.net/users/66 | 1970 | 1,252 |
https://mathoverflow.net/questions/1972 | 17 | Can someone explain, explicitly, how to, given a reductive complex algebraic group construct the Langlands dual group? I know it is a group with the cocharacters of G as its characters, but how does one go about writing down what group it is?
| https://mathoverflow.net/users/622 | Langlands Dual Groups | You can construct the dual group in a combinatorial manner as follows: Reductive groups are classified by their root datum. There is an obvious duality on the set of all root data, and the dual group is the reductive group with the dual root datum.
You can see [Wikipedia](http://en.wikipedia.org/wiki/Root_datum) for ... | 14 | https://mathoverflow.net/users/425 | 1991 | 1,267 |
https://mathoverflow.net/questions/697 | 2 | Hi,
I was wondering if anyone could point me to any sources regarding the convergence of iterated affine transformation, i.e. sequences where {a\_n} is a set of affine transforms and the sequence:
a\_n (a\_{n\_1}(...a\_1)...) converges to something as n->infinity. (Preferably another affine transform).
I know my ... | https://mathoverflow.net/users/16 | Convergence of Affine Transformations | Hi,
could you perhaps specify what kind of space your transformations are acting on? Before you do that, let me try to still share some things...
If it's a Riemannian (sub)manifold, e.g., Euclidean plane, then your problem fits well within the framework of dynamical systems, in particular "discrete-time" dynamical ... | 1 | https://mathoverflow.net/users/992 | 1998 | 1,270 |
https://mathoverflow.net/questions/2004 | 8 | Let $K$ be a number field and suppose $K$ contains no $p$-power roots of unity. Let $\mathcal{P}$ be a prime of $K$ *above the rational prime $p$*. Can someone prove or disprove the assertion that the local field $K\_{\mathcal{P}}$ will contain no $p$-power roots of unity?
| https://mathoverflow.net/users/1018 | p-power roots of unity in local fields | Looks to me like this is false. Let $K = \mathbb{Q}(z)/(z^p-1-p^2)$. This is an extension of degree $p$, so it is disjoint from the p-th cycloctomic field, and hence does not contain a $p$-th root of $1$. Thus, it also can not contain a $p^k$-th root of 1.
Now, let's see how $z^p - 1 - p^2$ factors in Qp. There is al... | 8 | https://mathoverflow.net/users/297 | 2011 | 1,277 |
https://mathoverflow.net/questions/2014 | 86 | There is a standard problem in elementary probability that goes as follows. Consider a stick of length 1. Pick two points uniformly at random on the stick, and break the stick at those points. What is the probability that the three segments obtained in this way form a triangle?
Of course this is the probability that ... | https://mathoverflow.net/users/143 | If you break a stick at two points chosen uniformly, the probability the three resulting sticks form a triangle is 1/4. Is there a nice proof of this? | Here's what seems like the sort of argument you're looking for (based off of a trick Wendel used to compute the probability the convex hull of a set of random points on a sphere contains the center of the sphere, which is really the same question in disguise):
Connect the endpoints of the stick into a circle. We now ... | 117 | https://mathoverflow.net/users/405 | 2016 | 1,280 |
https://mathoverflow.net/questions/1967 | 19 | I was talking this morning to a colleague who thinks about combinatorial Hopf algebras. He mentioned several rings, which are of interest in combinatorics, for which he didn't know whether a Hopf structure existed. I was able to rule out several by the following result:
If A is a finitely generated commutative algebr... | https://mathoverflow.net/users/297 | Hopf algebra reference | Oort has an elementary proof that group schemes in char. 0 are reduced -- see MR0206005.
| 8 | https://mathoverflow.net/users/986 | 2023 | 1,285 |
https://mathoverflow.net/questions/1912 | 7 | Sorry for a loaded question.
I'm not an expert on those things, but I do know that a fibration gives rise to the representations of pointed fundamental group of the base on the cohomology of the fiber.
What are the properties of this map for different classes of fibrations? I think it's known what the image of this... | https://mathoverflow.net/users/65 | Properties of monodromy of a fibration? | A **small clarification** on bhargav's answer: in algebraic geometry we only have quasi-unipotency of the *local* monodromy in one-parameter families (which is what bhargav is talking about); or in multi-parameter families but only near a normal crossing point of the discriminant. Global monodromies are reductive and l... | 16 | https://mathoverflow.net/users/439 | 2025 | 1,286 |
https://mathoverflow.net/questions/2022 | 46 | I never really understood the definition of the conductor of an elliptic curve.
What I understand is that for an elliptic curve E over ℚ, End(E) is going to be (isomorphic to) ℤ or an order in a imaginary quadratic field ℚ(√(-d)), and that this order is uniquely determined by an integer f, the conductor, so that End(... | https://mathoverflow.net/users/362 | Definition and meaning of the conductor of an elliptic curve | The conductor of the curve and the conductor of the order in the endomorphism ring are not equal in the CM case; it's just unfortunate terminology. For example, y^2 = x^3 - x has complex multiplication by the maximal order Z[i] (conductor = 1) of Q(i), but it certainly doesn't have everywhere good reduction.
The cond... | 24 | https://mathoverflow.net/users/1018 | 2026 | 1,287 |
https://mathoverflow.net/questions/689 | 11 | Could someone give an overview, or just some examples, of "finiteness conditions" for simplicial sheaves/presheaves and/or simplicial schemes? Any answer or comment about this would be interesting, but I am interested in particular in the following two things:
1) I once heard Toen say something about this, and that o... | https://mathoverflow.net/users/349 | Finiteness conditions on simplicial sheaves/presheaves | I don't know what Toën was talking about, but I suspect that it was about finiteness conditions for Artin stacks: the problem is that the usual finiteness conditions we look at for schemes (like the notion of constructibility for l-adic sheaves) do not extend to stacks in a straightforward way, which gives some trouble... | 16 | https://mathoverflow.net/users/1017 | 2027 | 1,288 |
https://mathoverflow.net/questions/807 | 22 | As is well known, the universal covering space of the punctured complex plane is the complex plane itself, and the cover is given by the exponential map.
In a sense, this shows that the logarithm has the worst monodromy possible, given that it has only one singularity in the complex plane. Hence we can easily visuali... | https://mathoverflow.net/users/362 | Describing the universal covering map for the twice punctured complex plane | Others have already given a satisfactory qualitative description as a modular function under a suitable congruence group. Since the quotient in question is necessarily genus zero, there are explicit formulas for such functions.
I was mistaken in my comment to Tyler's answer. The function I provided there is invariant... | 9 | https://mathoverflow.net/users/121 | 2034 | 1,294 |
https://mathoverflow.net/questions/2039 | 4 | k is a perfect field. X and Y are two regular varieties over k. Does their fiber product over k remain to be regular?
Note: When k is algebraically closed it's true by Jacobian criterion. When k is not perfect there's counter-example.
| https://mathoverflow.net/users/2008 | Does the fiber product of two regular varieties over perfect field remain regular? | The answer is yes. Indeed, over a perfect field the notions of smooth and regular coincide so it follows from the fact that base change and composition preserve smoothness.
| 6 | https://mathoverflow.net/users/310 | 2042 | 1,299 |
https://mathoverflow.net/questions/1977 | 84 | This is a somewhat long discussion so please bear with me. There is a theorem that I have always been curious about from an intuitive standpoint and that has been glossed over in most textbooks I have read. Quoting [Wikipedia](http://en.wikipedia.org/wiki/Level_set#Level_sets_versus_the_gradient), the theorem is:
> ... | https://mathoverflow.net/users/812 | Why is the gradient normal? | The gradient of a function is normal to the level sets because it is **defined** that way. The gradient of a function is **not** the natural derivative. When you have a function $f$, defined on some Euclidean space (more generally, a Riemannian manifold) then its derivative at a point, say $x$, is a function $d\_xf(v)$... | 112 | https://mathoverflow.net/users/45 | 2049 | 1,303 |
https://mathoverflow.net/questions/1886 | 22 | Suppose we have an infinite matrix A = (aij) (i, j positive integers). What is the "right" definition of determinant of such a matrix? (Or does such a notion even exist?) Of course, I don't necessarily expect every such matrix to have a determinant -- presumably there are questions of convergence -- but what should the... | https://mathoverflow.net/users/913 | Infinite matrices and the concept of "determinant" | There is a class of linear operators that have a determinant. They are, for some strange reason, known as "operators with a determinant".
For Banach spaces, the essential details go along these lines. Fix a Banach space, X, and consider the **finite rank** linear operators. That means that T: X → X is such that Im(T)... | 31 | https://mathoverflow.net/users/45 | 2052 | 1,306 |
https://mathoverflow.net/questions/2046 | 14 | I felt like following up on [Kate's question](https://mathoverflow.net/questions/544/why-are-subfactors-interesting). There were some good motivational answers there.
Given a pair of [factors](http://en.wikipedia.org/wiki/Von_Neumann_algebra#Factors) M < N, there is a standard way to construct a 2-category whose obje... | https://mathoverflow.net/users/121 | How do I describe a fusion category given a subfactor? | Here are some partial answers:
1- Usually the fusion category is the category of bifinite correspondences, i.e. Hilbert spaces with actions of $N$ and $M$ whose module dimensions are finite. Jones has a result saying that a bifinite correspondence is irreducible if and only if the algebraic module of bounded vectors ... | 8 | https://mathoverflow.net/users/351 | 2057 | 1,310 |
https://mathoverflow.net/questions/2038 | 8 | Suppose $k$ is an algebraically closed field, and $X$, $Y$ are two normal varieties over $k$. Is the product $X \times Y$ necessarily still normal?
| https://mathoverflow.net/users/2008 | Does the fiber product of two normal varieties remain normal? | The answer is yes.
In general one can define a normal morphism of schemes $f:X \rightarrow Y$ to be a flat morphism such that for every $y \in Y$ the fibre over $y$ is geometrically normal.
Then we have the following theorem on normality and base change (see EGA Ch 2 IV 6.14.1)
Let $g: Y' \rightarrow Y$ be a norm... | 9 | https://mathoverflow.net/users/310 | 2058 | 1,311 |
https://mathoverflow.net/questions/724 | 16 | Pretty much exactly what it says on the tin. Let G be a connected graph; then the Tutte polynomial T\_G(x,y) carries a lot of information about G. However, it obviously doesn't encode everything about the graph, since there are examples of non-isomorphic graphs with the same Tutte polynomial.
My question is, what inf... | https://mathoverflow.net/users/382 | What is the Tutte polynomial encoding? | No-one so far has mentioned matroids. The Tutte polynomial encodes some of the information from the cycle matroid of the graph. Two graphs with the same cycle matroid (and number of vertices) have the same Tutte polynomials. So if a graph property is not determined by the cycle matroid (and the number of vertices) then... | 20 | https://mathoverflow.net/users/1028 | 2061 | 1,314 |
https://mathoverflow.net/questions/2065 | 2 | I have a problem in computing (i.e. classify) a factor group.
For example The group Z\*Z\*Z/<(3,6,9)> is isomorphic to Z\_3\*Z\*Z. I can show this by contructing a homomorphism f
f(a,b,c) = ( a mod 3 , 2\*a - b, 3\*a - c )
and then show that Ker(f) = <(3,6,9)>.
It is not hard to see that Im(f) = Z\_3\*Z\*Z.
Bu... | https://mathoverflow.net/users/818 | Computing a Factor Group | Since 4 and 3 are coprime, you can obtain every integer as 4a-3b for some a and b, and thus the image is isomorphic to Z/(9) x Z.
In general, for each factor you get the quotient of Z by the ideal generated by the gcd of the coefficients in your expression.
**EDIT** Sorry for the confusion, wrote too quickly, hope ... | 0 | https://mathoverflow.net/users/914 | 2067 | 1,319 |
https://mathoverflow.net/questions/1939 | 8 | Let R be a commutative ring, and $A$ an $R$-algebra (possibly non-commutative). Then $A$ is separable if it is finitely generated (f.g.) projective as an $(A \otimes\_R A^{\mathrm{op}})$-algebra. Suppose further that $A$ is f. g. projective as an $R$-module. Does this imply that $A$ is a (symmetric) Frobenius algebra?
... | https://mathoverflow.net/users/184 | Separable and finitely generated projective but not Frobenius? | Theorem 4.2 of [On separable algebras over a commutative ring](https://projecteuclid.org/journals/osaka-journal-of-mathematics/volume-4/issue-2/On-separable-algebras-over-a-commutative-ring/ojm/1200691953.full) says that the answer is always yes.
| 3 | https://mathoverflow.net/users/345 | 2070 | 1,321 |
https://mathoverflow.net/questions/2071 | 11 | It is remarked in Shafarevich's Basic Algebraic Geometry 1 that Rees and Nagata constructed examples of quasiprojective varieties such that the ring of regular functions is not finitely generated, but I cannot find the source he is referring to. Can anyone give such examples here? Does that mean we can't really say any... | https://mathoverflow.net/users/nan | Non-finitely generated ring of regular functions | It's a theorem that a quasi-projective variety is affine if and only if it is Stein (we're working over C, say) and its ring of functions is finitely generated. So find a Stein manifold that isn't affine, and that will do it.
And, after a bit of looking, it appears that Vakil may have rediscovered the Rees and Nagata... | 8 | https://mathoverflow.net/users/622 | 2073 | 1,323 |
https://mathoverflow.net/questions/1937 | 8 | Let pi: \bar{Mg,1} \to \bar{M\_g} be natural projection of compactified moduli stacks of curves and let omega be the relative dualizing sheaf. Then the Hodge class \lambda of \bar{M\_g} is the first chern class of the pushforward \pi\_\*(ω). Among other things the hodge class, together with the boundary divisors, freel... | https://mathoverflow.net/users/2 | Why is the Hodge class of \bar{M_g} big and nef? | Some multiple of lambda is defined on the coarse moduli space and this is the pullback of an ample bundle on \bar{A\_g}, the Satake-Baily-Borel compactification of A\_g. Since \bar{M\_g} maps birationally onto its image in \bar{A\_g}, it follows that lambda is nef and big, in fact also semi-ample (some multiple is base... | 5 | https://mathoverflow.net/users/519 | 2075 | 1,325 |
https://mathoverflow.net/questions/2040 | 39 | People who talk about things like modular forms and zeta functions put a lot of emphasis on the existence and form of functional equations, but I've never seen them used as anything other than a technical tool. Is there a conceptual reason we want these functional equations around? Have I just not seen enough of the th... | https://mathoverflow.net/users/290 | Why are functional equations important? | There are many reasons that functional equations are important. Some background: To most varieties/schemes occurring in arithmetic geometry, you can associate a zeta function/L-function. There are two main ways of constructing these, either from l-adic cohomology, or from counting solutions in various finite fields. Us... | 28 | https://mathoverflow.net/users/349 | 2085 | 1,331 |
https://mathoverflow.net/questions/2083 | 26 | When is a Stein manifold a complex affine variety? I had thought that there was a theorem saying that a variety which is Stein and has finitely generated ring of regular functions implies affine, but in the comments to my answer [here](https://mathoverflow.net/questions/2071/non-finitely-generated-ring-of-regular-funct... | https://mathoverflow.net/users/622 | Stein Manifolds and Affine Varieties | Charlie, it is funny answering this way but here it is.
The criterion you are thinking about is a criterion that is relative to an embedding. It says that if $X$ is a quasi-affine complex normal variety, whose associated analytic space $X^{an}$ is Stein, then $X$ is affine if (and only if) the algebra $\Gamma(X,\mat... | 26 | https://mathoverflow.net/users/439 | 2087 | 1,333 |
https://mathoverflow.net/questions/2054 | 9 | I've been asking myself this question all the time. Let's say you are given a large set of time series data. Your task is to find out patterns that are meaningful or that you can use for future trend prediction.
The issue now is, how do you know for sure that the patterns you extract are valid, in the sense that they... | https://mathoverflow.net/users/807 | Data Mining-- how do you know whether the pattern you extract is valid? | A good way to do this is called 'cross-validation'. The data can be divided into three disjoint sets: the training set, the test set and the validation set.
Different models are developed using the training set. A reasonable way to do this is to take different subsets of points from the training set uniform randomly ... | 13 | https://mathoverflow.net/users/812 | 2096 | 1,340 |
https://mathoverflow.net/questions/959 | 11 | David Corfield asked the following questions yesterday: Is the
$n$-dimensional Fourier transform of $\exp(-\|x\|)$ always non-negative,
where $\|\cdot\|$ is the Euclidean norm on $\Bbb R^n$? What is its support?
I want to ask a more general question: what happens when $\|\cdot\|$ is the
$p$-norm, for arbitrary $p\in ... | https://mathoverflow.net/users/586 | Fourier transform of $\exp(-\|x\|_p)$: more general question | Okay, I think I do have an answer now. I'm borrowing arguments from the proof of Lemma 2.27 in the book "Fourier Analysis in Convex Geometry" by A. Koldobsky (apparently not available online at all). That lemma states that the Fourier transform of the function (on $\Bbb R$) $\exp(-|x|^p)$ is positive everywhere for $p ... | 11 | https://mathoverflow.net/users/1044 | 2106 | 1,349 |
https://mathoverflow.net/questions/665 | 1 | Recall that for k a field, a finite dimensional k-algebra A is called *symmetric* if it is isomorphic to its dual as a bimodule of itself. Which is to say, there's a trace map t:A -> k such that t(ab)=t(ba) and for any nonzero a, there is a b such that t(ab) is not 0.
The most popular examples of symmetric algebras a... | https://mathoverflow.net/users/66 | Are cyclotomic Khovanov-Lauda-Rouquier algebras symmetric? | This is now proven both as Theorem 1.7 in my paper "[Knot invariants and higher representation theory I](https://arxiv.org/abs/1001.2020)" and by Kang and Kashiwara in "[Categorification of Highest Weight Modules via Khovanov-Lauda-Rouquier Algebras](https://arxiv.org/abs/1102.4677)."
| 2 | https://mathoverflow.net/users/66 | 2109 | 1,351 |
https://mathoverflow.net/questions/2045 | 1 | Can you help me understand the class of problems solvable by a nondetermimistic Turing machine with an oracle for SAT running in polynomial time?
| https://mathoverflow.net/users/1027 | How can one characterize NP^SAT? | Surely this class, being $\text{NP}^\text{NP}$, is by definition equal to $\Sigma\_2^p$. In particular, if the Polynomial Hierarchy (PH) does not collapse, then it does not contain $\Pi\_2^p$.
| 5 | https://mathoverflow.net/users/1046 | 2112 | 1,353 |
https://mathoverflow.net/questions/2118 | 7 | The rationals are clearly dense in the real number system, i.e. for every pair a < b of real numbers there exists a rational number p/q s.t. a < p/q < b. I conjecture the same to be true with p and q both primes. Any idea of how one could prove it? It should depend on some strong result on the distribution of prime num... | https://mathoverflow.net/users/1049 | Density of a subset of the reals | Yes. Take q sufficiently big and fixed (in terms of a and b). Then the question is, is there some prime p between qa and qb? Use the prime number theorem to estimate pi(qb) - pi(qa) > 0, where q is chosen to be big enough so that the main term is bigger than the error terms. QED.
| 6 | https://mathoverflow.net/users/1050 | 2120 | 1,358 |
https://mathoverflow.net/questions/2077 | 17 | Let $SX$ be the suspension of CW complex. What are some results available to determine the homotopy groups of $SX$?
| https://mathoverflow.net/users/1034 | How to determine the homotopy groups of the suspension of a space? | This, in general, an incredibly difficult problem. Even we just want to compute the *rational* homotopy groups of the suspension of $X$ and $X$ is simply connected, where we can do everything using rational homotopy theory to reduce things to commutative DGAs, this starts to involve things like free Lie algebras and th... | 29 | https://mathoverflow.net/users/360 | 2121 | 1,359 |
https://mathoverflow.net/questions/1951 | 19 | One could try to apply the Eilenberg-Moore spectral sequence to the pullback diagram • → X ← •, obtaining a spectral sequence TorH•(X, R)(R, R) => H•(ΩX, R), but could there be differentials or extension problems which differ for different spaces X with the same cohomology ring?
| https://mathoverflow.net/users/126667 | Does the cohomology ring of a simply-connected space X determine the cohomology groups of ΩX? | Tyler's comment to my earlier answer seems to give a solution; he suggests comparing the space $T=(S^3\vee S^3)\cup\\_{[x,[x,y]]} e^8$ with a wedge $S^3\vee S^3\vee S^8$. It's probably easier to think about homology with the Pontryagin product. Homology of loops on on the wedge will be a tensor algebra on classes in 2,... | 14 | https://mathoverflow.net/users/437 | 2122 | 1,360 |
https://mathoverflow.net/questions/383 | 70 | In undergraduate differential equations it's usual to deal with the Laplace transform to reduce the differential equation problem to an algebraic problem.
The Laplace transform of a function $f(t)$, for $t \geq 0$ is defined by $\int\_{0}^{\infty} f(t) e^{-st} dt$.
How to avoid looking at this definition as "magical"? ... | https://mathoverflow.net/users/273 | Motivating the Laplace transform definition | What is also very interesting is that the Laplace transform is nothing else but the continuous version of power series - see this insightful video lecture from MIT:
<https://ocw.mit.edu/courses/18-03-differential-equations-spring-2010/resources/lecture-19-introduction-to-the-laplace-transform/>
| 53 | https://mathoverflow.net/users/1047 | 2141 | 1,369 |
https://mathoverflow.net/questions/2107 | 4 | Let $B$ be a curve (integral but not necessarily smooth) and let $\pi: C --> B$ be a family of curves such that each fiber is a rational curve with $g$ many elliptic tails attached.
Let $\omega$ be the relative dualizing sheaf.
>
> **Question**: Why is the pushforward $\pi\_\* \omega$ trivial (as a vector bundl... | https://mathoverflow.net/users/2 | Question about a family of semistable curves | I don't think this is quite right. Here is the right statement: let $E\_1, ..., E\_g$ be the tails, with maps $q\_i: E\_i \longrightarrow B$. Then
$\pi\_\* \omega\_{C/B} = \bigoplus (q\_i)\_\* \omega\_{E\_i/B}$.
So, if your tails don't vary with B, this bundle is trivial.
Explanation: $\omega\_{C/B}$ can be descr... | 8 | https://mathoverflow.net/users/297 | 2145 | 1,370 |
https://mathoverflow.net/questions/2128 | 18 | Let L be the lattice of Young diagrams ordered by inclusion and let Ln denote the nth rank, i.e. the Young diagrams of size n. Say that lambda > mu if lambda covers mu, i.e. mu can be obtained from lambda by removing one box and let C[L] be the free vector space on L. The operators
U lambda = summu > lambda mu
D la... | https://mathoverflow.net/users/290 | Young's lattice and the Weyl algebra | **EDIT** (3/16/11): When I first read this question, I thought "hmmm, Weyl algebra? Really? I feel like I never hear people say they're going to categorify the Weyl algebra, but it looks like that's what the question is about..." Now I understand what's going on. Not to knock the OP, but there's a much bigger structure... | 12 | https://mathoverflow.net/users/66 | 2152 | 1,374 |
https://mathoverflow.net/questions/2150 | 17 | Are filtered colimits exact in all abelian categories?
In Set, filtered colimits commute with finite limits. The proof carries over to categories sufficiently like Set (i.e. where you can chase elements round diagrams), in particular A-Mod where A is a commutative ring. This implies that filtered colimits are exact i... | https://mathoverflow.net/users/1046 | Exactness of filtered colimits | Here's a dumb counterexample. If C is an abelian category, so is Cop. In Cop, filtered colimits are filtered limits in C. And, of course, there are many examples of abelian categories (such as abelian groups) where filtered limits aren't exact.
Of course, your question is really: when is an abelian category C suffici... | 21 | https://mathoverflow.net/users/437 | 2158 | 1,379 |
https://mathoverflow.net/questions/2084 | 10 | The volume in the orthogonal group is measured by the Haar measure, which is the up to scaling unique measure that is invariant under the group operation. I consider the usual metric that is induced by the spectral norm $|M| = \max |Mx|$ where $x$ ranges over all vectors of length 1 and the vector norm is the Euclidean... | https://mathoverflow.net/users/1038 | What is the volume of a $\delta$-ball in the orthogonal group $O(n)$? Is there a (simple) lower bound? | The volume of the delta-ball of the special orthogonal group can be computed exactly by applying the Weyl integration formula: (Without loss of generality, we assume that the delta-ball is around the unit group element).
a. One notices (Again due to the invariance under the Haar measure) that the characteristic funct... | 6 | https://mathoverflow.net/users/1059 | 2168 | 1,388 |
https://mathoverflow.net/questions/2132 | 8 | Trying to understand answer to [this question](https://mathoverflow.net/questions/1988/regulators-of-number-fields-and-elliptic-curves).
What is the **(Beilinson) higher regulator** of a number field?
| https://mathoverflow.net/users/65 | What is the Beilinson regulator? | Here is an attempt to answer, but I hope that someone else can give a better explanation.
As Rob H. pointed out in his answer to the previous question, the survey of Nekovar is very nice, and it is also available online [here](http://people.math.jussieu.fr/~nekovar/pu/mot.pdf).
About your question: The Beilinson re... | 10 | https://mathoverflow.net/users/349 | 2187 | 1,400 |
https://mathoverflow.net/questions/1652 | 19 | I try to keep a list of standard ring examples in my head to test commutative algebra conjectures against. I would therefore like to have an example of a ring which is normal but not Cohen-Macaulay. I've found a few in the past, but they were too messy to easily remember and use as test cases. Suggestions?
| https://mathoverflow.net/users/297 | Simple example of a ring which is normal but not CM | Another family of examples is given by the homogeneous coordinate rings of
irregular surfaces (ie 2-dimensional $X$ such that $H^1({\mathcal O}\_X) \neq 0$);
these surfaces cannot be embedded in any way so that their homogeneous coordinate rings
become Cohen-Macaulay. Elliptic scrolls (such as the one in the previous a... | 22 | https://mathoverflow.net/users/5771 | 2194 | 1,406 |
https://mathoverflow.net/questions/2093 | 9 | I'm interested in graph families which are sparse, and by sparse I mean the number of edges is linear in the number of vertices. |E| = O(|V|).
Besides non-trivial minor-closed families of graphs (these turn out to be sparse), I don't know any other families. Can anyone suggest any interesting graph families (which are ... | https://mathoverflow.net/users/1042 | Interesting families of sparse graphs? | 1) Graphs of degree at most d. There are a myriad results known about them. Among them there are regular graphs (graphs in which every vertex is of degree exactly d).
2) Graphs of bounded degeneracy. These are the graphs in which every subgraph has a vertex of degree at most d. For example, stars have unbounded degre... | 10 | https://mathoverflow.net/users/806 | 2195 | 1,407 |
https://mathoverflow.net/questions/2127 | 5 | The aim of transforming the Black-Scholes PDE is of course to find a form where an relatively easy solution exists. Most of the steps seem to be straightforward - please use this reference:
<https://planetmath.org/AnalyticSolutionOfBlackScholesPDE>
...all but one, actually the last one where a convection-diffusion eq... | https://mathoverflow.net/users/1047 | Transformation of the Black-Scholes PDE into the diffusion equation - shift of coordinate system | I'm afraid the Planetmath page put my browser into an infinite reload loop, so I can't help you with the formalism there.
I would recommend instead looking at the change of variables in [the Wikipedia article](http://en.wikipedia.org/wiki/Black-Scholes#Derivation). The last time I checked it, it seemed to work.
**E... | 5 | https://mathoverflow.net/users/121 | 2199 | 1,410 |
https://mathoverflow.net/questions/2146 | 73 | There are some things about geometry that show why a motivic viewpoint is deep and important. A good indication is that Grothendieck and others had to invent some important and new algebraico-geometric conjectures *just to formulate the definition of motives*.
There are things that I know about motives on some level,... | https://mathoverflow.net/users/65 | What's the "Yoga of Motives"? | So this is a crazy question, but I will try to give at least a partial answer. This [question about the Beilinson regulator](https://mathoverflow.net/questions/2132/what-is-the-beilinson-regulator/2187#2187) is also relevant, and this is also an attempt to reply to the comments of Ilya there. I apologize for simplifyin... | 58 | https://mathoverflow.net/users/349 | 2206 | 1,414 |
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