parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/3764 | 28 | Let f be a complex-valued function of one real variable, continuous and compactly supported. Can it have a Fourier transform that is not Lebesgue integrable?
| https://mathoverflow.net/users/1386 | Does there exist a continuous function of compact support with Fourier transform outside L^1? | Here's a sketch of what I think is an example of the sort you want. Consider a trapezoidal function Tδ, supported on [-1,1], which is 1 on [-1+δ , 1-δ ] and is defined on the remaining intervals by interpolation in the obvious way. Then as δ tends to zero, the Fourier transform of Tδ is going to tend to infinity in the... | 24 | https://mathoverflow.net/users/763 | 3790 | 2,524 |
https://mathoverflow.net/questions/3716 | 25 | What are the possible automorphism groups of a principally polarized abelian variety $(A,\lambda)$ of dimension $g,$ say an abelian surface ($g=2$) over the complex numbers or algebraic closure of a finite field? The fact that the moduli stack $A\_g$ is of finite diagonal (over the integers) implies that the automorphi... | https://mathoverflow.net/users/370 | What are the automorphism groups of (principally polarized) abelian varieties? | The standard proof of finiteness goes as follows: the polarization defines a positive involution \* on the endomorphism algebra, and so the automorphisms are the elements of End(A) tensor the reals that are in End(A) and satisfy a\*a=1. Thus the set of automorphisms is the intersection of a discrete set and a compact s... | 15 | https://mathoverflow.net/users/930 | 3792 | 2,526 |
https://mathoverflow.net/questions/3656 | 78 | Singular homology is usually defined via singular simplices, but Serre in his thesis uses singular cubes, which he claims are better adapted to the study of fibre spaces. This young man (25 years old at the time) seemed to know what he was talking about and has had a not too unsuccessful career since.
So my (quite co... | https://mathoverflow.net/users/450 | Cubical vs. simplicial singular homology | Others have mentioned the advantages to cubical sets and so I don't want to say much on those; I'll just mention some facts about the other direction. The main disadvantage that comes solely from the point of view of homology is that you have this irritating normalization procedure: you have to take the cubical chains ... | 40 | https://mathoverflow.net/users/360 | 3795 | 2,529 |
https://mathoverflow.net/questions/3798 | 2 | It is easy to see that the totally ordered group $\mathbb{Z}$ (the integers) with the natural order has no non-trivial automorphisms. Is this also true for $\mathbb{Z}^n$ with the lexicographical order?
| https://mathoverflow.net/users/717 | Automorphisms of the totally ordered group $\mathbb{Z}{^n}$ with lexicographical order | Here's a counterexample: on $\mathbb{Z}^2$, $f(x,y)=(x,y+x)$.
More generally, the order-preserving automorphisms of $\mathbb{Z}^n$ are exactly the upper triangular matrices with 1s on the diagonal (this should be easy to see by combining Charles's argument with my example in the case $n=2$, and then the generalizati... | 5 | https://mathoverflow.net/users/75 | 3800 | 2,531 |
https://mathoverflow.net/questions/3797 | 3 | Why are $n$-fold complete segal spaces or $(\infty, n)$-categories (which I'm unsure of how to distinguish from omega-categories) important for $n \geq 3$? Why are they "badly behaved" for $n \geq 3$? (Lurie refers to them this way in his thesis).
Also, I'm particularly interested to connections between $n$-fold comp... | https://mathoverflow.net/users/429 | omega-categories and n-fold complete segal spaces | $n$-fold complete Segal spaces are one *model* for $(\infty,n)$-categories; there are other models. More precisely, they are supposed to be a model for *weak* $(\infty,n)$-categories.
The distinction that I think you are asking about is between *weak* and *strict*. Strict $n$-categories can be easily defined by a rec... | 10 | https://mathoverflow.net/users/437 | 3805 | 2,534 |
https://mathoverflow.net/questions/735 | 6 | Let $w(x,y)$ be a group word in $x$ and $y$.
Let $x$ and $y$ now vary in $\operatorname{SL}\_n(K)$, where $K$ is a field. (Assume, if you wish, that $K$ is an algebraically complete field of characteristic bigger than a constant.)
I would like to know for which words $w$ the map
$$y \mapsto w(x,y)$$
isn't surje... | https://mathoverflow.net/users/398 | When is a map given by a word surjective? | The discussion has now moved to
[Surjective maps given by words, redux](https://mathoverflow.net/questions/2082/surjective-maps-given-by-words-redux)
| 0 | https://mathoverflow.net/users/398 | 3807 | 2,536 |
https://mathoverflow.net/questions/3819 | 71 | Complex analytic functions show rigid behavior while real-valued smooth functions are flexible. Why is this the case?
| https://mathoverflow.net/users/1354 | Why do functions in complex analysis behave so well? (as opposed to functions in real analysis) | Well, real-valued analytic functions are just as rigid as their complex-valued counterparts. The true question is why complex smooth (or complex differentiable) functions are automatically complex analytic, whilst real smooth (or real differentiable) functions need not be real analytic.
As Qiaochu says, one answer is... | 103 | https://mathoverflow.net/users/766 | 3830 | 2,549 |
https://mathoverflow.net/questions/3841 | 13 | Taking tori in symmetric products and "miraculously" proving that the Floer homology is independent of choices always seemed, well, miraculous. Some time ago Max Lipyanski explained to me the origins of this construction from gauge theory on surfaces, a la Atiyah-Floer conjecture, which I have then forgotten. What is t... | https://mathoverflow.net/users/375 | The "miracle" of Heegard Floer. | I think the crude answer is that there is (or maybe just should be) an extended 4 dimensional TQFT that assigns the Fukaya category of a symmetric product to a surface, and the usual Heegard-Floer Lagrangian to a 3 manifold. So, the usual definition of Heegard-Floer is the gluing formula for a Heegard splitting, and in... | 9 | https://mathoverflow.net/users/66 | 3843 | 2,559 |
https://mathoverflow.net/questions/3849 | 2 | Suppose G is a Lie group (or algebraic group) acting on a manifold (or scheme) X, and H⊆G is a subgroup. Let x,y∈X be points such that x is in the closure of the orbit H⋅y (but not in H⋅y itself). Then obviously x is in the closure of G⋅y, but can it happen that x is actually in the orbit G⋅y (not just in the closure)?... | https://mathoverflow.net/users/1 | Do subgroups respect the orbit-closure relation? | Sure, that can happen. G = PGL\_2, H is the torus, X is the Riemann sphere, x is the north pole, y is some point other than the two poles.
I've read that Kapranov paper recently, I'll see if I can find something more useful to say.
| 3 | https://mathoverflow.net/users/297 | 3853 | 2,564 |
https://mathoverflow.net/questions/3857 | 5 | Some of my friends and I were trying to discover a universal mapping property that characterizes the integers $\mathbb{Z}$ in the category of groups without referring to the underlying sets (So it is a no no to say it is the free group on a one element set). One of the big uses of $\mathbb{Z}$ is that it is a separator... | https://mathoverflow.net/users/1106 | Separators in the Category of Groups | According to your definition, the separators will be exactly those groups $G$ with a surjection to $\mathbb{Z}$. One direction: if $f(x)\neq g(x)$, then take the composition $G \twoheadrightarrow \mathbb{Z} \rightarrow A$ where the latter map sends the generator of $\mathbb{Z}$ to $x$. For the other direction, to disti... | 6 | https://mathoverflow.net/users/250 | 3861 | 2,569 |
https://mathoverflow.net/questions/3858 | 48 | Can someone indicate me a good introductory text on geometric group theory?
| https://mathoverflow.net/users/1049 | Introductory text on geometric group theory? | de la Harpe's book is quite nice and has an amazing bibliography, but it doesn't really prove any deep theorems (though it certainly discusses them!). Some other sources.
1) Bridson and Haefliger's book "Metric Spaces of Non-Positive Curvature". Very easy to read and covers a lot of ground.
2) Ghys and de la Harpe'... | 47 | https://mathoverflow.net/users/317 | 3868 | 2,572 |
https://mathoverflow.net/questions/3859 | 3 | I am wondering how *B-fields*, which are basic objects in Generalized Geometry, relate to the B-fields of [Ben's question and the answers to it](https://mathoverflow.net/questions/1726/how-should-i-think-about-b-fields).
In Generalized Geometry, the B-field is a (1,1)-form, and when it is closed it preserves the gene... | https://mathoverflow.net/users/1177 | Prevalence of B-fields | The short answer is that both B-fields are the same object!
The way the B-field comes to us from string theory it doesn't come alone, but comes together (among other fields) with the Riemannian metric. Due to the way both originate in the string, they are interrelated by what is called [T-duality](http://ncatlab.org/... | 3 | https://mathoverflow.net/users/381 | 3872 | 2,574 |
https://mathoverflow.net/questions/3847 | 11 | The Cartan structure equations for a connection and various associated 1-forms can be checked in a straightforward algebraic manner. But is there a geometric or global significance to the equations- can one visualize the proof?
| https://mathoverflow.net/users/344 | What is the geometric significance of Cartan's structure equations? | There is a nice grand story behind all this. I don't know if you like thinking that way, but things do clarify when one looks at it from a more general perspective of [oo-Lie algebroid valued differential forms](http://ncatlab.org/schreiber/show/%E2%88%9E-Lie+algebroid+valued+differential+forms) with [curvature](http:/... | 6 | https://mathoverflow.net/users/381 | 3874 | 2,576 |
https://mathoverflow.net/questions/3871 | 25 | For a compact space $K$, the maximal ideals in the ring $C(K)$ of continuous real-valued functions on $K$ are easily identified with the points of $K$ (a point defines the maximal ideal of functions vanishing at that point).
Now take $K=\mathbb{R}$. Is there a useful characterization of the set of maximal ideals of $... | https://mathoverflow.net/users/25 | Maximal ideals in the ring of continuous real-valued functions on ℝ | Peter Johnstone's book [Stone Spaces](https://books.google.com/books?id=66Njdgsk3ukC&lpg=PR7&pg=PA144#v=onepage&q=&f=false) (p. 144) proves that for any X, maximal ideals in $C(X)$ are the same as maximal ideals in $C\_b(X)$ (bounded functions), i.e. the Stone-Cech compactification $\beta X$. Indeed, if I is a maximal ... | 32 | https://mathoverflow.net/users/75 | 3876 | 2,577 |
https://mathoverflow.net/questions/3863 | 2 | Some functions are not represented by their power series even when they are continuous and have all the necessary derivatives. What's the best characterization of these functions? Explanations at any level are welcome.
| https://mathoverflow.net/users/812 | What functions are not represented by their power series? | A smooth function is characterized as being analytic if its derivatives
on any closed interval have a certain growth rate. See "Alternate Characterizations"
in the
[Wikipedia article on analytic functions.](http://en.wikipedia.org/wiki/Analytic_function)
| 6 | https://mathoverflow.net/users/1345 | 3881 | 2,581 |
https://mathoverflow.net/questions/2883 | 6 | D. Orlov proved that any equivalence of bounded derived categories *F:Db(X) -> Db(Y)* is a Fourier-Mukai transform, when *X* and *Y* are smooth projective varieties. Is there any example of such equivalence, which is not a Fourier-Mukai transform (it is not an integral transform)?
| https://mathoverflow.net/users/1220 | Equivalence of derived categories which is not Fourier-Mukai | Schlichting gave an example of two categories of singularities which are derived equivalent but whose K-groups are not isomorphic. Dugger and Shipley (arXiv:0710.3070) expanded on this example and noted that it gives two dga's which are derived equivalent but not by an integral transform.
Otherwise, Lunts and Orlov's... | 9 | https://mathoverflow.net/users/1404 | 3891 | 2,589 |
https://mathoverflow.net/questions/3877 | 1 | Maybe I'm being dense here, but can someone give me a subset of the set of all languages which is uncountable and the subset is easy to describe? (Some natural subset -- not like "take the set of all languages and remove a few.")
For instance, I thought of the set of recursive or recursively enumerable languages, but... | https://mathoverflow.net/users/1042 | A subset of all languages which is uncountable? | One way to take the question is to ask for a set of languages, for instance a complexity class, which is uncountable "for a good reason". In other words, that people study the class for some substantially different reason, and it's clearly convenient for it to be uncountable.
Probably the most common example is the c... | 5 | https://mathoverflow.net/users/1450 | 3903 | 2,596 |
https://mathoverflow.net/questions/3913 | 2 | Let g,h be positive integers. Let E be an elliptic curve, C be a genus h curve, and D be a genus g-h-1 curve. Let c,d,e be points on (resp.) C,D, and E.
Let f:CC --> E-e be the family whose fiber over a point e' is the curve obtained by glueing C to E together at the points c and e and D to E at the points d and e'.... | https://mathoverflow.net/users/2 | Triviality of the Hodge bundle for a special family of semistable curves | Under (the extension of) Torrelli this curves maps to one point in Ag. On the other hand the hodge class on Mg minus D0 is a pullback (under the extension of Torelli) of the hodge class on Ag.
| 2 | https://mathoverflow.net/users/404 | 3914 | 2,602 |
https://mathoverflow.net/questions/3911 | 13 | SGA 3, expose 12, remark 1.6 says that one can easily construct a group scheme over a discrete valuation ring with generic fiber Gm and special fiber Ga.
What is such an example?
| https://mathoverflow.net/users/2 | Constructing a degeneration (as a group scheme) of G_m to G_a | R[x,y]/(y^2-x^3-\pi x^2) gives the coordinate ring of a bad cubic curve, where \pi is a uniformizer in R. Remove the origin (which is the one singular point), and projectivize the curve by adding a point at infinity, so your group has an identity.
The generic fiber (treating \pi as a unit) is then the smooth part of ... | 15 | https://mathoverflow.net/users/121 | 3918 | 2,606 |
https://mathoverflow.net/questions/3864 | 6 | I know the interior point method works both for Linear Programming (LP) and semidefinite programming (SDP). My question is, can the other popular method for solving LP, namely the simplex method, be extended to SDP? If not, what is the barrier?
| https://mathoverflow.net/users/1401 | Simplex method for SDP? | *[the first part of this answer is similar to Dinakar Muthiah's]*
When optimizing a linear function on a convex set, it can always be assumed that the optimal solution lies on an extreme point of the feasible region (if there are several optimal solutions, at least one is an extreme point).
In the case of linear p... | 3 | https://mathoverflow.net/users/1184 | 3923 | 2,609 |
https://mathoverflow.net/questions/3912 | 5 | Let X,Y to be mappings from the sample space Ω to R and suppose Y is measurable with respect to σ(X), the smallest σ-field that makes X measurable.
Does it follow that there exists some Borel-measurable function f: R → R such that
Y=f(X)?
| https://mathoverflow.net/users/1407 | question on sigma-fields | The answer is yes. The proof is quite standard.
**1.** If Y=1\_A, where A is in \simga(X), then by the definition of \sigma(X) there exists a Borel set B such that A = X^{-1}(B) and therefore
Y(\omega) = 1\_A(\omega) = 1\_B(X(\omega)) = f(X(\omega)),
where we put f:= 1\_B (of course f is now a Borel function).
**... | 6 | https://mathoverflow.net/users/1302 | 3925 | 2,611 |
https://mathoverflow.net/questions/3920 | 20 | When constructing proofs using [natural deduction](http://en.wikipedia.org/wiki/Natural_deduction) what does it mean to say that an assumption or premise is *discharged*? In what circumstances would I want to, or need to, use such a mechanism?
The reason I'm asking this question is that many texts on logic use this t... | https://mathoverflow.net/users/107 | What does it mean to 'discharge assumptions or premises'? | As I understand it, to discharging a premise or assumption is the opposite of introducing it: you absorb it (for example) into the antecedent of an implication --- this means that it is no longer an assumption. A trivial example:
P 1. Assume P
\_\_
P 2. From 1
\_\_
P->P 3. Discharging 1
Thus I have conclude... | 8 | https://mathoverflow.net/users/1222 | 3930 | 2,613 |
https://mathoverflow.net/questions/3280 | 15 | A Leibniz algebra L may be thought of as a noncommutative generalisation of a Lie algebra. One drops the requirement that the bracket be alternating and substitutes the Jacobi identity for the Leibniz identity
$$ [x,[y,z]] = [[x,y],z] + [y,[x,z]] $$ for all $x,y,z \in L$
**Remark**. What is being defined above is a... | https://mathoverflow.net/users/394 | Hopf algebra structure on the universal enveloping algebra of a Leibniz algebra? | Do you know the paper [of Loday and Pirashvili](http://www-irma.u-strasbg.fr/~loday/PAPERS/98LodayPira%28linmaps%29.pdf)? They discuss what, in their opinion, should replace the notion of a Hopf algebra in Leibniz setting, "Hopf algebras in the category of linear maps".
| 10 | https://mathoverflow.net/users/1306 | 3934 | 2,617 |
https://mathoverflow.net/questions/3927 | 19 | I really should know the answer to this, but I don't, so I'll ask here.
A *Q-curve* is an elliptic curve E over Q-bar which is isogenous to all its Galois conjugates. A Q-curve is *modular* if it's isogenous (over Q-bar) to some factor of the Jacobian of X\_1(N) for some N>=1 (here X\_1(N) is the compact modular curv... | https://mathoverflow.net/users/1384 | Are Q-curves now known to be modular? | Yes, this is a consequence of Serre's conjecture. The canonical reference is probably Corollary 6.2 of Ribet's paper on Q-curves:
<http://math.berkeley.edu/~ribet/Articles/korea.pdf>
| 18 | https://mathoverflow.net/users/nan | 3936 | 2,619 |
https://mathoverflow.net/questions/3928 | 2 | Let X be a continuous stochastic process. I know that (t>s)
P(|X(t) - X(s)|>δ) < |t-s|/δ
Is it possible to say anything (e.g. estimate the decay of the tail) about
Y=sup\_{s \in [0,1]} |X(s)|?
I suspect that the answer is no (it would be an easy question if we have |t-s|^{1+a}, for any a>0).
I wonder what a... | https://mathoverflow.net/users/1302 | Suprema of stochastic processes | The answer is, indeed, "No" because there is an unbounded with probability $1$ stochastic process that satisfies the given inequality, namely, $X(t)=0.1\log|t-w|$ where $w$ is equidistributed on $[0,1]$. Truncating it at high level $L$, we get a continuous process such that $E|X(t)|$ is uniformly bounded but the suprem... | 5 | https://mathoverflow.net/users/1131 | 3940 | 2,622 |
https://mathoverflow.net/questions/2888 | 15 | It's not hard to count the number of permutations in a given conjugacy class of Sn. In particular, the number of permutations in Sn whose cycle decomposition has ci i-cycles is n!/(Πi=1n ci!ici). It's also not too hard to see that this is maximized for the conjugacy class that leaves one element fixed and permutes the ... | https://mathoverflow.net/users/1060 | Injective proof about sizes of conjugacy classes in S_n | For any cycle decomposition, we can uniquely order the cycles from smallest length to largest length, breaking ties between cycles of the same length in some fixed arbitrary way (say by maximal elements). Let us do this for concreteness.
Suppose there was an (injective) way to "join" and A-cycle and a B-cycle togethe... | 5 | https://mathoverflow.net/users/nan | 3952 | 2,631 |
https://mathoverflow.net/questions/3951 | 52 | I always have trouble memorizing theorems. Does anybody have any good tips?
| https://mathoverflow.net/users/812 | Memorizing theorems | As far as possible, you should turn yourself into the kind of person who does not have to remember the theorem in question. To get to that stage, the best way I know is simply to attempt to prove the theorem yourself. If you've tried sufficiently hard at that and got stuck, then have a quick look at the proof -- just e... | 145 | https://mathoverflow.net/users/1459 | 3957 | 2,636 |
https://mathoverflow.net/questions/3971 | 12 | Let $E/\mathbf{Q}\_p$ be an elliptic curve having split multiplicative reduction. Then Tate uniformization gives a surjective homomorphism of $p$-adic analytic groups $G\_m \to E$, with infinite cyclic kernel. Is there an analogue of this fact for $E$ having nonsplit multiplicative reduction, perhaps replacing Gm with ... | https://mathoverflow.net/users/367 | Tate uniformization of nonsplit semistable elliptic curves | A form of this is contained in Silverman, second book, Chapter V, Corollary 5.4. I guess
that the image of Gm' in E (at the level of Q\_p-points) may have index 2.
| 4 | https://mathoverflow.net/users/1253 | 3985 | 2,657 |
https://mathoverflow.net/questions/2082 | 8 | I asked some time ago:
Let $w(X,Y)$ be a word in $X$ and $Y$ (i.e., an element in the free group on $X$ and $Y$). Let the variables $x$ and $y$ now range among elements of $SL\_n(K)$, $K$ an algebraically closed field.
For which $w$ is it the case that, for $x$ generic, the image of the map given by
$y \rightarro... | https://mathoverflow.net/users/398 | Surjective maps given by words, redux | You need some conditions on K to make sense of the question. For instance, y ⟼ y2 is not surjective when K = ℝ and n = 2, because you cannot reach [[-1,0],[0,-2]]. Of course, you might have meant surjectivity in the sense of algebraic groups rather than in the sense of set-theoretic groups. But I think that that just a... | 3 | https://mathoverflow.net/users/1450 | 3991 | 2,661 |
https://mathoverflow.net/questions/3988 | 3 | Let's say I want to prove that a closed subgroup of GL(n,R) or GL(n,C) is a Lie group, with an atlas given by exponential of matrices (restricted to an appropriate subalgebra of gl(n)), without using any manifold or Lie theory. Can you provide the necessary argument? Maybe it's trivial, but I can't see it at the moment... | https://mathoverflow.net/users/1049 | Closed subgroups of GL(n) | Well, the crux of the matter is to cook up the linear subspace of $gl(n)$ that will eventually be the subalgebra.
If $H$ is the subgroup: set $h=\{X\in gl(n): (\forall t \in \mathbb{R}) \, exp(tX) \in H \}$.
You need to show two not completely obvious things:
1. $h$ is a linear subspace
2. $exp(h)$ is a nbhd of $... | 8 | https://mathoverflow.net/users/1143 | 3996 | 2,665 |
https://mathoverflow.net/questions/3997 | 15 | I've been learning game theory on my own and was just curious how it connected with previous things I've learned. So are there any interesting connections between Game Theory and Algebraic Topology? or possibly other branches of topology?
| https://mathoverflow.net/users/695 | Are there any interesting connections between Game Theory and Algebraic Topology? | One example is in the concept of a Nash equilibrium, whose existence can be proved using various (topological) fixed point theorems. (Google "nash equilibrium proof" for a wide variety of examples... the main topological machinery that comes up is the Kakutani fixed point theorem or the plain-vanilla Brouwer theorem.) ... | 19 | https://mathoverflow.net/users/1227 | 3999 | 2,667 |
https://mathoverflow.net/questions/3426 | 6 | This is a question that has haunted me for some time. In the domain of time series you always talk about trends and mean reversion. But at least to me these concepts are either defined axiomaticly within the data generating process (like the drift component in a geometric Brownian motion) or feel more like a heuristic ... | https://mathoverflow.net/users/1047 | Rigorous definition, detection and test for trending vs. mean-reverting behaviour of stochastic processes | In the abstract, you're looking to see when the derivative of the previsible part of a martingale (a la the Doob-Meyer decomposition) is nonzero. Finding trends with perfect accuracy amounts to performing the D-M decomposition, but of course nobody can do this in practice. Although there are a lot of ad hoc approaches ... | 2 | https://mathoverflow.net/users/1847 | 4003 | 2,670 |
https://mathoverflow.net/questions/4001 | 2 | How can I compute the SRB measure for the cat map? Also any pointers to references for obtaining Markov partitions and recurrence times would be lovely. Thanks
| https://mathoverflow.net/users/1847 | The Arnold cat map | Unless I'm misinterpreting the question, the SRB measure is just Lebesgue measure... the cat map is hyperbolic, preserves area, and is topologically transitive. See Theorem 3.10 and the following remarks here:
<http://books.google.com/books?id=uu-qeVBvQNEC&pg=PA141#v=onepage&q=&f=false>
| 4 | https://mathoverflow.net/users/1227 | 4004 | 2,671 |
https://mathoverflow.net/questions/3845 | 1 | Let's use the notation of `[A=>B]` for `Hom(A, B)`. Take a 1-dimensional algebraic torus `G``m` and higher-dimensional torus `T` and let's live in the category of commutative algebraic groups over `k`.
Out of four expressions like `[G``m``=> [G``m``=>T]]` etc. half give back `T` `(*)`, others the *dual torus* `T``V`,... | https://mathoverflow.net/users/65 | How to make commutative algebraic groups strongly dualizable? | Since Ilya asked I'll write out in a bit more detail the case of finitely generated abelian groups. There is not (as far as I know) any reasonable notion of strong duality on the abelian category of finitely generated abelian groups in the sense you ask for. Indeed, Z is the tensor unit and any torsion group gets kille... | 1 | https://mathoverflow.net/users/310 | 4010 | 2,675 |
https://mathoverflow.net/questions/4011 | 33 | Because I have heard the phrase "totally ordered abelian group", I imagine there should be non-abelian ones. By this I mean a group with a [total ordering](https://en.wikipedia.org/wiki/Total_order) (not to be confused with a well-ordering) which is "bi-translation invariant": a < b should imply cad < cbd.
Does anyon... | https://mathoverflow.net/users/84526 | What's a non-abelian totally ordered group? | This concept is usually called biorderability (there is also left- and right-orderability). There are many examples, such as free groups and surface groups. Most spectacularly, the pure braid groups are biorderable, while the full braid groups are left orderable but not biorderable. The left ordering on the braid group... | 33 | https://mathoverflow.net/users/317 | 4015 | 2,677 |
https://mathoverflow.net/questions/4028 | 8 | Many special functions including the gamma function have a [duplication formula](http://en.wikipedia.org/wiki/Duplication_formula) of some sorts. In the case of the gamma function it reads:
>
> Gamma(2z) = Gamma(z) Gamma(z+1/2) 22z-1/Gamma(1/2)
>
>
>
On the other hand, there is no algebraic relation between Ga... | https://mathoverflow.net/users/359 | No simple duplication formula for factorials? | It's equivalent to show that there is no polynomial relationship f({2n choose n}, n!) = 0. On the other hand, we know that {2n choose n} ~ 4^n/sqrt{n} asymptotically and n! grows much faster.
Terence Tao once remarked that if a sufficiently simple duplication formula were known for the factorial then Wilson's theore... | 10 | https://mathoverflow.net/users/290 | 4032 | 2,687 |
https://mathoverflow.net/questions/4023 | 38 | Any suggestions on a good text to use for teaching an introductory Real Analysis course? Specifically what have you found to be useful about the approach taken in specific texts?
| https://mathoverflow.net/users/1462 | Text for an introductory Real Analysis course. | Stephen Abbott, [*Understanding Analysis*](http://books.google.com/books?id=7t1ZhUAc5yMC)
Strongly recommended to students who are ony getting to grips with abstraction in mathematics. Find a review [here](http://www.maa.org/press/maa-reviews/understanding-analysis-0).
| 32 | https://mathoverflow.net/users/532 | 4034 | 2,689 |
https://mathoverflow.net/questions/2826 | 20 | I have long been curious about equivalent forms of the Riemann hypothesis for automorphic L-functions.
In the case of the ordinary Riemann hypothesis, one gets a very good error term for the prime number theorem, one has the formulation involving the Mobius mu function which is a result to the effect of the parity of... | https://mathoverflow.net/users/683 | Equivalent forms of the Grand Riemann Hypothesis | Well, suppose pi is a cuspidal automorphic representation of GL(n)/Q. This has the structure of a tensor product, indexed by primes p, of representations pi\_p of the groups GLn(Qp). The Satake isomorphism tells us that at almost all primes, each pip is determined by a conjugacy class A(p) in GLn(C). In this language, ... | 13 | https://mathoverflow.net/users/1464 | 4037 | 2,692 |
https://mathoverflow.net/questions/3235 | 8 | Let $F$ be a free semigroup (say, $2$-generated) which is embedded in a group $G$, and suppose that $G$ (as a group) is generated by $F$. The most simple such situation would be when $G$ is a free group, but there are lot of groups, besides free ones, which could occur in this situation (for example, $G$ could be solva... | https://mathoverflow.net/users/1223 | Subgroups of a group generated by a free semigroup | I think that Henry's solution doesn't work either. In that example, I get at2a = ta3t = a6b3t2. But I think that this construction can be fixed.
Consider the group of affine linear maps f(x) = αx+β over the reals ℝ. Let a act by multiplication by α, where α is transcendental, let b act by adding 1, let F be the semig... | 11 | https://mathoverflow.net/users/1450 | 4041 | 2,696 |
https://mathoverflow.net/questions/4054 | 2 | Suppose $f: M\to N$ is a smooth map between two smooth manifolds, with $M$ compact and connected, and suppose there is a dense subset of $f(M)$ where each fiber is connected, then each fiber of $f$ is connected.
If it helps, you can just consider the case where the set of regular values is dense in $f(M)$ and the fib... | https://mathoverflow.net/users/1468 | Can connectedness of fibers of a smooth map be checked on a dense set? | Perhaps I've misunderstood the question, but it looks like it's false.
Let M={(x,y)∈ℝ²|(x,y)≠(0,0)}, N=ℝ, and define f(x,y)=x. This is a smooth map of smooth manifolds, with the fibers over ℝ-{0} connected, but the fiber over 0 disconnected.
**Edit:** Wayne has added the hypothesis that M is compact. I think the st... | 1 | https://mathoverflow.net/users/1 | 4056 | 2,707 |
https://mathoverflow.net/questions/234 | 8 | A $k$-component link defines a map $T^k \rightarrow \operatorname{Conf}\_k S^3$. Does the homotopy type of this map capture the Milnor invariants?
Some special cases:
* $k=2$, no, it's null homologous, but you can look instead at the map $T^2 \rightarrow \operatorname{Conf}\_2 R^3$, which captures linking number.
*... | https://mathoverflow.net/users/3 | A $k$-component link defines a map $T^k\rightarrow \operatorname{Conf}_k S^3$. Does the homotopy type capture Milnor's invariants? | It's been a while since I've thought about this but I think Koschorke answered much of your question back in 1997 "A generalization of Milnor's mu-invariants to higher-dimensional link maps" Topology 36 (1997), no 2. 301--324. Scanning through the paper I see he recovers many of the mu invariants but not all. He lists ... | 8 | https://mathoverflow.net/users/1465 | 4058 | 2,708 |
https://mathoverflow.net/questions/4064 | 9 | Given a toric variety, is it easy to see if a crepant resolution exists? If so, how can it be explicitly constructed?
| https://mathoverflow.net/users/28 | Crepant resolutions of toric varieties | Whether or not a resolution is crepant only depends on the hypersurfaces in the exceptional locus -- to speak casually, it depends on which hypersurfaces you add. In the toric case, resolution corresponds to subdividing the fan, and the new hypersurfaces correspond to the new rays you add. In particular, if I can resol... | 8 | https://mathoverflow.net/users/297 | 4085 | 2,725 |
https://mathoverflow.net/questions/4086 | 12 | Given an infinite group which is finitely generated, is there a proper maximal normal subgroup?
| https://mathoverflow.net/users/996 | Does every finitely generated group have a maximal normal subgroup? | If you mean nontrivial maximal normal subgroup (not 1 or the whole group), then the answer is no.
Higman constructed a finitely generated infinite group $G$ with no subgroups of finite index. You then get a finitely generated group with no nontrivial normal subgroups by taking the quotient by a maximal normal subgrou... | 16 | https://mathoverflow.net/users/1335 | 4091 | 2,730 |
https://mathoverflow.net/questions/4083 | 21 | I'm currently taking a course in Hodge theory ... and I wonder if all the splittings in $\{i,-i\}$ Eigenvalue pairs come from the Galois group action (of the extension $\mathbb{R}\rightarrow\mathbb{C}$) - it seems to me like that (and I couldn't find such a statement in my textbook).
Is this true? If yes, is this a g... | https://mathoverflow.net/users/956 | Is Hodge theory somehow connected with a Galois group action Gal(C/R)? | You are correct: there is a connection to the Galois theory of $\mathbb{C}/\mathbb{R}$ here.
To give a Hodge structure on a real vector space $V$ -- i.e., a direct sum decomposition of its complexification into $(p,q)$ subspaces such that $H^{q,p}$ is the complex conjugate of $H^{p,q}$ -- is equivalent to giving an a... | 17 | https://mathoverflow.net/users/1149 | 4092 | 2,731 |
https://mathoverflow.net/questions/4075 | 53 | I'm not sure if the questions make sense:
Conc. primes as knots and Spec Z as 3-manifold - fits that to the Poincare conjecture? Topologists view 3-manifolds as Kirby-equivalence classes of framed links. How would that be with Spec Z? Then, topologists have things like virtual 3-manifolds, has that analogies in arithm... | https://mathoverflow.net/users/451 | Questions about analogy between Spec Z and 3-manifolds | The analogy doesn't quite give a number theoretic version of the Poincare conjecture. See Sikora, "Analogies between group actions on 3-manifolds and number fields"
(arXiv:0107210): the author states the Poincare conjecture as "S3 is the only closed 3-manifold with no unbranched covers." The analogous statement in numb... | 30 | https://mathoverflow.net/users/428 | 4094 | 2,733 |
https://mathoverflow.net/questions/3888 | 17 | Background reading: [John Stembridge's webpage](http://www.math.lsa.umich.edu/~jrs/other.html).
The idea is that when you want to prove a theorem for all root systems, sometimes it suffices to prove the result for the simply laced case, and then use the concept of folding by a diagram automorphism to deduce the gener... | https://mathoverflow.net/users/425 | Folding by Automorphisms | One very important use of this technique is the relation between Lie algebras / quantum groups and quiver varieties. I first saw something about this in Lusztig's book, Introduction to Quantum Groups; but see also [this arXiv paper](http://arxiv.org/abs/math.QA/0406073) Alistair Savage. Quiver varieties are important f... | 14 | https://mathoverflow.net/users/1450 | 4102 | 2,741 |
https://mathoverflow.net/questions/4062 | 17 | Let $G$ and $H$ be affine algebraic groups over a scheme $S$ of characteristic 0 and let $\textbf{Hom}\_{S,gp}(G,H)$ be the functor $T \mapsto \text{Hom}\\_{T,gp}(G,H)$
>
> **Theorem** (SGA 3, expose XXIV, 7.3.1(a)): Suppose G is reductive. Then
> $\textbf{Hom}\_{S,gp}(G,H)$ is representable by a scheme.
>
>
> ... | https://mathoverflow.net/users/2 | Can Hom_gp(G,H) fail to be representable for affine algebraic groups? | $\operatorname{Hom}(\mathbb{G}\_a, \mathbb{G}\_m)$ is not representable.
Let $R$ be a $\mathbb{Q}$-algebra. I claim that $\operatorname{Hom}(\mathbb{G}\_a, \mathbb{G}\_m)(\operatorname{Spec} R)$ is {Nilpotent elements of $R$}. Intuitively, all homs are of the form $x\mapsto e^{nx}$ with $n$ nilpotent.
More precisel... | 26 | https://mathoverflow.net/users/297 | 4105 | 2,742 |
https://mathoverflow.net/questions/4104 | 4 | By the GAGA principle we know that a holomorphic vector bundle E->X is analitically isomorphic to an algebraic one, say F->X, and by definition F is locally trivial in the Zariski topology. But since the isomorphism between E and F is analytic, I fail to see if this implies that E is Zariski locally trivial too.
I ho... | https://mathoverflow.net/users/828 | Is a holomorphic vector bundle on a projective variety locally trivial in the Zariski topology? | You get (analytic) trivializations of E over Zariski-open sets just by composing a trivialization of E with the isomorphism between E and F. Of course, you do not get algebraic trivializations, but for this you would need an algebraic structure on E in the first place.
| 7 | https://mathoverflow.net/users/1476 | 4110 | 2,747 |
https://mathoverflow.net/questions/4117 | 18 | Any topological group $G$ has a classifying space, whose loopspace is a (homotopy) group which is homotopy equivalent to $G$ in a way that preserves the group structure. More generally, if $G$ is an $A\_\infty$-group (a space with a binary operation which satisfies the group axioms up to coherent homotopy), it similarl... | https://mathoverflow.net/users/75 | When can you desuspend a homotopy cogroup? | Whilst the n-lab page on [co-H-spaces](http://ncatlab.org/nlab/show/co-H-space) could be described as a little meagre, it does nonetheless contain a reference to a paper in the Handbook of Algebraic Topology. Various parts of this book are in the "preview" in google books, in particular page 1153 which contains the mag... | 9 | https://mathoverflow.net/users/45 | 4128 | 2,758 |
https://mathoverflow.net/questions/4132 | 2 | I know that this is on the boundaries of what's allowed, but hopefully someone'll answer before it gets closed!
What (periodic) function has Fourier series the harmonic series? I really want the even (cosine) terms to be the harmonic series and no odd terms.
Edit: so that the record is perfectly clear, what I wante... | https://mathoverflow.net/users/45 | What function has fourier series the harmonic series? | It's a standard series computation to show that
$$
\sum\_{n \ge 1} \frac{x^n}{n} = \log \frac{1}{1 - x}
$$
Now substitute $x = e^{i t}$ and take the real part.
(As an aside, the reason I write the identity this way is that this is the version which is combinatorially significant.)
| 11 | https://mathoverflow.net/users/290 | 4133 | 2,762 |
https://mathoverflow.net/questions/4138 | 17 | Let $X$ be a complex analytic space. It is a 'well known fact' that the categories of local systems on $X$ (i.e. locally constant sheaves with stalk $C^n$), and of (holomorphic) vector bundles on $X$ with flat connection, are equivalent. I've been looking for a proof of this, but every reference I can find merely says ... | https://mathoverflow.net/users/1481 | Why are local systems on a complex analytic space equivalent to vector bundles with flat connection? | The important point of the proof is that either of these objects can be locally trivialized with transition functions on each double overlap given by a constant element of GL(n). So, given a local system, you just build the vector bundle with flat connection that has the same transition functions, and vice versa.
**E... | 12 | https://mathoverflow.net/users/66 | 4140 | 2,766 |
https://mathoverflow.net/questions/4125 | 35 | Rings of functions on a nonsingular algebraic curve (which, over $\mathbb{C}$, are holomorphic functions on a compact Riemann surface) and rings of integers in number fields are both examples of Dedekind domains, and I've been trying to understand the classical analogy between the two. As I understand it, $\operatornam... | https://mathoverflow.net/users/290 | If Spec Z is like a Riemann surface, what's the analogue of integration along a contour? | For a variety $X$ over a finite field, I guess one can take $\ell$-adic sheaves to replace differential forms. Then the local integral around a closed point $x$ (like integral over a little loop around that point) is the trace of the local Frobenius $\operatorname{Frob}\_x$ on the stalk of sheaf, the so-called naive lo... | 34 | https://mathoverflow.net/users/370 | 4141 | 2,767 |
https://mathoverflow.net/questions/3312 | 16 | For $f: X → Y$ a morphism of schemes, does anybody know conditions for the existence of an adjunction $(f\_!,f^!)$ between the module-categories (*not the quasicoherent*), where $f\_!$ is direct image with proper support and $f^!$ is its *right* adjoint? Can this ever happen at all on the level of the module categories... | https://mathoverflow.net/users/733 | When does direct image with proper support have a right adjoint? | I assume the question holds in contexts where we can glue open immersions and proper morphisms to produce $f\_!$ for $f$ separated of finite type. In particular, we shall have $f\_!=f\_\ast$ for $f$ proper.
Non derived setting ---
If $f:X\rightarrow Y$ is proper, one can ask if $f\_\ast$ has a right adjoint.
Note tha... | 18 | https://mathoverflow.net/users/1017 | 4152 | 2,773 |
https://mathoverflow.net/questions/3701 | 14 | $\DeclareMathOperator\SL{SL}\DeclareMathOperator\Sp{Sp}$Suppose that $F/Q$ is a number field.
Using automorphic forms, Borel computed the ($R$-) stable cohomology of $\SL\_n(O\_F)$, and as a result, computed $K\_i(O\_F)\otimes Q$. Nowadays, using work of Suslin, Voevodksy, Rost, etc, one has an almost "complete" unde... | https://mathoverflow.net/users/nan | Stable homology of arithmetic groups | The algebraic K-groups of Z are to the homology of SLn(Z) as the **Hermitian K-groups** of Z are to the homology of Sp2g(Z). There is a paper by Berrick and Karoubi [here](https://faculty.math.illinois.edu/K-theory/0649/BerrickK.pdf "Hermitian K-theory of the integers") in which they discuss, and make some calculations... | 4 | https://mathoverflow.net/users/318 | 4158 | 2,775 |
https://mathoverflow.net/questions/4167 | 2 | Say I'm working in the space of linear transformations from $\mathbb R^n$ to $\mathbb R^n$ and I've picked a basis so I can identify with any operator a component matrix in $\mathbb R^{n\times n}$. Transposing an operator swaps components $(i,j)$ and $(j,i)$. In this setting, the transpose operation is itself a linear ... | https://mathoverflow.net/users/1195 | What are the components of a transpose operator from $\mathbb R^{n\times n}$ to $\mathbb R^{n\times n}$? | A linear map from R^{n\*n} to itself can be written as a matrix where its rows and columns are both indexed by pairs (i,j) where i,j are in {1,2,...,n}. The transpose operation is given by the matrix which has a 1 in position ((i,j),(j,i)) and a zero elsewhere. For the 2x2 case we can order the basis on R^(2\*2} as (1,... | 5 | https://mathoverflow.net/users/1233 | 4171 | 2,783 |
https://mathoverflow.net/questions/4156 | 4 | Some branches of math seem to have reasoning which is more global. There is a lot of efficiency in the proofs because the reasoning transfers easily between proofs. For other branches of math, a lot of truths seem to be more local. The proofs tend to have lots of sub-cases and exceptions. There are fewer general princi... | https://mathoverflow.net/users/812 | Why do branches of math vary in proof styles and what category are different branches in? | The only thing this reminds me of is [Tim Gowers's nice article](http://www.dpmms.cam.ac.uk/~wtg10/2cultures.ps) on the two cultures of mathematics, in which he compares and contrasts "geometry" (very broadly defined) and combinatorics.
The categories in the article don't exactly match up with the categories in the q... | 5 | https://mathoverflow.net/users/1463 | 4178 | 2,787 |
https://mathoverflow.net/questions/4183 | 14 | I just saw a post like this one, but particularly for statistical mechanics, I thought I'd ask the question in general.
Where does a mathematically trained person go to learn mathematical physics? By that I mean, what books or manuscripts are demanding in the area of mathematical maturity but not particularly demandi... | https://mathoverflow.net/users/429 | Mathematical Physics? (Particularly computational) | A classic reference is Courant and Hilbert ([volume 1](http://rads.stackoverflow.com/amzn/click/0471504475), [volume 2](http://rads.stackoverflow.com/amzn/click/0471504394)).
| 2 | https://mathoverflow.net/users/136 | 4191 | 2,793 |
https://mathoverflow.net/questions/4103 | 2 | Where can I learn more about shear matrices?
The Wikipedia article is not enough, and sadly it does not have any references.
I understand they are linear transformations. Do they form a group? How do they look like for n-dimensional vectors? How many independent shears can I do in n-dimensions? Are they related to ... | https://mathoverflow.net/users/1475 | Shear transformations | It might be helpful to note that, in two dimensions, a shear transformation is exactly one whose Jordan canonical form is $\begin{pmatrix}1&1\\0&1\end{pmatrix}$ or $\begin{pmatrix}1&0\\0&1\end{pmatrix}$ if you include the identity as a shear transformation. These are exactly the transformations whose eigenvalues are al... | 1 | https://mathoverflow.net/users/126667 | 4193 | 2,795 |
https://mathoverflow.net/questions/4187 | 3 | If $A$ is a C\*-algebra and $n$ is a normal element of $A$, then we have: (By Gelfand duality for example.)
$\operatorname{spec}( |N| ) = | \operatorname{spec}(N) | := \left\{ | \lambda | ; \lambda \in \operatorname{spec}(N) \right\}$
where we define: $|n|:=(n^\*n)^{1/2}$. My question is, does the converse also hol... | https://mathoverflow.net/users/1488 | Normal operators and it's spectrum in C*-algebras | You can take a normal operator $N$ with spectrum filling the unit disk and take its tensor product with any operator $T$ of norm less than $1$ thus effectively hiding the non-normal component's contribution to the spectrum in both $A$ and $|A|$.
| 5 | https://mathoverflow.net/users/1131 | 4202 | 2,804 |
https://mathoverflow.net/questions/4137 | 5 | Suppose M is a von Neumann algebra.
Consider a monoidal category of bimodules over M.
Here a bimodule is a Hilbert space with two normal representations of M.
The monoidal structure is given by Connes' fusion.
Alternatively we can take right W\*-correspondences (right Hilbert W\*-modules over M with a normal left actio... | https://mathoverflow.net/users/402 | One-parameter semigroups of bimodules | Such structures have been investigated at depth (welcome to the club!). Let me try to answer some of your questions.
1) First, let me suggest that you look at Bill Arveson's book on this subject, [Noncommutative Dynamics and E-semigroups](http://www.springer.com/math/analysis/book/978-0-387-00151-7). Arveson is consi... | 6 | https://mathoverflow.net/users/1193 | 4211 | 2,813 |
https://mathoverflow.net/questions/4119 | 3 | Lists of stellations of polyhedrons are given particular rules like in the book [*The Fifty Nine Icosahedra*](http://en.wikipedia.org/wiki/The_fifty_nine_icosahedra) which follows "Miller's Rules".
There seems to be no "correct" ruleset to use, so [more stellations are still being discovered](http://www.steelpillow.... | https://mathoverflow.net/users/441 | Are there infinite sets of stellations of polyhedra? | Let me quote from "In search of the lost icosahedra" the paper I mentioned above:
"Stellations of a polyhedron are obtained by extending some of its edges or faces until they intersect at a distance from the original polyhedron. One way of studying stellations is to consider the planes in which the faces of the polyh... | 2 | https://mathoverflow.net/users/1098 | 4213 | 2,815 |
https://mathoverflow.net/questions/1666 | 19 | This question is pretty technical, but there are some very smart people here.
Fix a quiver Q, WITH oriented cycles. Let k[[Q]] be the completed path algebra. (Like the path algebra, but we allow formal infinite sums.) My question is about what behavior I should expect when I equip Q with a generic choice of potential... | https://mathoverflow.net/users/297 | When should I expect a quiver with potential to be rigid? | As I understand it, the quotient C(Q) = k[[Q]]/[,] is the (completed) vector space of formal linear combinations of oriented circuits in Q. Define a *detour* in Q to be an edge e and an oriented path p with the same source and sink as e. For every detour (e,p), there is a detour operator D(e,p). Given a circuit s, D(e,... | 6 | https://mathoverflow.net/users/1450 | 4215 | 2,816 |
https://mathoverflow.net/questions/4190 | 5 | Consider the following question:
Input: Two graphs G1 and G2
Question: Is the cycle matroid M(G1) isomorphic to the cycle matroid M(G2)
What is the complexity of this question?
It is well known that the two cycle matroids are isomorphic if and only if the graphs are "2-isomorphic" which means that there is a se... | https://mathoverflow.net/users/1492 | Complexity of determining if two graphs have same cycle matroid? | The following paper seems to show that this problem is polynomial equivalent to graph isomorphism (see section 5):
<http://arxiv.org/abs/0811.3859>
| 4 | https://mathoverflow.net/users/1028 | 4230 | 2,824 |
https://mathoverflow.net/questions/4224 | 81 | Is there a relationship between the eigenvalues of individual matrices and the eigenvalues of their sum? What about the special case when the matrices are Hermitian and positive definite?
I am investigating this with regard to finding the normalized graph cut under general convex constraints. Any pointers will be ver... | https://mathoverflow.net/users/1501 | Eigenvalues of matrix sums | The problem of describing the possible eigenvalues of the sum of two hermitian matrices in terms of the spectra of the summands leads into deep waters. The most complete description was conjectured by Horn, and has now been proved by work of Knutson and Tao (and others?) - for a good discussion, see the [Notices AMS ar... | 72 | https://mathoverflow.net/users/763 | 4238 | 2,830 |
https://mathoverflow.net/questions/4243 | 5 | This should be a trivial question for people who know Gödel's 1st incompleteness theorem. I quote the statement the theorem from wikipedia: "Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated forma... | https://mathoverflow.net/users/1508 | Godel's 1st incompleteness theorem - clarification. | It means true in the usual model. For 1st order logic we have Godel's Completeness theorem, which guarantees that if something is true in every model, then it is actually provable in the theory.
| 6 | https://mathoverflow.net/users/828 | 4245 | 2,836 |
https://mathoverflow.net/questions/4260 | 17 | I've heard from multiple sources now that one's CV should include grants you've applied for, even if you didn't receive them or won't find out if you've received them until after your CV goes out. I haven't had much luck finding this in other people's CVs, though. I'd like to have confirmation of this from someone who'... | https://mathoverflow.net/users/699 | Curriculum vitae: including grants you've applied for, not received (or not yet received). | You have a certain amount of leeway as to what kind of material you wish to include in your CV. I have seen things on CVs that were of little or no interest [edit: here I actually meant "of interest to me as a potential hirer", not personal interest]: high school honors, nonmathematical awards, etc. It doesn't make me ... | 13 | https://mathoverflow.net/users/1149 | 4265 | 2,846 |
https://mathoverflow.net/questions/4254 | 2 | If X is an algebraic scheme, K\_0(X) has a filtration by taking the subgroups generated by coherent sheaves whose support as at most dimension k. The associated graded groups are the quotients, and there exists a natural map from the k-th Chow group A\_k(X) to the k-th graded part Gr\_k K\_0(X), just by mapping [V] to ... | https://mathoverflow.net/users/473 | k-th Chow Group and k-th graded part of K_0 ismorphic for DM-stacks? | Maybe you might have a look at Toën's paper:
"On motives for Deligne-Mumford stacks", IMRN No. 17 (2000), 909-928.
He defines Chow groups of a Deligne-Mumford stack X *with coefficients in the characters* of X, and proves that the corresponding graded ring of Chow groups is isomorphic to K\_0 (see the Remark following ... | 3 | https://mathoverflow.net/users/1017 | 4270 | 2,848 |
https://mathoverflow.net/questions/4269 | -3 | Is there a proof for no proof ?
| https://mathoverflow.net/users/1519 | what is logic without a proof system | If I understand your question you are asking can something be proved as unprovable? If so I'd suggest that Gödel's incompleteness theorems for a starting place.
>
> "Gödel's incompleteness theorems state
> that any effectively generated formal
> theory in which all arithmetic truths
> can be proved is inconsiste... | 0 | https://mathoverflow.net/users/1214 | 4271 | 2,849 |
https://mathoverflow.net/questions/4235 | 71 | I'm wondering what the relation of category theory to programming language theory is.
I've been reading some books on category theory and topos theory, but if someone happens to know what the connections and could tell me it'd be very useful, as that would give me reason to continue this endeavor strongly, and know w... | https://mathoverflow.net/users/429 | Relating category theory to programming language theory | The most immediately obvious relation to category theory is that we have a category consisting of types as objects and functions as arrows. We have identity functions and can compose functions with the usual axioms holding (with various caveats). That's just the starting point.
One place where it starts getting deepe... | 90 | https://mathoverflow.net/users/1233 | 4274 | 2,852 |
https://mathoverflow.net/questions/4180 | 45 | So, lots of people work on the Geometric Langlands Conjecture, and there have been a few questions around here on it (admittedly, several of them mine). So here's another one, tagged community wiki because there isn't really a "right" answer: what does GLC imply? Lots of big conjectures have well known consequences (Ri... | https://mathoverflow.net/users/622 | Consequences of Geometric Langlands | OK, this is a very broad question so I'll be telegraphic.
There is a sequence of increasingly detailed conjectures going by the name GL -- it's really a "program" (harmonic analysis of $\mathcal{D}$-modules on moduli of bundles) rather than a conjecture -- and only the first of this sequence has been proved (and only f... | 61 | https://mathoverflow.net/users/582 | 4275 | 2,853 |
https://mathoverflow.net/questions/4276 | 24 | It has been many years since I first read Categories for the Working Mathematician, but I still have a question about one of the first exercises. Question 5 in section 1.3 asks you to find two different functors $\mathsf{T}: \mathsf{Groups} \to \mathsf{Groups}$ with object function $\mathsf{T}(\mathsf{G}) = \mathsf{G}$... | https://mathoverflow.net/users/1106 | Two functors from Grp to Grp? | This is an "[evil](http://ncatlab.org/nlab/show/evil)" question, which deserves an evil answer. Pick your favorite pair of an object G0 of Groups and a nontrivial automorphism φ of G0. Define the functor T : Groups → Groups by T(G) = G, T(f) = [φ ∘] f [∘ φ-1] where we compose with φ if the target of f is G0 and compose... | 33 | https://mathoverflow.net/users/126667 | 4278 | 2,854 |
https://mathoverflow.net/questions/4268 | 10 | Suppose $f$ is a weight $2$ level $N$ cusp form. When can we realize the mod-$\ell$ representation of $f$ in a form of weight $2$ and level $Np^3$, where $p$ is some prime not dividing $N$? I assume that, if a simple criterion exists at all, it is a condition on the mod-$\ell$ representation of $f$ restricted to inerti... | https://mathoverflow.net/users/1464 | Level raising by prime powers | Presumably you want the form (let me call it g) of level Np^3 to be new at p, otherwise it's trivial.
Let me also assume ell isn't p.
If the form g is new at p, and has level Gamma0(p^3) at p, then the ell-adic representation attached to g will have conductor p^3. But this is a bit of a problem, because the conduct... | 10 | https://mathoverflow.net/users/1384 | 4282 | 2,856 |
https://mathoverflow.net/questions/1168 | 4 | This question is related to [my earlier, even more open-ended question](https://mathoverflow.net/questions/406/how-is-tropicalization-like-taking-the-classical-limit) on tropilcalization. I will give some background and ask my question at the end.
On **R**, consider the family of commutative, associative operations ⊕... | https://mathoverflow.net/users/78 | "Wick rotation" of tropical geometry | There has been very little activity on this question, so I'm going to take it off the unanswered list. In particular, in [a related question](https://mathoverflow.net/questions/4228/), kilimanjaro linked to [this paper](http://arxiv.org/abs/math.GM/0507014), which answers some of my questions and includes many referenc... | 1 | https://mathoverflow.net/users/78 | 4284 | 2,858 |
https://mathoverflow.net/questions/4259 | 8 | The following topic came up in conversation with my office-mate Lionel: Let $p$ be a fixed prime, $c$ a fixed positive real parameter and $n$ a large number. Consider a random $(0,1)$ matrix with entries in $Z/p$, where the probability of a $0$ is $1-\frac{c}{n}$ and that of a $1$ is $\frac{c}{n}$. As $n\rightarrow \in... | https://mathoverflow.net/users/297 | Singularity of sparse random matrices | A few observations:
-As n tends to infinity, the function corank/n is highly concentrated for each n -- for example, we can think of exposing the matrix minor by minor (looking at the upper left kxk matrix for k increasing towards n). Since changing what happens at each level of exposure can only affect the rank of t... | 7 | https://mathoverflow.net/users/405 | 4290 | 2,864 |
https://mathoverflow.net/questions/4216 | 21 | From Wiener's tauberian theorem we know that linear combinations of translates of f \in L^1(R) are dense in L^1(R) if and only if the Fourier transform of f never vanishes. It is also known that linear combinations of translates of f \in L^2(R) are dense in L^2 if and only if the Fourier transform of f is nonzero almos... | https://mathoverflow.net/users/630 | Is there an L^p tauberian theorem? | Actually this is a well known question. N. Lev and A. Olevskii have shown the following theorem:
**Theorem (Lev, Olevskii)** Given any 1 < p < 2 one can find two vectors in $l^1(Z)$, such that one is cyclic in $l^p(Z)$ and the other is not, but their Fourier transforms have an identical set of zeros.
The same resul... | 17 | https://mathoverflow.net/users/881 | 4291 | 2,865 |
https://mathoverflow.net/questions/4153 | 2 | Suppose we have a Hamiltonian action of a torus $T = T^m = R^m/Z^m$ on a compact, connected symplectic manifold $M$. According to the convexity theorem, we know every fiber of the momentum map $\mu: M\to R^m$ is connected. My question here is about the proof.
We assume the Hamiltonian action is effective without loss... | https://mathoverflow.net/users/1468 | Convexity Theorem of Hamiltonian actions - the connectedness part | I think the key point is that given a Hamiltonian torus action, the components of the moment map are Morse-Bott functions which have *even dimensional* critical submanifolds all with *even index*.
For example, suppose you have a Hamiltonian circle action on X. If p is fixed by the action, the circle acts on the tang... | 2 | https://mathoverflow.net/users/380 | 4303 | 2,875 |
https://mathoverflow.net/questions/3887 | 4 | In what sense does Lusztig's quantum Frobenius, defined on a quantum enveloping algebra, generalise the classical Frobenius mapping on a variety over a finite field?
| https://mathoverflow.net/users/1095 | Quantum Frobenius | There is one sense in which I'd say that Lusztig's Frobenius morphism is a generalization of the Frobenius morphism on a variety: In [this](http://arxiv.org/abs/math/0005246) paper, Kumar and Littelmann show that Lusztig's quantum Frobenius morphism induces a Frobenius morphism on a quantized analog of the multicone ov... | 2 | https://mathoverflow.net/users/1528 | 4319 | 2,887 |
https://mathoverflow.net/questions/4318 | 3 | Sorry if this question is too simple.
I once read, on a number theory textbook - forget the title, in one of the problems list that all Pythagorean triplets when multiplied are divisible by 60.
I proved that using the generating functions (is this the correct name? I got the name from my Discrete Mathematics textbo... | https://mathoverflow.net/users/1538 | Proof of "if $a^2 + b^2 = c^2$ then $abc$ is divisible by 60" | 1. Generating functions is not really the right name. I would say "parameterization."
2. These formulas were known to the Babylonians. The simple proof goes as follows: assume WLOG that a, b, c are relatively prime. Since a, b, c cannot all have the same parity, WLOG b, c have different parity. Then a^2 = (c + b)(c - b... | 14 | https://mathoverflow.net/users/290 | 4320 | 2,888 |
https://mathoverflow.net/questions/4310 | 5 | Let's say we have a sequence $T(n)$ with the corresponding generating function
$$A(t) = \sum\_{n = 0}^\infty T(n) t^n$$
Is there some relationship between the two functions $A(t)$ and $A(t^2)$? And for that matter is there some generalization for any integer power or $t$?
**Edit:** I'm actually trying to solve fo... | https://mathoverflow.net/users/1447 | Generating-functions: is there a relationship between a generating function and the corresponding squared generating function | Alright, so on the one side, you have this:
$$A(t)+(1+t)A(t^{2})=\sum\_{n=0}^{\infty}T(n)t^{n}+\sum\_{n=0}^{\infty}T(n)t^{2n}+\sum\_{n=0}^{\infty}T(n)t^{2n+1}$$
On the other side, you have:
$$\frac{t}{1-t^{2}}=\sum\_{n=0}^{\infty}t^{2n+1}$$
Equating the coefficients of $x^{2k}$, you have the relation: $T(2k)+T(... | 10 | https://mathoverflow.net/users/855 | 4328 | 2,894 |
https://mathoverflow.net/questions/4335 | 5 | At the risk of asking an uninformed question...
Imagine an ant on a compact two-dimensional surface embedded in 3-space. The ant is placed at a point on the surface with random orientation. Once placed and oriented, the ant will walk along a local geodesic path.
Are there examples of such 2D surfaces where we are g... | https://mathoverflow.net/users/774 | A walk on a compact 2D surface embedded in 3-space that never returns home | Silly answer: any plane in 3-space will work (so the answer is "yes").
You probably want a compact surface. These always admit geodesic cycles (so the answer in this case is "no").
| 6 | https://mathoverflow.net/users/121 | 4339 | 2,899 |
https://mathoverflow.net/questions/2828 | 11 | Let $\mathfrak{s} = \mathfrak{s}\_0 \oplus \mathfrak{s}\_1$ be a real Lie superalgebra. (The ground field does not matter much, but at least one formula will not work as written if the characteristic is 2 or 3.) Recall that this means that there is a bilinear 2-graded bracket $[-,-]$ with three components
(a) $\mathf... | https://mathoverflow.net/users/394 | 3/4-Lie superalgebras: how much of a theory can one develop? | At least in the semisimple case, it doesn't seem like there can be a theory of 3/4-Lie superalgebras other than a fairly predictable set of examples within Cartan-Weyl theory. I don't know how to do all of the relevant calculations, but it is easy to see roughly how they would go.
Suppose that the even part $s\_0$ is... | 5 | https://mathoverflow.net/users/1450 | 4344 | 2,902 |
https://mathoverflow.net/questions/4331 | 19 | * Is the wedge product of two harmonic forms on a compact Riemannian manifold harmonic? I'm looking for a counter-example that the textbooks say exists.
* I would like to see a counter example that is on a complex manifold, Ricci-flat (or Einstein) manifold or both, if it is at all possible.
* In general, I'm trying to... | https://mathoverflow.net/users/1544 | Is the wedge product of two harmonic forms harmonic? | It is easy to construct examples on Riemann surfaces of genus $>1$. Take any surface like this. Let $A$ and $B$ be two harmonic $1$-forms, that are not proportional. Then $A \wedge B$ is non-zero, but it vanishes at some point, since both $A$ and $B$ have zeros. At the same time a harmonic $2$-form on a Riemann surface... | 32 | https://mathoverflow.net/users/943 | 4355 | 2,906 |
https://mathoverflow.net/questions/4374 | 0 | Define a sequence $(C\_n)$ by $C\_0=1$, $C\_1=1$, and $C\_{n+1} = \sum\_{r=0}^n C\_r C\_{n-r}$ for $n\geq 2$. What is the simplest explicit formula for $C\_n$?
| https://mathoverflow.net/users/1553 | What is the simplest non-recursive formulation for the following recursive function? | You want [Catalan number](http://en.wikipedia.org/wiki/Catalan_number)
| 5 | https://mathoverflow.net/users/nan | 4375 | 2,919 |
https://mathoverflow.net/questions/4381 | 2 | I know how to pronounce Dijkstra's name correctly (hear it here: <http://en.wikipedia.org/wiki/Edsger_W._Dijkstra>).
But I'd like to know how people usually say his name. I've heard it in many different ways throughout my career, and since I'm teaching a course on graphs and Dijkstra's algorithm, I don't want to teac... | https://mathoverflow.net/users/1172 | Pronunciation: Dijkstra | I've always heard it basically the same way as Wikipedia, except with an American accent. Basically "dike' struh", with the accent on the first syllable as indicated, and struh is the same as in Strunk and strum.
| 11 | https://mathoverflow.net/users/1450 | 4385 | 2,925 |
https://mathoverflow.net/questions/4379 | 15 | Is/should there be a theory of finite solvable extensions over a given base field? Could it be based on/use class field theory? Assume the base field isn't a local field.
| https://mathoverflow.net/users/1555 | Solvable class field theory | As FC says, since solvable extensions are built up out of abelian extensions, class field theory is certainly relevant and helpful in understanding the structure of solvable extensions. On the other hand, to do this in a systematic way requires understanding class field theory of each number field in a tower "all at on... | 13 | https://mathoverflow.net/users/1149 | 4386 | 2,926 |
https://mathoverflow.net/questions/4329 | 73 | During a talk I was at today, the speaker mentioned that if you truncate the Taylor series for $e^x - 1$, you'll get lots of roots with nonzero real part, even though the full Taylor series only has pure imaginary roots.
If you plot the roots of truncations of $e^x - 1$ (or check out the ready-made plots in [this *Ma... | https://mathoverflow.net/users/1096 | Roots of truncations of $ e^x - 1$ | I finally got around to googling this a bit, and I immediately came up with <http://www.mai.liu.se/~halun/complex/taylor/> which describes the same phenomenon for the exponential function itself. Briefly, if Pn is the Taylor polynomial of ex, then the zeroes of Pn(nx) pile up on the curve |ze1-z|=1, |z|≤1. They credit ... | 31 | https://mathoverflow.net/users/802 | 4413 | 2,949 |
https://mathoverflow.net/questions/4347 | 72 | The question is about the function f(x) so that f(f(x))=exp (x)-1.
The question is open ended and it was discussed quite recently in the comment thread in Aaronson's blog here <http://scottaaronson.com/blog/?p=263>
The growth rate of the function f (as x goes to infinity) is larger than linear (linear means O(x)), ... | https://mathoverflow.net/users/1532 | f(f(x))=exp(x)-1 and other functions "just in the middle" between linear and exponential | Let me see if I can summarize the conversation so far. If we want $f(f(z)) = e^z+z-1$, then there will be a solution, analytic in a neighborhood of the real axis. See either [fedja's](https://mathoverflow.net/questions/4347/ffxexpx-1-and-other-functions-just-in-the-middle-between-linear-and-expo/4398#4398) Banach space... | 31 | https://mathoverflow.net/users/297 | 4423 | 2,956 |
https://mathoverflow.net/questions/4422 | 6 | **Background**: When proving that the group of $k$-isogenies $\mathrm{Hom}\_k(A,B)$ between two abelian varieties is finitely generated, one first shows that the Tate map $$\mathbb{Z}\_\ell\otimes\_{\mathbb{Z}} M \to \mathrm{Hom}\_{\mathbb{Z}\_\ell}(T\_\ell A,T\_\ell B)$$ is injective. Since each Tate module is free of... | https://mathoverflow.net/users/412 | Is a torsion-free abelian group finitely generated, if all of its localizations at primes $p$ are finitely generated over $\mathbb{Z}_p$? | I don't think so. Consider the $\mathbb{Z}$-module $M$ be the additive subgroup of the rationals consisting of rationals with square-free denominator.
| 21 | https://mathoverflow.net/users/1384 | 4425 | 2,957 |
https://mathoverflow.net/questions/4361 | 17 | Are there any general results on the (integral) cohomology of manifolds that are fibrations over the circle? Any literature references much appreciated.
| https://mathoverflow.net/users/1552 | Cohomology of fibrations over the circle | The above Mayer-Vietoris argument gives the cohomology of a fiber bundle over a circle in a concrete fashion. For sake of "mathematical culture", I thought I'd mention what happens for fiber bundles over a higher dimensional sphere (this is also a good excuse for me to test drive the new latex support).
For fiber bun... | 19 | https://mathoverflow.net/users/317 | 4428 | 2,959 |
https://mathoverflow.net/questions/4427 | 17 | A $\mathbb{Z}/2\mathbb{Z}$-graded algebra is said to supercommute if $xy = (-1)^{|x| |y|} yx$; in other words, odd elements anticommute. Why is this the "right" definition of supercommutativity? (Put another way, why is this the natural tensor product structure on super vector spaces?) Answers from both a categorical o... | https://mathoverflow.net/users/290 | What is the conceptual significance of supercommutativity? | The categorical answer is that (in characteristic zero) this is the only way that you can make a suitable symmetric tensor category, other than by using group representations. There is a Tannakian theorem of Deligne to this effect in the algebraic setting.
One of the physical answers is equivalent to the categorical ... | 21 | https://mathoverflow.net/users/1450 | 4430 | 2,961 |
https://mathoverflow.net/questions/4395 | 12 | Some physicists have told me that if you think about an extended n-dimensional TQFT $F$, then the decategorification is given by $F'(X)=F(X\times S^1)$, which I believe they call "compactification on a circle." Is there any way to make this statement precise?
| https://mathoverflow.net/users/66 | What do decategorification and "compactification on a circle" have to do with each other? | In a general extended TQFT Z, the assignment $Z(X \times S^1)$ is the "dimension" of Z(X), in the following sense. Write the circle as an incoming arc followed by an outgoing arc. The incoming arc is a morphism (coevaluation) from the unit (Z(empty set)) to Z(X) tensor its dual $Z(X^{op})=Z(X)^\*$, followed by a morphi... | 11 | https://mathoverflow.net/users/582 | 4446 | 2,970 |
https://mathoverflow.net/questions/4113 | 11 | Consider the transcendental extension Q(t) of the field of rationals.
To Q(t) adjoin the root of the polynomial x^5+t^5=1. The resulting
field Q(t)[x] is a radical extension of Q(t). Is it true that the
only solutions to the equation X^5+Y^5=1 in the field Q(t)[x] are
(0,1), (1,0), (t,x), (x,t), (1/t,-x/t) and (-x/t... | https://mathoverflow.net/users/1169 | a question on function fields | First, replace Q by the complex numbers C.
Write A = C[x,y]/(x^n+y^n-1). Then the field you write down, call it K, is the fraction field of A, which is the function field of the Fermat Curve.
Finding a solution (X,Y) to x^n + y^n = 1 is equivalent to finding a map A --> K, where one sends x to X and y to Y. Any ma... | 12 | https://mathoverflow.net/users/nan | 4451 | 2,972 |
https://mathoverflow.net/questions/3929 | 8 | The Newlander-Nirenberg theorem states that an almost complex structure is integrable if and only if the Nijenhuis tensor vanishes.
I heard that this statement is not true in infinite dimensions, since for example the Loop space of a Riemannian 3-manifold is counterexample. (In fact, I think NN fails for Fréchet manif... | https://mathoverflow.net/users/675 | Infinite dimensional Newlander-Nirenberg theorem | As far as I understand, in a paper of Petyi [On the ∂-equation in a Banach space. Bull. Soc. Math. France 128 (2000), no. 3, 391–406.] it is shown that the NN theorem does not hold for Banach manifolds in general. However, as you may know, the NN theorem has an easy proof when the almost complex structure is assumed to... | 6 | https://mathoverflow.net/users/1116 | 4501 | 3,007 |
https://mathoverflow.net/questions/4519 | 3 | Are the sites similar to mathoverflow in other sciences related to mathematics? statistics, computer science, physics, economics, etc?
Let me explain what I mean by "similar": those are sites devoted to posing questions and answers,in these areas. I do not insist on the precise format of "mathoverflow" (reputation po... | https://mathoverflow.net/users/1532 | Something like mathoverflow in other sciences | Here is a brief list of science-related sites that run on the same platform as Math Overflow:
* Science
+ [Science Stack](http://sciencestack.com/)
+ [asksci.com](http://asksci.com/)
* Physics
+ [physics.stackexchange.com](https://physics.stackexchange.com/)
+ [PhysicsOverflow](http://physicsoverflow.org/)
* Elec... | 10 | https://mathoverflow.net/users/19 | 4524 | 3,027 |
https://mathoverflow.net/questions/4504 | 15 | Let's fix a finite field F and consider abelian varieties of dimension g over F. Can we say how many isogeny classes there are? Is it even clear that there's more than one isogeny class? For g=1, and some fixed F with characteristic not 2 or 3, we could probably just write down all the Weierstrass equations and count i... | https://mathoverflow.net/users/88 | Can we count isogeny classes of abelian varieties? | Let q be the order of your finite field. Then the category of abelian varieties over $\mathbb{F}\_q$ up to isogeny is semisimple - any object is isogenous to a product of simple ones in an essentially unique way, so this reduces your question to one about simple objects.
For simple abelian varieties over $\mathbb{F}\... | 10 | https://mathoverflow.net/users/360 | 4525 | 3,028 |
https://mathoverflow.net/questions/4527 | 1 | Does this even make sense what I translated into english?
PS. I am probably gonna delete this question eventually
| https://mathoverflow.net/users/462 | Is it that only with normal matrices, the transition matrix to its [del: inherent] [ins: own] basis is unitary? | I am not sure I understand the question, but if you are talking about linear algebra over the complex numbers, then it is true that normal matrices (those which commute with their hermitian adjoint) are precisely those which can be diagonalised by a unitary transformation.
(This is proven in Herstein's *Topics in algeb... | 2 | https://mathoverflow.net/users/394 | 4528 | 3,029 |
https://mathoverflow.net/questions/1922 | 46 | One can build a projective plane from $\Bbb R^n$, $\Bbb C^n$ and $\Bbb H^n$ and is then tempted to do the same for octonions. This leads to the construction of a projective plane known as $\Bbb OP^2$, the Cayley projective plane.
What are the references for the properties of the Cayley projective plane? In particular... | https://mathoverflow.net/users/798 | What is the Cayley projective plane? | As I recall, the Cayley projective plane is painful to build, but it is a 2-cell complex, with an 8-cell and a 16-cell. The cohomology is Z[x]/(x^3) where x has degree 8, as you would expect. Its homotopy is unapproachable, because it is just two spheres stuck together, so you would pretty much have to know the homotop... | 40 | https://mathoverflow.net/users/1698 | 4532 | 3,033 |
https://mathoverflow.net/questions/4434 | 22 | Let g(z) be an entire function of a complex variable z. Does there exist an entire function f(z) such that f(z+1)-f(z)=g(z)? As I learned several years back, the answer to this [is apparently 'yes'](http://www.math.niu.edu/~rusin/known-math/01_incoming/coho_func), but I have not felt satisfied with the proof because it... | https://mathoverflow.net/users/806 | Elementary solutions to f(z+1)-f(z)=g(z) in entire functions | It took me some time to find a solution that satisfies both requirements:
a) If should be based on the power series expansion
b) It should use no tools heavier than contour integration.
So, let $g(z)=\sum a\_ k z^k$. We know that $a\_ k$ decay faster than any geometric progression. We want analytic functions $F\_... | 32 | https://mathoverflow.net/users/1131 | 4542 | 3,041 |
https://mathoverflow.net/questions/4547 | 36 | There is a definition of Iwahori-Hecke algebras for Coxeter groups in terms of generators and relations and there is a definition of Hecke algebras involving functions on locally compact groups. Are these two concepts somehow related? I think I read somewhere that the Hecke algebra with functions includes Iwahori-Hecke... | https://mathoverflow.net/users/717 | Definitions of Hecke algebras | A Hecke algebra describes the most reasonable way to convolve functions or measures on a homogeneous space. Suppose that you have seen the definition of convolution of functions on a vector space, or on a discrete group --- the latter is just the group algebra of the group or some completion. Then how could you reasona... | 65 | https://mathoverflow.net/users/1450 | 4555 | 3,052 |
https://mathoverflow.net/questions/4561 | 38 | Y.I. Manin mentions in a [recent interview](http://golem.ph.utexas.edu/category/2009/11/interview_with_manin.html#c028992 "link to n-category discussion")
the need for a “codification of efficient new intuitive tools, such as … the “brave new algebra” of homotopy theorists”. This makes me puzzle, because I thought that... | https://mathoverflow.net/users/451 | What is the "intuition" behind "brave new algebra"? | I hardly know how to begin to reduce this subject to some kind of intuitive ideas, but here are some thoughts:
\* Adding homotopy to algebra allows for generalizations of familiar algebraic notions. For instance, a *topological commutative ring* is a commutative ring object in the category of spaces; it has addition ... | 42 | https://mathoverflow.net/users/437 | 4565 | 3,058 |
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