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https://mathoverflow.net/questions/6883 | 5 | Suppose we have a quadratic eigenvalue problem $\lambda^2 M + \lambda C + K$. Under what conditions is the following statement true: If $\lambda$ is an eigenvalue, so is $1/\lambda$?
Here, $M$, $C$, and $K$ are square matrices (not necessarily full rank). This is of interest to me since I have such systems for which ... | https://mathoverflow.net/users/1074 | Under what conditions do eigenvalues of a quadratic eigenvalue problem come in reciprocal pairs? | Consider the linearization
$$
L(\lambda)=\lambda\begin{pmatrix}M&0\cr
0&1
\end{pmatrix}+\begin{pmatrix}C&K\cr
-1&0\end{pmatrix}=\lambda A+B.
$$
The eigenvalues of the original problem coincide with those for the linearization.
Now, the eigenvalues of the linearized problem are the roots of the polynomial $\det(\lambd... | 4 | https://mathoverflow.net/users/1704 | 6905 | 4,710 |
https://mathoverflow.net/questions/6912 | 13 | As anyone who follows the arxiv, I notice every now and then "proofs" and "disproofs" of Riemann Hypothesis. I looked on several such articles, and it seemed to me quite nonsense, but I didn't make the effort to find a mistake. My question is whether someone reads these "proofs"?
BTW, I wanted to refer to some of th... | https://mathoverflow.net/users/2042 | On the proofs (and disproofs) of Riemann Hypothesis | Whenever someone claims a proof (or disproof) of a big conjecture, many people leap to the question of whether the proof is correct. The problem then is it that it takes an enormous amount of work to confirm that a proof is correct. Even a clear mistake in a proof could be reparable. Moreover, attempted proofs have inf... | 46 | https://mathoverflow.net/users/1450 | 6916 | 4,718 |
https://mathoverflow.net/questions/6889 | 29 | I've been learning about Dedekind zeta functions and some basic L-functions in my introductory algebraic number theory class, and I've been wondering why some functions are called L-functions and others are called zeta functions. I know that the zeta function is a product of L-functions, so it seems like an L-function ... | https://mathoverflow.net/users/1355 | What is the difference between a zeta function and an L-function? | Let me say first that a Dedekind zeta function is *always* a product of Artin L-functions. It is the structure of the Galois closure which is relevant here. Let me give a nice example which is indicative of the general case. Let $p(x) \in \mathbb{Z}[x]$ be an irreducible cubic, and let $\alpha$ be a root of $p$. Then $... | 24 | https://mathoverflow.net/users/1464 | 6927 | 4,723 |
https://mathoverflow.net/questions/6928 | 47 | What papers should we read to start? What basic knowledge do we need to understand the question? What is this area really about? And what are people researching on it?
| https://mathoverflow.net/users/2110 | How do we study Iwasawa theory? | Iwasawa theory has its origins in the following counterintuitive insight of Iwasawa: instead of trying to describe the structure of any particular Galois module, it is often easier to describe every Galois module in an infinite tower of fields at once.
The specific example that Iwasawa studied was the $p$-Sylow subgr... | 62 | https://mathoverflow.net/users/1018 | 6933 | 4,726 |
https://mathoverflow.net/questions/6345 | 5 | How does Hermite normal form (over $Z$) vary in families? I.e. if I have an $n\times m$ matrix $M$ whose entries are integral polynomials in some integral variable $x$, how does the Hermite normal form of the integral matrix $M(p)$ (obtained by setting $x$ equal to $p$) vary as a function of $p$? What about the special... | https://mathoverflow.net/users/1672 | Hermite normal form in families | Hi "DC". I think that I have worked out that the Hermite normal form is a "trichotomous quasipolynomial" in the variable $p$. If $f:\mathbb{Z} \to \mathbb{Z}$ is a function, then my definition is that $f$ is a trichotomous quasipolynomial if it is a quasipolynomial for $x \gg 0$, possibly a different quasipolynomial fo... | 3 | https://mathoverflow.net/users/1450 | 6938 | 4,731 |
https://mathoverflow.net/questions/6950 | 50 | This question might not have a good answer. It was something that occurred to me yesterday when I found myself in a pub, needing to do an explicit calculation with 2-cocycles but with no references handy (!).
Review of group cohomology.
---------------------------
Let $G$ be a group acting (on the left) on an abeli... | https://mathoverflow.net/users/1384 | Why is the standard definition of cocycle the one that _always_ comes up?? | The difference between f(1,g) and f(g,1) is generally an issue of whether mathematicians give preference to "domains" or "ranges" of maps.
Here is one way that you could think of this. I can write EG for a category whose objects are objects are elements of G, and where each pair of objects has a unique map between th... | 18 | https://mathoverflow.net/users/360 | 6952 | 4,740 |
https://mathoverflow.net/questions/6941 | 8 | Article [Exotic $\mathbb{R}^4$](http://en.wikipedia.org/wiki/Exotic_R4) on Wikipedia says that there is at least one maximal smooth structure on $\mathbb{R}^4$, that is such an atlas on $\mathbb{R}^4$ that any other smooth $\mathbb{R}^4$ can be embedded into it. Is the construction of such a maximal exotic $\mathbb{R}^... | https://mathoverflow.net/users/896 | Maximal exotic $\mathbb{R}^4$ | The paper by Freedman and Taylor mentioned by Carsten Schultz (above or below) is indeed the place to find the explicit construction.
Very roughly, the idea of the construction is as follows. Recall that a Casson handle is, among other things, a smooth 4-manifold which is homeomorphic (but not diffeomorphic) to the s... | 12 | https://mathoverflow.net/users/284 | 6965 | 4,751 |
https://mathoverflow.net/questions/6964 | 18 | A question I got asked I while ago:
If $T$ is a triangle in $\mathbb R^2$, is there a function $f:T\to\mathbb R$ such that the integral of $f$ over each straight segment connecting two points in the boundary of $T$ not on the same side is always $1$?
(Of course, you can change $T$ for your favorite convex set... a... | https://mathoverflow.net/users/1409 | Egalitarian measures | In general, no. For the double integral $\iint\_T f(x,y)\,dx\,dy$ will be the height on any side, as is seen by turning the triangle with one side parallel to an axis and performing the integral. So at least, $T$ has to be equilateral. I don't know the answer in that case.
**Edit:** Wait, wait – the same trick works ... | 17 | https://mathoverflow.net/users/802 | 6967 | 4,753 |
https://mathoverflow.net/questions/6980 | 4 | Let $\overline{\rho}\_{\Delta,\ell}$ be the mod-$\ell$ representation associated to Ramanujan's $\Delta$-function. It is well-known that (the semisimplification of) this representation is reducible if, say, $\ell=5$ or $\ell=691$. Is there a general name for primes like this? Serre calls them (in a more general context... | https://mathoverflow.net/users/1464 | A name for primes where residual Galois representations are reducible | "Eisenstein" ?
| 4 | https://mathoverflow.net/users/1125 | 6993 | 4,770 |
https://mathoverflow.net/questions/6929 | 0 | Let's say a series of 10 bits is output randomly. Now lets do that 256 times. I'd like to find out what the expected number of streaks of 1s or 0s are for each of the possible sizes 1-10.
For example, the stream 0111001011 has 3 1-bit streaks, 2 2-bit streaks, and 1 3-bit streak. Note that even though the 3-bit strea... | https://mathoverflow.net/users/2111 | Chances of streaks in small bit-streams | I'm going to call your "streams" "strings" instead, because "streams" looks too much like "streaks" to me. This becomes a much easier problem if we translate it into an enumeration problem. Since each of the 210 possible bit strings occur with equal probability, it suffices to count the total number of streaks of lengt... | 3 | https://mathoverflow.net/users/2121 | 6995 | 4,772 |
https://mathoverflow.net/questions/6925 | 4 | I'm trying to get a better handle on characteristic subgroups, and many nice examples are given with some sort of "natural" definition. For example, it's clear that the center, torsion subgroup, and commutator subgroup of a given group are all characteristic, just because of the way they are defined. How can we formali... | https://mathoverflow.net/users/913 | How can we formalize the naturality of certain characteristic subgroups? | The commutator subgroup of a group is given by a functor on the category whose objects are groups and whose morphisms are all homomorphisms. We can say the similar statement for the torsion subgroup of an abelian group, and this is why the subgroups are fully invariant and not just characteristic. The center is not ful... | 6 | https://mathoverflow.net/users/121 | 6996 | 4,773 |
https://mathoverflow.net/questions/6984 | 1 | Can you suggest an introduction to structural equation modeling for math majors and mathematicians?
| https://mathoverflow.net/users/812 | Introduction to Structural Equation Modeling | There seem to be a number of topics in science that use mostly standard mathematics, but bury it with a lot of new terminology and non-mathematical ideas. Maybe there is no reasonable way to avoid or even criticize this, since mathematicians also make up new terms all the time. Still, it can be weird and frustrating to... | 3 | https://mathoverflow.net/users/1450 | 6997 | 4,774 |
https://mathoverflow.net/questions/6990 | 4 | The Fourier transform of periodic function $f$ yields a $l^2$-series of the functions coefficients when represented as countable linear combination of $\sin$ and $\cos$ functions.
* In how far can this be generalized to other countable sets of functions? For example, if we keep our inner product, can we obtain anothe... | https://mathoverflow.net/users/2082 | Generalize Fourier transform to other basis than trigonometric function | It is not what you want, but may be worth mentioning. There is a huge branch of abstract harmonic analysis on (abelian) locally compact groups, which generalizes Fourier transformation on reals and circle. The main point about sin and cos (or rather complex exponent $e^{i n x}$) is that it is a character (continuous ho... | 3 | https://mathoverflow.net/users/896 | 6999 | 4,775 |
https://mathoverflow.net/questions/6922 | 14 | I am looking for references related to the terms "Harish-Chandra pair" and "Harish-Chandra modules", and also to the term "category O". I know what these are, or I think I do (a Harish-Chandra pair is a pair (Lie algebra; subgroup) with the subgroup acting in the Lie algebra, satisfying some natural conditions). The qu... | https://mathoverflow.net/users/2106 | References for Harish-Chandra pairs and modules, category "O"? | Harish-Chandra pairs and their use in localization is discussed in Beilinson-Bernstein *A proof of Jantzen Conjectures*, available on [Joseph Bernstein's web page](http://www.math.tau.ac.il/~bernstei/Publication_list/Publication_list.html). See in particular sections 1.8 and 3.3.
Other sources:
1. Beilinson-Drinfel... | 4 | https://mathoverflow.net/users/121 | 7001 | 4,776 |
https://mathoverflow.net/questions/6981 | 5 | I have a heavily symmetric regular graph whose automorphisms I know. I remove one subgraph and insert another one in a consistent manner; for example, this could be a Delta-Y transformation (replacing a node with a complete subgraph). I'd like to compute the automorphisms of the new graph using the automorphisms of the... | https://mathoverflow.net/users/2122 | How are graph automorphisms are affected by transformations? | I do not think there are any results that relate the automorphism group of a graph after
subgraph replacement to the group of the original graph.
If the original graph was "heavily symmetric", you would expect the group of the new graph to be smaller - most local operations would destroy vertex transitivity, for exa... | 6 | https://mathoverflow.net/users/1266 | 7005 | 4,778 |
https://mathoverflow.net/questions/6987 | 10 | What is the definition of a singular value over a finite field $\mathcal{F}$ of a matrix ${\bf A}$ in $\mathcal{F}^{m\times n}$? Is there a geometric intuition in the same manner as with the real case where the eigenvalues are the radii of the ellipse $\frac{\|{\bf A}{\bf x}\|^2}{\|{\bf x}\|^2}$?
| https://mathoverflow.net/users/2124 | Singular value decomposition over finite fields? | There is no definition of a singular value of a matrix over a finite field. You could
define it to be a non-zero eigenvalue of $A^TA$, but this does not really work as you might expect.
Over the reals, the eigenvalues of $A^TA$ are non-negative and the smallest singular value
is a measure of how close $A$ is to being... | 15 | https://mathoverflow.net/users/1266 | 7012 | 4,783 |
https://mathoverflow.net/questions/7004 | 5 | I have \$3. I flip a coin. If I get heads, I get \$1. If I get tails, I lose \$1. The game stops when I have \$0 or \$7. What is the probability I get \$7?
I solved this by creating a system of linear equations, where $P\_0 = 0$, $P\_7 = 1$, and $P\_x = 0.5 \cdot P\_{x-1} + 0.5 \cdot P\_{x+1}$. Solving them, I got $P... | https://mathoverflow.net/users/1646 | Intuitive explanation to Probability question | I really like Vigleik's answer, but I'll throw in yet another way to look at your original problem. Px = (Px-1+Px+1)/2 is an example of a (discrete) harmonic function; i.e., a function whose value is the average of the adjacent values. In this case, Px is a harmonic function on a chain graph. For purposes of intuition,... | 7 | https://mathoverflow.net/users/2121 | 7015 | 4,786 |
https://mathoverflow.net/questions/6979 | 16 | What is etale descent? I have a vague notion that, for example, given a variety $V$ over a number field $K$, etale descent will produce (sometimes) a variety $V'$ over $\mathbb{Q}$ of the same complex dimension which is isomorphic to $V$ over $K$ and such that $V(K)=V'(\mathbb{Q})$. Is this at all right? How does one d... | https://mathoverflow.net/users/1464 | What is etale descent? | Let $L/K$ be a Galois field extension and consider a variety $Y$ over $L$. The theory of (Galois) descent addresses the question whether $Y$ can be defined over $K$.
More precisely, the question is: "does there exist a variety $X$ over $K$ such that $Y = X \times\_{Spec(K)} Spec(L)$".
Now assume such $X$ does exist. ... | 16 | https://mathoverflow.net/users/2132 | 7024 | 4,791 |
https://mathoverflow.net/questions/7037 | 0 | Theorem: (SRL)
For every $\epsilon>0$ and integer $m\geq 1$ there is an $M$ such that every graph $G$, with $|G|\geq m$ has an $\epsilon$-regular partition $V(G)=V\_0\cup\ldots\cup V\_k$ for some $m\leq k\leq M$.
Can someone explain to me why this statement is not trivial? For instance, what stops me choosing $M$ la... | https://mathoverflow.net/users/2011 | Szemeredi's Regularity Lemma | Quantifier error. You have to fix your M before you are given a graph G; whereas your approach would require one to have the graph G at hand, before choosing M.
| 5 | https://mathoverflow.net/users/763 | 7038 | 4,799 |
https://mathoverflow.net/questions/6982 | 5 | Can you suggest a book that has a thorough introduction to Singular Value Decomposition?
| https://mathoverflow.net/users/812 | Thorough Introduction to Singular Value Decomposition | I find Numerical Linear Algebra by N. Trefreten and D.Bau an extremely well-written book. It not only introduces the Singluar value decomposition but explains applications and history.
| 4 | https://mathoverflow.net/users/2011 | 7047 | 4,805 |
https://mathoverflow.net/questions/7046 | 1 | What i understand about strata for the nullcone is this: (from Mumford's "Geometric Invariant Theory" and Hesselink's paper "Desingularizations of Varieties of Nullforms")
**ADDED BY DAVID SPEYER** In this setting, we are studying a reductive group $G$ acting on a vector space $V$. We write $T$ for a maximal torus of... | https://mathoverflow.net/users/2623 | Strata for the nullcone (from Hesselink's paper) | Example of how the multiplicity can be greater than $1$: Let $\mathbb{G}\_m$ act on $\mathbb{A}^2$ by $t: (x,y) \mapsto (t^2x, t^3 y)$. Let $v$ be any element of $\mathbb{A}^2$ not on the coordinate axes, for example, $(1,1)$. So $\mathbb{G}\_m$ maps to $\mathbb{A}^2$ by $t \mapsto (t^2, t^3)$. This extends to a map $\... | 5 | https://mathoverflow.net/users/297 | 7049 | 4,807 |
https://mathoverflow.net/questions/7025 | 22 | Many commutative algebra textbooks establish that every ideal of a ring is contained in a maximal ideal by appealing to Zorn's lemma, which I dislike on grounds of non-constructivity. For Noetherian rings I'm told one can replace Zorn's lemma with countable choice, which is nice, but still not nice enough - I'd like to... | https://mathoverflow.net/users/290 | When can we prove constructively that a ring with unity has a maximal ideal? | I suspect that the most general reasonable answer is a ring endowed with a constructive replacement for what the axiom of choice would have given you.
How do you show in practice that a ring is Noetherian? Either explicitly or implicitly, you find an ordinal height for its ideals. Once you do that, an ideal of least ... | 7 | https://mathoverflow.net/users/1450 | 7062 | 4,816 |
https://mathoverflow.net/questions/6589 | 5 | There is a well known theorem that says that the functor associating to a perverse sheaf $F$ on $X$ the data $(F|\_U,\phi\_f(F),can:\psi\_f(F) \to \phi\_f(F),var:\phi\_f(F)\to \psi\_f(F)(-1))$ where $U = X \setminus (f=0)$ is an equivalence of categories.
In dimension 1, this gives that a perverse sheaf on $(\mathbb... | https://mathoverflow.net/users/1985 | Hypercube decomposition of perverse sheaves | In some sense there is no canonical way to extract these vector spaces. Algebraically this is because projective objects usually have automorphisms. Geometrically because these vector spaces are the stalks of local systems on a manifold with no natural base point. But the manifold and the local system are canonical.
... | 3 | https://mathoverflow.net/users/1048 | 7065 | 4,818 |
https://mathoverflow.net/questions/7071 | 5 | In reading the paper of Green and Tao on arithmetic progressions within the primes, I became very interested in the notion of a k-pseudorandom measure discussed in that paper.
A measure here is a function $\nu:\mathbf{Z}\_N\to\mathbf{R}$ such that $\mathbf{E}\nu=1+o(1)$, and it is k-pseudorandom if it obeys the ($k2... | https://mathoverflow.net/users/385 | k-pseudorandom measures | Linear forms condition says that these functions are morally the functions that are close to $1$ in appropriate $U^k$ norm. What I mean is that $U^k$ norms are a special kind of linear forms, and so linear forms condition implies proximity to $1$ in $U^k$, on one hand. On the other hand,if one controls $\nu-1$ in $U^t$... | 5 | https://mathoverflow.net/users/806 | 7077 | 4,827 |
https://mathoverflow.net/questions/7018 | 35 | One of my favorite results in algebraic geometry is a classical result of AX (see <http://terrytao.wordpress.com/2009/03/07/infinite-fields-finite-fields-and-the-ax-grothendieck-theorem/>) I'll recall the version of the theorem that I learned in an undergraduate class in model theory.
An algebraic map $F: \mathbb{C}^... | https://mathoverflow.net/users/2089 | Model theoretic applications to algebra and number theory(Iwasawa Theory) | It's hard for me to think of an area of algebra that applied model theorists **haven't** touched recently. I have not heard of any logicians working on Iwasawa theory, but it wouldn't surprise me if there are some.
Diophantine geometry: [here is a survey article by Thomas Scanlon](http://math.berkeley.edu/~scanlon/pa... | 34 | https://mathoverflow.net/users/93 | 7092 | 4,836 |
https://mathoverflow.net/questions/7059 | 30 | In short: what does Labesse-Langlands say?
Slightly more precise: what are the cuspidal automorphic representations of $SL\_2(\mathbf{A}\_{\mathbf{Q}})$, together with multiplicities? Let's say that I have a complete list of the cuspidal automorphic representations of $GL\_2/\mathbf{Q}$ and I want to try and deduce w... | https://mathoverflow.net/users/1384 | Overview of automorphic representations for $SL(2)/{\mathbf{Q}}$? | Warning: not an expert, so could be major mistakes in this.
The multiplicity is one for every element in the global packet, in the non-CM case. In the CM case,
half of the packet has multiplicity one and the other half has multiplicity zero.
For the general story, I'd suggest looking at Arthur's conjectures, whi... | 11 | https://mathoverflow.net/users/1253 | 7096 | 4,839 |
https://mathoverflow.net/questions/7091 | 13 | Let K/Q be a field, probably not a finite extension. Is it possible for a polynomial to be irreducible over K but have a root in every completion of K? What about all but finitely many completions?
This question is related to the question "Can a non-surjective polynomial map from an infinite field to itself miss only... | https://mathoverflow.net/users/2024 | Irreducible polynomial over number field with roots in every completion? | For a finite extension $K/\mathbf{Q}$, the answer is no. Suppose that $f$ is an irreducible polynomial with coefficients in $K$ and splitting field $L$. If $G$ is the Galois group of $L/K$, then the polynomial $f$ gives rise to a faithful transitive permutation representation $G \rightarrow S\_d$, where $d$ is the degr... | 24 | https://mathoverflow.net/users/nan | 7100 | 4,841 |
https://mathoverflow.net/questions/5994 | 16 | If G is a graph then its adjacency matrix has a distinguished Peron-Frobenius eigenvalue x. Consider the field Q(x). I'd like a result that says that if G is a "random graph" then the Galois group of Q(x) is "large" with high probability. (Well, what I really want is just that the Galois group is non-abelian, but I'm g... | https://mathoverflow.net/users/22 | Number theoretic spectral properties of random graphs | Just in case anyone else is still thinking about this question...
The answer is the following. Either:
1. The eigenvalues of $G\_n$ are all of the form $\zeta + \zeta^{-1}$ for roots of unity $\zeta$, and the graphs $G\_n$ are subgraphs of the Dynkin diagrams
$A\_n$ or $D\_n$.
2. For sufficiently large $n$, the lar... | 19 | https://mathoverflow.net/users/nan | 7102 | 4,843 |
https://mathoverflow.net/questions/7099 | 8 | I know at least two definitions of *correspondence*, and my question might as well be about both of them.
1. Let $X,Y$ be objects in your favorite category. A *correspondence* is a *span*, namely a diagram $X \leftarrow Z \rightarrow Y$.
2. Let $X,Y$ be objects in your favorite category with products. A *corresponden... | https://mathoverflow.net/users/78 | How should I think about correspondences? | You can think of replacing your category by its category of spans as a kind of "linearization" of the category. For instance if you start with the category of finite sets, a correspondence between X and Y is the same thing as a map from the free commutative monoid on X to the free commutative monoid on Y. (Here we have... | 9 | https://mathoverflow.net/users/126667 | 7105 | 4,845 |
https://mathoverflow.net/questions/7114 | 47 | As compared to classes of graphs embeddable in other surfaces.
Some ways in which they're exceptional:
1. Mac Lane's and Whitney's criteria are algebraic characterizations of planar graphs. (Well, mostly algebraic in the former case.) Before writing this question, I didn't know whether generalizations to graphs emb... | https://mathoverflow.net/users/382 | Why are planar graphs so exceptional? | (I think that the question of why planar graphs are exceptional is important. It can be asked not only in the context of graphs embeddable on other surfaces. Let me edit and elaborate, also borrowing from the remarks.)
**Duality:** Perhaps duality is the crucial property of planar graphs. There is a theorem asserting... | 34 | https://mathoverflow.net/users/1532 | 7116 | 4,852 |
https://mathoverflow.net/questions/7129 | 3 | I think it's clear that commutative semigroups S that are also bands, i.e. $e^2 = e$ for all e, correspond to finite posets (consider the elements of the semigroups as sets, where the intersection of two sets is their product), satisfying the condition that there is a unique minimal element in this finite poset (to mak... | https://mathoverflow.net/users/2623 | Representations of finite commutative band semigroups | Firstly, if you're only interested in finite semigroups, I suggest amending the title of the question ;)
Commutative semigroups in which ever element is idempotent are called *semilattices*, and are a special case of the more general notion of *inverse semigroup*. They have been much studied, although their infinite-... | 6 | https://mathoverflow.net/users/763 | 7131 | 4,862 |
https://mathoverflow.net/questions/7126 | 2 | Here's a question I've been thinking about, it's a curiosity that I don't know how to answer. There could be a simple counterexample, or it could be true and I don't know how difficult it would be to prove.
If we fix $m$, is it always possible to find a sufficiently large $n$ satisfying the conditions of the followi... | https://mathoverflow.net/users/2623 | Embedding group algebra $F[S_m X S_n]$ into a group algebra $F[S_{m+n}]$ | **NOTE:** *This answer was given in response to an earlier version of the question above, and so probably seems completely irrelevant if you can't see the old version. (More strikethrough and less deletion in editing, please?)*
Firstly, I'm going to assume you want $x'$ to be nonzero. Secondly, do you want $x$ and $x... | 3 | https://mathoverflow.net/users/763 | 7132 | 4,863 |
https://mathoverflow.net/questions/7134 | 7 | Why doesn't the join operation on the category of simplicial sets commute up to unique isomorphism? I mean, aren't products and coproducts commutative up to isomorphism? That leads me to conclude at first glance that the join is commutative, but it's not. Recall, given two simplicial sets $S$ and $S'$, we define the jo... | https://mathoverflow.net/users/1353 | Joins of simplicial sets | Implicit in the index of the coproduct is that you're writing J as an ordered disjoint union of I and I', where I comes first.
EDIT: Some elaboration.
For a simplicial set $T$, let's write $T\\_n$ for the "n-simplices", i.e. the value of on the ordered set $\{0,1,...,n\}$; these together with the maps between them ... | 9 | https://mathoverflow.net/users/360 | 7138 | 4,866 |
https://mathoverflow.net/questions/7112 | 15 | This is an extension of [this question](https://mathoverflow.net/questions/7080/definition-of-the-symmetric-algebra-in-arbitrary-characteristic-for-graded-vector/) about symmetric algebras in positive characteristic. The title is also a bit tongue-in-cheek, as I am sure that there are multiple "correct" answers.
Let ... | https://mathoverflow.net/users/78 | Which is the correct universal enveloping algebra in positive characteristic? | The notions do indeed diverge in positive characteristic: there is the enveloping algebra, and then (in the case that $\mathfrak g$ is the Lie algebra of an algebraic group G) there is also the hyperalgebra of G, which is the divided-power version you mention. In characteristic 0 these two algebras coincide, but in pos... | 24 | https://mathoverflow.net/users/1528 | 7142 | 4,870 |
https://mathoverflow.net/questions/7133 | 41 | Is there a classification of finite commutative rings available?
If not, what are the best structure theorem that are known at present?
All I know is a result that every finite commutative ring is a direct product of local commutative rings (this is correct, right?) in some paper which computes the size of the genera... | https://mathoverflow.net/users/2623 | Classification of finite commutative rings | Yes, a finite ring $R$ is a finite direct sum of local finite rings. As a first step, for each prime $p$ there is a subring $R\_p$ of $R$ corresponding to the elements annihilated by the powers of $p$. $\require{enclose} \enclose{horizontalstrike}{R\_p\ \style{font-family:inherit;}{\text{is then an}}\hspace{-7mm}}$
$\e... | 17 | https://mathoverflow.net/users/1450 | 7146 | 4,874 |
https://mathoverflow.net/questions/7153 | 49 | I'm not sure whether this is non-trivial or not, but do there exist simple examples of an affine scheme $X$ having an open **affine** subscheme $U$ which is not principal in $X$? By a principal open of $X = \mathrm{Spec} \ A$, I mean anything of the form $D(f) = \{\mathfrak p \in \mathrm{Spec} \ A : f \notin \mathfrak ... | https://mathoverflow.net/users/1107 | Open affine subscheme of affine scheme which is not principal | Let X be an elliptic curve with the identity element O removed. Let U=X-P where P is a point of infinite order. Then U is affine by a Riemann-Roch argument. Now suppose that U=D(f). Then on the entire elliptic curve, the divisor of f must be supported at P and O only. This implies that P is a torsion point, a contradic... | 54 | https://mathoverflow.net/users/425 | 7185 | 4,909 |
https://mathoverflow.net/questions/7192 | 10 | There is a simple, intuitive "construction" of twisted K-theory if we are allowed to ignore that many things only hold up to homotopy. We know that maps to $K(Z,2)$ give line bundles on a space and that $K(Z,2)$ forms a group corresponding to the tensor product of line bundles. Line bundles also act as endomorphisms of... | https://mathoverflow.net/users/947 | Constructing Twisted K-theory | The answer is yes if you're working on the level of $\infty$-categories (and I'm pretty sure no if you're working on the level of homotopy categories). In other words, in the $\infty$-world there's no problem talking about a principal bundle for K(Z,2)=BBZ on any space, and they're indeed classified by maps (in the $\i... | 12 | https://mathoverflow.net/users/582 | 7198 | 4,920 |
https://mathoverflow.net/questions/7152 | 10 | Suppose $A \rightarrow B$ is a faithfully flat map of rings. Then the Amitsur complex is exact:
$0 \rightarrow A \rightarrow B \rightarrow B \otimes\_A B \rightarrow \dots$
(the second map is $id \otimes 1 - 1 \otimes id$, and the subsequent maps are alternating sums of the different ways of putting in a 1.)
That... | https://mathoverflow.net/users/791 | intuition about the "section after base-change" for flat descent and exactness of the Amitsur complex | I think it feels like magic because there's something tautological going on. The story should really culminate with the definition of "faithfully flat" rather than begin with it.
As you suggested, let's consider the case where you cover an affine scheme Spec(A) by finitely many basic open affines Spec(Af). The cover ... | 7 | https://mathoverflow.net/users/1 | 7204 | 4,925 |
https://mathoverflow.net/questions/7212 | 7 | Is there a good place to learn about the structure of **moduli stack** of flat $G$-bundles on an algebraic curve?
Of course, we're just studying representations of a group $\pi\_1(X)\to G$ modulo some conjugation (that's why it should be a stack). Since this is very similar to **Galois representations** in number the... | https://mathoverflow.net/users/65 | Moduli space of flat bundles | You have to be a bit careful here. Over $\mathbb{C}$ the stack of representations of $\pi\_{1}(X)$ in $G$ and the stack of flat algebraic $G$-bundles on $X$ are isomorphic as complex analytic stacks but are *not* isomorphic as algebraic stacks. In fact the algebraic structure on the stack of flat $SL\_{n}(\mathbb{C})$ ... | 16 | https://mathoverflow.net/users/439 | 7221 | 4,939 |
https://mathoverflow.net/questions/7229 | 15 | I am a combinatorist by training and I am interested in learning about the connections between combinatorics and Schubert varieties. The theory of Schubert varieties seems to be a difficult area to break into if one has not already studied it in graduate school. I don't have a formal course in Algebraic Geometry, but I... | https://mathoverflow.net/users/2176 | Learning About Schubert Varieties | I would suggest Part III of Fulton's Young Tableaux book (of which you should skip Part I) as the best starting point for learning about Schubert varieties.
One can get very far in this subject with a naive 19th century view of algebraic geometry, especially if one is willing to occasionally accept without proof a fe... | 17 | https://mathoverflow.net/users/3077 | 7234 | 4,948 |
https://mathoverflow.net/questions/7095 | 22 | There is a fact that I should have learned a long time ago, but never did; I was reminded that I did not know the answer by Qiaochu's excellent series of posts, the most recent of which is [this one](http://qchu.wordpress.com/2009/11/28/the-noetherian-condition-as-compactness/).
Let $X$ be a topological space. I can ... | https://mathoverflow.net/users/78 | Which is the correct ring of functions for a topological space? | I work with infinite dimensional manifolds so am extremely distrustful of anything that requires some sort of compactness condition. Most of the time, it's just too restrictive.
Consider a really nice simple space: the coproduct of a countably infinite number of lines, $\sum\_{\mathbb{N}} \mathbb{R}$ (coproduct taken... | 10 | https://mathoverflow.net/users/45 | 7255 | 4,960 |
https://mathoverflow.net/questions/835 | 6 | This is a somewhat vague question which came up MSRI a few days ago: Suppose I have a family of curves over a one dimensional base, given in a computationally explicit way. For example, maybe I have a family F\_t(x,y,z) of homogenous polynomials, whose coefficients are polynomials in t, and which cut out smooth curves ... | https://mathoverflow.net/users/297 | Algorithms for semistable reduction of families of curves | If your curves are in P^n (specifically in P^2 - as in your example), I think there is something you can do: project your curves from a general P^{n-2} to P^1. This means that you
are now looking for a limit in a Hurwitz scheme. This can be broken into two problems:
* looking for the limit on the underlying M\_{0,n}
... | 4 | https://mathoverflow.net/users/404 | 7268 | 4,970 |
https://mathoverflow.net/questions/7264 | 3 | Can one formulate those version of Weak Lefschetz that uses tubular neighbourhoods purely in terms of cohomology of (some) algebraic varieties?
Theorem in 5.1 of Part II in Goresky-MacPherson's "Stratified Morse theory" implies (in particular) that:
for a smooth projective P (over the field of complex numbers), X open ... | https://mathoverflow.net/users/2191 | On algebraic tubular neighbourhoods and Weak Lefschetz | regarding the comparison theorem between (Z/l^n-cohomology of) etale and complex-analytic tubular neighborhoods, my feeling is that it should hold, but won't follow formally from the comparison theorem for varieties.
one can at least get the comparison map, i think, as follows (this also addresses your question about... | 1 | https://mathoverflow.net/users/2200 | 7296 | 4,993 |
https://mathoverflow.net/questions/7247 | 17 | I recently had the idea that maybe measure spaces could be viewed as sheaves, since they attach things, specifically real numbers, to sets...
But at least as far as I can tell, it doesn't quite work - if $(X,\Sigma,\mu)$ is a measure space and $X$ is also given the topology $\Sigma$, then we do get a **presheaf** $M:... | https://mathoverflow.net/users/1916 | measure spaces as presheaves? | For a sheaf-theoretical interpretation of measure theory, measure spaces are the wrong objects, you want *measure algebras* and then consider certain Grothendieck topologies on a Boolean algebra.
For measure algebras, check out volume 3 of FRemlin's 5-volume opus dedicated to measure theory. For more sheaf-theoretica... | 17 | https://mathoverflow.net/users/2562 | 7307 | 5,002 |
https://mathoverflow.net/questions/7283 | 37 | In a workshop about the geometry of $\mathbb{F}\_1$ I attended recently, it came up a question related to a mysterious but "not-so-secret-anymore" seminar about... an hypothetical Topological Langlands Correspondence!
I had never heard about this program; I have found this page via Google:
<http://www.math.jhu.edu/... | https://mathoverflow.net/users/1234 | Topological Langlands? | I would cautiously venture that there is not really something we could call a topological Langlands program to outsiders at this point. My objection is to the final word - we don't really know what we're doing. For example, I don't think we even have a conjecture at this point relating representations of something to s... | 35 | https://mathoverflow.net/users/360 | 7312 | 5,006 |
https://mathoverflow.net/questions/7308 | 1 | It is well know that the genus three non orientable surface, N3, has only periodic and reducible auto-homeomorphisms, meanwhile the surface N4 is the first non orientable surface with pseudo Anosov maps. Also, recently profesor B.Szepietowski gave the MCG presentation of N4, from where, I calculated that there are seve... | https://mathoverflow.net/users/2196 | N_3 and N_4 periodic and pseudo Anosov auto-homeomorphisms | Just to lend some context to the above question: the mapping class group of the two-torus is naturally isomorphic to GL(2, Z). If we restrict to orientation preserving homeomorphism the mapping class group is SL(2, Z). The periodic mapping classes (isotopy classes of homeomorphisms) are exactly those with trace less th... | 2 | https://mathoverflow.net/users/1650 | 7314 | 5,007 |
https://mathoverflow.net/questions/7252 | 5 | As I understand, when we have a nilpotent cone, or a nullcone of a Lie group representation, what seems to be done in a lot of the literature (e.g. Achar&Henderson-"Orbit closures in the enhanced nilpotent cone") is to compute the intersection cohomology sheaves and find polynomials that determine the dimensions of var... | https://mathoverflow.net/users/2623 | What cohomology theories would be interesting for nilpotent cones/nullcones? | I think most questions of interest regarding the nilpotent cone have to do with categories of equivariant sheaves on it - either equivariant perverse sheaves, like IC complexes of orbits, or equivariant coherent sheaves, like structure sheaves of orbit closures. So cohomologies that help elucidate the structure of thes... | 2 | https://mathoverflow.net/users/582 | 7319 | 5,008 |
https://mathoverflow.net/questions/7317 | 4 | It turns out that joins of simplicial sets are fairly easy to define, but hard to manage. In lots of cases, we'd like to compute what a join is, does it look like a horn?, a boundary?, etc? and identify it as such, so we can figure out when our morphisms from the join have certain nice properties like being anodyne, ha... | https://mathoverflow.net/users/1353 | Computation of Joins of Simplicial Sets | Since the join of simplicial sets is associative and $\Delta^m = \Delta^0 \star \cdots \star \Delta^0$ ($m+1$ times), we should start by trying to understand things like $\Lambda^n\_j \star \Delta^0$, a.k.a. the "final" cone on $\Lambda^n\_j$. It's not too hard to see that this is the subcomplex of $\Delta^{n+1}$ consi... | 5 | https://mathoverflow.net/users/126667 | 7323 | 5,009 |
https://mathoverflow.net/questions/7315 | 4 | So I've been skimming Bowen's 1972 paper "Symbolic Dynamics for Hyperbolic Flows" hoping it would give me some insight into how to build a Markov family for the cat flow (i.e., the Anosov flow obtained by suspension of the cat map with unit height). For the sake of completeness, the cat flow $\phi$ is obtained as follo... | https://mathoverflow.net/users/1847 | Proper families for Anosov flows | Take Adler-Weiss on $0\times\mathbb T^2$, $1/3\times\mathbb T^2$ and $2/3\times\mathbb T^2$. Take neighborhoods of this tripled Adler-Weiss. Then this collection would satisfy all the properties with $\alpha=1/3$.
I am not sure why are you particularly interested in suspension flow, everything is determined by the ba... | 32 | https://mathoverflow.net/users/2029 | 7325 | 5,011 |
https://mathoverflow.net/questions/7322 | 4 | Given an undirected connected graph, our goal is to remove some edges to make the graph disconnected. The constraint is that **each** node of the graph can not lose more than $m$ edges incident to it. I want to find the minimum $m$ for which the goal is achievable. Is there any efficient algorithm to compute this minim... | https://mathoverflow.net/users/1401 | A graph connectivity problem (restated) | It appears to be NP-complete even when m=1: see The Complexity of the Matching-Cut Problem, Maurizio Patrignani and Maurizio Pizzonia, WG 2001, <http://dx.doi.org/10.1007/3-540-45477-2_26>
| 5 | https://mathoverflow.net/users/440 | 7334 | 5,016 |
https://mathoverflow.net/questions/7326 | 12 | Let (X,x) be a pointed projective variety. Then there exists an abelian variety V which is universal for maps of pointed varieties $(X,x) \to (A,e\_A)$, called the [albanese variety](http://en.wikipedia.org/wiki/Albanese_variety). When X is a curve, the variety V is isomorphic to the Jacobian of X (in higher dimensions... | https://mathoverflow.net/users/2 | When is an Albanese variety principally polarized? | In general it could happen that the Albanese variety does not admit a principal polarization at all. For instance the Albanese variety of an abelian variety is the Abelian variety itself. So choose $X$ to be some abelian variety that has no principal polarization and you will get an example.
On the other hand it can... | 14 | https://mathoverflow.net/users/439 | 7336 | 5,017 |
https://mathoverflow.net/questions/7329 | 16 | I recently discovered [The College Mathematics Journal](http://www.maa.org/pubs/cmj.html) and enjoyed reading through some of the articles on fun applications of mathematics. I'd like to send some of the articles to my younger sister, a high school sophomore, but unfortunately most of them require calculus, a subject s... | https://mathoverflow.net/users/2205 | Math journal for high school students? | An excellent journal published by the University of New South Wales (my alma mater!) is [Parabola](http://www.parabola.unsw.edu.au/content/about-us), aimed at interested secondary students.
| 14 | https://mathoverflow.net/users/3 | 7355 | 5,034 |
https://mathoverflow.net/questions/7318 | 28 | $\newcommand{\bb}{\mathbb}\DeclareMathOperator{\gal}{Gal}$
Before stating my question I should remark that I know almost nothing about etale cohomology - all that I know, I've gleaned from hearing off hand remarks and reading encyclopedia type articles. So I'm looking for an answer that will have some meaning to an eta... | https://mathoverflow.net/users/683 | Etale cohomology and l-adic Tate modules | IMO, the scenario is closer to your (a). I'll sketch an explanation of the duality between $H^1(E,\mathbf{Z}\_l)$ and the dual to the Tate module. We have $H^1(E,\mathbf{Z}\_l)=\text{Hom}(\pi\_1(E),\mathbf{Z}\_l)$,
where that $\pi\_1$ means etale fundamental group with base point the origin $O$ of $E$. Thus the isom... | 37 | https://mathoverflow.net/users/271 | 7359 | 5,036 |
https://mathoverflow.net/questions/7365 | 5 | I know that for $X$ a connected space, $THH(\Sigma^\infty \Omega X) = \Sigma^\infty \Lambda X$, the suspension spectrum of the free loop space of $X$. The computation can be carried out in spaces and then transferred to spectra via $\Sigma^\infty$. What is $TC(\Sigma^\infty \Omega X)$? Can it also be computed from some... | https://mathoverflow.net/users/126667 | What is $TC(\Sigma^\infty \Omega X)$? | The TC spectrum, at a prime $p$, of this is the homotopy pullback of a diagram
$S^1 \wedge (\Sigma^\infty\_+ \Lambda X)\_{hS^1} \to \Sigma^\infty\_+ \Lambda X \leftarrow \Sigma^\infty\_+ \Lambda X$
after $p$-completion. Here the left-hand map is the $S^1$-transfer from homotopy orbits back to the spectrum and the r... | 5 | https://mathoverflow.net/users/360 | 7382 | 5,050 |
https://mathoverflow.net/questions/7432 | 14 | Let $(A,m\_A)$ and $(B,m\_B)$ be noetherian local rings and $f:A\rightarrow B$ a local homomorphism. Let $F = B/m\_AB$ be the fiber ring and assume that
$$\mathrm{dim}(B) = \mathrm{dim}(A) + \mathrm{dim}(F).$$
The following Theorem (23.1 in Matsumura's CRT) is really quite a miracle:
Theorem: If $A$ is regular and ... | https://mathoverflow.net/users/2215 | Generalizing miracle flatness (Matsumura 23.1) via finite Tor-dimension | The "Theorem" isn't true with both rings just normal, or just CM, or even normal *and* CM. Let $A = k[[x,y,z]]/(xz-y^2) \cong k[[a^2,ab,b^2]]$ and let $B = k[[a,b]]$, with $f$ the natural inclusion. The dimensions add up as they must, since $f$ is module-finite. In this case finite flat dimension is the same as finite ... | 8 | https://mathoverflow.net/users/460 | 7438 | 5,094 |
https://mathoverflow.net/questions/7439 | 3 | Any non-singular projective variety over $\mathbb{C}$ is easily seen to be a smooth manifold. Presumably the same is not true for algebraic varieties - one would not expect varieties with singular points to have a smooth structure. But do there exist non-singular varieties that are not smooth manifolds?
| https://mathoverflow.net/users/1977 | Algebraic Varieties which are also Manifolds | Every non-singular algebraic variety over $\mathbb C$ is a smooth manifold. See for instance:
<http://en.wikipedia.org/wiki/Manifold>
under "Generalizations of Manifolds".
In fact, Arminius' suggested answer in the comments seems to give a proof of this fact, and I'll attempt to flesh it out a small amount. Every alg... | 7 | https://mathoverflow.net/users/4 | 7448 | 5,101 |
https://mathoverflow.net/questions/7454 | 13 | The complex structure on a complex manifold pulls back to what's called a CR structure on any real codimension 1 submanifold. The structure induced on a submanifold of higher codimension is a CR structure if a non-degeneracy condition holds. It's possible to describe these structures intrinsically, without reference to... | https://mathoverflow.net/users/1048 | What are CR manifolds like? | CR does stand for Cauchy-Riemann.
CR structures on 3 dimensional manifolds arise as the boundaries of complex (or almost-complex) 4 manifolds; if these boundaries are strictly *pseudo-convex* (i.e. convex in "holomorphic directions") the CR structure on the 3-manifold is a contact structure (if the boundary is only (... | 13 | https://mathoverflow.net/users/1672 | 7455 | 5,106 |
https://mathoverflow.net/questions/7446 | 15 | I am taking a first course on Algebraic Geometry, and I am a little confused at the intuition behind looking at schemes over a fixed scheme. Categorically, I have all the motivation in the world for looking at comma categories, but I would like to make sense of this geometrically.
Here is one piece of geometric motiv... | https://mathoverflow.net/users/1106 | Intuition about schemes over a fixed scheme | This is going to be perhaps vague, but I'l try to write down the idea.
I've been told that in doing scheme theory, most of the time what is studied is not a scheme but a morphism of schemes $f:X\rightarrow S$. The studied of such a maps can be enriched allowing a chance of "base" $S$. Such a scheme $S$ can be crazy, b... | 5 | https://mathoverflow.net/users/1547 | 7461 | 5,111 |
https://mathoverflow.net/questions/7470 | 23 | There is a nice formula for the area of a triangle on the 2-dimensional sphere;
If the triangle is the intersection of three half spheres, and has angles $\alpha$, $\beta$ and $\gamma$, and we normalize the area of the whole sphere to be $4\pi$ then the area of the triangle is
$$
\alpha + \beta + \gamma - \pi.
$$
The ... | https://mathoverflow.net/users/2229 | Is there a neat formula for the volume of a tetrahedron on $S^3$? | [On the volume of a hyperbolic and spherical tetrahedron](http://www.f.waseda.jp/murakami/papers/tetrahedronrev3.pdf), by Murakami and Yano. The volume is obtained as a linear combination of dilogarithms and squares of logarithms. The origin of their formula is really interesting: Asymptotics of quantum $6j$ symbols. (... | 27 | https://mathoverflow.net/users/1450 | 7475 | 5,121 |
https://mathoverflow.net/questions/7373 | 11 | Let's suppose we have a Scheme $X$ over the the field $k$, where such a field can be though to be either $\mathbb{C}$ or a finite field $\mathbb{F}\_q$. Then having this in mind,
Where do we find some representative examples where Geometry governs arithmetic? That is to say, examples where the geometry (or topology) o... | https://mathoverflow.net/users/1547 | Geometry Vs Arithmetic of schemes | Let's start with the most elementary example: **projective space** $\mathbb P^n$. It's not hard to see that that the number of points on it is always $q^n + q^{n-1} + \dots + q + 1.$
Note that this is because $\mathbb P^n$ can be always decomposed into simpler pieces: $\mathbb A^n \cup \mathbb A^{n-1}\cup\dots\cup \m... | 5 | https://mathoverflow.net/users/65 | 7501 | 5,138 |
https://mathoverflow.net/questions/7507 | 10 | Recall that a module is called
1. semisimple if every submodule is a direct summand
2. pure semisimple if every pure submodule is a direct summand
There is quite a bit of work on semisimple and pure semisimple modules, of course.
My question is
What is a module called if every submodule is pure?
and
What... | https://mathoverflow.net/users/1698 | When is every submodule pure? | Hmmm, I seem to have found the answer to this question. In
[Regular and semisimple modules](https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-65/issue-2/Regular-and-semisimple-modules/pjm/1102866791.full)
by Cheatham and Smith in 1976, they call a module regular if every submodule is pure. Re... | 8 | https://mathoverflow.net/users/1698 | 7512 | 5,143 |
https://mathoverflow.net/questions/7521 | 0 | What's your usual action online in order to browse math journals? Like check Arxiv or MathSciNet. Any other good link directs you to most updated articles in major math journals. Or the traditional way of browsing periodical section of your library is still a better way to get a glimpse of current development in math. ... | https://mathoverflow.net/users/1947 | A website linking to most major math journals? | The AMS Digital Mathematics Registry has a huge list at <http://www.ams.org/dmr/JournalList.html>
| 3 | https://mathoverflow.net/users/532 | 7529 | 5,155 |
https://mathoverflow.net/questions/6749 | 27 | The salamander lemma is a lemma in homological algebra from which a number of theorems quickly drop out, some of the more famous ones include the snake lemma, the five lemma, the sharp 3x3 lemma (generalized nine lemma), etc. However, the only proof I've ever seen of this lemma is by a diagram chase after reducing to R... | https://mathoverflow.net/users/1353 | A proof of the salamander lemma without Mitchell's embedding theorem? | There's [a proof of the snake lemma without elements](https://math.colorado.edu/%7Ejonathan.wise/papers/snake.pdf) (a non-elementary proof?) on my website.
**Edit:** I added a section about the salamander lemma.
**Much later edit:** As Charles Rezk points out below ([1](https://mathoverflow.net/questions/6749/a-pro... | 39 | https://mathoverflow.net/users/32 | 7531 | 5,157 |
https://mathoverflow.net/questions/7532 | 0 | For a power series $f(z) = \sum\_{i=0}^{\infty} a\_i z^i$ with $a\_1$ nonzero, [Lagrange's inversion formula](http://en.wikipedia.org/wiki/Lagrange_inversion_theorem) gives an explicit way to compute the Taylor coefficients of the inverse function.
Is there any analogous formula for Laurent series?
| https://mathoverflow.net/users/83 | Inversion of Laurent series | The Lagrange inversion formula is meant to give you the Taylor series expansion of $f^{-1}$ at the point $f(0)$. If $f$ has a Laurent series instead, then it means that $f(0) = \infty$ and that $f$ is meromorphic. The Taylor series at $\infty$ of $f^{-1}$ then doesn't particularly mean anything unless you change to a d... | 5 | https://mathoverflow.net/users/1450 | 7533 | 5,158 |
https://mathoverflow.net/questions/7535 | 0 | Let A = $\{a\_1,...,a\_n\}$ be a set of numbers. We can assume all elements of A are integers.
Is there any efficient way to partition A into two sets B = $\{b\_1,...,b\_k\}$ and C = $\{c\_1,...,c\_l\}$ such that $|(b\_1...b\_k) - (c\_1...c\_l)|$ is minimal?
Is the problem anything easier if we let A be a set of st... | https://mathoverflow.net/users/1737 | Paritioning a set of numbers A into two sets B,C so that abs(prod(B) - prod(C)) is minimal | I suspect that it's NP-hard even to check whether you can get prod(B) - prod(C) = 0, although there's a problem with the obvious argument that I don't know off the top of my head how to fix.
"Reduction" from subset sum: If you have a set S of integers, replace each integer $k \in S$ with $2^k$. Then this new set can ... | 2 | https://mathoverflow.net/users/382 | 7540 | 5,162 |
https://mathoverflow.net/questions/7539 | 2 | I understand the ordinary Springer correspondence gives a bijection between orbits in the nilpotent cone for the adjoint representation and irreducible representations of the Weyl group, through action of the Weyl group on the top intersection cohomology. (I'm still learning about this, I know a very vague picture).
M... | https://mathoverflow.net/users/2623 | Springer corresponding for nullcones other than the standard nilpotent cone | Let me make another guess as to what you are seeing:
The Lie/algebraic group acts on the coordinate ring for the closure of the nilpotent orbit (since it acts on the nilpotent orbit). Take the weight 0 weight space of this coordinate ring; the Weyl group (realized as N(T)/T) acts on it since T acts trivially on this ... | 2 | https://mathoverflow.net/users/3077 | 7551 | 5,170 |
https://mathoverflow.net/questions/7556 | 9 | This is related to the [question](https://mathoverflow.net/questions/6704/how-to-think-about-cm-rings) I asked last time. This sounds a bit too specific, I hope this question is still acceptable on MO.
I am still not quite comfortable with the concept of depth, and there is this exercise in Matsumura's book that goes... | https://mathoverflow.net/users/nan | Some examples of depth | 1) Start with a regular local ring $R$. Take 2 ideals $I,J$ such that $I$ does not contain $J$, $R/I$ is CM and $\dim R/J <\dim R/I$. Then $A=R/(I\cap J)$ and $M=R/I$ work. In your example, $I=(x)$ and $J=(y,z)$. The reason is that CM means unmixed, so by having components of different dimensions one makes sure A is no... | 9 | https://mathoverflow.net/users/2083 | 7564 | 5,178 |
https://mathoverflow.net/questions/7561 | 7 | Let $E$ be a spectrum. Then $E \wedge E$ is a $\mathbb{Z}/2$-spectrum in the naivest possible sense, i.e., an object with $\mathbb{Z}/2$-action in the (∞,1)-category of spectra. Can I make it a $\mathbb{Z}/2$-spectrum in the less naive, but still not genuine, sense? (That is, a $\mathbb{Z}/2$-spectrum indexed on the tr... | https://mathoverflow.net/users/126667 | Naive Z/2-spectrum structure on E smash E? | Given a spectrum $E$ there is a "standard" lift of $E \wedge E$ to a $\mathbb{Z}/2$-spectrum using the basic technique you describe. One way to describe it as follows.
You can construct the category of genuine $\mathbb{Z}/2$-spectra (indexed on the full universe) via collections of $\mathbb{Z}/2$-spaces $X\\_n$ with ... | 4 | https://mathoverflow.net/users/360 | 7579 | 5,188 |
https://mathoverflow.net/questions/7541 | 16 | The following question was posed to me a while ago. No one I know has a given a satisfactory (or even a complete) proof:
Suppose that $M$ is an $n$ x $n$ matrix of non-negative integers. Additionally, suppose that if a coordinate of $M$ is zero, then the sum of the entries in its row and its column is at least $n$.
... | https://mathoverflow.net/users/2043 | Extremal question on matrices | The following looks too simple, so perhaps there's a mistake, but here goes.
Let $m$ be the smallest among all row sums *and* column sums. If $m\geq n/2$, we are done.
Otherwise, $m=cn$ with $c\lt 1/2$. Suppose it is a column which has sum $m$. This column has at least $n-m$ zeroes, and each of the corresponding ro... | 13 | https://mathoverflow.net/users/932 | 7581 | 5,190 |
https://mathoverflow.net/questions/7490 | 7 | Let R be a normal noetherian domain.
What is the difference between a finitely generated reflexive module and a finitely generated projective module?
Can anybody recommend any references or make any suggestions about this?
---
Finitely generated projective modules can be identified with idempotents matrix...... | https://mathoverflow.net/users/2040 | Differences between reflexives and projectives modules | Well, the answer is well known of course. For a finitely generated module over a commutative normal Noetherian domain TFAE
1. M is reflexive
2. M is torsion-free and equals the intersection of its localizations at the codimension 1 primes
3. M satisfies Serre's condition S2 and its support is Spec R.
4. M is the dual... | 16 | https://mathoverflow.net/users/1784 | 7588 | 5,193 |
https://mathoverflow.net/questions/7585 | 5 | Are any finitely generated reflexive module a second syzygy?
(I´m thinking especially in normal noetherian domains)
More general...
Are any divisorial lattice a second syzygy?
(I´m thinking especially in Krull domains)
| https://mathoverflow.net/users/2040 | Are any finitely generated reflexive module a 2nd syzygy? | Over a normal domain (in fact, you only need Gorenstein in codimension 1, being second syzygy and reflexive are equivalent). This is Theorem 3.6 of Evans-Griffith "Syzygies" book.
| 6 | https://mathoverflow.net/users/2083 | 7592 | 5,197 |
https://mathoverflow.net/questions/7304 | 41 | Is there an example of a complex bundle on $\mathbb CP^n$ or on a Fano variety (defined over complex numbers), that does not admit a holomorphic structure? We require that the Chern classes of the bundle are $(k,k)$ Hodge classes (which is automatic for $\mathbb CP^n$ or Fanos of dimension<4). If by any chance such exa... | https://mathoverflow.net/users/943 | Complex vector bundles that are not holomorphic | Here is the answer to the question, kindly explained to me by Burt Totaro.
EDITED. This is an OPEN PROBLEM.
0) Apparently in the case of $\mathbb CP^n$ existence of a complex bundle without holomorphic structure is still an OPEN PROBLEM. Though it is believed that there should be plenty of examples starting from $... | 34 | https://mathoverflow.net/users/943 | 7596 | 5,200 |
https://mathoverflow.net/questions/7586 | 22 | Here $\zeta(s)$ is the usual Riemann zeta function, defined as $\sum\_{n=1}^\infty n^{-s}$ for $\Re(s)>1$.
Let $A\_n=${$s\;:\;\zeta(s)=n$}. The behaviour of $A\_0$ is basically just the Riemann hypothesis; my question concerns $A\_n$ for $n\neq0$.
1) Is determining this just as hard as the Riemann hypothesis?
2)... | https://mathoverflow.net/users/385 | When does the zeta function take on integer values? | Regarding 3), this "Big Picard" stuff is serious overkill.
Think like an undergraduate real analysis student:
The p-series $\zeta(p)$ converges for real $p > 1$, whereas $\zeta(1)$ = sum of the harmonic series = oo.
An easy argument using (e.g.) the integral test shows that
$$lim\_{p \rightarrow \infty} \zeta(p... | 29 | https://mathoverflow.net/users/1149 | 7600 | 5,204 |
https://mathoverflow.net/questions/7603 | 21 | From [a post](https://mathoverflow.net/questions/7374/the-jouanolou-trick/7602#7602) to [The Jouanolou trick](https://mathoverflow.net/questions/7374/the-jouanolou-trick/):
>
> Are all **topologically trivial** (contractible) **complex algebraic varieties** necessarily affine? Are there examples of those not birati... | https://mathoverflow.net/users/65 | Topologically contractible algebraic varieties | No. Counterexamples were first constructed by Winkelmann, as quotients of $\mathbb A^5$ by algebraic actions of $\mathbb G\_{\text{a}}$. I learned this from Hanspeter Kraft's very nice article available here:
[Challenging problems on affine $n$-space](http://www.numdam.org/numdam-bin/item?id=SB_1994-1995__37__295_0).... | 26 | https://mathoverflow.net/users/1048 | 7606 | 5,208 |
https://mathoverflow.net/questions/7441 | 7 | Not sure how to tag this one so feel free to edit and add tags.
When I initially started graduate school my choice for an area of study was quite nebulous. I had only figured out enough to know that I wanted to do some work involving a lot of category theory. So when I applied to schools I figured I could find some ... | https://mathoverflow.net/users/nan | Choice of adviser | I strongly recommend finding a good advisor (someone who you get along with, who has compatible understanding of how hands-on the advisor will be, who will keep you funded, who can get you a postdoc, who actually wants a student etc.) over choosing a particular subfield. There's too little correlation between what you ... | 15 | https://mathoverflow.net/users/22 | 7611 | 5,213 |
https://mathoverflow.net/questions/7604 | 7 | As is well known, the line bundles over \**CP*\*$^1$ are indexed by the integers. My question is how are the line bundles over \**CP*\*$^n$, $n > 1$, and \**Gr*\*$(n,k)$ indexed? Moreover, do there exist any other interesting classifications of line bundles over spaces (I remember something about Atiyah and elliptic cu... | https://mathoverflow.net/users/1648 | Indexing the line bundles over a Grassmannian. | Algebraic line bundles on a smooth variety $X$ are classified by the Picard group $Pic(X) = H^1(X, \mathbf O\_X^\*)$. This is an exercise in Hartshorne's book, basically every line bundle is mapped to it's gluing cocycle.
The group $Pic(X)$ is also equal to the group $CH^1(X)$ of divisors modulo rational equivalence. T... | 18 | https://mathoverflow.net/users/2260 | 7614 | 5,216 |
https://mathoverflow.net/questions/7523 | 21 | This question is closely related to [this one](https://mathoverflow.net/questions/124/is-every-finite-dimensional-lie-algebra-the-lie-algebra-of-an-algebraic-group).
Ado's theorem states that given a finite-dimensional Lie algebra $\mathfrak g$, there exists a faithful representation $\rho\colon\mathfrak g \to \mathf... | https://mathoverflow.net/users/2164 | Is every finite-dimensional Lie algebra the Lie algebra of a closed linear Lie group? | I think that the answer is yes. It looks like you can prove it by relying on a convenient proof of Ado's theorem.
Procesi's book, "Lie groups: an approach through invariants and representations", has the following theorem preceding the proof of Ado's theorem:
**Theorem 2.** Given a Lie algebra $L$ with semismiple p... | 9 | https://mathoverflow.net/users/1450 | 7623 | 5,222 |
https://mathoverflow.net/questions/7626 | 13 | This seems like an obvious fact, but I'm not sure what the necessary meaning of "nice" is to get a result like this. I'm wondering if there is a theorem of the form:
For any <1> field extension $K/F$, a map from $\phi:K\rightarrow F$ that satisfies <2> is the field norm (or trace).
where <1> could be something lik... | https://mathoverflow.net/users/1916 | Are the field norm and trace the unique "nice" maps between fields? | The field norm and trace exist when $K$ is a finite algebraic extension of $F$. In this case, an element $\alpha \in K$ can be interpreted as an $F$-linear map on $K$ by multiplication. The field norm is just the determinant of $\alpha$ as a linear map, while the trace is the trace of $\alpha$ as a linear map. This yie... | 27 | https://mathoverflow.net/users/1450 | 7628 | 5,223 |
https://mathoverflow.net/questions/7624 | 0 | I am puzzled with the following discrete logarithm problem:
Given positive integers `b, c, m` where `(b < m) is True` it is to find a positive integer `e` such that
```
(b**e % m == c) is True
```
where two stars is exponentiation (e.g. in Ruby, Python or ^ in some other languages) and % is modulo operation. Usi... | https://mathoverflow.net/users/2266 | The Discrete Logarithm problem | The question is not phrased to our taste at mathoverflow, but the user has a point that this particular Wikipedia page is under-developed. As David Speyer suggests, it is a very different problem for very large primes than for small ones. For small primes the simplest algorithms described in Wikipedia are probably the ... | 5 | https://mathoverflow.net/users/1450 | 7629 | 5,224 |
https://mathoverflow.net/questions/7569 | 2 | How would one classify the strata for the standard nilpotent cone for $GL\_{k}(\mathbb{C})$, using the definition from Hesselink's paper "[Desingularizations of Varieties of Nullforms](https://doi.org/10.1007/BF01390087)"? I know that they correspond to partitions / nilpotent orbits etc, but from first principles why a... | https://mathoverflow.net/users/2623 | Classifying strata for the adjoint representation of GL from first principles | I'm going to give a partial answer here for two reasons: (1) I am lazy and (2) this is starting to feel a little homeworky to me. Obviously, no one would assign this material as homework, but part of reading a math paper is taking the time to work out lots of simple examples and see how the definitions work. I feel lik... | 3 | https://mathoverflow.net/users/297 | 7636 | 5,229 |
https://mathoverflow.net/questions/7567 | 3 | Let $R\to S$ be a ring map such that $S$ is projective over $R$ (I am willing to assume $S=R[X\_1,...,X\_n]$). Let $M,N$ be finite $S$-modules. Let $P\in Spec R$ such that $M\_P$ is $R\_P$-flat. Under what condition can one say that $Ext^1\_R(M,N)\_P=0$?
This is trivial if $M$ is finite over $R$, but in general $Ext... | https://mathoverflow.net/users/2083 | When can one localize Ext? | I would suggest having a look at the article ["Compactifying the Picard Scheme"](http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6W9F-4CRY60R-1H3&_user=2459438&_rdoc=1&_fmt=&_orig=search&_sort=d&_docanchor=&view=c&_acct=C000057302&_version=1&_urlVersion=0&_userid=2459438&md5=deca64ad5b5501727ed918a10bc1ffd9)
... | 5 | https://mathoverflow.net/users/2215 | 7651 | 5,237 |
https://mathoverflow.net/questions/7650 | 36 | This is a follow up to Harrison's question: [why planar graphs are so exceptional](https://mathoverflow.net/questions/7114/why-are-planar-graphs-so-exceptional). I would like to ask about (and collect answers to) various notions, in graph theory and beyond graph theory (topology; algebra) that generalize the notion of ... | https://mathoverflow.net/users/1532 | Generalizations of Planar Graphs | I guess one possible generalization could be: an $m$-dimensional [stratified space](https://en.wikipedia.org/wiki/Topologically_stratified_space) (i.e. "manifold with singularities") which is embeddable in $2m$-dimensional Euclidean space. Every smooth manifold can be so embedded (by Whitney's theorem), but singulariti... | 17 | https://mathoverflow.net/users/25 | 7652 | 5,238 |
https://mathoverflow.net/questions/7639 | 21 | I'm looking for a reference which has the first statement of the twin prime conjecture. According to wikipedia, nova, and several other quasi-reputable resources it is Euclid who first stated it, but according to Goldston
<http://www.math.sjsu.edu/~goldston/twinprimes.pdf>
it was stated nowhere until de Polignac. I... | https://mathoverflow.net/users/2043 | Twin Prime Conjecture Reference | I don't have it to hand right at this moment, but Narkiewicz' *The Development of Prime Number Theory* is excellent on just this kind of question. It is a historiomathematical survey of prime number theory up to 1910, and also has discussions of later developments directly related to work done before 1910. It is histor... | 9 | https://mathoverflow.net/users/3304 | 7659 | 5,240 |
https://mathoverflow.net/questions/7320 | 35 | We know that presheaves of any category have enough projectives and that sheaves do not, why is this, and how does it effect our thinking?
[This](https://mathoverflow.net/questions/5378/when-are-there-enough-projective-sheaves-on-a-space-x) question was asked(and I found it very helpful) but I was hoping to get a bet... | https://mathoverflow.net/users/348 | Heuristic explanation of why we lose projectives in sheaves. | This is pretty much Dinakar's answer from a different view point: He says that it is too easy for a sheaf morphism to be an epi, so, since there are so many epis, it is now a stronger requirement that for *every* epi we find a lift - so strong that is not satisfied most of the times. I just want to call attention to th... | 26 | https://mathoverflow.net/users/733 | 7661 | 5,241 |
https://mathoverflow.net/questions/7656 | 71 | Defining $$\xi(s) := \pi^{-s/2}\ \Gamma\left(\frac{s}{2}\right)\ \zeta(s)$$ yields $\xi(s) = \xi(1 - s)$ (where $\zeta$ is the Riemann Zeta function).
Is there any conceptual explanation - or intuition, even if it cannot be made into a proof - for this? Why of all functions does one have to put the Gamma-function the... | https://mathoverflow.net/users/733 | Why does the Gamma-function complete the Riemann Zeta function? | To the best of my understanding, the answer is yes, and this uniform way consists of some integration over the local field. This is explained in John Tate's dissertation. One starts with a certain smooth rapidly decreasing function, for which one takes the characteristic function of the p-adic integers in the nonarchim... | 57 | https://mathoverflow.net/users/2106 | 7662 | 5,242 |
https://mathoverflow.net/questions/7645 | 4 | Let $\alpha\in(0,1)$ and $\eta\in\Lambda\_0^\alpha(\mathbb{R})$ be a compactly supported Hölder continuous function of order $\alpha$. I would like to show that, for any $n\in\mathbb{N}$, it is possible to decompose $$\eta=f+g$$
in such a way that $f\in C^n(\mathbb{R})$ and $||f||\_{C^n}=O(R^C)$, and $g\in L^\infty(\m... | https://mathoverflow.net/users/2274 | Decomposition of Hölder continuous functions | I have carried out the suggestion in the last paragraph of Yemon Choi's answer. Choose $\phi\in C^\infty(\mathbb{R})$, $\phi\ge0$ and $\int\_{\mathbb{R}}\phi(x)dx=1$, and let $\phi\_R(x)=R\phi(Rx)$. Define
$$ f=\phi\_R\star\eta,\quad g=\eta-f.$$
Then it is easy to see that
$$ \|f\|\\_{C^n}=O(R^{n-\alpha}),\quad \... | 3 | https://mathoverflow.net/users/1168 | 7671 | 5,247 |
https://mathoverflow.net/questions/7647 | 4 | Given a polyhedron consists of a list of vertices (`v`), a list of edges (`e`), and a list of surfaces connecting those edges (`s`), how to break the polyhedron into a list of tetrahedron?
I have a convex polyhedron.
| https://mathoverflow.net/users/807 | Break polyhedron into tetrahedron | If I understand your question correctly, you're saying that the given information is the face structure of a 3-dimensional convex polytope, and you would like a subdivision of the polytope into tetrahedra.
Here is one way to proceed. First, subdivide all the faces into triangles. Then pick your favourite vertex $v\_... | 8 | https://mathoverflow.net/users/468 | 7672 | 5,248 |
https://mathoverflow.net/questions/7666 | 21 | Lax functors of bicategories were introduced at the very inception of bicategories, and I'm trying to get a better feel for them. They are the same as ordinary 2-functors, but you only require the existence of a coherence morphism, not an isomorphism. The basic example I'm looking at are when you have a lax functor fro... | https://mathoverflow.net/users/184 | Lax Functors and Equivalence of Bicategories? | First of all, for any two bicategories A and B, there is a bicategory $Fun\_{x,y}(A,B)$ where x can denote either strong, lax, or oplax functors, and y can denote either strong, lax, or oplax transformations. There's no problem defining and composing lax and oplax transformations between lax or oplax functors, and the ... | 18 | https://mathoverflow.net/users/49 | 7702 | 5,270 |
https://mathoverflow.net/questions/7689 | 17 | I was in a lecture not long ago given by C. Teleman and at some point he said "Well, since Riemann-Roch is an index problem we can do..."
Then right after that he argued in favour of such a sentence. Could anyone tell me what did he mean exactly?. That is to say, in this case what is elliptic operator like, what is t... | https://mathoverflow.net/users/1547 | Why is Riemann-Roch an Index Problem? | Here is a sketch of the argument as I learned it in a complex analysis class: For a Riemann surface $X$ and a holomorphic line bundle $L$, we want
$$\text{dim}H^0(X,L)-\text{dim}H^0(X,L\otimes\Lambda^{0,1})=c\_1(L)+\frac{1}{2}\chi(X)$$
You have an operator $\overline{\partial}$ (differentiation with respect to $d\over... | 22 | https://mathoverflow.net/users/88 | 7710 | 5,277 |
https://mathoverflow.net/questions/7668 | 6 | If I remembered correctly. There are some work done by C.M.Ringel,he defined so called Ringel-Hall algebra on abelian category and then show that Ringel-hall algebra is isomorphic to positive part of quantized enveloping algebra. Some others generalized to triangulated(derived category of coherent sheaves on projective... | https://mathoverflow.net/users/1851 | What is the geometric meaning of reconstruction of quantum group via Ringel Hall algebra | I tend to think of the geometric content of Hall algebra construction as reflecting the categorification of quantum groups as studied by Lusztig, Rouquier, Khovanov-Lauda, etc. You can read [Lusztig's original paper](http://www.ams.org/mathscinet-getitem?mr=1035415), though I feel like a sadist even suggesting that. I ... | 5 | https://mathoverflow.net/users/66 | 7724 | 5,288 |
https://mathoverflow.net/questions/7733 | 12 | I seem to remember written or said somewhere that at some point Thurston decided to stop writing down his theorems in order not to repel mathematicians from his field (maybe this is not correct?). I am really curious if now 25-30 years later there is some nice source, book, or notes, where it is possible to learn some ... | https://mathoverflow.net/users/943 | The work of Thurston | There are several sources for Thurston's hyperbolization theorem, some published, some not.
Off the top of my head:
1) M.Kapovich, Hyperbolic manifolds and discrete groups.
2) J. Hubbard's Teichmuller theory volume II (not yet published)
3) J. Morgan, H. Bass (eds). The Smith conjecture. (English)
Papers pres... | 12 | https://mathoverflow.net/users/1465 | 7735 | 5,296 |
https://mathoverflow.net/questions/7677 | 2 | Start with a distribution $\mu$ on [n], and drop m balls into these n+1 slots independently and according to the distribution &mu. That is, we have iid random variables x 1 through x m distributed according to &mu. Assume that m is close to the same size as n.
I call a collection of tuples (a(i), b (i)), with all... | https://mathoverflow.net/users/2282 | Parity, Balls and Boxes | if $m=\alpha n$ and $\mu$ is uniform, it seems like a basic sub-additivity argument shows that $\frac{M\_n}{n}$ converges almost surely to a constant $C\_{\alpha}$, where $M\_n$ is the cardinal of a maximal partial match of $[n]$. To see that, put a Poisson process $P$ on the real line with intensity $\alpha$ and say t... | 2 | https://mathoverflow.net/users/1590 | 7745 | 5,305 |
https://mathoverflow.net/questions/7746 | 2 | Please, any information on the periodic mapping classes of the genus two orientable surface, $O\_2$, will be greatly thanked. We had been studying the topological structure of 3d surface bundles and reintrepreting them as a circle bundles over orbifolds.
In the <http://web.archive.org/web/20070316045651/http://www.s... | https://mathoverflow.net/users/2196 | Periodic mapping classes of the genus two orientable surface | If you want to enumerate the finite-order automorphisms (up to conjugacy) I suggest the following exercise. The associated 3-manifold is Seifert fibred. So determine how the genus 2 surface is sitting in the Seifert manifold (horizontal incompressible surface).
This will give you a formula relating the various branc... | 4 | https://mathoverflow.net/users/1465 | 7747 | 5,306 |
https://mathoverflow.net/questions/7750 | 12 | Where can I find the most direct and simplest presentation of what geodesics on a (complex) Grassmannian look like? I know how to do it from scratch, but, if I want to provide a reference to, say, a graduate student in EE who doesn't want to deal with any unnecessary abstract mathematical machinery, what should I point... | https://mathoverflow.net/users/613 | Geodesics on a Grassmannian | Grassmanians are symmetric spaces, and symmetric spaces are "geodesic orbit spaces", that is, their geodesics are orbits of their group of isometries. Your Grassmanians, in particular, are of the form $SU(p+q)/SU(p)\times SU(q)$. If $g$ is the Lie algebra of the big group and $h\subseteq g$ the Lie algebra of the subgr... | 12 | https://mathoverflow.net/users/1409 | 7757 | 5,314 |
https://mathoverflow.net/questions/7751 | 2 | Let $V$ be a vector space over $\mathbb R$, and $a: V\otimes V\to \mathbb R$ a symmetric bilinear pairing. Recall that the *Morse index* of $a$ is the maximal dimension of any subspace $V\_- \subseteq V$ on which $a$ is negative-definite.
If $V$ is infinite-dimensional, it can be very hard to check every negative-def... | https://mathoverflow.net/users/78 | How can I measure the Morse index in infinite dimensions? | Let $V\_-$ be this subspace of $V$ defined in the weak sense that it is negative-definite and not extendible. There is a linear map $b:V \mapsto V\_-^\*$ given by the formula $b(v) = a(v,\cdot)$, where the right side is interpreted as a dual vector on $V\_-$. Since $V\_-$ is finite-dimensional by hypothesis, $\dim V\_-... | 3 | https://mathoverflow.net/users/1450 | 7761 | 5,317 |
https://mathoverflow.net/questions/7715 | 16 | I am starting on a Phd program and am supposed to read Colliot Thelene and Sansuc's article
on R-equivalence for tori. I find it very difficult and although I have some knowledge over schemes , I am completely baffled by this scalar restriction business of having a field extension $K/k$ , a torus over $K$ and "restrict... | https://mathoverflow.net/users/2292 | What is "restriction of scalars" for a torus? | As said, the sought after concept is also known as Weil restriction. In a word, it is the algebraic analogue of the process of viewing an $n$-dimensional complex variety as a $(2n)$-dimensional real variety.
The setup is as follows: let $L/K$ be a finite degree field extension and let $X$ be a scheme over $L$. Then t... | 28 | https://mathoverflow.net/users/1149 | 7765 | 5,320 |
https://mathoverflow.net/questions/7709 | 2 | Let $V$ be a vector space over $\mathbb R$. A *symmetric bilinear pairing* on $V$ is a linear map $a: V\otimes V \to \mathbb R$. Because $\mathbb R$ is characteristic not-two, I will freely confuse symmetric bilinear pairings with quadratic forms; if $v\in V$, I will write $av^2$ for $a(v\otimes v)$; and $av\_1v\_2$ fo... | https://mathoverflow.net/users/78 | Splitting a space into positive and negative parts | Here's another counter-example, taken from *Loop Groups* (p 128).
Consider the space of continuous functions on the circle and define
$$
f(\theta) = \sum\_{k \gt 1} \frac{\sin k \theta}{k \log k}
$$
The positive part of this function is
$$
f\_+(\theta) = \frac{1}{2 i} \sum\_{k \gt 1} \frac{e^{i k \theta}}{k \lo... | 1 | https://mathoverflow.net/users/45 | 7768 | 5,323 |
https://mathoverflow.net/questions/7687 | 40 | **Background**
For definiteness (even though this is a categorical question!) let's agree that a *vector space* is a finite-dimensional real vector space and that an *associative algebra* is a finite-dimensional real unital associative algebra.
Let $V$ be a vector space with a nondegenerate symmetric bilinear form ... | https://mathoverflow.net/users/394 | Clifford algebra as an adjunction? | Disqualifier: this isn't a complete answer.
There's a basic "chalk and cheese" problem here. The "categories" that you are comparing are of two different types, although they do seem similar on the surface. On the one hand you have an honest algebraic category: that of associative algebras. But the other category (wh... | 14 | https://mathoverflow.net/users/45 | 7769 | 5,324 |
https://mathoverflow.net/questions/7508 | 22 | Ok, it's time for me to ask my first question on MO.
Consider the affine curve $Y+Y^q=X^{q+1}$ over the finite field $\mathbf{F}\_q$. It's interesting because it has the largest number of points over $\mathbf{F}\_{q^2}$ possible relative to its genus, which is $q(q-1)/2$. In other words, this curve realizes the Weil ... | https://mathoverflow.net/users/271 | A hypersurface with many points | Here is a complete solution to the main question when $n$ and $q$ are both odd, and a partial solution for the other parities. The partial solution includes a reduction to the case $n=2$.
Let $\text{Tr}\_k$ denote the trace map from $\mathbb{F}\_{q^n}$ to $\mathbb{F}\_{q^k}$, assuming that $k|n$.
The equation is
$... | 8 | https://mathoverflow.net/users/1450 | 7781 | 5,331 |
https://mathoverflow.net/questions/7776 | 8 | **Question** Is there a nice universal property which captures the notion of "collection of all epimorphisms out of a given object". Of course I will have to consider two epimorphisms $X \rightarrow Y$ the same if they are isomorphic over $X$. The answer to the dual question is yes, at least in a topos: The power objec... | https://mathoverflow.net/users/1106 | Universal property for collection of epimorphisms | I think the natural meaning of "collection of all epimorphisms out of $X$" or "epimorphism classifier" in a category $\mathbf{S}$ would be: an object $E$, an object $Y\to E$ of $\mathbf{S}/E$, and an epimorphism $p\colon E\times X \twoheadrightarrow Y$ in $\mathbf{S}/E$, such that for any object $U$ and any epimorphism... | 7 | https://mathoverflow.net/users/49 | 7789 | 5,335 |
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