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Questions tagged [reference-request]

This tag is used if a reference is needed in a paper or textbook on a specific result.

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Are there universe-indexed spectra over simplicial sets?

In "Rings, Modules, and Algebras in Stable Homotopy Theory" Elmendorf, Kriz, Mandell and May introduce a notation of spectra indexed over an universe $\mathcal{U}$ as a collection of pointed ...
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4answers
4k views

References for logarithmic geometry

Hi everyone, I'm looking for a systematical introduction to (or treatment of) logarithmic structures on schemes. I am reading Kato's article ("Logarithmic structures of Fontaine-Illusie") at the ...
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1answer
443 views

cardinal equivalence: for each boolean formula, |quantifications| = |assignments|. [closed]

Cardinal Equivalence Theorem For each boolean formula, |quantifications| = |assignments|. The set of valid quantifications has some cardinality, call that |Q(B)...
4
votes
1answer
312 views

Non-commutative versions of X/G

Let $X$ be a Riemannian manifold and let $G$ be a (at most countable, if that matters) discrete group acting properly and by isometries on $X$. Let $\mathcal{O}$ be the sheaf of analytic functions on ...
15
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4answers
1k views

Introductory text for the non-arithmetic moduli of elliptic curves

I'm looking for an introduction to the non-arithmetic aspects of the moduli of elliptic curves. I'd particularly like one that discusses the $H^1$ local system on the moduli space (whether it's $Y(1)$ ...
13
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0answers
741 views

Compact Symplectic Fano (strongly monotone) manfiolds

What are known examples of compact symplectic Fano manifolds, apart from those that come from algebraic geometry? We define symplectic Fano manifold as a symplectic manifold $(M,w)$, such that $[c_1(...
2
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2answers
2k views

Statement of Lagrange's theorem on determinants(elementary question).

Apologies for this elementary question; but I was unable to find a reference otherwise. Let $A, B, C$ be square matrices of the same dimension. Then, $$\begin{vmatrix} A & C \\\ 0 & B \end{...
11
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1answer
590 views

An arithmetic highest weight theory?

I apologize if these questions seem naive or loaded. Is there an analogous theory of highest weights for irreducible finite-dimensional representations of Lie algebras of algebraic group (or perhaps ...
3
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6answers
2k views

Source needed (at final-year undergrad level) for the double cover of SO(3) by SU(2)

This is a bit of an ill-defined question, and I feel I should have been able to resolve it by combining Google with a few library trips, but I'm having difficulty narrowing down the search results to ...
21
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11answers
7k views

Introduction to deformation theory (of algebras)?

So I know that the idea of deformation theory underlies the concept of quantum groups; I haven't found any single introduction to quantum groups that makes me fully satisfied that I have some kind of ...
13
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2answers
2k views

References for Artin motives

I find the following description of Artin motives in Wikipedia. Since these seem to be quite related to number theory, I am interested to learn more in that context. I request the experts available in ...
4
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2answers
417 views

Burnside ring and zeroth G-equivariant stem for finite G

Let $G$ be a finite group. The theorem that the Burnside ring $A(G)$ is isomorphic to the zeroth stable stem $\pi^{G}_0(S)$ is usually said to originate from Segal. I search for a reference of a proof ...
6
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1answer
875 views

What is the standard reference on “infinitesimal space” in algebraic geometry??

infinitesimal 'spaces' is a serious issue in noncommutative (and commutative) geometry: they serve as a base of a Grothendieck-Berthelot crystalline theory and are of big importance for the D-module ...
37
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18answers
59k views

Suggestions for a good Measure Theory book

I have taken analysis and have looked at different measures, but I am currently looking at realizing a certain problem in a different light and feel that I need a better background in various measures ...
10
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3answers
2k views

Looking for reference on gauge fields as connections.

Can anyone give me references where I would see a detailed exposition of how to translate gauge field theory as known to physicists into the language of connections. I am looking for a detailed ...
15
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6answers
3k views

CW-structures and Morse functions: a reference request

The following is probably well known, but I wasn't able to locate a reference in the literature. Let $f$ be a Morse function on a smooth compact manifold $M$ without boundary and let $\rho$ be a ...
48
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2answers
7k views

What is a good roadmap for learning Shimura curves?

I am interested in learning about Shimura curves. Unlike most of the people who post reference requests however (see this question for example), my problem is not sorting through an abundance of books ...
11
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5answers
2k views

Classical Enumerative Geometry References

I want to start out by making this clear: I'm NOT looking for the modern proofs and rigorous statements of things. What I am looking for are references for classical enumerative geometry, back before ...
4
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2answers
3k views

A telegram by Grothendieck to Serre

In an opinion piece which appeared in the AMS Notices of January 2010, John Wermer tells us that he once heard about a seminar given by Grothendieck which was described as "a telegram by Grothendieck ...
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5answers
4k views

Partial sums of multiplicative functions

It is well known that some statements about partial sums of multiplicative functions are extremely hard. For example, the Riemann hypothesis is equivalent to the assertion that $|\mu(1)+\mu(2)+\dots+\...
17
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6answers
3k views

English reference for a result of Kronecker?

Kronecker's paper Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten apparently proves the following result that I'd like to reference: Let $f$ be a monic polynomial with integer ...
10
votes
2answers
549 views

Reference request: The stable Kronecker ring is isomorphic to the ring of symmetric polynomials

Background For $\lambda$ any partition and $n$ a positive integer, write $\lambda[n]$ for the sequence $(n - |\lambda|, \lambda_1, \lambda_2, \ldots, \lambda_r)$. For $n$ large enough, this is a ...
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3answers
796 views

References for theorem about unipotent algebraic groups in char=0?

There is a textbook theorem that the categories of unipotent algebraic groups and nilpotent finite-dimensional Lie algebras are equivalent in characteristic zero. Indeed, the exponential map is an ...
10
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9answers
2k views

Comprehensive reference for synthetic euclidean geometry

Euclidean geometry is a special case of the theory of Hilbert spaces; but in order to convince small children of basic facts, e.g. that the line segments from each of the vertices of a triangle to the ...
152
votes
48answers
53k views

Ways to prove the fundamental theorem of algebra

This seems to be a favorite question everywhere, including Princeton quals. How many ways are there? Please give a new way in each answer, and if possible give reference. I start by giving two: ...
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10answers
8k views

Uniformization theorem for Riemann surfaces

How does one prove that every simply connected Riemann surface is conformally equivalent to the open unit disk, the complex plane, or the Riemann sphere, and these are not conformally equivalent to ...
18
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9answers
10k views

Teichmuller Theory introduction

What is a good introduction to Teichmuller theory, mapping class groups etc., and relation to moduli space of curves or Riemann surfaces?
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6answers
2k views

“Every scheme as a sheaf” references?

I have sometimes hard time reading papers that are written in the language of schemes being replaced by the functors they represent (I have especially homotopy scheme theory in mind). I think the ...
15
votes
5answers
2k views

Reference for this theorem in representation theory?

Let $G$ be a finite group and $\chi$ be an irreducible character of $G$ (characteristic zero algebraically closed base field). If $H$ is the kernel of $\chi$ then the irreducible representations of $G/...
15
votes
4answers
1k views

Equivalence of ordered and unordered cech cohomology.

Given a topological space X and a finite cover X = $\cup X_i$, one can define Cech cohomology of a sheaf of abelian groups F with respect to the cover $\{X_i\}$ in two different ways: (Ordered): ...
13
votes
4answers
5k views

Who invented the gamma function?

Who was the first person who solved the problem of extending the factorial to non-integer arguments? Detlef Gronau writes [1]: "As a matter of fact, it was Daniel Bernoulli who gave in 1729 the ...
30
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3answers
4k views

Matrix factorizations and physics

I have heard during some seminar talks that there are applications of the theory of matrix factorizations in string theory. A quick search shows mostly papers written by physicists. Are there any ...
16
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4answers
2k views

Characteristic classes in generalized cohomology theories?

Hello, 'ordinary' Stiefel-Whitney classes are elements of the singular cohomology ring and are constructed using the Thom isomorphism and Steenrod squares. So I think they should exist for any (...
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4answers
3k views

How far is Lindelöf from compactness?

A while ago I heard of a nice characterization of compactness but I have never seen a written source of it, so I'm starting to doubt it. I'm looking for a reference, or counterexample, for the ...
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3answers
3k views

References for Donaldson-Thomas theory and Pandharipande-Thomas theory?

I'm looking for good introductory references for Donaldson-Thomas theory and Pandharipande-Thomas theory. I know a bit about Gromov-Witten theory, but almost nothing about Donaldson-Thomas and ...
22
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6answers
4k views

What are some good resources for mathematical translation?

I am currently in the process of translating a lecture on the étale topology by John Hubbard from French into English (and from transparencies into Beamer). For the most part, the translation is ...
5
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1answer
389 views

Request for reference: Banach-type spaces as algebraic theories.

Sparked by Yemon Choi's answer to Is the category of Banach spaces with contractions an algebraic theory? I've just spent a merry time reading and doing a bit of reference chasing. Imagine my delight ...
5
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2answers
267 views

Truncated exact sequence of homotopy groups

This is a question about a name of a very useful lemma, that permits one in particular to show that smooth birational complex projective varieties have isomorphic fundamental groups. If this lemma ...
319
votes
78answers
144k views

Proofs without words

Can you give examples of proofs without words? In particular, can you give examples of proofs without words for non-trivial results? (One could ask if this is of interest to mathematicians, and I ...
8
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1answer
986 views

Elementary questions in arithmetic geometry

In many theories there is a rough divide between elementary problems that can be solved with "one's hands", and "deep results that require powerful tools". For example, I am told that Hodge theory is ...
3
votes
5answers
1k views

Martingales and Betting Strategies

Does anyone know of a good introduction to the theory of martingales and betting strategies from the point of view of statistics and/or probability theory? I'm looking for something basic, with lots ...
18
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15answers
14k views

Learning Topology

EDIT (Harry): Since this question in its original form was poorly stated (asked about topology rather than graph theory), but we have a list of Topology books in the answers, I guess you should go ...
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6answers
2k views

Reference for Learning Global Class Field Theory Using the Original Analytic Proofs?

Hi Everyone! I'm wondering if anyone knows of a reference for learning global class field theory using the original analytic proofs developed in the 1920s and 1930s. Almost every book I can find ...
12
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0answers
529 views

References for a certain generalization of Hochschild cohomology?

Let $C$ be an algebra. Let $E = C^{\otimes 2n}$ be the tensor product (over the ground field) of $2n$ copies of $C$. [EDIT: Or better, $E = C\otimes C^{op}\otimes C\otimes C^{op}\cdots\otimes C \...
3
votes
4answers
956 views

Examples of divisors on an analytical manifold

I am trying to understand divisors reading through Griffith and Harris but it is difficult to come up with any particular interesting example. I have browsed through Hartshone's book but everything is ...
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3answers
3k views

Twin Prime Conjecture Reference

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 ...
18
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0answers
1k views

Two conjectures by Gabber on Brauer and Picard groups

In a paper I need to make reference to 2 conjectures by Gabber (see Conjectures 2 and 3, page 1975) http://www.mfo.de/programme/schedule/2004/32/OWR_2004_37.pdf 1) Let $R$ be a strictly henselian ...
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0answers
1k views

Complexes of representations with complementary central charges

This is another question asking for references. There is an important phenomenon of correspondence between (complexes of) representations of infinite-dimensional Lie algebras with the complementary ...
9
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5answers
801 views

References/literature for pushouts in category of commutative monoids? [ed. - amalgams]

This is more of a request for pointers to relevant literature than a question per se. I am, erm, looking at a paper which uses a kind of iterated pushout construction to obtain a commutative monoid ...
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6answers
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What books should I read before beginning Masaki Kashiwara's “Sheaves on Manifolds”

I am a beginner trying to learn about sheaves. I am trying to read Masaki Kashiwara's book "Sheaves on Manifolds", but I find it is not easy for me to understand. What other books should I read first,...