This tag is for questions where the poster seeks references (books, articles, etc.) about a particular subject. Please do not use this as the only tag for a question.

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322 views

Eigenprojection as Contour Integral over Resolvent

Let $H$ be a Hilbert space and let $A \in L(H)$ be a bounded linear operator. Assume that $\lambda$ is an eigenvalue of $A$ and assume further that $C_\lambda$ is a simple closed curve in the complex ...
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193 views

Characteristic Class exercises

I tagged this as "homework" because my supervisor told me I need to be better at computing characteristic classes. The classic examples I can think of are tangent bundles and tautological line ...
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0answers
108 views

Direct limits of completely positive maps on $C^*$-algebras vs. operator systems

I believe I've heard, as part of the "lore," that the category (operator systems, completely positive maps) has direct limits, whereas the category ($C^*$-algebras, completely positive maps) does not. ...
4
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0answers
193 views

Riemannian Connection (Very basic question)

We know that a connection $\nabla$ in a manifold M hashas the purpose of performing the same role as the covariant derivative of vector fields of surfaces in $\mathbb{R}^3$. Such analogies are ...
4
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69 views

Any local algebraic group is birationally equivalent to an algebraic group

In this paper, page $6$ the authors state the following: By Weil’s theorem $[17]$, any local algebraic group is birationally equivalent to an algebraic group. Where $[17]$ A.Weil. On ...
4
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179 views

Proving NP-completeness (hardness) exercises

I am looking for a list of exercises that can be done to practice polynomial time reductions to prove NP-hardness of problems. I know there are hundreds (thousands?) of problems proven to be NP-hard. ...
4
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0answers
129 views

Non-closed subgroups of Lie groups

The following are my impressions from playing with a line of irrational slope $\gamma$ in the standard torus $S^1\times S^1=\mathbb{R}^2/\mathbb{Z}^2$, $\mathbb{R}\hookrightarrow ...
4
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0answers
84 views

Ways to think about one-relator groups

What are some intuitive ways to think about one-relator groups? I am aware of the Freiheitsatz, and Bass-Serre theory. What I'm interested in are ways people who work extensively with one-relator ...
4
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0answers
200 views

Clarification: intersection of a finite number of subgroups of finite index and Poincaré

From Scott's book Group Theory $1.7.10.$ (Poincaré) The intersection of a finite number of subgroups of finite index is of finite index. My question is: Did Poincaré prove the Theorem as stated ...
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359 views

Interchanging the order of limits

Would you advise me on the references of Pringsheim Convergence about interchanging the order of limits? Where can I find the most general statement?
4
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236 views

Reference for Martingale version of Riesz representation theorem / Riemann–Stieltjes integral

(Please let me know if this is more appropriate as a MathOverflow question.) I can work out most of the following martingale generalization to the Riesz representation theorem and the ...
4
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0answers
201 views

How does a Riemannian metric “naturally induce” distances in the Grassmannian bundle

When reading about Pesin theory I've run into the necessity of defining a metric on the Grassmannian bundle of a compact Riemannian manifold $M$. More specifically a fiber at $x \in M$ in the ...
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191 views

Reference: Geometric group theory

H. Bass has studied existence of lattices on trees. Can someone suggest a (readable) reference for lattices on graphs?
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66 views

References about the symmetric products of a stack

I would like to know references about a construction of the symmetric product (or the moduli space of effective divisors) $X^{(d)}$ of a stack $X$. I am currently thinking about the following case: ...
3
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0answers
36 views

Pre-requisites and references for $K3$ surfaces

I would like to know the "roadmap" to study $K3$ surfaces. Perhaps, my background might be helpful: I am an undergraduate student, who knows the basics of Differential Geometry, Topology, Complex ...
3
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0answers
30 views

Sharkovskii's theorem in two dimensions?

Question A weak form of Sharkovskii's theorem in $1$D dynamical systems states that, if a continuous function $f:I\to I$ does not include a periodic point of least period $2$ on $I$, then ...
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46 views

Pushing forward vector bundles on a plane curve via projection from a point

Let $C \subset \mathbb{P}^2$ be a smooth plane curve, $P \in \mathbb{P}^2$ is point not on $C$, consider projection from this point $$ \pi :\mathbb{P}^2 - \{P\} \to \mathbb{P}^1, $$ and restrict this ...
3
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0answers
69 views

Real Analysis texts: Royden versus Stein & Shakarchi. Which is better? (and other suggestions welcome)

I am taking an introductory "graduate" analysis class and am comparing Analysis books that cover measure theory. I have had an "advanced calculus" class that covered the standard topics. I am having ...
3
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0answers
42 views

What resources (books, videos, etc) to help develop math thinking skills?

I am a applied mathematics major and have taken basic undergraduate coursework up to multi-variable calculus, ordinary differential equations, and elementary linear algebra. I am currently taking ...
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49 views

Number of lines needed to pass through every region of a map

The webpage http://what-if.xkcd.com/113 explores the fewest number of lines needed so that every state in the US has at least one line going through it. (actuallly great circles on a sphere) Can you ...
3
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0answers
32 views

Irrational roots of unity?

Is it possible to take irrational roots of unity? For example, say I wanted to solve $f(x)=(x+1)^{\sqrt{2}}=1$. I found that one solution is the obvious $x=0$, and another one can be written nicely as ...
3
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71 views

I have one month preparation. Please suggest some books.

I have one month for my GRE subjective Mathematics test. I am from India. I have learnt $75\%$ of the syllabus in my UG and high school mentioned in the ETS. I am starting today, will I be able to ...
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33 views

Cambridge Maths Tripos Papers

Does anyone know where I can find Cambridge Maths Tripos Papers for the 1980s?
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0answers
35 views

Books with “project”-like questions

I'm looking for a big list of resources for advanced undergraduate - beginning graduate (and even beyond, really) with a particular feature. Namely, I really like solving "project"-like problems that ...
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0answers
89 views

Why is there no collection about all mathematical theorems and open questions?

I really would like to have a giant math collection which is sorted according to the Mathematics Subject Classification, but with more then 3 levels, and it should contain all known theorems and also ...
3
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0answers
56 views

How to approach real analaysis

I'm just starting first year in university in Europe and here there there is no Calculus, instead you jump right into Analysis. The trouble is, for some time I self-studied through US style books and ...
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0answers
53 views

Relations between the Eisenstein series and the hypergeometric series

It is known that $$E_4(\tau) = {}_{2}F_{1}\left(\frac{1}{12}, \frac{5}{12}; 1; \frac{1}{J(\tau)}\right)^4$$ and $$E_6(\tau) = {}_{2}F_{1}\left(\frac{1}{12}, \frac{7}{12}; 1; \frac{1}{1 - ...
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38 views

Generalization of small set/large set

A small set is a subset of the positive integers, such that the infinite sum of the reciprocals of the members of the set converges. Conversely, the sum of the reciprocals of a large set diverges. ...
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39 views

References on the relations between Top, Diff and PL

I have heard many times informal statements like "differentiable and pl manifolds are essentially the same for such and such dimensions", but I would like to know what they mean exactly and how such ...
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35 views

Where can I find some articles of Weil.

Where can I find the articles of Weil: Variétés abéliennes et courbes algébriques Sur les courbes algébriques et les variétés qui s'en déduisent. on Internet?
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87 views
+50

Have any authors suggested mathematics-wide prefixes for “missing a quotient” and/or “missing an identity”?

The prefixes in the following terms both mean: "missing the obvious quotient by the obvious equivalence relation." seminorm pseudometric Similarly, the prefixes in the following terms both mean: ...
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0answers
137 views

Any comments on Lax's “Calculus with Applications, 2e”

There's a new calculus book titled Calculus with Applications by Peter Lax (2nd edition of an old one). I really liked his linear algebra and functional analysis books, and I was wondering if this ...
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33 views

Arithmetic properties of the partition function

Ramanujan mentioned in his paper in 1920 that "it appears that there are no equally simple properties for any moduli involving primes other than these three" $p(5n+4)\equiv0 \mod 5$ $p(7n+5)\equiv0 ...
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81 views

Complex structures on Riemann surfaces

Let $M$ be a Riemann surface and $[\alpha] \in H^{0,1}(M; T^{1,0} M) \simeq H^0(M;K^2)$. Considering $\alpha$ as a map $T^{0,1} M \to T^{1,0} M$, the bundle $$ \{v + \alpha(v) \mid v \in T^{0,1} M\} ...
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0answers
67 views

Expository Papers on Extenders

Extenders are discussed in many set theory text books. Here I am looking for some expository "papers" which are focused on this subject and its connection with forcing and large cardinals. More ...
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0answers
38 views

Parameter-dependent integral: Is the following statement true?

Is the following statement true? If so, could anyone provide a reference? Suppose $f(x, \alpha)$ is continuous on $(a, b) \times \{\alpha_0\}$. If there exists $g(x)$ which is continuous on $(a, b)$, ...
3
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0answers
55 views

Conjectured Value of Ramsey Number $R(3,10)$

It is known that the value of the Ramsey number $R(3,10)$ is either 40, 41, or 42. Have any experts in the field offered a conjecture as to which it might be?
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65 views

Resources about infinite primes of form $n^2 + 1$

Where can one find existing work on the following problem? Prove there are infinitely many primes of the form $n^2 + 1$. Resources about related work would also be appreciated.
3
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0answers
78 views

Regularity of a Weak Solution to Fokker-Planck Equation

Suppose that $\rho \in L^1(\mathbb{R}^n \times (0,T))$ for every $T < \infty$ is a weak solution of the PDE \begin{align} \partial_t\rho &= \Delta \rho + \text{div}(\rho\nabla\Psi(x))\\ \rho(t ...
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0answers
87 views

Is Courant's Introduction to Calculus and Analysis still up-to-date?

I just found this marvelous book and I think that it's the best book in this category, but I'm worried that it is not up-to-date. I've heard that Hardy's A Course of Pure Mathematics has some switched ...
3
votes
0answers
99 views

Developing intuition in algebraic geometry through differential geometry?

I'm interested in algebraic geometry (I am working through Ravi Vakil's notes and also have worked with curves and general varieties in the past), and have seen some basic definitions from ...
3
votes
0answers
58 views

$h^{p,q}$ of projective space

How can we calculate the Hodge number $h^{p,q}= \dim H^p(\mathbb{P^n},\Omega^q_{\mathbb{P}^n})$ of projective space? Is there a reference for that?
3
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0answers
59 views

Equations in the semiring of f.g. modules

Let $R$ be a commutative ring. Then we may consider the semiring $G(R)$ of isomorphism classes of finitely generated $R$-modules with $+ = $ direct sum, $* = $ tensor product, $0 = $ zero module, $1 = ...
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0answers
46 views

Reference request: certain special LFTs

$\newcommand{\cs}{\operatorname{cs}}$ My actual question is at the bottom. Let $$ a\diamond b = \frac{a+b}{1+ab} $$ and $$ a\dagger=\dfrac{1-a}{1+a}. $$ Then $$ a\dagger\dagger = a $$ and ...
3
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0answers
74 views

Am I missing out by not knowing another language?

A bunch of famous mathematicians, e.g. Kolmogorov, `Bourbaki,' Laplace, Lebesgue etc. wrote in foreign languages and I have seen peripherally that lots of new results are published in French. ...
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62 views

How to practice applied mathematics calculation skill

As a natural science student in university, you may encounter so many problems that might require a deep understanding in integrating skills and series calculation. But as many of the college students ...
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0answers
51 views

Actual Definition of the term: Hopf Band?

Sorry if this is too trivial: I need an actual working definition of the term: Hopf band. I see references to it in many searches, but never an actual precise definition. All I know so far is that ...
3
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0answers
114 views

Differentiation of norm in Banach space (explanation of text needed)

Let $Y$ be uniformly smooth Banach space. Consider the convex $C^1$ functional $\Phi:Y \to \mathbb{R}$ defined $$\Phi(y) = \frac{1}{q}\Vert y \Vert^q_{Y}.$$ Its derivative $\varphi:Y \to Y'$ is a ...
3
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0answers
98 views

Books for preparing for B.Sc(HONS.) MATHEMATICS. Thanks

Don't close this question i am in real need of some reference. I am preparing for B.Sc(HONS.) entrance and i want reference advice. Please help.
3
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0answers
69 views

Saddle point method: a rigorous proof?

I am trying to prove in a fully rigorous way the Saddle Point method for holomorphic functions of 1 complex variable. In books I find only complicated general statements or non-rigorous proofs. Hence ...