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1500 questions
63
votes
12 answers
Advanced Differential Geometry Textbook
I tried this post on StackExchange with no luck. Hopefully the experts at MathOverflow can help.
In algebraic topology there are two canonical "advanced" textbooks that go quite far beyond the usual graduate courses.
They are Switzer Algebraic…
63
votes
4 answers
Proof that pi is transcendental that doesn't use the infinitude of primes
I just taught the classical impossible constructions for the first time, and in finding my class a reference for the transcendence of pi, I found a dearth of distinct proofs. In particular, those that I read all require the existence of infinitely…
Barry
- 1,501
63
votes
7 answers
How to prove this determinant is positive?
Given matrices
$$A_i= \biggl(\begin{matrix}
0 & B_i \\
B_i^T & 0
\end{matrix} \biggr)$$
where $B_i$ are real matrices and $i=1,2,\ldots,N$, how to prove the following?
$$\det \big( I + e^{A_1}e^{A_2}\ldots e^{A_N} \big) \ge 0$$
This seems to be…
Lei Wang
- 845
63
votes
11 answers
Theorem versus Proposition
As a non-native English speaker (and writer) I always had the problem of understanding the distinction between a 'Theorem' and a 'Proposition'. When writing papers, I tend to name only the main result(s) 'Theorem', any auxiliary result leading to…
MRA
- 217
63
votes
4 answers
Is there a good way to think of vanishing cycles and nearby cycles?
Once in a while I run into literature that invokes vanishing cycle machinery with a cryptic sentence like, "this follows from a standard vanishing cycle argument." Is there a good way to look at vanishing cycles, nearby cycles, and specialization? …
S. Carnahan
- 45,116
63
votes
12 answers
How much of differential geometry can be developed entirely without atlases?
We may define a topological manifold to be a second-countable Hausdorff space such that every point has an open neighborhood homeomorphic to an open subset of $\mathbb{R}^n$. We can further define a smooth manifold to be a topological manifold…
Harry Gindi
- 19,374
63
votes
3 answers
Are there infinitely many integer-valued polynomials dominated by $1.9^n$ on all of $\mathbb{N}$?
The original post is below. Question 1 was solved in the negative by David Speyer, and the title has now been changed to reflect Question 2, which turned out to be the more difficult one. A bounty of 100 is offered for a complete solution.
Original…
Vesselin Dimitrov
- 13,703
63
votes
4 answers
What do theta functions have to do with quadratic reciprocity?
The theta function is the analytic function $\theta:U\to\mathbb{C}$ defined on the (open) right half-plane $U\subset\mathbb{C}$ by $\theta(\tau)=\sum_{n\in\mathbb{Z}}e^{-\pi n^2 \tau}$. It has the following important transformation property.
Theta…
Aleksandar Bahat
- 1,033
63
votes
11 answers
Has mathoverflow yet led to mathematical breakthroughs?
Some people ask questions here out of simple curiosity. But some ask them because they are working on a research project, come up with a question they need to know the answer to, and think that the answer is probably known. In the past, one had to…
gowers
- 28,729
63
votes
19 answers
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 that have been used / discovered / et cetera in…
Andrew
- 97
62
votes
11 answers
Elementary results with p-adic numbers
I'm giving a talk for the seminar of the PhD students of my math departement. I actually work on Berkovich spaces and arithmetic geometry but, of course, I cannot really talk about that to an audience that includes probabilists, computer scientists…
Daniele Turchetti
- 1,022
- 1
- 11
- 10
62
votes
6 answers
Simplest examples of nonisomorphic complex algebraic varieties with isomorphic analytifications
If they are not proper, two complex algebraic varieties can be nonisomorphic yet have isomorphic analytifications. I've heard informal examples (often involving moduli spaces), but am not sure of the references.
What are the simplest examples of…
Ravi Vakil
- 3,837
62
votes
5 answers
Does homology have a coproduct?
Standard algebraic topology defines the cup product which defines a ring structure on the cohomology of a topological space. This ring structure arises because cohomology is a contravariant functor and the pullback of the diagonal map induces the…
JoeG
- 621
62
votes
4 answers
who fixed the topology on ideles?
I am teaching a course leading up to Tate's thesis and I told the students last week, when defining ideles, that the first topology that was put on the ideles was not so good (e.g., it was not Hausdorff; it's basically the profinite topology on the…
KConrad
- 49,546
62
votes
2 answers
Thomason's "open letter" to the mathematical community
In 1989, Bob Thomason left his CNRS position in Orsay and moved to Paris VII. It was during this period that he composed his "Open Letter" to the mathematical community. The letter explained Thomason's reasons for leaving the United States.
I lost…
John Klein
- 18,574