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1500 questions
45
votes
2 answers
Definition of "finite group of Lie type"?
The list of finite simple groups of Lie type has been understood for half a century, modulo some differences in notation (and identifications between some of the very small groups coming from different Lie types). Call this collection of…
Jim Humphreys
- 52,369
45
votes
8 answers
An "advanced beginner's" book on algebraic topology?
It has so happened that I have come this far knowing nothing on the subject of algebraic topology (as in homology theories of topological spaces and their applications). I've decided to finally read up on that during the summer.
Seemingly, however,…
Igor Makhlin
- 3,493
45
votes
17 answers
Good algebraic number theory books
I have just finished a master's degree in mathematics and want to learn everything possible about algebraic number fields and especially applications to the generalized Pell equation (my thesis topic), $x^2-Dy^2=k$, where $D$ is square free and $k…
Jason Smith
- 171
45
votes
4 answers
The origin of sets?
The history of set theory from Cantor to modern times is well documented. However, the origin of the idea of sets is not so clear. A few years ago, I taught a set theory course and I did some digging to find the earliest definition of sets. My notes…
François G. Dorais
- 43,723
45
votes
5 answers
Liouville's theorem with your bare hands
Liouville's theorem from complex analysis states that a holomorphic function $f(z)$ on the plane that is bounded in magnitude is constant. The usual proof uses the Cauchy integral formula. But this has always struck me as indirect and…
Jonah Sinick
- 6,942
45
votes
2 answers
Combinatorial interpretation of ${i\choose n}$, where $i^2=-1$
At MIT all departments have numbers, and math is 18. Last year MIT
math majors produced a tee shirt that said ${i\choose 18}$ ("I choose
18") on the front, and on the back
$$ \frac{34376687+1499084559i}{14485008384}. $$
With the more natural…
Richard Stanley
- 49,238
44
votes
9 answers
Journals and other sources with "easy reading" papers?
Some time ago the journal "Algebra and Analysis" (English translation is published in
"St. Petersburg Mathematical Journal") had a special section which was called "easy readings for professional mathematicians", which tried to present in…
Alexander Chervov
- 23,944
44
votes
11 answers
Algorithm for finding the volume of a convex polytope
It's easy to find the area of a convex polygon by division into triangles, but what is the optimal way of finding the volume of higher-dimensional convex bodies? I tried a few methods for dividing them into simplices, but gave it up and went with a…
Xerxes
- 441
44
votes
11 answers
Resources for learning practical category theory
I've been doing functional programming, primarily in OCaml, for a couple years now, and have recently ventured into the land of monads. I'm able to work them now, and understand how to use them, but I'm interested in understanding more about their…
Michael Ekstrand
- 519
44
votes
4 answers
What motivates modern algebraic geometry for a combinatorial/constructive algebraist?
This is, basically, me trying to generalize "Why should I care for sheaves and schemes?" into a reasonable question. Whether successfully, time will tell, but let me hope that if not the question, then at least the answers might be of use not just…
darij grinberg
- 33,095
44
votes
6 answers
Book on mathematical "rigorous" String Theory?
I've been looking high and low for a mathematical book on String Theory. The only book I could find was "A Mathematical Introduction to String Theory" by Albeverio, Jost, Paycha and Scarlatti. I only stumbled upon this because I really like Jost's…
Michael Kissner
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44
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2 answers
Categorical definition of the ideal product within the category of rings
This is an extension of this question. Let $I,J$ be ideals of a ring $R$; every ring is commutative and unital here. Is it possible to define $R \to R/(I*J)$ out of $R \to R/I$ and $R \to R/J$ in categorical terms within the category of rings? To be…
Martin Brandenburg
- 61,443
44
votes
6 answers
Why does one think to Steenrod squares and powers?
I'm studying Steenrod operations from Hatcher's book. Like homology, one can use them only knowing the axioms, without caring for the actual construction. But while there are plenty of intuitive reasons to introduce homology, I cannot find any for…
Andrea Ferretti
- 14,454
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44
votes
7 answers
What's an example of a space that needs the Hahn-Banach Theorem?
The Hahn-Banach theorem is rightly seen as one of the Big Theorems in functional analysis. Indeed, it can be said to be where functional analysis really starts. But as it's one of those "there exists ..." theorems that doesn't give you any…
Andrew Stacey
- 26,373
44
votes
8 answers
What makes a theorem *a* "nullstellensatz."
I know what the (Hilbert) Nullstellensatz says. A MathSciNet search on "nullstellensatz" turns up nearly 200 papers, with only a minority offering either new proofs or new applications of the classic theorem. The others use "nullstellensatz" in a…
David Feldman
- 17,466