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Results tagged with big-list
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user 2083
Questions designed to generate a "big list" of certain results, examples, conjectures, etc. via many individual answers, each contributing one or a few instances. Such a question should typically be in Community Wiki mode (CW); after asking, please, flag for moderators attention requesting the question to be made CW.
16
votes
Elementary / Interesting proofs of the Nullstellensatz
I found this "geodesic" proof by Munshi, written by May (I like the very last paragraph (:).
40
votes
3
answers
2k
views
Good user manuals for technical topics?
This question is motivated by this (highly recommended) comment by Emerton on Terry Tao's post "Learn and relearn your field". In particular, the following paragraphs:
In particular, the first couple …
8
votes
The half-life of a theorem, or Arnold's principle at work
There is the Hilbert-Burch Theorem, which gives structure of Cohen-Macaulay ideals having projective dimension one in a regular or polynomial ring. The named authors published their results about 80 y …
7
votes
Most intricate and most beautiful structures in mathematics
I heard good things about F_un!
39
votes
Suggestions for special lectures at next ICM
The Weapons of Math Destruction would make an interesting and timely topic for such a lecture.
12
votes
How helpful is non-standard analysis?
Let $k$ be an algebraically closed field of characteristic $0$. Let $T_n$ be the set of all possible log canonical threshold of a pair $(X,Y)$ where $X/k$ is a smooth variety and $Y \subseteq X$ is a …
- 30.6k
12
votes
Your favorite surprising connections in mathematics
The connection between rational homotopy theory and local algebra has been very useful, I was told. See Section 3 of this survey by Kathryn Hess and the references therein, especially Anick's countere …
34
votes
Noteworthy, but not so famous conjectures resolved recent years
The homological conjectures in commutative algebra using perfectoid methods. A survey on many recent developments written by André can be found here.
6
votes
What formal properties should resolution of singularities have?
It should exist in all characteristics, even mixed characteristic!
This sounds somewhat cheeky, but I was fairly serious. To algebraists, and the OP is one last time I met him, a purely ring-the …
- 30.6k
26
votes
Short exact sequences every mathematician should know
Given a finitely generated module $M$ over a commutative Noetherian ring $R$, there is a short exact sequence $$0\to M_1 \to R^n \to M\to 0$$
where you map $1$ in each $R$ to a generator of $M$ and $M …
31
votes
How professional mathematicians deal with discouragement?
Pour yourself a beer and reflect on how crazy it is that 1) modern life actually needs knowledge about stuff like "elliptic curves over finite fields" and 2) you are lucky enough to make a living thin …
15
votes
Gaining intuition for how submodules behave
Doing all exercises in Atiyah-MacDonald, like BCnrd suggested, is surely the ideal way to learn about this and much more. Let me offer a couple of practical tips to get you started:
A surprisingly ef …
- 30.6k
30
votes
Every mathematician has only a few tricks
Those of us who are old enough may remember http://www.tricki.org/
Localize + complete, taking a hypersurface section, and using the socle are useful tricks in commutative algebra.