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Results tagged with big-list
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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.
171
votes
Most memorable titles
"Hodge's general conjecture is false for trivial reasons."
79
votes
Applications of the Chinese remainder theorem
Parallel computation: Suppose you have a huge computation to do that involves adding, multiplying and subtracting integers. Possibly also dividing but, if so, only division by numbers in a finite set …
65
votes
Strengthening the induction hypothesis
Here is a bit of advice that took me a while to learn:
You don't need to know what you are proving when you start to write a proof by induction.
The following method isn't needed for easy prob …
59
votes
Applications of the Chinese remainder theorem
Secret sharing. Suppose we have $N$ people. We want any $k+1$ of them to be able to launch a missile attack, but no $k$ of them to have this power.
Solution: Choose some large prime $p$ and a random …
56
votes
What should be learned in a first serious schemes course?
As you should, you prove the Nullstellensatz early on, as the statement that the closed points of $\mathbb{A}^n_k$ are in bijection with $k^n$, for $k$ an algebraically closed field. I wonder whether …
51
votes
What should be learned in a first serious schemes course?
I found differentials hard to understand when I learned this material. Here are two things that helped me which I think are not in your notes:
(1) The description of the Zariski tangent space to $X$ …
41
votes
Suggestions for good notation
Writing $\int_{x=0}^{2 \pi} \sin x dx$ rather than $\int_0^{2 \pi} \sin x dx$ can be very useful when there are integrals stacked several layers deep. EG
$$\int_{x=-\infty}^{\infty} \int_{y=-\infty}^ …
34
votes
Facts from algebraic geometry that are useful to non-algebraic geometers
If $p_1$, $p_2$, ..., $p_m$ are polynomials in $n$ variables, with $m>n$, then there is a polynomial $q$ such that $q(p_1, p_2, \ldots, p_m)$ is identically zero.
30
votes
Examples of common false beliefs in mathematics
I'm not sure how common this is, but it confused me for years. Let $f : \mathbb{C} \to \mathbb{C}$ be an analytic function and $\gamma$ a path in $\mathbb{C}$. In your first class in complex analysis, …
29
votes
Examples of common false beliefs in mathematics
I'm not sure that anyone holds this as a conscious belief but I have seen a number of students, asked to check that a linear map $\mathbb{R}^k \to \mathbb{R}^{\ell}$ is injective, just check that each …
26
votes
Shortest/Most elegant proof for $L(1,\chi)\neq 0$
My preferred proof is to consider $Z(s) := \prod_{\chi} L(\chi, s)$, where the product is over all characters of $\mathbb{Z}/n$, including the trivial character. It is easy to see that $L(s, \chi)$ is …
- 156k
25
votes
Interesting results in algebraic geometry accessible to 3rd year undergraduates
I saw Brian Conrad give an excellent one hour talk to undergraduates where he proved that there do not exist nonconstant, relatively prime, polynomials $a(t)$, $b(t)$ and $c(t) \in \mathbb{C}[t]$ such …
- 156k
24
votes
Undergraduate Level Math Books
Concrete Mathematics, Graham, Knuth and Patashnik. Extremely useful, very good exercises, and a sense of humor that appeals to me.
24
votes
PhD dissertations that solve an established open problem
June Huh's recent proof of Rota's conjecture (stated by Read in 1968 for graphical matroids and Rota in 1971 for all matroids) formed his 2014 Ph. D. thesis. For matroids over $\mathbb{C}$, this appea …
22
votes
Good algebraic number theory books
Many people have recommended Neukirch's book. I think a good complement to it is Janusz's Algebraic Number Fields. They cover roughly the same material. Neukirch's presentation is probably the slickes …