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Results tagged with soft-question
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user 18060
Questions that are about research in mathematics, or about the job of a research mathematician, without being mathematical problems or statements in the strictest sense. Do not use this tag for easy or supposedly easy mathematical questions.
80
votes
Accepted
Fibonacci series captures Euler $e=2.718\dots$
It follows from the identity
$$F_{n+k} = \sum_{j=0}^k {k \choose j} F_{n-j}$$
which is obtained by applying the standard recurrence $k$ times to the left side, each time splitting up each term int …
- 149k
42
votes
Accepted
How and when do I learn so much mathematics?
The other answers have some good general advice. Let me try to say something that is specific to the topics of analytic number theory, and number theory generally.
First, there is no such thing as tr …
37
votes
Modern results that are widely known, yet which at the time were ignored, not accepted or cr...
The Selberg integral was proved in a 1944 paper of Selberg, after being stated without proof in a 1941 paper. The paper was in Norwegian, and was also in a journal that would have been of little inter …
32
votes
Why is game theory formulated in terms of equilibrium instead of winning strategies?
There's a few issues that need to be distinguished here. First, one can distinguish the question of how you find the winning strategy from the question of how you define what the winning strategy even …
- 149k
31
votes
Pressure to defend the relevance of one's area of mathematics
First let me try to answer the question in an "ideal world", where (in particular) set theory is treated like any other branch of mathematics, and then let me discuss how we might fall short of it.
I …
31
votes
What are possible applications of deep learning to research mathematics?
I have some thoughts on a level of generality that is a bit higher than the question asks for:
One obstacle that faces applications of supervised machine learning to predict properties of mathematica …
31
votes
Theorems with many distinct proofs
The Riemann hypothesis for curves over finite fields has a number of distinct proofs.
Weil gave two proofs, one based on the Jacobian and one based on intersection theory on the product $C \times C$.
…
29
votes
Examples of "unsuccessful" theories with afterlives
Motives and the standard conjectures were developed by Grothendieck to prove the last of the Weil conjectures. They failed at this as none of the standard conjectures were proven - despite some progre …
28
votes
Accepted
How do you check that your mathematical research topic is original?
(1) It depends a lot on the field. In fields that rely on specialized techniques discovered relatively recently or known only to a few, or fields where the questions involve recently-introduced objec …
26
votes
The probabilistic method outside of discrete mathematics
One application of the probabilistic method in topology was found by Melanie Matchett Wood and myself:
Let $H$ be the finite group $(\mathbb Z/15) \rtimes Q_8$, where generators $i$ and $j$ of $Q_8$ a …
21
votes
Why not a Stacks project for Homotopy Theory?
According to the about page of Kerodon:
Kerodon is an online textbook on categorical homotopy theory and related mathematics. It currently consists of a single chapter, but should grow (slowly) ov …
- 149k
20
votes
Lunch seminars for PhD students
Princeton has a Graduate Student Seminar which, at least when I was there, was a lunch seminar offering pizza. It is part of the pure math department but I think people from the program for applied an …
17
votes
Why does mathematics seem to have a polarity bias?
Coproducts of sets are introduced earlier in mathematical education than products of sets, under the name "union" or "disjoint union". Also, addition is of course introduced earlier than multiplicatio …
- 149k
11
votes
What does keep you "doing what you do"?
This question seems to assume that after doing a PhD and continuing your career in mathematics you will keep struggling with the same things. I don't think this is quite right.
In a PhD the main thin …
- 149k
9
votes
How to think of algebraic geometry in characteristic p?
The basic strategy you give is correct: To first approximation, algebraic geometry over $\overline{\mathbb F_p}$ is like algebraic geometry over $\mathbb C$ except for extra phenomena. Usually, I thin …
- 149k