How to be successful as a research mathematician? Follow your gut
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Re: How to be successful as a research mathematician? Follow your gut
#2That's not very actionable advice. I've found this article [0] from Terence Tao very insightful:
[A]ctual solutions to a major problem tend to be arrived at by a process more like the following (often involving several mathematicians over a period of years or decades, with many of the intermediate steps described here being significant publishable papers in their own right):
1. Isolate a toy model case x of major problem X.
2. Solve model case x using method A.
3. Try using method A to solve the full problem X.
4. This does not succeed, but method A can be extended to handle a few more model cases of X, such as x’ and x”.
5. Eventually, it is realised that method A relies crucially on a property P being true; this property is known for x, x’, and x”, thus explaining the current progress so far.
6. Conjecture that P is true for all instances of problem X.
7. Discover a family of counterexamples y, y’, y”, … to this conjecture. This shows that either method A has to be adapted to avoid reliance on P, or that a new method is needed.
8. Take the simplest counterexample y in this family, and try to prove X for this special case. Meanwhile, try to see whether method A can work in the absence of P.
(... 15 more steps)
Re: How to be successful as a research mathematician? Follow your gut
#3That's not very actionable advice. I've found this article [0] from Terence Tao very insightful: [A]ctual solutions to a major problem tend to be arrived at by a process more like the following (often involving several mathematicians over a period of years or decades, with many of the intermediate steps described here being significant publishable papers in their own right): 1. Isolate a toy model case x of major pro…
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Re: How to be successful as a research mathematician? Follow your gut
#4That's not very actionable advice. I've found this article [0] from Terence Tao very insightful: [A]ctual solutions to a major problem tend to be arrived at by a process more like the following (often involving several mathematicians over a period of years or decades, with many of the intermediate steps described here being significant publishable papers in their own right): 1. Isolate a toy model case x of major pro…
Overheard at the watercooler: The only way a CS guy knows how to prove an algorithm works/does not work is to find counterexamples.
Re: How to be successful as a research mathematician? Follow your gut
#5About as useful as Feynman's algorithm for solving problems:
1. Write down the problem. 2. Think very hard. 3. Write down the solution.
Re: How to be successful as a research mathematician? Follow your gut
#6That's not very actionable advice. I've found this article [0] from Terence Tao very insightful: [A]ctual solutions to a major problem tend to be arrived at by a process more like the following (often involving several mathematicians over a period of years or decades, with many of the intermediate steps described here being significant publishable papers in their own right): 1. Isolate a toy model case x of major pro…
Overheard at the watercooler: The only way a CS guy knows how to prove an algorithm works/does not work is to find counterexamples.
Seems hard to prove that an algorithm works by finding counterexamples.
Re: How to be successful as a research mathematician? Follow your gut
#7That's not very actionable advice. I've found this article [0] from Terence Tao very insightful: [A]ctual solutions to a major problem tend to be arrived at by a process more like the following (often involving several mathematicians over a period of years or decades, with many of the intermediate steps described here being significant publishable papers in their own right): 1. Isolate a toy model case x of major pro…
Polya‘s „How to solve it“ might also be worth a look.
Re: How to be successful as a research mathematician? Follow your gut
#8Earlier quoted context omitted.
Overheard at the watercooler: The only way a CS guy knows how to prove an algorithm works/does not work is to find counterexamples.
Seems hard to prove that an algorithm works by finding counterexamples.
Just express it as an n-state Turing machine and see if it halts within BB(n) steps. /s
Re: How to be successful as a research mathematician? Follow your gut
#9Being a successful research mathematician, although I am not one, sounds nowadays like more about figuring out the market for your skills among multitudes of opaque research projects so I imagine following your gut you might just get into a good post if you're lucky.
Re: How to be successful as a research mathematician? Follow your gut
#10That's not very actionable advice. I've found this article [0] from Terence Tao very insightful: [A]ctual solutions to a major problem tend to be arrived at by a process more like the following (often involving several mathematicians over a period of years or decades, with many of the intermediate steps described here being significant publishable papers in their own right): 1. Isolate a toy model case x of major pro…
From the number of math channels I follow, the common advice I see for handling a complex problem is to start with a simpler version of it and solve it. Try simplifying it in a few different ways, find a few different ways to solve each simplification, and then see which ways of solving the problem might work to solve the more complex problem. With experience one gets better at finding the right simplification with less effort/time invested.