“The function ei(kx-ωt) represents a wave that travels at a speed of ω/k, which we have just shown to be equal to -k2. Therefore, the different plane-wave components of the solution travel at different speeds: the higher the angular frequency,… — Timothy Gowers Copy Share Image
“the simple algebraic equation ω+k3 = 0. This is called the dispersion relation of (1): with the help of the Fourier transform it is not hard to show that every solution is a superposition of solutions of the form ei(kx-ωt),… — Timothy Gowers Copy Share Image
Although the prime numbers are rigidly determined, they somehow feel like experimental data. — Timothy Gowers Copy Share Image
“if, for each t, we write u(t) for the function from to that takes x to u(x, t), then it describes how the function… — Timothy Gowers Copy Share Image
What a mathematical proof actually does is show that certain conclusions, such as the irrationality of , follow from certain premises, such as the… — Timothy Gowers Copy Share Image
“ut = -uux - uxxx. Why is it that this equation gives rise to the remarkable stability of the solutions that was observed experimentally… — Timothy Gowers Copy Share Image
“It is a remarkable phenomenon that children can learn to speak without ever being consciously aware of the sophisticated grammar they are using.” — Timothy Gowers Copy Share Image
“The function ei(kx-ωt) represents a wave that travels at a speed of ω/k, which we have just shown to be equal to -k2. Therefore,… — Timothy Gowers Copy Share Image
It is obvious that mathematics needs both sorts of mathematicians, theory-builders and problem-solvers. — Timothy Gowers Copy Share Image
“differential equations that can be solved in “closed form,” that is, by means of a formula for the unknown function f, are the exception… — Timothy Gowers Copy Share Image
This attitude [the abstract method in mathematics] can be encapsulated in the following slogan: a mathematical object is what it does. — Timothy Gowers Copy Share Image
At the other end of the spectrum is, for example, graph theory, where the basic object, a graph, can be immediately comprehended. One will… — Timothy Gowers Copy Share Image
... the atlas is a manifold. This is a typical mathematician's use of the word "is", and should not be confused with the normal use. — Timothy Gowers Copy Share Image
Here is something Category-Theorists like: it is trivial, but not trivially trivial. — Timothy Gowers Copy Share Image
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