Algebraic number

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Diophantine Approximation and the Discovery of Transcendental Numbers

Delivered by Anton Mosunov on Friday January 20, 2017

In 1844, Joseph Liouville discovered transcendental numbers — those numbers that are not roots of polynomials with rational coefficients. Nowadays, we see Liouville’s discovery as the foundation of Diophantine approximation, a fascinating subarea of number theory, which studies how well algebraic numbers can be approximated by the rationals. In my talk, I will prove the classical result of Liouville and explain further advances in the area, such as Thue’s theorem and the celebrated theorem of Roth, which enabled its discoverer to receive the Fields medal in 1958.

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Almost Integers and Pisot Numbers

Sometimes some non-integers like $e^{π} - π = 19.9990999791...$ are curiously close to integers. While often this is just a numerical coincidence, sometimes there is a clear mathematical reason for certain numbers to be almost integers. Pisot numbers, which are special types of algebraic integers, provide a systematic means to construct almost integers that approximate integers at an exponential rate. Of course, Pisot numbers are interesting on their own right and an interested speaker can also venture into the connections to Beta expansions (see the 2nd and 3rd references below), or fractals (see the 4th reference below).

Required Background: Basic analysis at the level of 147 and 138, algebra at the level of 145. Any other background depends on the direction the speaker takes with this topic.

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algebra algebraic number number theory recreational mathematics