Terence Tao is widely considered to be one of the greatest mathematicians in history. He won the Fields Medal and the Breakthrough Prize in Mathematics, and has contributed to a wide range of fields from fluid dynamics with Navier-Stokes equations to mathematical physics & quantum mechanics, prime numbers & analytics number theory, harmonic analysis, compressed sensing, random matrix theory, combinatorics, and progress on many of the hardest problems in the history of mathematics.
What is the reason that 1/x is not lebesgue integrable where as 1/x^2 is integrable. You can use any theorems: monotone convergence, dominated convergence...
If you’ve done any 3D programming, you’ve likely encountered the zoo of techniques and representations used when working with 3D rotations. Some of them are better than others, depending on the situation. Based on CMU 15-462 course materials by Keenan Crane.
In 1959, the English writer and physicist C P Snow delivered the esteemed Rede Lecture at the University of Cambridge. Regaled with champagne and Marmite sandwiches, the audience had no idea that they were about to be read the riot act. Snow diagnosed a rift of mutual ignorance in the intellectual world of the West. On the one hand were the ‘literary intellectuals’ (of the humanities) and on the other the (natural) ‘scientists’: the much-discussed ‘two cultures’. Snow substantiated his diagnosis with anecdotes of respected literary intellectuals who complained about the illiteracy of the scientists but who themselves had never heard of such a fundamental statement as the second law of thermodynamics. And he told of brilliant scientific minds who might know a lot about the second law but were barely up to the task of reading Charles Dickens, let alone an ‘esoteric, tangled and dubiously rewarding writer … like Rainer Maria Rilke.’
June Huh wasn’t interested in mathematics until a chance encounter during his sixth year of college. Now his profound insights connecting combinatorics and geometry have led to math’s highest honor.
In our journey so far, we’ve seen how certain prime numbers split in two complex domains simultaneously, and how these prime factorizations can be encoded in matrices that transform into physical…
Bézier curves are widely used for defining vector graphics. They are basically polynomial parametric curves, but given in the Bernstein basis, which enables us to define the curve using control points.
Warning: Math, Handwaving I spent a lot of time doodling in middle school in lieu of whatever it is middle schoolers are supposed to be doing. Somewhere between the Cool S’s and Penrose triangles I stumbled upon a neat way to fill up graph paper by repeatedly combining and copying squares. I suspected there was more to the doodle but wasn’t quite sure how to analyze it. Deciding to delegate to a future version of me that knows more math, I put it up on the wall behind my desk where it has followed me from high school to college to the present day.
Due to the rather broad spectrum of topics within the IMPRS, the curriculum consists of a core curriculum to be attended by all students and a variety of more specialized lectures and courses. The heart of our teaching program certainly is the Ringvorlesung. Each semester the Ringvorlesung focuses on one field and is usually delivered by scientific members of the IMPRS who introduce different approaches and visions within this field.
Basic ideas pre-dating need for Measure Theory and Lebesgue Integration
Fourier's solution to the Heat Equation, forces introduction of Fourier Series, issues of convergence, interchanging series and integrals
Cauchy's wrong proof that limits of continuous functions are continuous
Vitali's construction of non-measurable sets, assuming the Axiom of Choice (independence from Zermelo-Frankel axioms due to Godel and Cohen)
Polynomials are equations involving a variable raised to powers, such as the degree two polynomial: 1 + 4x – 3x2 = 0.
The equations are fundamental to math as well as science, where they have broad applications, like helping describe the movement of planets or writing computer programs.
However, a general method for solving "higher order" polynomial equations, where x is raised to the power of five or higher, has historically proven elusive.