Euler discovered how to assign a finite value to certain divergent sums, such as 1 + 2 + 4 + 8 + … that he determined should equal -1. How does one make sense of this?!? The 19th century mathematicians discovered a systematic approach to such madness. Contemporary number theorists have yet another approach, using an exotic number system that is not taught in schools. Dr. Conrad spoke to the Forum on how to calculate in this exotic world, and use it to understand a remarkable concrete fact about ordinary numbers.
Continue reading “Brian Conrad, Stanford University”
Of Planets, Stars and Eternity
After Newton’s great achievements, it seemed to scientists and philosophers of the Enlightenment that we should be able to predict everything about the future of the physical world from a knowledge of its present state. One of the areas where that should be easiest is to predict the future of the solar system, and indeed our ability to predict the motions of the planets in the short term is extremely good. But what about the long term? Can we even say whether our planets will someday be thrown far from the sun by the cumulative forces of gravitational perturbations?
This problem has occupied mathematicians since the 19th century, and has led to great advances in our understanding of dynamical systems — but the original question remains open. Prof. Villani spoke to the Forum about the study of the long time behavior of such systems from, from the Solar system itself to galaxies and related questions from fluid mechanics.
Continue reading “Cédric Villani, Institut Henri Poincaré”
The Mathematics of Doodling
Doodling has many mathematical aspects: patterns, shapes, numbers, and more. Not surprisingly, there is often some sophisticated and fun mathematics buried inside common doodles. I’ll begin by doodling, and see where it takes us. It looks like play, but it reflects what mathematics is really about: finding patterns in nature, explaining them, and extending them. By the end, we’ll have seen some important notions in geometry, topology, physics, and elsewhere; some fundamental ideas guiding the development of mathematics over the course of the last century; and ongoing work continuing today. Continue reading “Ravi Vakil, Stanford University”
Proving Prime Patterns
The sequence of primes begins 2, 3, 5, 7, 11, 13, 17, 19… and seems at first somewhat irregular, even random. But looking at lists of thousands of primes some patterns seem to appear, such as the persistence of twin primes (pairs of primes differing by just 2). Are there really any persistent patterns? Is there a formula for the primes?
In this talk we will review some of what is known and what most mathematicians believe but none can prove. We will also discuss some wild speculations. Finally, I will explain how to apply some of the latest and most exciting discoveries to prove that a few of the apparent patterns are indeed persistent.
Continue reading “Andrew Granville, Université de Montréal”
The P vs. NP Problem
Dr. Avi Wigderson spoke about “The P vs. NP Problem” at the first Math Lovers Forum event in October 2013. Slides from the lecture on efficient computation, Internet security, and the limits of human knowledge are available as a PDF download, and a video of another lecture on the same theme presented at ETH Zürich in 2012 can be viewed on YouTube. Continue reading “Avi Wigderson, Institute for Advanced Study”