Mathematics - Theories

by Margaret Allen

Scientists have made many discoveries about the origins of our 13.7 billion-year-old universe. But many scientific mysteries remain. What exactly happened during the Big Bang, when rapidly evolving physical processes set the stage for gases to form stars, planets and galaxies? Now astrophysicists using supercomputers to simulate the Big Bang have a new mathematical tool to unravel those mysteries, says Daniel R. Reynolds, assistant professor of mathematics at SMU.

Read more: New Mathematical Model Aids Simulations of Early Universe

Mathematics - Theories

by Larry Hardesty

Constantinos Daskalakis applies the theory of computational complexity to game theory, with consequences in a range of disciplines.

Computer scientists have spent decades developing techniques for answering a single question: How long does a given calculation take to perform? Constantinos Daskalakis, an assistant professor in MIT’s Computer Science and Artificial Intelligence Laboratory, has exported those techniques to game theory, a branch of mathematics with applications in economics, traffic management — on both the Internet and the interstate — and biology, among other things. By showing that some common game-theoretical problems are so hard that they’d take the lifetime of the universe to solve, Daskalakis is suggesting that they can’t accurately represent what happens in the real world.

Read more: What Computer Science Can Teach Economics

Mathematics - Theories

We don't have any trouble coping with three dimensions – or four at a pinch. The 3D world of solid objects and limitless space is something we accept with scarcely a second thought. Time, the fourth dimension, gets a little trickier. But it's when we start to explore worlds that embody more – or indeed fewer – dimensions that things get really tough.

Read more: Beyond Space and Time : Fractals, Hyperspace and More

Mathematics - Theories

Jean Bourgain
Institute for Advanced Study

Although the concept of randomness is ubiquitous, it turns out to be difficult to generate a truly random sequence of events. The need for “pseudorandomness” in various parts of modern science, ranging from numerical simulation to cryptography, has challenged our limited understanding of this issue and our mathematical resources. In this talk, Professor Jean Bourgain explores some of the problems of pseudorandomness and tools to address them.

Watch the video

Mathematics - Theories

Optimization

The problem of finding an unconstrained minimizer of an n-dimensional function, f, may be stated as follows:

given f: Rn R, find x* (an element of Rn) such that f(x*) is a minimum of f.

For example:    f (x) = (x0 – 3)4 + (x1 - 2)2
x* = [3, 2]

Read more: Optimization