School of Mathematics

Constructive Type Theory and Homotopy

Steve Awodey
Institute for Advanced Study
December 3, 2010

In recent research it has become clear that there are fascinating connections between constructive mathematics, especially as formulated in the type theory of Martin-Löf, and homotopy theory, especially in the modern treatment in terms of Quillen model categories and higher-dimensional categories. This talk will survey some of these developments.

Patterns, Universality and Computational Algorithms

Nigel Goldenfeld
University of Illinois at Urbana-Champaign
December 1, 2010

Can we use computational algorithms to make accurate predictions of physical phenomena? In this talk, intended for non-experts, I will give examples where complicated space-time phenomena can be exquisitely captured with simple computational algorithms, that not only produce patterns resembling those seen in experiment, but also make accurate predictions about probes of dynamics and spatial organisation, such as correlation functions. I use examples from condensed matter physics, as well as from geophysics.

Self-Correction, Distance Estimation and Local Testing of Codes

Dana Moshkovitz
Massachusetts Institute of Technology
November 29, 2010

We construct linear codes of almost-linear length and linear distance that can be locally self-corrected on average from a constant number of queries:

1. Given oracle access to a word $w\in\Sigma^n$ that is at least $\varepsilon$-close to a codeword $c$, and an index $i\in [n]$ to correct, with high probability over $i$ and over the internal randomness, the local algorithm returns a list of possible corrections that contains $c_i$.