1. Testing Juntas: A Powerpoint Presentation, and a .
2. Learning Juntas: a
3. Beckner’s Thesis: a
Basic knowledge in Linear Algebra and Normed Spaces.
Program of the course:
Influence of variables, norms, Fourier representation. Voting systems: dictatorship, juntas, monotonicity.
The space of Boolean functions, the Fourier-Walsh representation, and some formulas connecting them.
Testing linear codes:
Testing the Hadamard code and the Long-code, and relations with Noise-Sensitivity.
How to check mathematical proofs without reading them.
Testing whether a given Boolean function is a J-junta, in time polynomial in J.
Average sensitivity and juntas:
The Bonami-Beckner Inequality, a theorem of Friedgut, extensions to the biased case.
Monotonicity and symmetry:
A lemma of Margulis, A theorem of Kahn, Kalai and Linial, threshold phenomena for graph-properties, first passage percolation.
Linear and almost-linear functions:
Almost linear functions are dictatorships. The linear weight is bounded by the expectation.
Rationality and social choice:
The only rational preference selection is a dictatorship
functions are juntas.
Low-degree bounded functions are
13. Vertex-Cover and Max-Cut.