# Lectures by Faculty

## Polyfolds V

Helmut Hofer
Institute for Advanced Study
April 5, 2012

## Polyfolds IV

Helmut Hofer
Institute for Advanced Study
April 4, 2012

## Polyfolds III

Helmut Hofer
Institute for Advanced Study
April 4, 2012

## Polyfolds II

Helmut Hofer
Institute for Advanced Study
April 4, 2012

## Polyfolds I

Helmut Hofer
Institute for Advanced Study
April 4, 2012

## Nodal Lines of Maass Forms and Critical Percolation

Peter Sarnak
Institute for Advanced Study
March 20, 2012

We describe some results concerning the number of connected components of nodal lines of high frequency Maass forms on the modular surface. Based on heuristics connecting these to a critical percolation model, Bogomolny and Schmit have conjectured, and numerics confirm, that this number follows an asymptotic law. While proving this appears to be very difficult, some approximations to it can be proved by developing number theoretic and analytic methods. The work report on is joint with A. Ghosh and A. Reznikov.

## Local Correction of Codes and Euclidean Incidence Geometry

Avi Wigderson
Institute for Advanced Study
March 5, 2012

A classical theorem in Euclidean geometry asserts that if a set of points has the property that every line through two of them contains a third point, then they must all be on the same line. We prove several approximate versions of this theorem (and related ones), which are motivated from questions about locally correctable codes and matrix rigidity. The proofs use an interesting combination of combinatorial, algebraic and analytic tools.

Joint work with Boaz Barak, Zeev Dvir and Amir Yehudayoff

## On Zaremba's Conjecture on Continued Fractions

Jean Bourgain
Institute for Advanced Study
February 14, 2012
Zaremba's 1971 conjecture predicts that every integer appears as the denominator of a finite continued fraction whose partial quotients are bounded by an absolute constant. We confirm this conjecture for a set of density one.