The PCP theorem (Arora et. al., J. ACM 45(1,3)) says that every
NP-proof can be encoded to another proof, namely, a
probabilistically checkable proof (PCP), which can be tested by a
verifier that queries only a small part of the PCP. A
natural...
Fix a metric (Riemannian or Finsler) on a compact manifold M.
The critical points of the length function on the free loop space
LM of M are the closed geodesics on M. Filtration by the length
function gives a link between the geometry of closed...
A few years ago Ichino-Ikeda formulated a quantitative version
of the Gross-Prasad conjecture, modeled after the classical work of
Waldspurger. This is a powerful local-to-global principle which is
very suitable for analytic and arithmetic...
Incidence geometry is a part of combinatorics that studies the
intersection patterns of geometric objects. For example, suppose
that we have a set of L lines in the plane. A point is called
r-rich if it lies in r different lines from the set. For a...
olynomials are a special class of functions. They are useful in
many branches of mathematics, often in problems which don't mention
polynomials. We discuss two examples: polynomials in
error-correcting codes and polynomials in geometric
inequalities...
In 2007, Zeev Dvir shocked experts by giving a one-page proof of
the finite field Kakeya problem. The new idea in the proof was to
introduce high degree polynomials into a problem about points and
lines. This idea has led to progress on several...
