This introductory lecture will describe results about counting rational points on certain non-algebraic sets and sketch how they can be used to attack certain problems in diophantine geometry and functional transcendence.
This talk is mainly concerned with the investigation of small set expansion on the Grassmann Graphs, a study that was motivated by recent applications to Probabilistically Checkable Proofs and hardness of approximation.
The mechanism of the explosion of massive stars remains uncertain. I will
discuss aspects of the critical condition for explosion, the observed
supernova diversity, and the connection to gamma-ray bursts and
super-luminous supernovae. I will focus on the first few seconds after
explosion, during the "proto-neutron star" cooling epoch, when a wind
driven by neutrino heating emerges from the cooling neutron star into the
overlying massive stellar progenitor, powering the explosion. I will
Matrix rank is well-known to be multiplicative under the Kronecker product, additive under the direct sum, normalized on identity matrices and non-increasing under multiplying from the left and from the right by any matrices. In fact, matrix rank is the only real matrix parameter with these four properties.