Description
Lattices are fundamental objects in physics, mathematics and computer science. Starting from a cubic lattice, say, we can perturb the structure by linear transformations (shearing, stretching, rotating) to obtain a whole family of lattices. I will discuss the resulting "space of lattices", the dynamics of group actions on this space, natural probability measures, as well as some fascinating applications to long-standing problems in various areas of mathematics and mathematical physics. My plan is to tell you about kinetic transport in crystals and quasicrystals (the Lorentz gas), pseudo-random properties of simple arithmetic sequences, knapsack problems, diameters of random Cayley graphs and (time permitting) subtle lattice point counting problems in hyperbolic geometry.