Professor Richard Stanley (MIT)
Let P be a polygon in the plane with integer vertices. Suppose that the area of P is A and that P has I interior lattice points and B lattice points on the boundary. Alexander Pick showed that A = (2I + B − 2)/2. What happens in higher dimensions? This question gave rise to a beautiful theory of lattice points in polytopes. After surveying the highlights of this theory, including connections with commutative algebra, we will discuss some recent work related to integrality, positivity, and period collapse.
You can learn more about Professor Stanley at http://www-math.mit.edu/~rstan/