Some applications of the graph of supersingular elliptic curves over a finite field

The graph of supersingular elliptic curves over a finite field connected by isogenies has many applications in computational number theory. In this talk we look at some old (in number theory) and new (in cryptography) applications of these graphs. In particular, we discuss new constructions of secure hash functions and pseudorandom number generators from these graphs. We will also study the interesting graph-theoretic properties of this graph. If time permits, I will sketch a generalization of these graphs to graphs of superspecial abelian varieties. The new results in this talk are from joint work with Eyal Goren and Kristin Lauter.
In my defense: Integral points on elliptic curves
In a talk not completely unrelated to my thesis defense, I will examine the question For which elliptic curves E, which integral points P on E, and which integers n is nP also an integral point? Most (if not all) of the attention will be devoted to congruent number curves.