Abstract: In 1992, Connes conjectured that the Wodzicki residue of the inverse square of the Atiyah-Singer-Lichnerowicz Dirac operator D equals the Einstein- Hilbert functional of general relativity, up to a constant multiple. After giving a description of the fundamental actions in Mathematics and Physics, including the Yang-Mills action, Polyakov action and Einstein-Hilbert action, I will describe the fundamental steps involved in the purely computational proof given by Daniel Kastler in 1994. In addition, I will present a more straightforward and direct proof given by Coiai and Spera in 2000 that only involves an application of the Riemann-Zeta function and the second Seeley de-Witt coefficient of the heat kernel expansion of $\Delta = D^{2}$.

Abstract: The Skolem-Mahler-Lech theorem is a beautiful result in number theory, which asserts that if a complex-valued sequence f(n) satisfies a linear recurrence (e.g., the Fibonacci numbers) then the set of natural numbers n for which f(n)=0 is a finite union of arithmetic progressions along with a finite set.  We'll show that this has a geometric analogue in which one has a complex variety $X$, an automorphism $g\colon X\to X$, and a point $x$ in $X$ and one wishes to know when $g^n(x)$ lies in some fixed subvariety $Y$.  We'll then discuss the positive characteristic case.  In positive characteristic, the conclusion to the Skolem-Mahler-Lech theorem need not hold and we'll talk about work of Harm Derksen, which shows that one can express the zero sets of linear recurrence using what are called finite-state machines, and we'll ask whether Derksen's result has a similar geometric analogue.