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 $\Delta = D^{2}$.

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.

We present some universality results concerning harmonic functions on trees. One of these results relates to approximation by martingales on the boundary of the tree. We discuss topological genericity, algebraic genericity and spaceability.

This is a report on a paper by Lapidus and Sarhad titled "Dirac Operators and Geodesic Metric on the Harmonic Sierpinski Gasket and Other Fractal Sets". First I will define the graph approximation of the Sierpinski gasket. Then I will talk about Kusuoka's measurable Riemannian geometry on Sierpinski gasket and introduce counterparts of Riemannian volume, Riemannian metric and Riemannian energy in that setting. Thereafter harmonic functions on Sierpinski gasket will be introduced as energy minimizing functions. Using those functions we can define harmonic gasket. I will also talk about Kigami's geodesic metric on harmonic gasket. Then I will construct two spectral triples on harmonic gasket and we will see that those two triples induce the same spectral metric as Kigami's geodesic metric.

For both elliptic and transversally elliptic operators, we have seen that the equivariant index defines a virtual $G$-representation, where $G$ is a compact Lie group. This representation can be expressed as a sum of irreducible representations with multiplicities. In the elliptic case, this sum is finite. In the transversally elliptic case, the sum is infinite, but the index still defines a distributional character on $G$.

The aim of this talk is to give an overview of how the de Concini-Procesi-Vergne machinery (associated to the infinitesimal index) can be used to give a formula for the multiplicities of the irreducible representations within the equivariant index. This will be more of a ``big picture'' talk that attempts to tie together some of the particular results we've encountered so far, without getting too much into the details. The main reference will be Vergne's paper on the Euler-Maclaurin formula for the multiplicity function (arXiv:1211.5547).

I will describe a project with June Huh on Chern-Schwarz-MacPherson classes of some varieties associated with matroids, and combinatorial inequalities that result.