Superconformal algebras are superextensions of the Virasoro algebra. They play an important role in the string theory and conformal field theory.

Central extensions of contact superalgebras $K(2)$ and $K'(4)$ are known to physicists as the $N = 2$ and the big $N = 4$ superconformal algebras. A remarkable property of $\widehat{K}'(4)$ and the exceptional superconformal algebra $CK_6$ is that they admit embeddings into the Lie superalgebras of pseudodifferential symbols on the circle, extended by
$N = 2, 3$ odd variables. Associated to these embeddings, there are ``small" irreducible representations of these superalgebras and their realizations in matrices of size $2^N$ over a Weyl algebra. The general construction of such matrix realizations is connected with the spin
representation of $\mathfrak{o}(2N + 1, \mathbb{C})$.

We also obtain a realization of the family of simple exceptional finite-dimensional Lie superalgebras $D(2; 1; \alpha)$, related to $K(4)$ in matrices over a Weyl algebra.

In 1643, Ren\'{e} Descartes discovered a formula relating curvatures of circles in Apollonian circle packings, constructed by Apollonius of Perga in 200 BC. This formula has recently led to a connection between the construction of Apollonius and orbits of a certain so-called \emph{thin} subgroup $\Gamma$ of $\textrm{GL}_4(\mathbb Z)$. This connection is key in recent results on the arithmetic of Apollonian packings, which I will describe in this talk. A crucial ingredient in the proofs is the spectral gap coming from families of expander graphs
associated to $\Gamma$ -- this gap is far less understood in the case of thin groups than that of non-thin groups. Motivated by this problem, I will then discuss the ubiquity of thin groups and present results on thinness of monodromy groups of hypergeometric equations in the case where these groups act on hyperbolic space.

The Finite Element Method is a powerful tool for approximating the solutions of a large class of PDE's. Traditional FEM requires the function spaces to be linear. We discuss an extension of FEM, Geodesic Finite Elements, which allows the functions being approximated to have range in a non-linear Riemannian Manifold. We realize the set of pseudo Riemannian metrics as a symmetric space in order to pull back the matrix exponential onto it, thus endowing it with a Riemannian metric. Once done, we are able to discretize a class of pseudo-Riemannian manifolds. This has possible applications in non-linear hyperbolic PDE's.

In recent years, it has become interesting from a number-theoretic point of view to be able to determine whether a finitely generated subgroup of $GL_n(\mathbb Z)$ is a so-called thin group. In general, little is known as to how to approach this question. In this talk we discuss this question in the case of hypergeometric monodromy groups, which were studied in detail by Beukers and Heckman in 1989. We will convey what is known, explain some of the difficulties in answering the thinness question, and show how one can successfully answer it in many cases where the group in question acts on hyperbolic space. This work is joint with Meiri and Sarnak.

The Global Positioning System involves about 24 satellites
orbiting the earth each sending a unique signal. Your GPS receiver
receives the signal from at least 4 of them and from these signals it
tells you where you are within 10 meters (if your GPS can handle
WAAS the accuracy is 3 meters). The calculations done by your
receiver involve general relativity, error correction and geometry.