\indent I will talk about the fundamental theorem of affine sieve (joint with Sarnak). The main black box in the proof of this result will be also explained. It is a theorem on a necessary and sufficient condition for a finitely generated subgroup of SL(n,Q) under which the Cayley graphs of such a group modulo square free integers form a family of expanders (joint with Varju).

\indent We consider group-invariant CR mappings from spheres to hyperquadrics. Given a finite subgroup $\Gamma \subset U(n)$, a construction of D'Angelo and Lichtblau yields a target hyperquadric $Q(\Gamma)$ and a canonical map $h_{\Gamma} : S^{2n-1}/\Gamma \to Q(\Gamma)$. For every $\Gamma \subset SU(2)$, we determine this hyperquadric $Q(\Gamma)$, that is, the numbers of positive and negative eigenvalues in its defining equation. For families of cyclic and dihedral subgroups of $U(2)$, we study these numbers asymptotically as the order of group tends to infinity. Finally, we explore connections with invariant theory and representation theory.

\indent Whereas many physical systems can be identified as Hamiltonian dynamical systems, mechanical systems under rolling and sliding constraints, even simple ones such as a rolling penny, skateboard, and sleigh, are non-Hamiltonian. This is due to the fact that those constraints are nonholonomic (non-integrable). Nonholonomic constraints destroy some nice features of Hamiltonian dynamical systems, most importantly symplecticity, while retaining some Hamiltonian properties, such as energy conservation. Many concepts and ideas in Hamiltonian dynamics have been generalized from the differential-geometric point of view to incorporate nonholonomic constraints and also to explain the "almost Hamiltonian" behavior of nonholonomic systems. In this talk, I will show how to generalize Hamilton--Jacobi theory to nonholonomic systems and its application to exactly integrating the equations of motion, touching on the basic geometric concepts of nonholonomic and Hamiltonian systems and also the tools and techniques used to reconcile the differences between them.

\indent Let $M_0$ be a closed subset of $R^n+1$ that is a smooth hypersurface except for a finite number of isolated singular points. Suppose that $M_0$ is asymptotic to a regular cone near each singular point.

Can we flow $M_0$ by mean curvature?

Theorem $(n<7)$: there exists a smooth mean curvature evolution starting at $M_0$ and defined for a short time $0<t<eps$.

Such an initial $M_0$ might arise as the limit of a smooth mean curvature evolution defined earlier than $t=0$. Thus, the result allows us to flow through singularities in some cases.

We use a monotonicity formula that complements the monotonicity formula of Huisken. The method applies to other geometric heat flows as well.

\indent Have you ever wanted to access a database in such a way that even the database can't tell what you're looking for? Better yet, don't answer that. Regardless, we will investigate how an accommodating database can allow such a query. We'll also discuss possible connections to circumventing the PATRIOT Act (don't tell the FBI!) and furthering space research.