We define and study the secondary Chern-Euler class for a general submanifold of a Riemannian manifold. Using this class, we define and study the index for a vector field with non-isolated singularities on a submanifold. As an application, we give conceptual proofs of a result of Chern.

We study a second order nonlinear system driven by the vector $p$-Laplacian, with a multivalued nonlinearity and defined on the positive time semi-axis $\mathbb{R}_+.$ Using degree theoretic techniques we solve an auxiliary mixed boundary value problem defined on the finite interval $[0,n]$ and then via a diagonalization method we produce a solution for the original infinite time-horizon system.

The number of cyclic cubic fields with a given conductor and a given index is determined.

For a given irrational number $\theta$, we define Toeplitz operators with symbols in the irrational rotation algebra ${\mathcal A}_\theta$, and we show that the $C^*$-algebra $\mathcal T({\mathcal A}_\theta)$ generated by these Toeplitz operators is an extension of ${\mathcal A}_\theta$ by the algebra of compact operators. We then use these extensions to explicitly exhibit generators of the group $KK^1({\mathcal A}_\theta,\mathbb C)$. We also prove an index theorem for $\mathcal T({\mathcal A}_\theta)$ that generalizes the standard index theorem for Toeplitz operators on the circle.