The Chern classes for degeneracy loci of quivers are natural generalizations of the Thom-Porteous-Giambelli formula. Suppose that $E,F$ are vector bundles over a manifold $M$ and that $s : E\to F$ is a vector bundle homomorphism. The question is, which cohomology class is defined by the set $\Sigma\sb k(s)\subset M$ consisting of points $m$ where the linear map $s(m)$ has corank $k$? The answer, due to I. Porteous, is a determinant in terms of Chern classes of the bundles $E,F$. We can generalize the question by giving more bundles over $M$ and bundle maps among them. The situation can be conveniently coded by an oriented graph, called a quiver, assigning vertices for bundles and arrows for maps.

We give a new method for calculating Chern class formulae for degeneracy loci of quivers. We show that for representation-finite quivers this is a special case of the problem of calculating Thom polynomials for group actions. This allows us to apply a method for calculating Thom polynomials developed by the authors. The method–reducing the calculations to solving a system of linear equations–is quite different from the method of A. Buch and W. Fulton developed for calculating Chern class formulae for degeneracy loci of $A\sb n$-quivers, and it is more general (can be applied to $A\sb n$-, $D\sb n$-, $E\sb 6$-, $E\sb 7$-, and $E\sb 8$-quivers). We provide sample calculations for $A\sb 3$- and $D\sb 4$-quivers.

We prove that if $A\subseteq\{1,\ldots N\}$ has density at least $(\log \log N)\sp {-c}$, where $c$ is an absolute constant, then $A$ contains a triple $(a, a+d,a+2d)$ with $d=x\sp 2+y\sp 2$ for some integers $x,y$, not both zero. We combine methods of T. Gowers and A. Sárközy with an application of Selberg's sieve. The result may be regarded as a step toward establishing a fully quantitative version of the polynomial Szemerédi theorem of V. Bergelson and A. Leibman.

We discuss some natural maps from a unitary group ${\rm U}(n)$ to a smaller group ${\rm U}(n-m)$. (These maps are versions of the Livšic characteristic function.) We calculate explicitly the direct images of the Haar measure under some maps. We evaluate some matrix integrals over classical groups and some symmetric spaces. (Values of the integrals are products of $\Gamma$-functions.) These integrals generalize Hua integrals. We construct inverse limits of unitary groups equipped with analogues of Haar measure and evaluate some integrals over these inverse limits.

We analyze the limit of the $p$-form Laplacian under a collapse, with bounded sectional curvature and bounded diameter, to a smooth limit space. As an application, we characterize when the $p$-form Laplacian has small positive eigenvalues in a collapsing sequence.

We prove two conjectures of J. Tits on the unipotent elements of semisimple algebraic groups defined over a field with positive characteristic.

The wave kernel for a class of second-order subelliptic operators is explicitly computed. This class contains degenerate elliptic and hypo-elliptic operators (such as the Heisenberg Laplacian and the Grušin operator). Three approaches are used to compute the kernels and to determine their behavior near the singular set. The formulas are applied to study propagation of the singularities. The results are expressed in terms of the real values of a complex function extending the Carnot-Caratheodory distance, and the geodesics of the associated sub-Riemannian geometry play a crucial role in the analysis.