this paper is a slightly revised version of an introduction into singularity theory corresponding to a series of lectures given at the ``advanced school and conference on homological and geometrical methods in representation theory'' at the international centre for theoretical physics (ictp), miramare - trieste, italy, 11-29 january 2010. we show how to associate to a triple of positive integer...

To an integral lattice L in the euclidean space (R, (, )), one associates the set of characteristic vectors v ∈ R with (v, x) ≡ (x, x) mod 2Z for all x ∈ L. They form a coset modulo 2L, where L = {v ∈ R | (v, x) ∈ Z ∀x ∈ L} is the dual lattice of L. Recall that L is called integral, if L ⊂ L and unimodular, if L = L. For a unimodular lattice, the square length of a characteristic vector is cong...

Let G be a simple graph, and consider an orientation of the edges of G. Where V is the vertex-set of G, let A = (A,, : u, w E V) be the integral antisymmetric (0, +_ 1)-matrix such that A,,, = 1 if and only if VW is an edge of G directed from u to w. The orientation of G is said to be unimodular if for every WC V the determinant of the submatrix (A,, : u, w E W) is 0 or 1. For example, it is kn...

An integral square matrix A is called principally unimodular (PU if every nonsingular principal submatrix is unimodular (that is, has determinant \1). Principal unimodularity was originally studied with regard to skew-symmetric matrices; see [2, 4, 5]; here we consider symmetric matrices. Our main theorem is a generalization of Tutte's excluded minor characterization of totally unimodular matri...

Random mechanisms have been used in real-life situations for reasons such as fairness. Voting and matching are two examples of such situations. We investigate whether desirable properties of a random mechanism survive decomposition of the mechanism as a lottery over deterministic mechanisms that also hold such properties. To this end, we represent properties of mechanisms—such as ordinal strate...

In this paper, we discuss several relations between the existence of invariant volume forms for Hamiltonian systems on Poisson-Lie groups and unimodularity structure. particular, prove that vector fields a Lie group endowed with unimodular structure preserve multiple any left-invariant group. Conversely, also if there exists function such identity element is nondegenerate singularity associated...

We consider unimodular random rooted trees (URTs) and invariant forests in Cayley graphs. We show that URTs of bounded degree are the same as the law of the component of the root in an invariant percolation on a regular tree. We use this to give a new proof that URTs are sofic, a result of Elek. We show that ends of invariant forests in the hyperbolic plane converge to ideal boundary points. We...

A unimodular complex surface is a complex 2-manifoldX endowed with a holomorphic volume form Υ. A strictly pseudoconvex real hypersurface in X inherits not only a CR-structure but a canonical coframing as well. In this article, this canonical coframing is defined, its invariants are discussed and interpreted geometrically, and its basic properties are studied. A natural evolution equation for s...