In a joint work (in progress) with J. Heller, following Edinin-Graham and Totaro's construction for equivariant Chow groups we construct a Borel-style G-equivariant algebraic cobordism for G-schemes, where G is a lineal algebraic group over a field of characteristic zero. We will discuss some basic properties and some computations for some groups G.

In the first lecture we will introduce some basic notions to study the behaviour of polynomial mappings, we will see what can happened at infinity and how the bifurcation set is related to the set of asymptotic critical values. Then, in the next two lectures we will focus on the asymptotic variety of polynomial mappings, i.e. on the set of points at which such a map fails to be proper. The geometry of this set and how we can get explicit description of it will be discussed.

A function f : Rⁿ -> R is said to be piecewise polynomial if there exist finitely many polynomials f_i in n variables such that for every point a in Rⁿ we have f(a) = f_i(a) for at least one f_i. The celebrated Pierce-Birkhoff conjecture asserts that every piecewise polynomial function f on Rⁿ can be obtained from a finite collection of polynomials by iterating the operations of maximum and minimum. This is equivalent to saying that there exists a finite collection f_{ij} of polynomials such that f = max limits_i (min limits_j f_{ij}). In this lecture, I will describe an approach to proving this conjecture proposed by J. Madden in the nineteen eighties and which we continue to  develop more recently with F. Lucas, D. Schaub and J. Madden. Our key tool is  the real spectrum of a ring; a large part of the lecture will be devoted to introducing the real spectrum.

Toric spaces have associated to them Borel constructions with respect to the actions of various tori. The cohomology (corresponding to complex-oriented theories) can be related to Stanley-Reisner rings, rings of piecewise polynomials and associated quotients. Examples exist for which the distinction between a ring of piecewise polynomials and the Stanley-Reisner ring mirrors that between the true orbit space and the Borel construction. The discussion will include also a short overview of work in progress on the KO-theory of these spaces.

The material is based on joint work with Matthias Franz and Nigel Ray and touches on additional joint work with Martin Bendersky, Fred Cohen and Sam Gitler.