Georgia Institute of Technology College of Sciences

One of the classical problems in scissors congruence is
this: given two polytopes in $n$-dimensional Euclidean space, when is
it possible to decompose them into finitely many pieces which are
pairwise congruent via translations?  A complete set of invariants is
provided by the Hadwiger invariants, which measure "how much area is
pointing in each direction."  Proving that these give a complete set
of invariants is relatively straightforward, but determining the
relations between them is much more difficult.  This was done by
Dupont, in a 1982 paper. Unfortunately, this result is difficult to
describe and work with: it uses group homological techniques which
produce a highly opaque formula involving twisted coefficients and
relations in terms of uncountable sums.  In this talk we will discuss
a new perspective on Dupont's proof which, together with more
topological simplicial techniques, simplifies and clarifies the
classical results.  This talk is partially intended to be an
advertisement for simplicial techniques, and will be suitable for
graduate students and others unfamiliar with the approach.

In this talk, I will discuss progress in our understanding of Legendrian surfaces. First, I will present a new construction of Legendrian surfaces and a direct description for their moduli space of microlocal sheaves. This Legendrian invariant will connect to classical incidence problems in algebraic geometry and the study of flag varieties, which we will study in detail. There will be several examples during the talk and, in the end, I will indicate the relation of this theory to the study of framed local systems on a surface. This talk is based on my work with E. Zaslow.