Moscow-Petersburg seminar on low-dimensional mathematics
September 5, 2003

Vladimir Chernov (Dartmouth)
"Affine linking numbers and their 
application to the study of front propagations".


Abstract: The linking number is the classical invariant of 
zero homologous objects $N_1$ and $N_2$ in a manifold $M$ 
such that $dim N_1+dim N_2+1=dim M$. We use the approach 
based on Vassiliev-Gusarov invariants to introduce a new 
``Affine linking invariant'' that generalizes linking 
numbers to the case of arbitrary nonzerohomologous $N_1$ 
and $N_2$.

The Causality Relation invariant $CR(W_1, W_2)$ of two 
fronts is the algebraic number of times one front has 
passed through the origin of the other before the other 
appeared. We show that affine linking numbers allow one to 
reconstruct the value of $CR(W_1, W_2)$ for all manifolds 
except of odd-dimensional rational homology spheres without 
any knowledge of the propagation law of fronts, and obtain 
other results relating affine linking to front propagation. 

The construction of the affine linking numbers is based on 
our generalization of Chas-Sullivan string homology bracket 
of mapping $S^1\to M$ to the case of mapping of garlands 
based on any manifold $P$ to $M$.

The talk is based on joint work with Yuli Rudyak, from the University 
of Florida Gainesville.