Geometry and Combinatorics Seminar - Alex Suciu (Math)

Speaker: Alex Suciu (Northeastern University)

"Sigma-invariants and tropical geometry"

The Bieri--Neumann--Strebel--Renz invariants Σq(X)Σq(X) of a connected, finite-type CW-complex XX are the vanishing loci for the Novikov--Sikorav homology of XX in degrees up to qq. These invariants live in the unit sphere inside H1(X,R)H1(X,R); this sphere can be thought of as parametrizing all free abelian covers of XX, while the ΣΣ-invariants keep track of the geometric finiteness properties of those covers. On the other hand, the characteristic varieties Vq(X)⊂H1(X,C∗)Vq(X)⊂H1(X,C∗) are the non-vanishing loci in degree qq for homology with coefficients in rank 11 local systems. After explaining these notions and providing motivation, I will describe a rather surprising connection between these objects, to wit: each BNSR invariant Σq(X)Σq(X) is contained in the complement of the tropicalization of V≤q(X)V≤q(X). I will conclude with some examples and applications pertaining to complex geometry, group theory, and low-dimensional topology.