- 6635 South Hall
Two integral lattices of the same dimension and signature are said to be in the same genus if they are equivalent over the p-adic integers for every prime p. Each genus is a union of integral equivalence classes of lattices and reflects a general failure of local-to-global methods to characterize integral equivalence. By somewhat of a folk theorem, two lattices are in the same genus if and only if they become integrally equivalent after a direct sum with the rank-2 unimodular even lattice with bilinear form f(x,y) = x1 y2 + x2 y1. I will discuss the proof of this theorem, which relies on the spinor genus and the strong approximation theorem on rational orthogonal groups. Stable equivalence with respect to sums of the odd unimodular lattice with bilinear form f(x,y) = x1^2 - x2^2 arises naturally in physics through the classification of edge phases of two-dimensional fermionic systems having the same bulk topological order. While each genus consists entirely of even or odd lattices, these equivalence classes are unions of even and odd genera. I will discuss in what sense the theorem can be extended to cover this more general type of equivalence.