Colloquium: Tuesday April 28. Speaker: Louis Rowen (Bar-Ilan). Title: Tropical Algebra.

Tropicalization involves passing to an ordered group M, usually taken to be (R,+) or (Q,+), and viewed as a semifield. Although there is a rich theory arising from this viewpoint, idempotent semi-rings possess a restricted algebraic structure theory, and also do not reflect important valuation-theoretic properties, thereby forcing researchers to rely often on combinatoric techniques.

Over the last few years, often jointly with Izhakian and Knebusch, we have studied an alternative structure, more compatible with valuation theory, that permits fuller use of algebraic structure in understanding the underlying tropical geometry. The idempotent max-plus algebra A of an ordered monoid M is replaced by R := L \times M, where L is a given indexing semiring (not necessarily with 0). In this case we say R layered by L. When L is trivial, i.e, L = {1}, R is the usual bipotent max-plus algebra. When L = {1, \infty} we recover the “standard” supertropical structure with its “ghost” layer. When L = {N} we can describe multiple roots of polynomials via a “layering function” s : R \to L.

Recently, two related structures have been developed, one by Sheiner related to the “exploded” algebra and the other by Perri involving homomorphisms of ordered groups.

In this talk, we describe these various algebraic approaches, focusing on supertropical linear algebra and a version of SL (done jointly with Izhakian and Niv).