# Combinatorics Seminar 2015

Date: Mon, Nov 16, 2015

Time: 16:30 - 17:30

Venue: Middle Meeting Room

Title: Arithmetic and polynomials over fuzzy rings

Abstract:

An important operation in abstract algebra is the process of constructing appropriate quotients of given algebraic structures, e.g. quotient groups of given groups relative to normal divisors or quotients of rings relative to (two-sided) ideals.

While this does not yield anything of interest when applied to fields (for lack of two-sided ideals), the search for "coordinate rings" for combinatorial geometries (the name Gian-Carlo Rota proposed for matroids) rather than for affine or projective geometries suggested that many interesting structures can arise when replacing fields by "fuzzy fields" that are obtained by constructing quotients of fields (or rings or even fuzzy rings) relative to subgroups of their multiplicative group by "brute force".

Particularly interesting "fuzzy fields" arise for the subgroup of all positive numbers in an ordered field or the subgroup of units in a valuation subring.

In the lecture, I will introduce this proposal and indicate how not only matroid theory can profit from it.

Duality theory for finite and infinite matroids with coefficients

Grassmann-Plücker relations and matroids with coefficients

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