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Abstract: The Tutte polynomial is a powerful invariant associated to any graph or matroid. Besides its combinatorial interest, Tutte polynomials also play a role in other areas such us knot theory and statistical physics.
In the talk we survey the main properties of Tutte polynomials, both of graphs and matroids, and discuss recent results. Particular attention is payed to computational complexity issues.
Colloquium - 16:00
Abstract: We report on recent work in progress. After giving a minimal amount of contextual information about lattice-ordered groups (l-groups for short), we show how to associate an l-group to an abstract simplicial complex, in much the same way as one constructs the Stanley-Reisner ring from the same data. We then characterise the class of l-groups arising in this manner --- they are exactly the finitely generated projectives. We next outline an algebraic theory of support functions of a unimodular fan entirely expressed in the language of l-groups. Building on this, we prove:
Main Result. Finitely generated projective l-groups endowed with a distinguished set of generators coordinatise smooth toric varieties.