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Khovanskii bases, Newton-Okounkov polytopes and tropical geometry

Banff International Research Station for Mathematical Innovation and Discovery

I will talk about a recent work with Chris Manon giving a direct relation between the theory of Newton-Okounkov bodies (which is concerned with full rank valuations on a graded algebra) and tropical geometry (which is concerned with rank 1 valuations). More specifically, we show that an algebra has a full rank valuation with a finitely generated value semigroup if and only if there is a "prime cone" in its tropical variety for some presentation of this algebra as a quotient of a polynomial ring. A key concept in our theory is that of a Khovanskii basis. Roughly speaking, it is an algebra analogue of a Grobner basis for an ideal. This approach unites "toric degenerations" arising in the context of Newton-Okounkov bodies and the ones coming from tropical geometry. Representation theory provides many interesting and natural examples of this theory, in particular, wonderful compactification makes an appearance.

Mathematics; Algebraic geometry; Differential geometry

video/mp4

Author affiliation: University of Pittsburgh

2017-08-09

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/62530

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/