Convexity and Commutative Algebra

Statement of Travel/Invitation Goals: The characterization of Khovanskii bases given in [KMb] provides a connection between two important ways to combinatorialize an algebraic variety, and suggests many open problems to explore. In a recent paper with Ilten [IM], Ilten and I prove that the coordinate ring of any rational complexity 1 T-variety has many Khovanskii bases. In fact, we show that it is possible to find embeddings of such a variety so that every cone in its tropical variety is a prime cone from the theorem above. We call such an embedding is called ”well-poised.” Linear subspaces, toric varieties, and the Grassmannian variety Gr2(n) can be shown to carry well-poised embeddings, and it would be interesting to find other such varieties. For example, one approach is to attempt to generalize my work with Ilten by investigating T-varieties whose Chow quotients carry a well-poised embedding. It can be shown that the Newton-Okounkov bodies of well-poised varieties are closely related by piecewise-linear ”mutation” maps, making them natural candidates for other combinatorial equipment like cluster structures. It is also interesting to know when an algebra R has at least one finite Khovanskii basis. In recent work with Kaveh and his graduate student Murata [KMM], we’ve shown that any graded integral domain of finite type dim(R) has a finite Khovanskii basis for a valuation of rank dim(R)..1. It would be interesting to find conditions which ensure this occurs at full rank (for example, the global section ring of any curve corresponding to a divisor which is not equivalent to a point has no such Khovanskii basis). Based on recent work of Postinghel and Urbinati [PU17], it should also be possible to show such a finite Khovanskii basis exists in the total coordinate ring of any Mori dream space. Along this line, certain Mori dream spaces have a Cox ring which admits a presentation as a hypersurface, these are natural candidates for well-poised varieties. In order to develop ideas about T varieties and the well-poised property it would be useful to follow up my work with Nathan Ilten by visiting Simon Frasier University in Vancouver. The broader connections between Khovanskii bases, tropicalization, valuation theory, and toric geometry should be aided considerably by regular visits to Kiumars Kaveh and his students at the University of Pittsburgh. One important consequence of the work in [KMb] is the connection between Khovanskii bases and tropical geometry. If a collection of rank 1 valuations share a common Khovanskii basis, then the tropicalization map defined by this basis is 1 .. 1 on that collection; this provides a method for verifying that a tropical variety has an expected combinatorial structure or modular interpretation. Conversely, tropical operations, in particular tropical modification, give a discrete way to modify the generating set of an algebra until it is possibly a Khovanskii basis. In the curve case, it should be possible to prescribe a sequence of modifications which produce a Khovanskii basis, based purely on the combinatorics of the tropical curve. Cueto and Markwig ([]) have used ideas similar to these on tropical curves. Visits to OSU in Columbus Ohio would allow us to explore these ideas further with Angelica Cueto. The idea of using valuations, their Khovanskii bases, and generalizations of these structures to explain combinatorial features of certain algebraic varieties has also been studied intensely by Peter Littelmann’s group at Köln (see e.g. [FFL17], [FFL11]). I will be giving a series of workshop talks on the relationship between tropical geometry and representation theory in March of 2018, and it would be useful to follow up with visits to Littelmann’s group in Köln. This group’s work on so-called birational sequences in a Lie algebra looks to be a powerful organizing tool for the tropical geometry and Newton-Okounkov theory of flag varieties, and could inform and extend my work on toric degenerations and the WZNW model if properly developed. It is also worth pursuing a tropical interpretation of the Littelmann path model for bases in representations. Work of Kapovich, Leeb, and Millson [KLM08] on this topic has a tropical flavor which could be explained by spherical tropicalization, [KMa]. Travel to Europe would also enable me to follow up my collaboration with Joachim Hilgert and Johan Martens. The goal of this work is to further understand the ramifications of toric degeneration in the symplectic category. To start, this means formulating the meaning of such a degeneration as a generalization of the notion of contraction of Hamiltonian space [HMM].