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1. Lorentzian Fans

   https://doi.org/10.1093/imrn/rnad239  

   Ross, Dustin (October 2023, International Mathematics Research Notices)

   Abstract We introduce the notion of Lorentzian fans, which form a special class of tropical fans that are particularly well-suited for proving Alexandrov–Fenchel-type inequalities. To demonstrate the utility of Lorentzian fans, we prove a practical characterization of them in terms of their two-dimensional star fans. We also show that Lorentzian fans are closed under many common tropical fan operations, and we discuss how the Lorentzian property descends to the underlying tropical variety, allowing us to deduce Alexandrov–Fenchel-type inequalities in the general setting of tropical intersection theory on tropical fan varieties. 

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2. Promotion for fans of Dyck paths

   Pappe, Joseph; Pfannerer, Stephan; Schilling, Anne; Simone, Mary_Claire (April 2023, Seminaire Lotharingien de Combinatoire)

   We construct an injection from the set of r-fans of Dyck paths of length n into the set of chord diagrams on [n] that intertwines promotion and rotation. This is done in two different ways, namely as fillings of promotion matrices and in terms of Fomin growth diagrams. Our analysis uses the fact that r-fans of Dyck paths can be viewed as highest weight elements of weight zero in crystals of type Br, which in turn can be analyzed using virtual crystals. On the level of Fomin growth diagrams, the virtualization process corresponds to the Roby–Krattenthaler blow up construction. Our construction generalizes to vacillating tableaux as well. We give a cyclic sieving phenomenon on r-fans of Dyck paths using the promotion action. 

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3. Tropical fans and normal complexes

   https://doi.org/10.1016/j.aim.2023.108981  

   Nathanson, Anastasia; Ross, Dustin (May 2023, Advances in Mathematics)
4. Skeletons and Fans of Logarithmic Structures

   https://doi.org/10.1007/978-3-319-30945-3  

   Abramovich, Dan; Chen, Qile; Marcus, Steffen; Ulirsch, Martin; Wise, Jonathan (January 2016, Nonarchimedean and Tropical Geometry)

   We survey a collection of closely related methods for generalizing fans of toric varieties, include skeletons, Kato fans, Artin fans, and polyhedral cone complexes, all of which apply in the wider context of logarithmic geometry. Under appropriate assumptions these structures are equivalent, but their different realizations have provided for surprisingly disparate uses. We highlight several current applications and suggest some future possibilities. 

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5. Small Gröbner fans of ideals of points

   https://doi.org/10.1142/S0219498820500875  

   Dimitrova, Elena; He, Qijun; Robbiano, Lorenzo; Stigler, Brandilyn (May 2020, Journal of Algebra and Its Applications)

   In the context of modeling biological systems, it is of interest to generate ideals of points with a unique reduced Gröbner basis, and the first main goal of this paper is to identify classes of ideals in polynomial rings which share this property. Moreover, we provide methodologies for constructing such ideals. We then relax the condition of uniqueness. The second and most relevant topic discussed here is to consider and identify pairs of ideals with the same number of reduced Gröbner bases, that is, with the same cardinality of their associated Gröbner fan. 

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