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1. Stellahedral geometry of matroids

   https://doi.org/10.1017/fmp.2023.24  

   Eur, Christopher; Huh, June; Larson, Matt (January 2023, Forum of Mathematics, Pi)

   Abstract We use the geometry of the stellahedral toric variety to study matroids. We identify the valuative group of matroids with the cohomology ring of the stellahedral toric variety and show that valuative, homological and numerical equivalence relations for matroids coincide. We establish a new log-concavity result for the Tutte polynomial of a matroid, answering a question of Wagner and Shapiro–Smirnov–Vaintrob on Postnikov–Shapiro algebras, and calculate the Chern–Schwartz–MacPherson classes of matroid Schubert cells. The central construction is the ‘augmented tautological classes of matroids’, modeled after certain toric vector bundles on the stellahedral toric variety. 

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2. Equivariant Log Concavity and Representation Stability

   https://doi.org/10.1093/imrn/rnab352  

   Matherne, Jacob P; Miyata, Dane; Proudfoot, Nicholas; Ramos, Eric (December 2021, International Mathematics Research Notices)

   Abstract We expand upon the notion of equivariant log concavity and make equivariant log concavity conjectures for Orlik–Solomon algebras of matroids, Cordovil algebras of oriented matroids, and Orlik–Terao algebras of hyperplane arrangements. In the case of the Coxeter arrangement for the Lie algebra $$\mathfrak{s}\mathfrak{l}_n$$, we exploit the theory of representation stability to give computer-assisted proofs of these conjectures in low degree. 

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3. An Improved Lower Bound for Matroid Intersection Prophet Inequalities

   https://doi.org/10.4230/LIPICS.ITCS.2023.95  

   Saxena, Raghuvansh R; Velusamy, Santhoshini; Weinberg, S Matthew (January 2023, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Tauman_Kalai, Yael (Ed.)

   We consider prophet inequalities subject to feasibility constraints that are the intersection of q matroids. The best-known algorithms achieve a Θ(q)-approximation, even when restricted to instances that are the intersection of q partition matroids, and with i.i.d. Bernoulli random variables [José R. Correa et al., 2022; Moran Feldman et al., 2016; Marek Adamczyk and Michal Wlodarczyk, 2018]. The previous best-known lower bound is Θ(√q) due to a simple construction of [Robert Kleinberg and S. Matthew Weinberg, 2012] (which uses i.i.d. Bernoulli random variables, and writes the construction as the intersection of partition matroids). We establish an improved lower bound of q^{1/2+Ω(1/log log q)} by writing the construction of [Robert Kleinberg and S. Matthew Weinberg, 2012] as the intersection of asymptotically fewer partition matroids. We accomplish this via an improved upper bound on the product dimension of a graph with p^p disjoint cliques of size p, using recent techniques developed in [Noga Alon and Ryan Alweiss, 2020]. 

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4. Foundations of Matroids, Part 1: Matroids without Large Uniform Minors

   https://doi.org/10.1090/memo/1536  

   Baker, Matthew; Lorscheid, Oliver (January 2025, Memoirs of the American Mathematical Society)

   Thefoundationof a matroid is a canonical algebraic invariant which classifies, in a certain precise sense, all representations of the matroid up to rescaling equivalence. Foundations of matroids arepastures, a simultaneous generalization of partial fields and hyperfields which are special cases of both tracts (as defined by the first author and Bowler) and ordered blue fields (as defined by the second author). Using deep results due to Tutte, Dress–Wenzel, and Gelfand–Rybnikov–Stone, we give a presentation for the foundation of a matroid in terms of generators and relations. The generators are certain “cross-ratios” generalizing the cross-ratio of four points on a projective line, and the relations encode dependencies between cross-ratios in certain low-rank configurations arising in projective geometry. Although the presentation of the foundation is valid for all matroids, it is simplest to apply in the case of matroidswithout large uniform minors. i.e., matroids having no minor corresponding to five points on a line or its dual configuration. For such matroids, we obtain a complete classification of all possible foundations. We then give a number of applications of this classification theorem, for example: We prove the following strengthening of a 1997 theorem of Lee and Scobee: every orientation of a matroid without large uniform minors comes from a dyadic representation, which is unique up to rescaling. For a matroid $M$without large uniform minors, we establish the following strengthening of a 2017 theorem of Ardila–Rincón–Williams: if $M$is positively oriented then $M$is representable over every field with at least 3 elements. Two matroids are said to belong to the samerepresentation classif they are representable over precisely the same pastures. We prove that there are precisely 12 possibilities for the representation class of a matroid without large uniform minors, exactly three of which are not representable over any field. 

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5. On Hodge-Riemann Cohomology Classes

   Julius Ross and Matei Toma (May 2023, Birational Geometry, Kähler–Einstein Metrics and Degenerations)

   Ivan Cheltsov, Xiuxiong Chen (Ed.)

   We prove that Schur classes of nef vector bundles are limits of classes that have a property analogous to the Hodge-Riemann bilinear relations. We give a number of applications, including (1) new log-concavity statements about characteristic classes of nef vector bundles (2) log-concavity statements about Schur and related polynomials (3) another proof that normalized Schur polynomials are Lorentzian. 

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