The stellahedral toric variety is a toric variety whose cohomology ring and K-ring are both closely related to matroids. We construct an integral isomorphism from the K-ring to the cohomology ring of the stellahedral toric variety, and use this to prove that three equivalence relations on matroids, valuative equivalence, numerical equivalence, and homological equivalence, coincide. Based on joint work with Chris Eur and June Huh.

Consider an effective smooth action of a compact torus on a connected closed smooth manifold X having isolated fixed points. We introduce the finite graded poset S(X) called the face poset of the action. If X is a toric variety or a quasitoric manifold, then S(X) is the face poset of the moment polytope X/T. However, S(X) is defined for actions of any complexity, in which case the local structure of S(X) is determined by the linear matroids of tangent weights.

If an action on X is equivariantly formal, we prove that the geometrical realization |S(X)| has some degree of acyclicity, depending on tangent weights. This statement gives a homological obstruction for particular actions to be equivariantly formal. As a motivating example, we study canonical conjugation actions on the manifolds of isospectral Hermitian matrices, having zeroes at prescribed positions. We prove a complete classification, which of these manifolds are equivariantly formal.

This talk is based on several works written jointly with V.Buchstaber, V.Cherepanov, M.Masuda, G.Solomadin, and K.Sorokin.