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We prove that the tropical surface of the root system A_{n−1} has degree n(n − 1)(n − 2)/2. 

Chow rings of toric varieties, which originate in intersection theory, feature a rich combinatorial structure of independent interest. We survey four different ways of computing in these rings, due to Billera, Brion, Fulton–Sturmfels, and Allermann–Rau. We illustrate the beauty and power of these methods by giving four proofs of Huh and Huh–Katz’s formula μ_k(M) = deg_M(α^{r−k}β^k) for the coefficients of the reduced characteristic polynomial of a matroid M as the mixed intersection numbers of the hyperplane and reciprocal hyperplane classes α and β in the Chow ring of M. Each of these proofs sheds light on a different aspect of matroid combinatorics, and provides a framework for further developments in the intersection theory of matroids. Our presentation is combinatorial, and does not assume previous knowledge of toric varieties, Chow rings, or intersection theory. This survey was prepared for the Clay Lecture to be delivered at the 2024 British Combinatorics Conference. 

We prove that the number of tropical critical points of an affine matroid (M,e) is equal to the beta invariant of M. Motivated by the computation of maximum likelihood degrees, this number is defined to be the degree of the intersection of the Bergman fan of (M,e) and the inverted Bergman fan of N=(M/e)*, where e is an element of M that is neither a loop nor a coloop. Equivalently, for a generic weight vector w on E-e, this is the number of ways to find weights (0,x) on M and y on N with x+y=w such that on each circuit of M (resp. N), the minimum x-weight (resp. y-weight) occurs at least twice. This answers a question of Sturmfels. 

The theory of matroids or combinatorial geometries originated in linear algebra and graph theory, and has deep connections with many other areas, including field theory, matching theory, submodular optimization, Lie combinatorics, and total positivity. Matroids capture the combinatorial essence that these different settings share. In recent years, the (classical, polyhedral, algebraic, and tropical) geometric roots of the field have grown much deeper, bearing new fruits. We survey some recent successes, stemming from three geometric models of a matroid: the matroid polytope, the Bergman fan, and the conormal fan. 

Muchos llegamos a la ciencia desde la curiosidad, en búsqueda del entendimiento y de la belleza. Con el tiempo descubrimos que la ciencia es una herramienta tremendamente poderosa. El posible éxito y aplicabilidad de nuestro proyecto científico nos enfrentan a preguntas éticas que no podemos ignorar. En una sociedad profundamente desigual, ¿quién tiene acceso a las tecnologías que puedan resultar? ¿Quién se beneficia y quién se perjudica? ¿Cuál es nuestra responsabilidad como científicxs e ingenierxs? ¿Qué papel jugamos en la construcción de una sociedad más justa y equitativa? Esta es la reflexión de un investigador cuyo trabajo en matemática “pura” encontró aplicaciones inesperadas. 

We prove that the maximum likelihood degree of a matroid M equals its beta invariant β(M). For an element e of M that is neither a loop nor a coloop, this is defined to be the degree of the intersection of the Bergman fan of (M,e) and the inverted Bergman fan of N = (M/e)^⊥. Equivalently, for a generic vector w ∈ R^E−e, this is the number of ways to find weights (0, x) on M and y on N with x + y = w such that on each circuit of M (resp. N), the minimum x-weight (resp. y-weight) occurs at least twice.