Attached to an extension K/F of algebraically closed fields there is a combinatorial geometry called the "full algebraic matroid" A(K/F). It can be thought of as a geometric space whose "points" are algebraically closed fields between K and F of transcendence degree 1 over F, "lines" are algebraically closed intermediate fields of transcendence degree 2, and so on. Given six points in A(K/F) which satisfy certain matroid-theoretic conditions, Hrushovski's Group Configuration Theorem reconstructs an entire one-dimensional algebraic group in K. This construction was used by Evans and Hrushovski (1989) to detect subgeometries of A(K/F) which are projective planes and to prove that they are all coordinatized by a small list of possible skew fields. In this talk I will introduce the main ideas of their paper which draw on connections between model theory, algebraic groups, and classical projective geometry. I will then explain how these techniques can be developed further to prove that the recognition problem for algebraic matroids is algorithmically unsolvable. This is based on ongoing joint work with Geva Yashfe.

A positive geometry is, roughly speaking, a complex projective variety with a distinguished semialgebraic set in its real points, considered as the nonnegative part. Important to a positive geometry is a differential form, called the canonical form, that is compatible with the combinatorics of the boundary of the nonnegative part. Positive geometries are first studied by physicists and continue to have close connections to physics. On the other hand, del Pezzo surfaces are classical objects in algebraic geometry that are known for their rich combinatorics. In this talk, I will discuss del Pezzo surfaces and their moduli spaces and show that in many cases, they can be equipped with the structure of a positive geometry. Special attention will be given to the moduli space of marked del Pezzo surfaces of degree 3. This is joint work with Nick Early, Alheydis Geiger, Marta Panizzut, and Bernd Sturmfels.