The classical Sylvester-Gallai theorem says that if a finite set of points in the Euclidean plane has the property that the line joining any two points contains a third point from the set, then all the points must be collinear. More generally, a Sylvester-Gallai type configuration is a finite set of geometric objects with certain "local" dependencies. A remarkable phenomenon is that the local constraints give rise to global dimension bounds for linear SG-type configurations, and such results have found far reaching applications to complexity theory and coding theory.

In this talk we will discuss non-linear generalizations of SG-type configurations which consist of polynomials. We will discuss how the commutative-algebraic principle of Stillman uniformity can shed light on low dimensionality of SG-configurations. I’ll talk about recent progress showing that these non-linear SG-type configurations are indeed low-dimensional as conjectured by Gupta. This is based on joint work with R. Oliveira.

The famous primitive element theorem states that every number field K is of the form Q(a) for some element a in K, called a primitive element. In fact, it is clear from the proof of this theorem that not only there are infinitely many such primitive elements in K, but in fact most elements in K are primitive. This observation raises the question about finding a primitive element of small “size”, where the standard way of measuring size is with the use of a height function. We discuss some conjectures and known results in this direction, as well as some of our recent work on a variation of this problem which includes some additional avoidance conditions. Joint work with Lenny Fukshansky at Claremont McKenna College.

We discuss Hodge-Tate crystals on the absolute prismatic site of O_K, where K is a p-adic field. These are vector bundles defined over the Hodge--Tate structure sheaf. We first classify them by O_K-modules equipped with small endomorphisms. We then classify rational Hodge-Tate crystals by nearly Hodge--Tate C_p-representations. This is joint work with Yu Min and Yupeng Wang.

A numerical semigroup is a collection of nonnegative integers that includes zero, is closed under addition, and has a finite complement. The gap of a numerical semigroup is defined as the complement of the semigroup. In this talk, we observe the distribution of even and odd integers within the gaps of numerical semigroups.