An algebraic variety over a field k is said to be rational if its function field is a purely transcendental extension of k. In this talk, we work over the real numbers and study the rationality question for a class of conic bundle threefolds; the varieties we consider all become rational over the complex numbers, but this rationality construction does not in general descend to R. This talk is based on joint work with Sarah Frei, Soumya Sankar, Bianca Viray, and Isabel Vogt, and on joint work with Mattie Ji.

A weakening of Frobenius splitting, Quasi-F-Splittings have proven to be a vital invariant in the study of varieties in positive characteristic, with numerous applications to arithmetic and birational geometry. This weaker condition extends the application of Frobenius to study singularities of arithmetically supersingular varieties, encompassing a much broader class of examples.  In this talk I'll provide an overview of Quasi-F-Splittings and introduce a local analogue, Quasi-F-Purity.

The weakly étale ring maps -- those which are flat and have flat diagonal -- have gained attention recently for their role in defining the pro-étale site of a scheme. We will propose a new definition of weakly étale ring maps via a lifting property analogous to the one used to define formally étale ring maps. We will use a result of Gabber on the cohomology of Henselian pairs to deduce the equivalence of the two definitions.

We define two new versions of integral and Frobenius closures of ideals which incorporate an auxiliary ideal and a real parameter. These additional ingredients are commonly used to adjust old definitions of ideal closures in order to generalize them to pairs. In the case of tight closure, similar generalizations exist due to N. Hara and K. I. Yoshida, as well as A. Vraciu.

The Bernstein-Sato Polynomial is a classical \(D\)-module invariant that has been used to measure the singularities of a hypersurface \(f\). In this talk we will review some of the classical theory before considering the Bernstein-Sato polynomial over a numerical semigroup ring \(R\). We will see that in this case the Bernstein-Sato polynomial detects not just of an element in \(R\) but of the ambient numerical semigroup ring itself. 

Given a finitely generated module \(M\) over a noetherian local ring \(R\), one may assign to it a conical affine variety, called the cohomological support variety of \(M\) over \(R\). This theory was first developed by Luchezar Avramov for local complete intersection rings in 1989, and by the work of many has recently been extended to encompass all commutative noetherian local rings. Geometric properties of this variety encode important homological information about \(M\) as well as \(R\).

Note the non-standard day

This talk is about joint work with I. Swanson.

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A longstanding question in algebraic geometry is the classification of  reduced and 
irreducible local complete one--dimensional domains $R$ over an algebraically closed 
field $k$. It is known that such a ring is completely determined once it is known up 
to a "sufficiently high" power of its maximal ideal, where this sufficiently 
high power depends on the singularity degree $\delta$ of the ring.