TBA

TBA

Note the non-standard day

TBA

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.

In this talk we show that two curve singularities $(R, \mathfrak m)$ and $(R', \mathfrak m')$ 
are already isomorphic if there exists an isomorphishm 
$\varphi: R/ \mathfrak m^{j+1} \longrightarrow R'/ {\mathfrak m'}^{j+1}$ of 
$k$--algebras for some $j \geq 2 \delta +1$, and that the isomorphism may be chosen to agree with $\varphi 
\pmod{\mathfrak m^{j-2 \delta+1}}$. This strengthens a result of Hironaka, who obtained the
bound $3 \delta + 1$. 

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. We study their basic properties and give computationally effective calculations of the adjusted tight, Frobenius, and integral closures in the case of affine semigroup rings in terms of the convex geometry of the associated exponent sets. Finally, we study submodules of the fraction field of a domain defined in terms of our adjusted closures and the application of the new closures to an F-nilpotent property for ideal pairs. This is a joint work with Kyle Maddox and Lance Miller.