TBA

TBA

Note the non-standard day

TBA

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.

Consider a rooted d-regular tree with \ell layers, where each vertex is colored either blue or green. Starting from the root, the color propagates down the tree so that each child inherits its parent’s color but flips with probability 30%. Now, suppose you only observe the colors of the leaves—can you infer the color of the root?

This setting describes a broadcasting process on trees, where in general we have q possible 'colors' and a transition matrix specifying the probability that a child receives color a given that its parent has color b. The associated inference problem is known as the Tree Reconstruction Problem.

A classical result, the Kesten–Stigum bound, characterizes a sharp threshold: above the bound, simply counting the colors at the leaves provides enough information to make a reliable guess of the root color, whereas below it, counting reconstruction is impossible.

In our recent work, we identify the Kesten–Stigum bound as a threshold of computational complexity. Specifically, we show that while it may still be statistically possible to infer the root color below the bound, any algorithm achieving this must overcome a complexity barrier.

I will aim to make this talk accessible to a broad audience, beyond probability.

This is a joint work with Elchanan Mossel.

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. If time permits, we will discuss an example of a weakly étale morphism which does not lift along a surjective ring map. This is joint work with Johan de Jong.