Activities This Week

We will introduce operator spaces and systems, completely bounded and completely positive maps, and give some examples.

Modeling data by latent tree models is a powerful approach in multiple applications. A canonical example of this setting is the “tree of life”, where the evolutionary history of a set of organisms is inferred by their DNA. Generally, in latent tree models, the main task is to infer the structure of the tree, given only observations of its terminal nodes. While inferring a tree structure is a common task, in many applications, a robust algorithm for the recovery of large trees is still missing.

In this talk, we will see a new method for the recovery of latent tree models, which is based on spectral graph theory. We show that the hidden tree structure is strongly related to the spectral properties of a graph, defined over the terminal nodes of the tree. Finally, we see that while in terms of accuracy the method performs similarly to state-of-the-art methods, it is significantly more computationally efficient.

The Katz-Litt theorem gives an explicit recipe to describe Vologodsky integration on curves with semi-stable reduction in terms of Coleman integration on on the rigid analytic domains reducing to the smooth components of the reduction. In work with Mueller and Srinivasan we gave an alternative recipe, more closely related to our past work with Zerbes, which was proved to follow from the Katz-Litt theorem by Katz. In this talk I will describe this alternative recipe and prove it directly. This new proof is significantly simpler than the original proof.

Let F be a free group and K a field. The free group algebra K[F] bears a strong resemblance to F, making it an excellent tool in the study of free groups. For example, by a theorem due to Cohn and Lewin, one-sided ideals in K[F] are free as K[F]-modules, analogously to the Nielsen-Schreier theorem. I will discuss this resemblance, along with other motivations for our interest in K[F] arising from the theory of word measures. I will then present a new algorithm for deciding if a given element is part of some basis of a given ideal, similarly to what Whitehead’s algorithm performs in free groups. Based on joint work with Danielle Ernst-West and Doron Puder.