# Colloquia

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## Andrey Sarantsev – University of Nevada

### Autoregression Modeling of the American Stock Market

High price-to-earnings ratios indicate overvaluation of the stock market due to irrational exuberance (a phrase by Alan Greenspan). We develop new measures of earnings and study whether the market is currently in a bubble. We use linear regression and time series modeling.

## Danny Crytser – St Lawrence University

### Pure extension property for étale groupoid crossed products

A state on a $$C^*$$-algebra is a positive linear functional of norm $$1;$$ it's pure if it can't be written as a convex combination of other states. The GNS construction relates states to the representation theory of $$C^*$$-algebras and is crucial in the study of operator algebras. If $$A \subset B$$ is a unital inclusion of $$C^*$$-algebras, and every pure state on $$A$$ has a unique extension to a state on $$B$$, we say that $$A$$ has the extension property, first identified by Kadison and Singer. In this talk I'll discuss the extension property for inclusions of the form $$A \subset A \rtimes \mathcal{G}$$, where $$A \rtimes \mathcal{G}$$ is an étale groupoid crossed product. This covers inclusions arising from étale groupoids as well as discrete group crossed products.  This work builds off work of Zarikian and relates the extension property to a groupoid action on the spectrum. I'll also discuss the related question of the almost extension property, first defined by Nagy and Reznikoff.

## Juergen Kritschgau – Iowa State University

### Few H copies of F saturated graphs

A graph is $$F$$-saturated if it is $$F$$-free but the addition of any edge creates a copy of $$F$$. In this talk we will discuss the quantity $$\mathrm{sat}(n, H, F)$$ which denotes the minimum number of copies of $$H$$ that an $$F$$-saturated graph on $$n$$ vertices may contain. This parameter is a natural saturation analogue of Alon and Shikhelman's generalized Turán problem, and letting $$H = K_2$$ recovers the well-studied saturation function. We will focus on the cases where the host graph is either $$K_s$$ or $$C_k$$-saturated.

## Algebra SeminarAdam Chapman – Tel-Hai College in Israel

### The u-Invariant and Friends

The maximal dimension of an anisotropic quadratic forms over a given field is an important arithmetic field invariant known as the u-invariant.

We will discuss the computation of the u-invariant in certain cases, and propose a finer invariant that tells apart fields the identical u-invariant based on the linkage properties of quaternion algebras over the fields.

The talk is partially based on joint work with Jean-Pierre Tignol.

## Atish Mitra – Montana Tech

### The Space of Persistence Diagrams on n points Coarsely Embeds in Hilbert Space

TDA (Topological Data Analysis) is a rapidly developing field that uses ideas from geometry and topology to get qualitative and quantitative information about the structure of data (finite sets of points in a metric space). One of the tools is the idea of Persistent Homology, which takes a one-parameter family of topological spaces and creates a signature called the persistence diagram that encodes useful information about the data set. For using existing kernel methods for analyzing such persistence diagrams, one needs to know how close the various metrics on the space on persistence diagrams can be to an inner product structure.

Using the methods of coarse geometry, we prove that the space of persistence diagrams on n points (with either the Bottleneck distance or a Wasserstein distance) coarsely embeds into Hilbert space.  We also discuss various non-embeddability results when the number of points is not bounded.

This is joint work with Žiga Virk.

## Ellie Bayat-Mokhtari – University of Idaho

### Information theoretic-based Biological Network Analysis

Biological networks are complex and often contain nonlinear interactions among a usually large number of species, genes, nutrients, metabolites, .... Correlation coefficients are widely used to analyze “omics” data as measures of linear interactions. However, how would we detect dependence when data is non-linear? In this talk, I will use mutual information based graph theory to analyze microbiome network and introduce a method to find a partition between contaminants and true bacteria that minimizes the loss of information. Among all the possible partitions of a network, this can be considered an optimal partition for characterizing the underlying structures of the network. Time permitting, I will briefly, discuss other projects that I have been working on.

## Tracy Payne – Idaho State University

### N-Graded Filiform Nilpotent Lie Algebras

We will start with a general discussion of Lie algebras, and nilpotent Lie algebras in particular, giving some examples and motivations for their study. Next we will focus on the subclass of N-graded filiform nilpotent Lie algebras, giving definitions, and examples. Finally we describe some recent results on the classification of N-graded nilpotent Lie algebras done jointly with Cameron Krome and John Edwards using iterated central extensions and a Python program.

Available Dates:

April 6, 13, 20