# Colloquia

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## Dave Futer – Temple University

### Can you hear the shape of a 3-manifold?

In the 1960s, Marc Kac popularized the question, "Can you hear the shape of a drum?" In slightly more mathematical language, the question asks: "Can you determine the shape of a domain in the plane from the spectrum of frequencies at which it vibrates?" The study of this question has been extended to surfaces and manifolds of other dimensions.

We now know that the answer is usually "no." There exist planar domains (and surfaces, and 3-dimensional manifolds) that have the same spectrum but different shapes. However, essentially all known counterexamples are related by a rigid cut-and paste procedure called commensurability. I will explain how this works in the context of 3-dimensional manifolds, leading up to some recent joint work with Christian Millichap.

## Ellie Bayat Mokhtari – University of Montana

### Influenza-type illnesses and air pollutants of particulate matter < 2.5μm (PM2.5): an application of Archetypal Analysis to identify spatiotemporal structure

Particulate matter (PM2.5) readings are often included in air quality reports from environmental authorities as it can pose the most danger when it builds up in human respiratory system and increases the risk of respiratory infections and lung diseases. Understanding the spatio-temporal variability of  upper respiratory illness and its dependence upon air quality in Montana is an area of active research in the public health sphere.

Archetypal analysis (AA), Culter and Breiman 1994, is introduced as a method to decompose and characterize structures within spatio-temporal data. AA seeks to synthesize a set of multivariate observations through a few, not necessarily observed points (archetypes), which lie on the boundary of the data cloud. This method is new in climate science, although it has been around for more than two decades in pattern recognition.

The goal of this presentation is to examine the spatio-temporal variability of two sets of weekly influenza cases and PM2.5 across Montana between 2008-2018 through AA. Compared to other conventional methods, such as PCA, the results provide the direct link to the observations which facilitate the interpretation. The patterns exposed by AA in both cases are contrasted, as one data set is approximately spatially continuous (PM) and the other is not (Flu counts).

## Derek Williams – Montana State University

### Relationships Between Undergraduate Students' Engagement and Understanding

This presentation discusses results from a mixed methods study investigating student engagement, understanding of precalculus concepts, and associations between engagement, understanding, and instructional approaches as reported by community college precalculus students. Student- and classroom-level factors associated with precalculus students' engagement are identified, and task-based interviews reveal a relationship between affective and cognitive experiences. Implications for teaching, and the current/future directions of this research are shared.

## Leonard Huang – University of Nevada, Reno

### Generalizing a Real-Analysis Exam Problem: A Potpourri of Functional Analysis, Probability, and Topology

This talk is inspired by the following problem, which has tormented many a graduate student in real-analysis qualifying exams around the world:

Let $$(x_{n})_{n \in \mathbb{N}}$$ be a sequence in $$\mathbb{R}$$. If $$\displaystyle \lim_{n \to \infty} (2 x_{n + 1} - x_{n}) = x$$ for some $$x \in \mathbb{R}$$, then prove that $$\displaystyle \lim_{n \to \infty} x_{n} = x$$ also.

In the spirit of mathematical research, one may now ask: Is this result still true if we replace $$\mathbb{R}$$ by some other topological vector space? In this talk, we will show that the result is true for a wide class of topological vector spaces that includes all locally-convex ones, as well as some that are not locally convex, such as the $$L^{p}$$-spaces for $$p \in (0,1)$$. We will then construct, using basic probability theory, an example of a badly-behaved topological vector space for which the result is false.

## Tien Chih – Montana State University-Billings

### Homotopy in the Category of Graphs

Homotopy Theory is the study of bending spaces into each other. While this is an essential part of the study of Topology, it does not translate immediately to discrete settings such as Graphs. However, we can use the language of Categories to abstract ideas from homotopy, and apply them to Graph Theory.

We begin by discussing the basic definitions of graph homotopies first established by Anton Dochtermann in 2008. We then discuss new results in homotopy of graphs, including a way to find a unique representative for each homotopy class of graphs. This work was done in collaboration with Dr. Laura Scull of Fort Lewis College.