The winding number of a continuous function on the unit circle counts how many times a graph of the function loops around the origin. It is homotopy invariant and has applications to several areas of Mathematics.
Toeplitz operators with continuous symbol are bounded linear operators on the Hardy Space involving multiplication by a continuous function. The index of such a Toeplitz operator is closely connected to the winding number of its symbol.
This connection is examined and then extended for Toeplitz operators with crossed product symbols.

Personality psychologists often apply clustering techniques on questionnaire data to model personality structure. Inspired by this work, we apply techniques from topological data analysis (TDA) to understand the structure of this data. The data comes from Cattell’s Sixteen Personality Factor Questionnaire (collected by Bell, Rose, & Damon in 1972). Subjects were 969 adult male volunteers divided into three age groups: 25 to 34, 35 to 54, and 55 to 82. We use persistent homology (a TDA tool) to cluster the data and identify that personality structure is slightly different between the age groups. It is also curious to note that data from the youngest age group appears to have a topological “hole”, which raises questions of the psychological significance. This work suggests that additional research, including applying TDA tools to other questionnaire data sets can provide insights to the study of personality.

An algebraic integer is a complex number that is a root of a monic polynomial with integer coefficients. It is well-known that there is not always a single algebraic integer that can generate the ring of algebraic integers contained in a field extension of the rational numbers. The index of an algebraic integer is a natural number that measures how far a ring of integers is from having such a "primitive element." We investigate these indices in cubic fields and determine which natural numbers occur as indices in given families.

Background: There have been many food waste studies done in elementary schools around the country. Several studies have determined that main entrées contribute significantly to plate waste in elementary school food programs, but studies relating the use of condiments and their influence on food waste need further exploration.
Objectives: Determine the correlation between the addition of condiments and the amount of plate waste from a chicken entrée.
Methods: In Phase I, data was collected in an elementary afterschool meal program. Researchers evaluated plate waste for the chicken entrée once a week for a total of four weeks. Chicken entrée plate waste was evaluated by weight and visual assessment. The waste weight was compared to the weight of one serving of the chicken entrée. A photograph of the total plate waste was taken each week for visual comparison. Researchers compared the total number of servings prepared to the number of servings leftover. In Phase II of the study a condiment (ketchup) was added to the menu when the chicken entrée was served. A marketing campaign was implemented with flyers to advertise the addition of the condiment. For the remaining four weeks, plate waste was documented using the same methods utilized during Phase I.
Results: In Phase I, an average of 26.7% of chicken entrées was wasted. In Phase II, an average of 20.8% of chicken entrées was wasted. No statistically significant difference was found in the percentage of food leftover between Phase I and Phase II (p<0.06). After adjusting for differences in initial portion size, there was still no statistically significant difference in weight of entrée left over (p<0.3).
Conclusion: Though there was no significant difference, the amount of waste is large enough to draw attention to the problem of waste in school foodservice. More research is necessary to determine what factors are leading to food waste.

Located inside the Large Magellanic Cloud, fierce explosions called supernovae have thrown out massive amounts of gas in every direction. A portion of this gas is aimed toward the Milky Way and is on a crash course with our galaxy. We are observing this gas with the Wisconsin H-Alpha Mapper, which provides a window into how the gas is distributed. These observations show two periods of supernovae explosions that created two distinct gas winds. One of these winds is currently active while the other was produced roughly 300 Million years old. Studying these gas clouds will provide information on how massive these winds are and the rate at which they are produced. The ejected gas is headed toward the Milky Way could supply our galaxy with additional gas to form stars in the future.