Date of Award

Spring 4-29-2024

Document Type

Honors Project

Degree Name

Bachelor of Science


Chemistry and Physics

Department Chair or Program Director

Asper, Janet

First Advisor

Villalba, Desmond

Second Advisor

Helmstutler, Randall

Third Advisor

Makhija, Varun

Fourth Advisor

Konieczny, Janusz

Major or Concentration



In nature, symmetries play an extremely significant role. Understanding the symmetries of a system can tell us important information and help us make predictions. However, these symmetries can break and form a new type of symmetry in the system. Most notably, this occurs when the system goes through a phase transition. Sometimes, a symmetry can break and produce a tear, known as a topological defect, in the system. These defects cannot be removed through a continuous transformation and can have major consequences on the system as a whole. It is helpful to know what type of defect is produced when a symmetry breaks. Defects are differentiated based on their dimension: a domain wall, cosmic string, and magnetic monopole are 2-, 1-, and 0- dimensional defects, respectively. Depending on the type, each defect can have very different implications for the entire system, so we classify them. We do this using homotopy theory. The basic idea of this is to show equivalence of paths based on continuous transformations. As we noted earlier, a defect cannot be removed upon a continuous transformation, so homotopy theory can tell us about the defect because it will not be “homotopically” equivalent to the rest of the surface. Specifically, we compute zeroth, first, and second homotopy groups to classify defects. As our Universe evolved, it is reasonable to ask if there were any defects produced. We look at some proposed models of our Universe’s evolution and compute homotopy groups to classify defects produced in phase transitions.