Degrees

Ph.D., Rice University
M.A., Rice University
B.S., The University of Texas at Dallas, Mathematics with minors in Physics and Performing Arts

Personal Website

kuzbarymath.com

Research Interests

Geometric Topology, Low dimensional Topology, Knot Theory, Group Theory

My mathematical interests lie at the intersection of group theory and topology; I am curious about groups whose elements are topological spaces themselves, like three-manifolds or knots. The three-dimensional spaces and their subspaces I usually think about only form groups once they are considered up to four-dimensional equivalence relations.

Topology in dimension four is very strange. In higher dimensions there is a lot of space to move around; this is analogous to how a person walking around on a two-dimensional plane cannot walk through a very high wall, but a bird can fly three-dimensionally over the wall instead. Since there is less freedom in dimensions lower than four, the proof techniques must be quite different even to prove the same theorems which are true for all higher dimensions. We can think of four-dimensional spaces as the ``bridge” between high dimensional and low dimensional behavior, and as such they exhibit many properties unique to dimension four. For example, there is only one n-dimensional Euclidean space up to diffeomorphism (a smooth bijection with smooth inverse) in dimensions other than four, but there are uncountably many distinct smooth four-dimensional Euclidean spaces.  These odd properties of dimension four produce rich and mysterious structure in the groups I like to think about. 

More specifically, I am interested in link concordance and groups related to the study of link concordance (such as the knot concordance group, the pure braid group, and the string link concordance group). My work addresses these topics using generalizations of classical techniques related to combinatorial group theory as well as modern gauge theoretic tools from Heegaard Floer homology. Additionally, I am curious about questions involving mapping class groups of surfaces and the various flavors of homology cobordism group. I also enjoy learning about applications of topology and geometry to more applied fields as well as to music and art, and have worked on a project involving knot theory and knitting.

Teaching Interests

My teaching interests are concerned with how to create a collaborative environment in the classroom where students from a broad range of mathematical and personal backgrounds feel both intellectually challenged and supported to meet those challenges. I enjoy teaching a broad range of topics, including classes unrelated to my research area, and I work to make material accessible and relevant to students. In the classroom, I aim to cultivate an interactive environment with an inquiry-based approach and provide opportunities for students to explore the material themselves. Finally, I am particularly sensitive to the needs of students studying to become math educators themselves, as I am a certified high school math teacher in Texas and have trained extensively in K-12 pedagogy.

I find Federico Ardila’s axioms to be a useful foundation for my teaching philosophy:

Axiom 1. Mathematical talent is distributed equally among different groups, irrespective of geographic, demographic, and economic boundaries.

Axiom 2. Everyone can have joyful, meaningful, and empowering mathematical experiences.

Axiom 3. Mathematics is a powerful, malleable tool that can be shaped and used differently by various communities to serve their needs.

Axiom 4. Every student deserves to be treated with dignity and respect.

Awards and Honors

NSF Mathematical Sciences Postdoctoral Research Fellowship (Fall 2021-Summer 2024)

Georgia Tech Student Recognition of Excellence in Teaching: Class of 1934 CIOS Honor Roll (Fall 2020, Spring 2021, Fall 2022)

AAUW American Dissertation Fellowship (Fall 2018)

NSF Graduate Research Fellowship (Fall 2014-Spring 2019)

Rice University Bob E. and Lore Merten Watt Fellowship (Fall 2012)

NPSC Fellowship (declined for personal reasons) (Fall 2012)