Miriam Kuzbary

NSF Postdoctoral Fellow and Visiting Assistant Professor at Amherst College.
Tenure Track Assistant Professor appointment to begin Fall 2024.

---

Research Interests: Geometric 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.

I recieved my Ph.D. from Rice University under the supervision of Shelly Harvey and my postdoctoral mentor at Georgia Tech was Jen Hom.

Notes and Videos of Previous Talks:

Video.

CKVK* Seminar

Notes.

Virginia Topology Conference at UVA

Video.

Andrews University Undergraduate Colloquium