Daniel Alexander Ramras
My research is at the interface between algebra, geometry, and topology. In particular, my research aims to use ideas from geometry and topology to study algebraic objects, and vice versa.
One of the central themes in my work is the idea that complicated algebraic structures can be understood using matrix models called linear representations. These models consist of systems of matrices whose multiplication reflects the algebraic structure in question. For instance, modular (or clock) arithmetic can be modeled using simple 2x2 matrices called rotation matrices. Another important example is that quaternion arithmetic, which is widely used to describe rotations in 3-dimensional space, can be represented in terms of 2x2 matrices called special unitary matrices.
In certain contexts, it is possible to catalogue, or classify, all of the linear representations of a given algebraic object. But when studying infinite algebraic objects such as the symmetries of a crystal pattern, this is usually not the case. In these contexts, it can help to study not just individual representations, but also how one representation can be continuously deformed into another. This idea of deformation brings tools from geometry and topology into the picture, leading to new understanding and new questions.
The key background for this project is linear algebra (especially eigenvalues and eigenvectors) and experience with writing proofs. A basic course in group theory or topology would also be very helpful, although background on these topics could be developed during the course of the project.