Georgia Institute of Technology College of Sciences

After quickly recalling the established theory on the combinatorics of orthogonal matroids, we define and study basic properties of the extended rank function and the modular tuples in orthogonal matroids. We then prove a weak version of the path theorem concerning the connectivity of circuits. 

Next, we consider representations of orthogonal matroids over fields (and more generally, over tracts) by bases. We then give a few applications, purely using this basis approach, to the representation theory of orthogonal matroids. We also give a different way of representing orthogonal matroids by circuit functions, which is proved to be equivalent to the basis approach. This is based on joint work with Matthew Baker and joint work with Donggyu Kim. 

The final part of the thesis focuses on the rescaling classes of representations. We construct the foundation of an orthogonal matroid, which possesses the universal property that the set of rescaling classes of representations is in one-to-one correspondence with the set of morphisms from the foundation to the target field. We also give explicit generators and relations of the foundation and an algorithm for computations. 

Zoom link: https://gatech.zoom.us/my/tongjinmath?pwd=QzRDalp2ditGL2tVNUozWm1RK1UwUT09