Title:  
Modified geometries, Clifford algebras and graphs: their impact on discreteness, locality and symmetry

Abstract:  
A lot of mathematical structures used in physics meet the goals of empirical predictions yet don't stand up to various conceptual standards. One example of this is the way discrete models violate Lorentz symmetry, yet said violation is too small to be empirically observed and, therefore, is overlooked. Another example is Grassmann numbers that are not defined outside of integral, but since every calculation is ultimately based on the integral of said Grassmann numbers, empirical predictions can still be made. 

In my thesis I will explore various ways of modifying those structures so that they make sense on a more conceptual level. I will explore different models, along with what criteria each model meets. For example, the model involving Riemannian metric on a tangent bundle to Lorentzian manifold might address locality issues, but it fails to produce exact symmetries upon discretization. The matrix models produce exact symmetries, but it is not clear whether it would approximate the results from lattice models. 

Since my motivation is purely conceptual, I make no claim on improving the actual predictions. The validity of my proposals rests on the conjecture that their predictions approximate the predictions of more conventional models. In some cases, such as Berezin integral, I was able to prove thats the case. In other cases, such as causal set theory and matrix models, the most I could do was provide some qualitative arguments as to why one can hope for the approximations to work, and left the mathematically rigorous work for the future. 
 

This will be on Zoom:  https://unm.zoom.us/j/98058707305