My thesis consists of two projects I was a part of with my advisors and collaborators (defense slides).
In one, we (Kolaczyk, Lin, Rosenberg, W., Xu) studied the space of unlabeled networks, a quotient of Euclidean space by a natural subgroup of the symmetric group. When d=3, this space looks similar to the moduli space of triangles (though different, as edge weights are not subject to the condition a+b>c). In higher dimensions (more than three nodes), the edge permutation group induced by permuting the nodes is a proper subgroup, and the space is more complicated. We used the Fréchet mean, a generalization of the center of mass, to develop a notion of averaging networks, and establish a restricted set of existence and uniqueness criteria for such averages. Further, it is shown that central limit type theorems hold. This paper has been published in Annals of Statistics (vol. 48): https://projecteuclid.org/euclid.aos/1581930145
In the other, we (Szczesny, W., Williams) studied a relationship between a certain class of vertex algebras associated to Lie algebras, which describe symmetries of 2d conformal field theories, and factorization algebras in the formalism of Costello-Gwilliam. Factorization algebras, in a precise technical sense, generalize vertex algebras to global settings and allow one to compute invariants called conformal blocks in terms of factorization homology. In our work, we restrict to the local case where the base space is the complex plane rather than more general Riemann surfaces, and so only need to consider prefactorization algebras. This paper has been published in Advances in Mathematics (vol. 386): https://www.sciencedirect.com/science/article/abs/pii/S0001870821002383