Abstract: My research is focused on the existence of special Hermitian metrics on complex manifolds. In this talk, I will introduce some of the basic concepts and objects involved in complex differential geometry: Kahler forms, the Levi-Civita connection, the Ricci form, and the first Chern class. Then, I will give a very quick account of Yau's theorem before pivoting to the problem of how such a result might be generalized to the non-Kahler setting.

Contact person: Ioana Suvaina and Rares Rasdeaconu

Abstract: We will discuss some obstructions to the existence of "Bismut Hermitian-Einstein" metrics, providing counter-examples to a naive generalization of Calabi-Yau metrics. The obstruction will require us to introduce some concepts from generalized complex geometry. These tools will also prove to be very helpful for studying the flow equation. To attempt to overcome these obstructions, we will consider a generalization of the Kahler condition coming out of generalized complex geometry and some Monge-Ampere-type PDEs that arise in the quest for non-Kahler Calabi-Yau metrics in this setting.

Contact person: Ioana Suvaina and Rares Rasdeaconu

Abstract: Transfer systems are combinatorial objects that encode information about equivariant operations. More precisely, a transfer system encodes the transfers (or wrong-way maps) carried by algebras over certain equivariant operads. Thus, transfer systems allow us to use combinatorial tools to study equivariant homotopy theory. Compatible pairs of transfer systems, which are a pair of transfer systems satisfying certain conditions, correspond to multiplicative structures compatible with an underlying additive structure. In particular, compatible pairs are closely related to N_\infty-operads which encode commutative structures in equivariant homotopy theory. In this talk we introduce transfer systems, compatible pairs, and discuss how Rubin functors significantly aid our study of said transfer systems and their pairs. The work discussed in this talk is from two separate projects, the first is joint with DeMark, Hill, Kamel, Niu, Stoeckl, and Yan, and the second is joint with Darnall, Klanderman, Lewis, Shibata, and Trimble.

Contact person: Anna Marie Bohmann

Abstract: One of the primary tools of topological data analysis is persistent homology, where, in the most general setting, one associates to a dataset a filtration of chain complexes. Homology of this filtration then allows one to identify persistent topological features of the dataset. In this talk, based on joint work with Chris Kapulkin, we propose a new algorithm for computing persistent homology of a filtration of chain complexes. We suggest that persistence can be viewed as replacing a filtration with a quasi-isomorphic sequence of chain complexes whose differentials are all zero. This viewpoint leads to a new implementation for persistent homology, which we call RedZeD (for Reduction to Zero Differentials). The most common form of persistent homology is of Vietoris-Rips filtrations, and is the main motivation of this work. In this setting, our viewpoint allows for a further improvement to RedZeD. When implemented, we see an improvement in both time and memory over Ripser, the current fastest general algorithm for Vietoris-Rips persistence, developed by Ulrich Bauer.

Contact person: Chris Kapulkin