The finitistic dimension conjecture, which serves as a sufficient condition for numerous homological conjectures, has been a central topic of research in homological algebra. Despite over 60 years of study, the conjecture is only solved in special cases under the context of finite dimensional algebras. In this talk, we present three perspectives in studying the finitistic dimension conjecture: 1. the representation theory perspective, 2. the invariant perspective, and 3. the perspective of algebraic operations. These perspectives are closely connected, and their interaction often leads to new ideas for the conjecture. We also mention a potential geometric perspective for future study.

We study the universal compactified Jacobians of degree d, as constructed by Kass and Pagani over the moduli space of stable curves. These spaces compactify the universal degree-d Jacobian over the moduli of smooth curves, and depend on a choice of universal stability condition. We show that the Hodge numbers of these compactified Jacobians are independent of both the degree d and the choice of stability condition. This is joint work with Rahul Pandharipande, Dan Petersen, and Johannes Schmidt.

The theory of skein lasagna modules, initiated by Morrison-Walker-Wedrich, has been the subject of much interest in recent years. This theory takes as input Khovanov-Rozansky link homology and yields invariants of oriented 4-manifolds, which are generally very powerful yet hard to calculate. Most excitingly, this theory was used recently to detect exotic compact 4-manifolds by Ren-Willis, marking the first such result proven without the use of gauge theory. In this talk I will discuss an extension of skein lasagna theory for cornered 4-manifolds and describe an application to trisections of 4-manifolds, with an eye towards calculations. This is ongoing joint work with Slava Krushkal and Yangxiao Luo.