Mathematicians have discovered a new way to measure and compare algebraic structures—called homological stability—that reveals when large families of these objects become increasingly similar. This matters because algebras are fundamental mathematical objects that appear across pure mathematics and theoretical physics. Until five years ago, researchers did not know whether homological stability could apply to algebras at all. The discovery that it does opens a new avenue for understanding these structures, which underpin much of modern mathematics. The project has three concrete goals: finding new families of algebras that exhibit this stability, using homology to connect representation theory with group theory, and reviving the study of homological stability in Lie algebras—objects central to both mathematics and theoretical physics. This is fundamental science with no immediate practical application. But similar work on homological stability in other contexts has reshaped entire fields of mathematics. A deeper understanding of algebras could eventually influence areas from cryptography to quantum physics, though that lies decades ahead. For now, the value is in building the mathematical infrastructure that future discoveries will depend on.
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In mathematics we often want to study all the objects of a certain type. One way to do this is to find a way to "measure" the objects, so that we can use the measurements to tell the objects apart, understand them in more detail, and so on. Homology is one such way of measuring, and it can be applied to many different objects, from different areas of pure mathematics. If you have a sequence of objects, then the technique called homological stability, which belongs to the subject of algebraic topology, tells you that their homology gets more and more similar (so the objects are more and more alike) the further you go along the sequence. Homological stability doesn't apply to every sequence of objects, but there are many important cases where it does, and it is often one of the most important things that we know in those situations. Algebras are a kind of mathematical object that arise outside algebraic topology. New research over the last five years showed for the first time that homological stability can work for algebras: it applies in some important cases, the known methods often work, and new methods can be found in cases where the old ones fail. Homology is an important way to understand algebras, so homological stability for algebras is an important new development. This project will seize some of the opportunities it presents. We have three main objectives: A. We will find new families of algebras where homological stability applies. The majority of cases thus far are called diagram algebras. We will look instead at several important families of Hecke algebras. B. We will use homology as a bridge to bring ideas from representation theory (where algebras are studied) to the study of groups (where homology is used more often). We will study the homology of a large family of groups called Coxeter groups, and build a detailed picture of how the technique of modular reduction, taken from representation theory, applies in this different setting. C. We will study homological stability of Lie algebras, which are fundamental objects in mathematics and theoretical physics. This subject began in the 1980s, but little has happened since then, and the existing work is limited. Inspired by the recent progress in homological stability for algebras, we will revive the subject and push it into new territory. This project will show that homological stability of algebras is a widespread phenomenon; it will show that homology of algebras as a bridge for importing ideas from representation theory; and it will revive homological stability of Lie algebras as a working subject.
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