In this project, we aim to extend advanced mathematical tools to study more complex and unbounded shapes, and find new ways to calculate detailed properties of these shapes.
Index theory for non-compact spaces
Given a compact manifold (a bounded, edge-free shape), we can use index theory and the Atiyah-Singer index theorem to calculate important properties (invariants) of the shape. These invariants can give us a lot of information about the manifold. We can get even more detailed information by using a higher index that takes values in K-theory, which is a more advanced mathematical framework than just using integers. However, many interesting spaces in mathematics are not compact, but still have bounded volume. (For example the quotient of the upper half-plane by SL(2,Z).).
The classical Atiyah-Singer index theory has been extended to work with these non-compact spaces. Our goal is to also extend the higher, K-theoretic index theory to these non-compact spaces. Additionally, we aim to construct and compute higher versions of secondary invariants, such as eta invariants and analytic torsion, for these non-compact spaces. These secondary invariants provide even more detailed information about the spaces.
New insights
We expect the results of this project to be relevant they extend powerful mathematical tools to a broader class of spaces, which can lead to new insights and discoveries in geometry and topology. By understanding the properties of non-compact spaces with finite volume, we can solve more complex problems in mathematics and potentially find applications in number theory and theoretical physics, such as in the study of space-time.