In his proposal, the PI wishes to explore the central role of symmetries in various problems of geometry and topology. In more detail: The proposal includes the study of the equivariant Johnson- Wilson spectrum, denoted by ER(n), under the involution symmetry given by complex conjugation acting on the standard Johnson-Wilson spectrum. This theory ER(n) belongs to a family that starts with real K-theory. These theories have rich structure that has been explored by the PI in previous work. The power of this structure has been demonstrated by the PI and his collaborators in extending previously known results about the immersion dimension of real projective spaces. In his proposal the PI also studies infinite dimensional symmetry groups like Kac-Moody groups and symplectomorphism groups. Kac-Moody groups are infinite dimensional generalizations of compact Lie groups in terms of their structure and representation theory. The PI studies them from the standpoint of topology by studying their classifying spaces and the underlying equivariant K-theory. The PI also studies symplectormorphism groups of certain 4-manifolds. In earlier work, he and his coauthors have demonstrated that one can use homotopical techniques to completely understand these groups for rational ruled surfaces. The techniques used there appear to be applicable in much more generality which the PI plans to explore.
On a general level, the proposal attempts to solve geometric problems by first understanding the apparent symmetries in the problem and then applying algebraic techniques to the framework. For example, complex K- theory is an invariant constructed from families of complex vector spaces parametrized over a general space (a concept known as a complex vector bundle). Complex conjugation demonstrates an apparent symmetry of this structure and one obtains real K-theory by encoding this symmetry. This framework can be formalized into a very strong algebraic invariant the power of which was demonstrated almost 50 years ago to answer a fundamental question like which spheres have a group structure, or which euclidean spaces support the structure of a skew field. The PI studies a family ER(n) that generalizes real K- theory. In addition to this, the PI studies questions with apparent symmetries that are no longer finite, but still have fundamental geometric significance. For example, the infinite dimensional symplectomorphism group is defined as the symmetries of a smooth manifold which preserve a fixed symplectic form. The PI has shown that in some cases, this group is amenable to study using algebraic techniques.
