This two-year, $106,027 National Science Foundation Division of Mathematical Sciences Project Grant supports research into mapping class groups and transformation groups at the University of Maryland, College Park. The principal investigator and graduate students will develop new mathematical tools to study the structure of diffeomorphism groups and homeomorphism groups representing symmetries of manifolds. Previous work classified homomorphisms between such groups and proved conjectures regarding their dimensions. This award continues that work by analyzing gluing orbits of homomorphisms and determining which subgroups of mapping class groups admit sections under projection relationships. The team will also explore whether infinite torsion groups can act faithfully on high-dimensional manifolds. The research directly supports the NSF's Mathematical and Physical Sciences program and its goals of increasing scientific knowledge and understanding of major problems in these fields.