Enumerating classes and characters of p-groups

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Groups are the mathematical language of symmetry. Whenever scientists aim to describe symmetries - be it of a lattice, the set of solutions to certain equations, or space-time - they are likely to use the language of group theory. Much research in finite group theory is motivated by the desire to understand in how many ways a given group may be represented by matrices. In practice, it is an unfeasibly hard problem to classify all such representations. Can we at least enumerate the representations of interesting groups? In our joint project, Professor O'Brien of Auckland, New Zealand, and I are investigating enumerative questions regarding representations and conjugacy classes of p-groups. A group is a p-group if its size is given by the power of a prime number p. An advantage of working with p-groups is the availability of a geometric method to construct all representations, known as the Kirillov orbit method. In previous work, we have already utilized the Kirillov orbit method for p-groups to give a geometric description of characters and classes of these groups, in terms of algebraic varieties over finite fields. During the projected visit, we are aiming to refine our analysis to answer outstanding questions on important specific classes of p-groups. We will also organize our results in research publications.