Professor Monica Nevins (University of Ottawa) and Peter Latham, Heilbronn Research Fellow (King's College London, UK)
Department of Mathematics and Statistics
Coming from two radically different worlds of mathematics, North American Professor Monica Nevins has joined forces with European Postdoctoral Research Fellow Peter Latham to attempt to solve a major open problem in number theory.
For too long, a fundamental question in the representation theory of p-adic groups, called the unicity of types, has remained a mystery. These so-called “types” are small representations of compact groups that miraculously encode all of the information of a full infinite-dimensional representation of the p-adic group. In 2005, a French PhD student discovered that in some cases, you can replace each type with a unique maximal one, leading mathematicians to theorize that this is a general property of all p-adic groups. Last year, Prof. Nevins and her team applied new geometric tools to look further into this theory and promptly disproved it. Of course, what rises from the ashes is always more interesting and they have now discovered much more about the underlying mechanisms of types. In particular, they have made significant strides to proving a new theory about how types, in particular maximal types, can occur. This, in turn, has significant implications for the formulation of the inertial Langlands correspondence, a web of important and influential theories about connections between number theory and geometry, which happens to be one part of the ultimate goal in this field.
This collaboration is unique in the sense that two schools of thought, one North American, the other European, have dominated the theory of representations of p-adic groups; each developing its own set of tools to work on the classification of types. As Nevins and Latham learn each other’s “language”, they are discovering how to build on the strengths of each of the two separate worlds, and as such are sitting at the forefront of research in this area. Their work, though geared towards the goal of the unicity of types, encompasses a great deal more – they are building a set of tools that may bring them closer to the major objectives in this field. Immediate implications of this work are to number theory; however, it has the potential for much broader application in future. For example, the p-adic groups researched here complement real Lie groups, whose representation theory is fundamental to quantum mechanics. As our understanding of the quantum world evolves, so too does our need for more powerful mathematical tools to describe it.