Events of the Week

In 2008, P. Diaconis and I.M. Isaacs introduced a generalization of classical character theory, which they called supercharacter theory. Let X be a partitioning of the irreducible characters of a finite group G such that the trivial character is in its own block of this partition. Let Y be a partitioning of the conjugacy classes of G such that identity element of G is in its own block of this partition. Define the supercharacters of G to be the sums of the irreducible characters in each block in X, weighted by their degree. Define the superclasses of G to be the unions of the elements in each block in Y. If the values of the supercharacters are constant on the superclasses, this is called a supercharacter theory of G. As a supercharacter is determined by these partitions, the structure of a group allows for multiple supercharacter theories. In this talk we will compare the construction of two different supercharacter theories, one based on the degrees of the irreducible characters of G and one based on the size of the conjugacy classes of G. In particular, we will demonstrate necessary and sufficient conditions that will force these two supercharacter theories to coincide when the groups considered are semidirect products of cyclic groups.

Of great interest is to establish the large time dynamical behavior of a fluid or plasma in a non-equilibrium state, whether transitioning to turbulence or relaxing to neutrality, a resolution of which requires a deep understanding of resonances (mainly between particles and waves). This talk aims to provide a roadmap towards resolving the problem, plus the current state of the art.

We discover a neat linear relation between the mock modular generating functions of the level 1 and level N Hurwitz class numbers. This relation gives rise to a holomorphic modular form of weight 32 and level 4N for N > 1 odd and square-free. We extend this observation to a non-holomorphic framework and obtain a higher level analog of Zagier’s Eisenstein series as well as a preimage under the ξ-operator. All of these observations are deduced from a more general inspection of the weight 12 Maass–Eisenstein series of level 4N at its spectral point s = 34 . This idea goes back to Duke, Imamo¯glu and Tóth in level 4 and relies on the theory of so-called sesquiharmonic Maass forms. Furthermore, we connect the aforementioned results to a regularized Siegel theta lift as well as a regularized Kudla–Millson theta lift for odd prime levels, which builds on earlier work by Bruinier, Funke and Imamo¯glu.

This is joint work with Olivia Beckwith.

Title and abstract to be announced