Algebra, Geometry, and Number Theory Seminar

Abstracts

Candace Bethea - The degree in stable equivariant homotopy theory

I will talk about joint work with Kirsten Wickelgren defining a global degree in stable equivariant homotopy theory and showing it is equal to a sum of local equivariant degrees. We define what it means for an equivariant map of $G$-manifolds to be oriented relative to an equivariant ring spectrum, and show our definition of the equivariant degree recovers the classical equivariant degree definition of Segal for an equivariant map between representation spheres. I will also talk about an application of the local degree to counting orbits of rational plane cubics through a set of 8 general points which are invariant under a group action on $\mathbb{CP}^2$. Pre-talk at 2:30.

Michael Nelson - Homological Associativity of Differential Graded Algebras and Gröbner Bases

We investigate associativity of multiplications on chain complexes over commutative noetherian rings from two perspectives. First, we introduce a natural associator subcomplex and show how its homology can detect associativity. Second, we use Gröbner bases to compute associators.

Andreas Mono - A Modular Framework for Generalized Hurwitz Class Numbers

We discover a simple 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 \(\frac{3}{2}\) 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 Bruinier–Funke operator. All of these observations are deduced from a more general inspection of the weight \(\frac{1}{2}\) Maass–Eisenstein series of level \(4N\) at its spectral point \(s = \frac{3}{4}\). This idea goes back to Duke, Imamoğlu and Tóth in level 4 and relies on the theory of so-called sesquiharmonic Maass forms. This is joint work with Olivia Beckwith.

Fall Break - --

David Favero - TBD

Thanksgiving Break - --

PANTS Conference - Palmetto Number Theory Series

https://wt8zj.github.io/PANTS-XXXIX/

Last semester's seminar.