## Departmental PhD Thesis Exam – Ivan Livinskyi

Everyone is welcome to attend.  Refreshments will be served in the Math Lounge before the exam.

Wednesday, August 3, 2016
2:10 p.m.
BA6183

PhD Candidate:  Ivan Livinskyi
Supervisor:  Steve Kudla
Thesis title:  On the integrals of the Kudla-Millson theta series

****
Abstract:

The Kudla-Millson theta series $\theta_{km}$ of a pseudoeuclidean space $V$ of signature $(p,q)$ and lattice $L$ is a differential form on the symmetric space $\mathcal D$ attached to the pseudoorthogonal group $\mathrm{O}(p,q)$ that transforms like a genus $n$ Siegel modular form of weight $(p+q)/2$. Any integral of $\theta_{km}$ inherits the modular transformation law and becomes a nonholomorphic Siegel modular form. A special case of such integral is the well-known Zagier Eisenstein series $\mathcal{F}(\tau)$ of weight $3/2$ as showed by Funke.

We show that for $n=1$ and $p=1$ the integral of $\theta_{km}$ along a geodesic path coincides with the Zwegers theta function $\widehat{\Theta}_{a,b}$. We construct a higher-dimensional generalization of Zwegers theta functions as integrals of $\theta_{km}$ over geodesic simplices for $n\geq 2$.

If $\Gamma$ is a discrete group of isometries of $V$ that preserve the lattice $L$ and act trivially on the cosets $L^\ast/L$, then the fundamental region $\Gamma\backslash \mathcal D$ is an arithmetic locally symmetric space. We prove that the integral of $\theta_{km}$ over $\Gamma\backslash \mathcal D$ converges and compute it in some cases. In particular, we extend the results of Kudla to the cases $p=1$, and $q$ odd.

A copy of the thesis can be found here: Livinsky_Thesis