Jan

22

2019

The paramodular conjecture pairs abelian surfaces and paramodular forms,

the 2-dimensional counterparts of the elliptic curves and elliptic modular

forms that correspond in the modularity theorem. As with modularity, the

level of the form matches the conductor of the surface. The paramodular

correspondence is now established for some particular levels, including

the nonprime level 731, where a computational result was required to

complete the analytic argument. This talk will be a quick overview of

Jan

25

2019

Jan

28

2019

In this talk, we will construct an example of a closed Riemann surface $X$ that can be realized as a quotient of a triply periodic polyhedral surface $\Pi \subset \mathbb{R}^3$ where the Weierstrass points of $X$ coincide with the vertices of $\Pi.$ First we construct $\Pi$ by attaching regular octahedra in a periodic manner then consider the surface of this solid. Due to periodicity we can ﬁnd a compact quotient of this surface. The symmetries of $X$ allow us to construct hyperbolic structures and various translation structures on $X$ that are compatible with its conformal type.

Jan

28

2019

In classical differential geometry, a central question has been whether abstract surfaces with given geometric features can be realized as surfaces in Euclidean space. Inspired by the rich theory of embedded triply periodic minimal surfaces, we seek examples of triply periodic polyhedral surfaces that have an identifiable conformal structure. In particular we are interested in explicit cone metrics on compact Riemann surfaces that have a realization as the quotient of a triply periodic polyhedral surface.

Jan

29

2019