Codes and Expansions (CodEx) Seminar


Marc Christian Zimmermann (University of Cologne):
Critical lattices in the Gaussian core model

In this talk I discuss even unimodular lattices which are critical for potential energy with respect to Gaussian potential functions in the manifold of lattices having point density 1.

I will focus on our search for a local maximum and to this end the determination of the Morse index for even unimdoular lattices in small dimensions. For this we can use objects such as spherical designs, modular forms and root systems. All even unimodular lattices up to dimension 24 are critical, but none of them are local maxima. But it will turn out that hope is not lost when we venture into dimension 32.

(This is based on joint work with Arne Heimendahl, Aurelio Marafioti, Antonia Thiemeyer, and Frank Vallentin)

Building on the concepts used before I will give a brief outlook on two current projects: Firstly, how to find local maxima in smaller dimensions by abondoning even unimodular lattices, and secondly, how similar ideas can be used in an inhomogeneous version of energy, often referred to as lattice polarization.