Classification of eight dimensional perfect forms

Perfect Delaunay Polytopes in Low Dimensions

A lattice Delaunay polytope is known as perfect if the only ellipsoid, that can be circumscribed about it, is its Delaunay sphere. Perfect Delaunay polytopes are in one-to-one correspondence with arithmetic equivalence classes of positive quadratic functions on the n-dimensional integral lattice that can be recovered, up to a scale factor, from the representations of its minimum. We develop a structural theory of such polytopes and describe all known perfect Delaunay polytopes in dimensions one through eight. We suspect that this list is complete.

Classification of Algebraic Function Fields with Class Number One

In this paper we prove that there are exactly eight function fields, up to isomorphism, over finite fields with class number one.

On the Form of Odd Perfect Gaussian Integers

We extend the sum-of-divisors function to the complex plane via the Gaussian integers. Then we prove a modified form of Euler's classification of odd perfect numbers.

A classification of the commutative Banach perfect semi-fields of characteristic 1. Applications

We define and study the concept of commutative Banach perfect semi-field of characteristic 1 by using results from Connes-Consani. We prove a Gelfand-Naimark type theorem allowing to identify such a Banach semi-field F to the semi-field of all the continuous functions on a compact space which we define as the spectrum of F. We give many applications.

二元情形下的调和完美格

Tempered Perfect Lattices in the Binary Case

本文在平面上完整分类了调和完美格，这是刻画MacPherson-McConnell用于计算SL(2,Z)及其算术子群Hecke算子的调和完美格的第一步，结果依赖于虚二次域序的类域论。

Perfect forms and the cohomology of modular groups

For N=5, 6 and 7, using the classification of perfect quadratic forms, we compute the homology of the Voronoi cell complexes attached to the modular groups SL_N(\Z) and GL_N(\Z). From this we deduce the rational cohomology of those groups.