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AG Audience

English

30 Minutes

Saarbrücken

From the theory of modular forms it is known that the minimum of an even unimodular lattice of dimension n is always ≤2 ⌊n/24⌋ + 2. Lattices achieving this bound are called extremal. Of particular interest are extremal unimodular lattices in the so called “jump dimensions”, these are the multiples of 24. There are four even unimodular lattices known in the jump dimensions, the Leech lattice Λ, the unique even unimodular lattice in dimension 24 without roots, and three lattices called P48p, P48q, P48n, of dimension 48 which have minimum 6. It was a long standing open problem whether there exists an extremal 72-dimensional unimodular lattice. Many people tried to construct such a lattice, or to prove its non-existence. Most of these attempts are not documented, all constructed lattices contained vectors of norm 6. On August 11 this year I discovered such an extremal lattice.

Benjamin Doerr

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Kolkom 2010