# Lattices

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Lattices have many significant applications in pure mathematics, particularly in connection to Lie algebras, number theory and group theory. They also arise in applied mathematics in connection with coding theory, in cryptography because of conjectured computational hardness of several lattice problems, and are used in various ways in the physical sciences.

## Extremal Lattices in $\operatorname{II}_{20}(5^{+10})$

Two new twenty-dimensional five-modular lattices have been found; finally settling the question of the existence of such lattices.