Explicit methods for lattices over orders and applications

• Applicant: Professor Dr. Tommy Hofmann
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 539387714

Investigating number theoretic questions using algorithmic tools has a long tradition in mathematics, with Euclid and Gauß being among the prominent proponents of this idea. Since the 20th century, number fields (natural generalization of the rational numbers) play an important role in this area, which in recent has been intensified due to the connection to post-quantum cryptography. On the other hand, the focus has also shifted to arbitrary finite-dimensional algebras over global fields, which can be seen as a non-commutative generalization of number fields. As in the case of number fields, the focus is mainly on integral structures, which are given by orders and lattices. One of the main algorithmic challenges is the question of deciding, wether two given lattices are isomorphic (and finding an isomorphism in case it exists). We present a work program for the isomorphism problem of lattices, whose objectives are both theoretical and practical in nature. On the one hand, for the important class of Eichler algebras, the aim is to relate the complexity of the isomorphism problem to the classical situation of number fields. On the other hand, the goal is a practical algorithm, which applies to lattices and orders over arbitrary algebras. The aim is then to apply this algorithm to investigate topological spaces, since via the theory of higher homotopy groups, properties can be studied by means of lattice theoretic tools.

• DFG Programme: Research Grants