# Wintersemester 2018/19: Seminar Gitter und Codes

### Prof. Dr. Katrin WendlandDr. Santosh Kandel

Wann und wo: Di 14 - 16, SR 404, Ernst-Zermelo-Str. 1

Vorbesprechung: Mo, 16.07.2018, 14:15, SR119, Ernst-Zermelo-Str. 1
Um teilzunehmen, kommen Sie bitte in die Vorbesprechung des Seminares; eine Teilnehmerliste wird nicht vorab ausliegen.

Topic:
A lattice Γ of rank n in Rn is an additive subgroup of Rn of the form Γ=Ze1⊕...⊕Zen where (e1,...,en) is a basis of Rn. An example of a lattice in Rn is ZnRn. An important tool to study lattices, the so-called theta function of a lattice, comes from complex analysis. It is a holomorphic function on the complex upper half plane H and contains information about distributions of lattice points of fixed length. For example, if a lattice Γ is even, which means that the square of the length of x is an even integer for each x∈Γ, then the theta function can be used to count the number of lattice points of length √(2r) for each positive integer r. If an even lattice has the so-called unimodularity property, then the corresponding theta function becomes a modular form, which is a holomorphic function on H with certain symmetry properties. The theory of modular forms is useful in the classification of lattices, for instance, it can be used to show that there is a unique even unimodular lattice of rank 8 in R8 up to isomorphism.
The theory of lattices interacts deeply with coding theory. Here, by definition, a code is a certain fixed set whose elements are the "codewords". Choosing this "dictionary" and its mathematical properties conveniently can enable correction of transmission errors. As such, coding theory has many applications, for example in the telephone and satellite communication. There are some surprising parallels between the theory of lattices and coding theory. For example, the notion of unimodularity in the theory of lattices is analogous to the notion of self duality in coding theory, the theta function in the theory of lattices is analogous to the so-called weight numerator in coding theory and so on.
In this seminar, we will study lattices, codes and modular forms. We will also explore connections between them including the ones mentioned above.

Literatur:
Die Links führen auf Webseiten, von denen aus dem Universitätsnetz die jeweiligen Referenzen zugänglich sind. Falls kein Link gesetzt ist, finden Sie die Referenz in der Bibliothek des Mathematischen Institutes Freiburg.

Vortragsprogamm:
Das Vortragsprogramm finden Sie hier.
Die Vorträge können auf Deutsch oder auf Englisch präsentiert werden.

Tutorium: Dr. Santosh Kandel