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
Zn⊂Rn. 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
