tbranzov

In the first part of the talk, I shall explore the consequences of distinguishing the foundations of meaning and the foundations of truth in mathematical statements, or imagination and rigor as motors of mathematical development. The foundations of meaning can be sought in our largely unconscious perception of the world, which modern cognitive science is exploring.

In this talk, I will first introduce the basic facts and ideas of non-archimedean uniformization and discuss some applications in mirror symmetry if time is permitted.

In my talk, I will overview progress in the area and its connection with other fields: theoretical computer science, number theory, and analysis. In particular, I will discuss a joint work with Zilin Jiang confirming Fejes Toth's long-standing zone conjecture and recent results with Alexey Glazyrin and Roman Karasev on a polynomial plank problem, a far-reaching generalization of Bang's theorem.

In this talk, partly based on joint work with H. Seppanen, I will present a description of the GIT-classes of L-ample line bundles on X and some properties of the respective GIT-quotients. Under mild assumptions, we prove the existence of a quotient whose Cox ring is, up to a finite extension, isomorphic to the ring of L-invariants in the Cox ring of X. This is indeed a special property, as such a quotient inherits, a priori, only information about the ample line bundle with respect to which it is defined.

We employ the Jordan algebras for a succinct description of the dynamical conformal symmetries of integrable models.

The study of codes over the rings (ring-linear codes) attracted great interest after the work of Calderbank, Hammons, Kumar, Sloane, and Sole in the early 1990s. In this seminar, the basic theory of linear codes over finite commutative rings will be presented including the importance of codes over rings, various kinds of rings for ring-linear coding theory, the weight functions on finite rings, MacWilliams equivalence theorem and the connection between these codes and codes over fields via the Gray maps. Moreover, the cyclic codes over finite commutative rings will be considered. Finally, some well-known generalizations of cyclic codes such as negacyclic, quasi-cyclic, polycyclic, multivariable, polynomial and Abelian codes will be introduced.