Department of Algebraic and Diophantine Geometry

In the coming years, the scientific programme of the Department of Algebraic and Diophantine Geometry of AMU will be focused on the following directions of mathematical research:

  • Algebraic geometry in the sense of Grothendieck, including the geometry of schemes, algebraic varieties and motives, and particularly cohomologies and Galois groups representations associated to these geometric objects and the theory of \ell -representations of basic Grothendieck group.
  • Diophantine equations, including arithmetic of group schemes, abelian varieties and elliptic curves. The department staff study Mordell-Weil groups and integer points on abelian varieties with the use of the theory of heights and the theory of diophantine approximation.
  • Theory of automorphic forms and modular forms in its aspects applicable to the above-mentioned research directions. The Langlands programme provides for the relations of Galois groups representations obtained from the algebraic varieties cohomologies with the theory of Diophantine equations and describes it in a precise manner.
  • Computational algebraic geometry and computational number theory. In day-to-day research practice in the above-mentioned fields of mathematics such computational systems as MAGMA and SAGE, used for numerical verification of hypotheses and quantitative analysis of theorems, are often applicable. Currently and probably in the future the computational methods will play an important practical role in the case of the research topics addressed by the department staff .
  • Universal algebra and logic. Polish logic follows the programme of algebraization established by A. Tarski. In this research, the logics are replaced by the relevant classes of algebras (e.g. by the algebras of function of algebraic varieties) which belong to the area of universal algebra. Since the 1980s, the programme of R. McKenzie of research on finite algebras has been underway; the programmes of Tarski and McKenzie are complementary. Polish specialization are studies on lattices of non-classical logics; Professor Kazimierz Świrydowicz has effectively studied the lattice of relevant logics for years. These studies will be continued.

Seminar on Algebra, Geometry and Arithmetic
Seminar list

Seminar on automorphic forms and abelian varieties
Seminar list

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