Department of Algebraic and Diophantine Geometry
In the upcoming years, the scientific programme of the Department of Algebraic and Diophantine Geometry 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 group representations associated with these geometric objects and the theory of \ell -representations of the basic Grothendieck group.
- Diophantine equations, including the 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 as applicable to the above research topics. The Langlands programme considers the relationship of Galois group representations obtained from the algebraic variety 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 these fields of mathematics, such computational systems as MAGMA and SAGE, used for numerical verification of hypotheses and quantitative analysis of theorems, are often applicable. Computational methods currently play an important practical role in the research work of the department’s staff, and are expected to continue to do so in the future.
- Universal algebra and logic. Polish logic follows the programme of algebraization established by Alfred Tarski. In this research, logics are replaced by the relevant classes of algebras (e.g. algebras of functions of algebraic varieties) belonging to the field of universal algebra. Since the 1980s, McKenzie’s programme of research on finite algebras has been pursued; the programmes of Tarski and McKenzie are complementary. Polish specializations include the study of lattices of non-classical logics, as pursued successfully for many years by Professor Kazimierz Świrydowicz. Research in this area will be continued.