Algebraic geometry studies systems of polynomial equations that, on the one hand, can be described algebraically using commutative algebra as ideals in polynomial rings, and on the other hand can be interpreted geometrically as their solution sets—so-called algebraic varieties. This fruitful interplay between algebraic and geometric perspectives, lasting over two centuries, is effectively illustrated by Hilbert’s Nullstellensatz as a fundamental “dictionary” between the two fields.

In the mid-20th century, this dictionary was formalized by Grothendieck, leading to the development of scheme theory. The language of schemes enables a unified approach to algebra, geometry, number theory, and topology, connecting concepts that were previously located in separate branches of mathematics.

In parallel, key objects of algebraic geometry were explored algorithmically. This opened up a productive interplay between theoretical research, algorithmic implementation, and computer-aided experimentation. Such methods are fundamental today—not only in geometry, number theory, and cryptography but also in representation theory.

Our research group focuses, among other topics, on algebraic surfaces, invariant theory, asymmetric cryptanalysis, code-based cryptography, representation theory, algebraic combinatorics, and number theory. Additionally, connections to algebraic topology, higher category theory, and mathematical physics are investigated, especially regarding algebraic structures in field theories. Another focus lies in quantum geometry, where mathematical structures are developed to describe quantization processes of physical systems and their dualities using methods from complex algebraic geometry, representation theory, and number theory.