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# Algebra and Number Theory

Algebra is the study of structures with operations like addition and multiplication, subject to certain properties. Our research in this direction ranges from number theory, arithmetic geometry and algebraic K-theory, through representation theory, the foundations of topological data analysis, and some computer algebra for formally verifying mathematics.

In its most basic form, number theory studies properties of the integers. Examples include studying prime numbers and the solubility of polynomial equations in integers, which nowadays found applications in cryptography on the internet.  Looking at these and related questions from a geometrical point of view leads to arithmetic algebraic geometry.  On the other hand, algebraic K-theory can be viewed as a far-reaching generalisation of the notion of dimension of a vector space. Originally defined because of its convenient formal properties, it is now related to many fields in mathematics, including number theory, algebraic geometry, hyperbolic geometry, and even theoretical physics.

The representation theory of quivers is a type of 'higher linear algebra' where one studies quivers (directed graphs) of vector spaces and linear maps. Such representations are ubiquitous in mathematics and play integral roles in representation theory, cluster theory, dynamical systems, geometry (algebraic, differential, symplectic), string theory, and, more recently, topological data analysis.

## Sample publication

Çanakçı, İlke, and Philipp Lampe. "An expansion formula for type A and Kronecker quantum cluster algebras." Journal of Combinatorial Theory, Series A 171 (2020): 105132.

The article introduces an expansion formula for elements in quantum cluster algebras associated to type A and Kronecker quivers with principal quantization. Furthermore, we discuss a relation of our expansion formula to generating functions of BPS states.

## Researchers and their interests

• ### Magnus Bakke Botnan. Topological data analysis.

I work at the algebraic foundations of topological data analysis, where commutative diagrams of homology vector spaces are the central objects. My main research objective is to extract information from such representations that can be used in data analysis.

Webpage: https://www.few.vu.nl/~botnan/

• ### Ilke Canakci. Cluster algebras.

The main focus of my work stems from the theory of cluster algebras, a structure which appears in seemingly different disciplines such as representation theory, combinatorics, geometry, dynamical systems and string theory. Central to my work is establishing the structural properties of cluster algebras in the presence of surface models using the combinatorial tools therein and exploring the interplay of such models in quiver representations.

• ### Sander Dahmen. Diophantine equations and formalized mathematics.

My research in number theory focusses mostly on the explicit resolution of Diophantine equations. These are multivariate polynomial equations (with integer coefficients) where the solutions typically have to be integers, or sometimes rational numbers. Much of my work entails developing, applying, and combining a diverse range of both theoretical and computational techniques, most notably from arithmetic geometry, involving Abelian varieties, modular forms, and Galois representations. I am also actively involved in the formalization of mathematics. This is mostly in connection with number theory, considering again both theoretical and more computational aspects.

Webpage: https://www.few.vu.nl/~sdn249/

• ### Pol van Hoften. Langlands programme and Shimura varieties.

My research is part of the Langlands programme, which is a vast web of conjectures and results connecting number theory, algebraic geometry, and representation theory. More specifically, I am interested in the algebraic geometry of Shimura varieties modulo p, and more recently also in their p-adic analytic geometry.

Webpage: https://polvanhoften2.github.io/

• ### Rob de Jeu. Algebraic K-theory, arithmetic algebraic geometry, number theory.

My work is concentrated around algebraic K-theory. It involves making K-groups more explicit, computing regulators, and establishing relations between those and the values of L-functions at certain points. Since this area relates to and uses techniques from number theory and arithmetic algebraic geometry, I also do research in those fields without any link to algebraic K-theory. As some of my work is computer based there are also algorithmic aspects to my research.

Webpage: https://www.few.vu.nl/~rju300/

• ### Assia Mahboubi. Type theory and formalized mathematics.

My research interests revolve around the foundations and formalization of mathematics in type theory and the automated verification of mathematical proofs. In particular, I am interested in the new insights that one often gets on familiar mathematical objects when looking for their most adequate formal representation for the purpose of computer-aided proof checking. I also have a special interest for the interplay between computer algebra and formal proofs, and more generally for computer-aided mathematics.