Matěj Doležálek
Nationality: Czech Republic
Host: University of Konstanz
| Background: Matěj finished his bachelor studies at the Faculty of Mathematics and Physics of the Chalres University (MFF UK) in Prague, study programme General mathematics, and he has finished his master studies there, studying programme Mathematical structures. He also did a semester-long research internship at the University of Konstanz in the summer semester of 2023. |
| Master thesis: “Quaternion orders and quadratic forms” Abstract: A proof of Lagrange’s and Jacobi’s four-square theorem due to Hurwitz utilizes orders in a quaternion algebra over the rationals. Seeking a generalization of this technique to orders over number fields, we identify two key components: an order with a good factorization theory and the condition that all orbits under the action of the group of elements of norm 1 acting by multiplication intersect the suborder corresponding to the quadratic form to be studied. We use recent results on class numbers of quaternion orders and then find all suborders satisfying the orbit condition. Subsequently, we obtain universality and formulas for the number of representations by the corresponding quadratic forms. We also present a quaternionic proof of Götzky’s four-square theorem. |
| Research interests: algebraic geometry (in particular matroid invariants in toric geometry) and algebraic number theory (in particular universal quadratic forms over number fields). |
| Goal in TENORS: to study tensors as algebraic varieties and compute their invariants, with the hope that this will yield insights into the broad range of topics that tensor varieties generalize, from combinatorial invariants of graphs and matroids to algorithmic complexity of matrix multiplication. |
| Hobby: bouldering |
