Construction and Symmetries of k-Free Lattice Systems from Quadratic and Biquadratic Number Fields

Final Defense
Construction and Symmetries of k-Free Lattice Systems from Quadratic and Biquadratic Number Fields

by Kurt Anthony C. de los Santos
MS Mathematics Candidate

Date: Monday, 19 August 2024
Time: 3 pm
Venue: SECA 321 (MJR room)

Advisers:
Eden Delight P. Miro, PhD
Ateneo de Manila University
Mark L. Loyola, PhD
Ateneo de Manila University

Panelists:
Manuel Joseph C. Loquias, PhD (Critic Reader)
University of the Philippines Diliman
Reginaldo M. Marcelo, PhD
Ateneo de Manila University
Job A. Nable, PhD
Ateneo de Manila University

The study of algebraic B-free systems, whose general idea is the construction of shift dynamical systems from arbitrary number fields, has become an interesting field of research in the theory of aperiodic order. These number-theoretic shifts are known for having relatively simple symmetries despite having positive entropy. This research aims to discuss the current results involving the symmetry group Aut(X_B,Z_K) and the extended symmetry group Sym(X_B,Z_K) of a shift space X_B arising from a number field K. We shall discuss the classical examples of B-free systems and their symmetries, together with the current results related to k-free lattice systems from quadratic number fields. We also show detailed proofs involving the extended symmetries of the k-free lattice system from Q( ✓2) to highlight some relevent techniques. We will then extend these techniques by considering biquadratic number fields. In particular, we will determine the extended symmetries of the k-free lattice system from the biquadratic number field Q( ✓2,i), together with discussions on the SageMath and Wolfram Mathematica programs that we created to implement the number-theoretic computations involved.