NSERC CRC II in Number Theory and Arithmetic Geometry
Mathematics and Statistics
Numbers, fractals, and dynamical systems are ubiquitous in our everyday lives. Integers are whole numbers such as 0, 1, 2, 999, 2147483647, etc. that not only appear in every ancient civilization but also continue to play a fundamental role in our modern society through coding theory and data security. Fractals are stunning objects with repeating patterns such as the shapes of snowflake, tree roots, etc. A dynamical system consists of the state a(n) such as the coordinate, weight, temperature, etc. of an object after n units of time in which the state a(n) depends on the previous state a(n-1) by a predetermined rule. Dr. Nguyen’s research area involves all of the above subjects under the principle of unlikely intersections which suggests that two random objects in dynamics should have a small intersection. This principle gives rise to the Pollard’s rho algorithms for the factorization and the discrete logarithm problems which are crucial in cryptography. Dr. Nguyen's research group uses tools from algebra, number theory, and dynamics to obtain progress in the mentioned theme of unlikely intersections and related aspects. More specifically, they investigate the dynamics of d-dimensional algebraic objects such as affine spaces and tori. Such systems model the physical world when d=3 and the data are taken from the real numbers. On the other hand, when our data are taken from finite fields (i.e. certain structures with only finitely many elements in which we can perform addition, subtraction, multiplication, and division), their research has potential applications to information theory. View contact information.