Arithmetic dynamical systems, polynomials, and polytopes

A famous mathematical problem, which remains unsolved almost 80 years after its discovery by the German mathematician Lothar Collatz, is the so-called Collatz conjecture, also known as the 3n+1 problem: pick any positive integer and, if even, divide it by two; otherwise multiply it by three and add one. If this rule is applied repeatedly, eventually one always seems to reach the number one, no matter the start value. Despite the fact that there is no known counter-example to this phenomenon and despite the fact that it can be fully explained using no more than the four basic arithmetical operations, it remains unproven up to this day. The two rules which define the respective successor of an even or odd number are an example of a so-called arithmetic dynamical system and the 3n+1 problem essentially is a conjecture on its ultimate behavior. Trying other rules than those which define the 3n+1 problem gives examples of arithmetic dynamical systems for which the question for ultimate behavior ranges from trivial to impossible in difficulty. One of the goals of the project is to analyze and categorize these examples in order to get a better understanding of the true nature of this great discrepancy in difficulties. In doing so it turns out that there is a connection between arithmetic dynamical systems and so-called permutation polynomials which define dynamical processes with similar properties. Another topic of this project is the study of triangulations of convex polytopes. Just as every polygon in the plane can be decomposed into triangles and every polyhedron in the three-dimensional space can be decomposed into three-sided pyramids, every polytope (the high-dimensional analogue of polygons and polyhedra) can be decomposed into simplices (the high-dimensional analogue of triangles and three-sided pyramids). Such a decomposition is called a triangulation. A recent result answers the question under which circumstances an arbitrary polytope can be triangulated in a way such that all simplices of the triangulation contain a given subset of the vertices of the polytope. This result allows for the computation of the volumes of the members of a known family of polytopes which was not possible before. There are many other families of convex polytopes (such as the so- called Birkhoff polytopes) the volumes of which play important roles in different fields and which are not known in general up to now. In addition to the study of arithmetic dynamical systems, the computation of the volumes of such polytopes is another aim of this project, as well as the further development of the underlying geometric result.

Arithmetic dynamical systems, polynomials, and polytopes A famous mathematical problem, which remains unsolved almost 80 years after its discovery by the German mathematician Lothar Collatz, is the so-called Collatz conjecture, also known as the 3n+1 problem: pick any positive integer and, if even, divide it by two; otherwise multiply it by three and add one. If this rule is applied repeatedly, eventually one always seems to reach the number one, no matter the start value. Despite the fact that there is no known counter-example to this phenomenon and despite the fact that it can be fully explained using no more than the four basic arithmetical operations, it remains unproven up to this day. The two rules which define the respective successor of an even or odd number are an example of a so-called arithmetic dynamical system and the 3n+1 problem essentially is a conjecture on its ultimate behavior. Trying other rules than those which define the 3n+1 problem gives examples of arithmetic dynamical systems for which the question for ultimate behavior ranges from trivial to impossible in difficulty. One of the goals of the project is to analyze and categorize these examples in order to get a better understanding of the true nature of this great discrepancy in difficulties. In doing so it turns out that there is a connection between arithmetic dynamical systems and so-called permutation polynomials which define dynamical processes with similar properties. Another topic of this project is the study of triangulations of convex polytopes. Just as every polygon in the plane can be decomposed into triangles and every polyhedron in the three-dimensional space can be decomposed into three-sided pyramids, every polytope (the high-dimensional analogue of polygons and polyhedra) can be decomposed into simplices (the high-dimensional analogue of triangles and three-sided pyramids). Such a decomposition is called a triangulation. A recent result answers the question under which circumstances an arbitrary polytope can be triangulated in a way such that all simplices of the triangulation contain a given subset of the vertices of the polytope. This result allows for the computation of the volumes of the members of a known family of polytopes which was not possible before. There are many other families of convex polytopes (such as the so-called Birkhoff polytopes) the volumes of which play important roles in different fields and which are not known in general up to now. In addition to the study of arithmetic dynamical systems, the computation of the volumes of such polytopes is another aim of this project, as well as the further development of the underlying geometric result.