Number Theory

Start with any natural number. If it is even, divide it by two. If instead it is odd, multiply it by three and add one. Now repeat this process indefinitely. The Collatz conjecture asserts that no matter how large an initial number one starts with, this process eventually reaches the number one (and then loops back to one indefinitely after that). This conjecture has been tested for quintillions of initial numbers, but remains unsolved in general; it is perhaps one of the simplest to state problems in all of mathematics that remains open; it is also one of the most notorious "mathematical diseases" that can lure professional and amateur mathematicians alike into devoting hours of futile effort into trying to solve the problem. While it is itself mostly a curiosity, and the full resolution still remains well out of reach of current technology, the Collatz problem is a model example of the more general concept of a dynamical system, which occurs throughout mathematics and science; and so progress on the Collatz conjecture can shed some light on the more general problem of understanding dynamical systems. In this lecture we give some of the history of the Collatz conjecture and some of its variants, and also describe some recent partial results on the problem.

About Dr. Tao:

Terence Tao was born in Adelaide, Australia in 1975. He has been a professor of mathematics at UCLA since 1999. Tao's areas of research include harmonic analysis, PDE, combinatorics, and number theory. He has received a number of awards, including the Fields Medal in 2006, the MacArthur Fellowship in 2007, the Waterman Award in 2008, and the Breakthrough Prize in Mathematics in 2015. Terence Tao also currently holds the James and Carol Collins chair in mathematics at UCLA, and is a Fellow of the Royal Society and the National Academy of Sciences.

These are sequences formed by iterating the sum-of-proper-divisors function. For example: 12, 16, 15, 9, 4, 3, 1, 0. Of interest since Pythagoras, who remarked on the fixed point 6 (a perfect number) and the 2-cycle 220, 284 (an amicable pair), aliquot sequences were also one of Richard Guy's favorite subjects. The Catalan--Dickson conjecture asserts that every aliquot sequence is bounded (either terminates at zero or becomes periodic), while the Guy--Selfridge counter-conjecture asserts that many aliquot sequences diverge to infinity. It is interesting that Guy and Selfridge would make such a claim since no aliquot sequence is known to diverge, though the numerical evidence is certainly suggestive. The first case in doubt is the sequence beginning with 276. This talk will survey what's known about the problem and give evidence for and against the two countervailing views.

Richard Guy was a supporter of the database of integer sequences right from its beginning in the 1960s. This talk will be illustrated by sequences that he contributed, sequences he wrote about, and especially sequences with open problems that he would have liked but that I never got to tell him about.

Joy Morris (University of Lethbridge, Canada)

A natural way to understand groups visually is by examining objects on which the group has a natural permutation action. In fact, this is often the way we first show groups to undergraduate students: introducing the cyclic and dihedral groups as the groups of symmetries of polygons, logos, or designs. For example, the dihedral group $D_8$ of order 8 is the group of symmetries of a square. However, there are some challenges with this particular example of visualisation, as many people struggle to understand how reflections and rotations interact as symmetries of a square.

Every group G admits a natural permutation action on the set of elements of $G$ (in fact, two): acting by right- (or left-) multiplication. (The action by right-multiplication is given by $\left{t_g : g \in G\right}, where $t_g(h) = hg$ for every $h \in G$.) This action is called the "right- (or left-) regular representation" of $G$. It is straightforward to observe that this action is regular (that is, for any two elements of the underlying set, there is precisely one group element that maps one to the other). If it is possible to find an object that can be labelled with the elements of $G$ in such a way that the symmetries of the object are precisely the right-regular representation of $G$, then we call this object a "regular representation" of $G$.

A Cayley (di)graph $Cay(G,S)$ on the group $G$ (with connection set $S$, a subset of $G$) is defined to have the set $G$ as its vertices, with an arc from $g$ to $sg$ for every $s$ in $S$. It is straightforward to see that the right-regular representation of $G$ is a subset of the automorphism group of this (di)graph. However, it is often not at all obvious whether or not $Cay(G,S)$ admits additional automorphisms. For example, $Cay(Z_4, {1,3})$ is a square, and therefore has $D_8$ rather than $Z_4$ as its full automorphism group, so is not a regular representation of $Z_4$. Nonetheless, since a regular representation that is a (di)graph must always be a Cayley (di)graph, we study these to determine when regular representations of groups are possible.

I will present results about which groups admit graphs, digraphs, and oriented graphs as regular representations, and how common it is for an arbitrary Cayley digraph to be a regular representation.