Mathematics

As the underlying configuration behind many elegant finite structures, partial difference sets have been intensively studied in design theory, finite geometry, coding theory, and graph theory. Over the past three decades, there have been numerous constructions of partial difference sets in abelian groups with high exponent, accompanied by numerous very different and delicate techniques. Surprisingly, we manage to unify and extend a great many previous constructions in a common framework, using only elementary methods. The key insight is that, instead of focusing on one single partial difference set, we consider a packing of partial difference sets, namely, a collection of disjoint partial difference sets in a finite abelian group. Although the packing of partial difference sets has been implicitly studied in various contexts, we recognize that a particular subgroup reveals crucial structural information about the packing. Identifying this subgroup allows us to formulate a recursive lifting construction of packings in abelian groups of increasing exponent.

This is joint work with Jonathan Jedwab.

Speaker Bio

Shuxing Li received his Ph. D. degree in Mathematics from Zhejiang University, China, in 2016. From September 2016 to September 2017, he was a postdoctoral fellow at Department of Mathematics, Simon Fraser University. He was an Alexander von Humboldt Postdoctoral Fellow from October 2017 to September 2019, at Faculty of Mathematics, Otto von Guericke University Magdeburg, Germany. Since November 2019, he is a PIMS Postdoctoral Fellow at Department of Mathematics, Simon Fraser University. His research focuses on finite configurations with strong symmetry, which involves algebraic and combinatorial design theory, algebraic coding theory, and finite geometry. In 2018, he received the Kirkman Medal from the Institute of Combinatorics and its Applications in recognition of the excellence of his research.

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.

Richard started the Unsolved Problems in Combinatorial Games Column. I'll consider some of his favourites, talk about some developments, and add a few reminiscences.