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Mathematics of shuffling

Cheryl Praeger

University of Western Australia

June 23, 2020 - 2pm

Online streaming on YouTube: https://www.youtube.com/watch?v=S7KTCA_1ntk

Cheryl Praeger

University of Western Australia

June 23, 2020 - 2pm

Online streaming on YouTube: https://www.youtube.com/watch?v=S7KTCA_1ntk

Abstract

The crux of a card trick performed with a deck of cards usually depends on understanding how shuffles of the deck change the order of the cards. By understanding which permutations are possible, one knows if a given card may be brought into a certain position. The mathematics of shuffling a deck of 2n cards with two “perfect shuffles” was studied thoroughly by Diaconis, Graham and Kantor in 1983. I will report on our efforts to understand a generalisation of this problem, with a so-called “many handed dealer” shuffling kn cards by cutting into k piles with n cards in each pile and using k! possible shuffles. A conjecture of Medvedoff and Morrison suggests that all possible permutations of the deck of cards are achieved, as long as k is not 4 and n is not a power of k. We confirm this conjecture for three doubly infinite families of integers, but the conjecture remains open. We initiate a more general study of shuffle groups, which admit an arbitrary subgroup of shuffles.

This is joint work with Carmen Amarra and Luke Morgan.

Short bio

Cheryl Praeger is Emeritus Professor of Mathematics at the University of
Western Australia. In 2007 she won an Australian Research Council
Federation Fellowship and in 2010-14 she served as inaugural Director of
the Centre for the Mathematics of Symmetry and Computation at UWA. Her
mathematical research work has transformed our understanding of how
groups act on large complex systems, through new theories,
constructions, algorithms and designs, which exploit the classification
of the finite simple groups.
Professor Praeger received BSc and MSc degrees from the University of
Queensland, a DPhil degree from the University of Oxford in 1973, and
has received honorary doctorates from Universities in six countries on
three continents. She has served on the Executive of the International
Mathematical Union and on the Council and Executive of the Australian
academy of Science. She is a former Vice President of the International
Commission for Mathematical Instruction, and former Foreign Secretary of
the Australian Academy of Science. She is Fellow of the American
Mathematical Society, an Honorary member of the London Mathematical
Society, and she was the first woman to be President of the Australian
Mathematical Society of which she is now an Honorary Life Member. In
2019 she was awarded the Prime Minister’s Prize for Science in
recognition of her incredible contribution to mathematics research and
education in Australia and around the world.
Professor Praeger has published more than 400 journal articles and five
research monographs, many of them with her students (30 PhD students, 10
research masters students, 21 postdoctoral research associates) and
research colleagues. She has played an active role supporting and
mentoring young scientists, especially women.
More infos are available here:
https://research-repository.uwa.edu.au/en/persons/cheryl-praeger
https://en.wikipedia.org/wiki/Cheryl_Praeger

More infos are available here:

https://research-repository.uwa.edu.au/en/persons/cheryl-praeger

https://en.wikipedia.org/wiki/Cheryl_Praeger