Distinguished Lecture Series 2014: Persi Diaconis
17 Feb 2016, by in Events2 – 4 April 2014
Organised in collaboration with the School of Mathematics, University of Bristol, UK
Persi Diaconis, Stanford University, USA
Persi Diaconis is a world leading statistician working in probability, combinatorics and group theory. He is Professor of Mathematics and Mary Sunseri Professor of Statistics at Stanford University. He is the recipient of many honours and awards, among which are the MacArthur Fellowship, the Rollo Davidson Prize at the University of Cambridge, the Conant Prize and the Euler Prize of the AMS, and the Van Wijngaarden Award, Amsterdam.
Persi Diaconis is an exceptional communicator, and his public lectures are famous. He also had a very unusual career path for a mathematician. He was born in New York in 1945. After graduating from High School at 14, he left home to become a professional magician. When he was 24 years old, he went back to New York to study for a BSc in mathematics at City College, where he graduated in 1971. He then went on to obtain a PhD in Statistics in 1974 at Harvard University.
Lecture 1: The Mathematics of Shuffling Cards
I will explain why it takes about seven ordinary riffle shuffles to mix up a deck of 52 cards. The math involves group theory, Hopf algebras and lots in between. The theory has become law in Las Vegas and lots of other practical shuffling schemes remain to be studied.
Lecture 2: Shuffling Cards and Adding Numbers
When numbers are added in the usual manner, carries accrue along the way. These carries form a Markov chain with an “amazing” transition matrix. The same matrix comes up in character theory (Foulkes characters of the symmetric group) in algebraic geometry (Hilbert series of veronese embeddings) and in fractal properties of Pascal’s triangle mod two. It is also closely connected to riffle shuffling. In the end, we also learn some things about basic arithmetic.
This is joint work with Jason Fulman.
Lecture 3: Carries and Cocycles
When numbers are added in the usual way, ‘carries’ occur. Using balanced arithmetic can cut the carries down by a factor of two and nothing does better. Generalizing to other than groups, the carries are cocycles and one is led to the following problem: let H be a normal subgroup of a finite group G. Let X be coset representatives for H in G. As a measure of efficiency, let C(X) be the number of pairs x,y in X with xy in X divided by |X|^2. Thus if X can be chosen as a subgroup, C(X) = 1 and there are no carries. One of our theorems says that if C(X) is greater than 7/9 then there is a subgroup K with KH=G and K intersecting H only in the identity (so the extension splits). The many related topics make heavy use of additive combinatorics. Our best proof of the theorem above use approximate homomorphisms.
This is joint work with Fernando Shao and Kannan Soundarajan.
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