P+X
PROBABILITY + X: six lectures by Persi Diaconis
Probability has interactions with most areas of mathematics and science. The lectures that follow are based on these interactions. I hope that they will be accessible to advanced undergraduates and graduate students. I will try to give enough details so that the applications come into focus and show how these lead to new mathematics.
Lecture 1. X=group theory. (Tuesday, February 4th, 9h-11h, bât. 1R3 rdc, Amphi Schwartz)
Adding Numbers and Shuffling Cards
Abstract: When numbers are added in the normal way, « carries » occur along the way. These carries form a Markov chain with an « amazing » transition matrix. The same matrix occurs in the analysis of the usual method of the usual way of shuffling cards I will discuss the fact that « seven shuffles suffice to mix up 52 cards »). The matrix also occurs in areas of higher maths (fractals, Foulkes characters of the symmetric group, algebraic geometry). I will try to discuss all of this « in English ».
Lecture 2. X=additive combinatorics. (Tuesday, February 25th, 9h-11h, bât. 1R3 rdc, Amphi Schwartz)
An introduction to Additive Combinatorics
Abstract: Additive combinatorics is an emerging field of combinatorics and number theory. Led by stars like Gowers, Green and Tao, it has produced striking results (there are arithmatic progressions of arbitrary length in the primes). In this introductory talk, I take simple problems such as ‘carries’ in ordinary addition and show how basic ideas of approximate homomorphisms shed light on things. Typical theorem: let H be a normal subgroup of a finite group G. Choose coset representatives X for H in G. As an indication of how close X is to being a subgroup, let C(X) be # {x,y in X with xy in X} / |X|^2. In joint work with Soundarajan and Xiao we show that if C(X) is greater that 7/9 the X can be chosen as a subgroup (the extension splits).
Lecture 3. X=cooking. (Tuesday, March 11th, 9h-11h, bât. 1R3 rdc, Amphi Schwartz)
The Mathematics of Cooking Potatoes
Abstract: When food is stirred around in a frying pan, some ill-defined ergodic theorem explains that it gets roughly evenly browned and cooked. In joint work with Jean-Luc Thifault and Susan Holmes we have begun to make math of this.Typical theorem: picture n slices of potatoes around a circular pan. A spatula of radius d is inserted at random and those d slices are turned over. How long does it take until the up/down pattern becomes random? It turns out that it doesn’t depend on d; a tiny little spatula of radius one is as effective as a big spatula of radius n/2. I will also report some careful (and tasty) experimental work involving the heat equation.
Lecture 4. X= combinatorics. (Tuesday, April 15th, 9h-11h, bât. 1R3 rdc, Amphi Schwartz)
Understanding ‘Things’
Abstract: Most any mathematical object can be illuminated by asking what a typical object ‘looks like’. I will illustrate with the example of set partitions (there are 5 partitions of three things). What does a typical partitions look like? how many singletons, how many blocks, size of the largest block, number of crossings, … . These questions arose for me in understanding ‘supercharacters’ of the upper-triangular group. The topic provides a nice example of how the computer is changing mathematical work: In joint work with Chen,Kane and Rhodes, we define an algebra of statistics and show that the distribution of any of them has a nice formula (shifted Bell polynomials) which can be determined by exact computation for a few small n’s and then holds for all n.
Lecture 5. X=statistics. (Tuesday, May 13th, 9h-11h, bât. 1R3 rdc, Amphi Schwartz)
Understanding Exponential Random Graphs
Lecture 6. X=physics. (Tuesday, June 10th, 9h-11h, bât. 1R3 rdc, Amphi Schwartz)
Bose in Boxes
Abstract: Bose-Einstein condensation is a striking prediction whose experimental verification led to two Nobel prizes. It can be understood as a simple fact about dropping N balls into B boxes. In joint work with Chatterjee and Soundarajan, we determine the fluctuations of the condensate. Some strange new probability pops up. I here treat the case of non-interacting Bosons, extending this work to Fermions or problems with interactions is cutting edge research.