Supervisor: Dr Natasha Blitvic
Project description:
Many of the problems in mathematical physics, algebra, and probability boil down to counting certain types of words in which the order of the letters plays a key role. For example, consider the words formed of two symbols, A and B, such that the number of B’s from left to right cannot exceed the number of A’s, and the number of A’s from right to left cannot exceed the number of B’s. (For example, AABABB is a valid word, whereas ABBABA is not.) These words are enumerated by Catalan numbers, which encode surprisingly many important objects in mathematics: for instance, Catalan numbers give the moments of Wigner’s semicircle law (the free-probabilistic analogue of the Gaussian random variable), or the degree of the Grassmannian G(1,n+1) (the set of lines in (n+1)-dimensional projective space). This project explores other types of noncommutative words, which come with interesting and challenging combinatorics, and have profound meaning in mathematical physics, from noncommutative probability to quantum field theory. Interested candidates should have a strong combinatorics background, and an interest in branching out to mathematical physics and/or probability theory.
Further information:
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Fees and funding