(Please note the unusual time and place of this talk.)

I will highlight some connections between graph theory and additive combinatorics. I will explain how to prove the celebrated Green-Tao theorem, that the primes contain arbitrarily long arithmetic progressions. Following a graph theoretic viewpoint, we will see an answer to the question: what kinds of pseudorandomness are used in the proof of the Green-Tao theorem?

(Please note the unusual time of this talk.)

A sneak preview into the beauty and wonder of Combinatorial Physics. I will talk about Random Matrix Theory, heavy nuclei, Dyson's Threefold way, Macdonald polynomials, Heisenberg XX spin chain, localization and dualities.

It is well known that the number of non-isomorphic unit interval orders on [n] equals the n-th Catalan number. Using work of Skandera and Reed and work of Postnikov, we show that each unit interval order on [n] naturally induces a rank n positroid on [2n]. We call the positroids produced in this fashion unit interval positroids. We characterize the unit interval positroids by describing their associated decorated permutations, showing that each one must be a 2n-cycle encoding a Dyck path of length 2n.

There are many fundamental problems concerning sequences that arise in many areas of mathematics and computation. Typical problems include finding or avoiding patterns; testing or validating various 'random-like' behavior; analyzing or comparing different statistics, etc. In this talk, we will examine various notions of regularity or irregularity for sequences and mention numerous open problems.

We consider the problem of placing k queens on an n x n chessboard so that the number of unattacked squared is as large as possible. We focus on the domain where k is small relative to n. We are able to solve this problem by relating it to various related problems in additive combinatorics.

For any given graph H, we may define a natural corresponding functional ||·||_H. We then say that H is norming if ||·||_H is a semi-norm. A similar notion ||·||_r(H) is defined by || f ||_r(H) := || | f | ||_H and H is said to be weakly norming if ||·||_r(H) is a norm. Classical results show that weakly norming graphs are necessarily bipartite. In the other direction, Hatami showed that even cycles, complete bipartite graphs, and hypercubes are all weakly norming. Using results from the theory of finite reflection groups, we identify a much larger class of weakly norming graphs. This result includes all previous examples of weakly norming graphs and adds many more. We also discuss several applications of our results. In particular, we define and compare a number of generalisations of Gowers' octahedral norms and we prove some new instances of Sidorenko's conjecture. Joint work with Joonkyung Lee.

The collection of large Fourier coefficients of a function, whether they be called 'major arcs' or the 'large spectrum', are one way of representing the linearly structured component of a function, and as such plays an important role in many problems in additive combinatorics, analytic number theory, theoretical computer science, and beyond. In this talk I will discuss some results concerning what kind of additive structure such sets can have, and how this structure can be exploited to improve density increment arguments.

In this talk we will present several new results on forward exchange matroids, which are combinatorial structures that arise naturally in the theory of zonotopal algebra. In particular, a forward exchange matroid consists of an ordered matroid along with a choice of a certain type of order ideal on its bases. We will classify these objects by introducing an extension of Las Vergnas's external order on the bases of an ordered matroid. By identifying forward exchange matroids with order ideals in this extended ordering, we will derive an elegant recursive structure on these objects. This recursive structure may help to further unify the theory of the polynomial spaces constructed in zonotopal algebra.

(Please note the unusual time of this talk.)

The (type A) Hecke algebra H_n(q) is a certain module over Z[q^1/2, q^-1/2] which is a deformation of the group algebra of the symmetric group. The Z[q^1/2, q^-1/2]-module of its trace functions has rank equal to the number of integer partitions of n, and has bases which are natural deformations of those of the symmetric group algebra trace module. While no known formulas give the evaluation of these traces at the natural basis elements of H_n(q), there are some nice combinatorial formulas for the evaulation of certain traces at certain Kazhdan-Lusztig basis elements. We will also discuss the open problem of evaluating these traces at other basis elements.

It is well known (and a result of Goodman) that a random 2-colouring of the edges of the complete graph K_n contains asymptotically the minimum number of monochromatic triangles (K₃s). Erdős conjectured that this was also true of monochromatic copies of K₄, but his conjecture was disproved by Thomason in 1989. The question of determining for which small graphs Goodman's result holds true remains wide open. In this talk we explore an arithmetic analogue of this question: what can be said about the number of monochromatic additive configurations in 2-colourings of finite abelian groups?