Date

Speaker

Title (click for abstract)

Wednesday, January 25, 4:105:00 in 740 Evans

Yufei Zhao, University of Oxford

Pseudorandomness in graph theory and additive combinatorics
(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 GreenTao 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 GreenTao theorem?

January 30, 11:1012:00

Emmanuel Tsukerman, UC Berkeley

Combinatorial Physics: you only live twice
(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.

January 30, 12:101:00

Anastasia Chavez and Felix Gotti, UC Berkeley

Dyck paths and positroids from unit interval orders
It is well known that the number of nonisomorphic unit interval orders on [n] equals the nth 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 2ncycle encoding a Dyck path of length 2n.

February 6



February 13

Fan Chung, UCSD

Sequences: random, structured or something in between?
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 'randomlike' 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.

February 20

Presidents' Day


February 27

Daniel Kane, UCSD

Recent results on the queen packing problem
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.

March 6



March 13



March 20



March 27

Spring recess


April 3

David Conlon, University of Oxford

Finite reflection groups and graph norms
For any given graph H, we may define a natural corresponding functional ·_{H}. We then say that H is norming if ·_{H} is a seminorm. 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.

April 10

Thomas Bloom, University of Bristol

Additive structure of sets of Fourier coefficients
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.

April 17

Bryan Gillespie, UC Berkeley

Combinatorics of forward exchange matroids
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.

April 24, 11:1012:00

Mark Skandera, Lehigh University

Combinatorial evaluation of Hecke algebra traces
(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 KazhdanLusztig basis elements. We will also discuss the open problem of evaluating these traces at other basis elements.

April 24, 12:101:00

Julia Wolf, University of Bristol

Counting monochromatic structures in finite abelian groups
It is well known (and a result of Goodman) that a random 2colouring of the edges of the complete graph K_{n} contains asymptotically the minimum number of monochromatic triangles (K_{3}s). Erdős conjectured that this was also true of monochromatic copies of K_{4}, 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 2colourings of finite abelian groups?
