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TUCS Short Course: Partition Regularity of Matrices
Neil Hindman will give a short course and a distinguished lecture in Turku according to the following schedule:
TUCS short course (6 hours):
Tuesday 13.3. at 14-16 in XXIV
Wednesday 14.3. at 12-14 in XXIV
Thursday 15.3. at 14-16 in XXIII
FiDiPro Distinguished Lecture "Partition Regularity of Matrices":
Friday 16.3. at 10-12 in XXIII
Everybody welcome!
Mini-course
Day 1. I will cover basic results in semigroup theory that are relevant to the Stone-Cech compactification of a discrete semigroup. This will include results that follow from the existence of a minimal left ideal with an idempotent, including the structure theorem for the smallest two sided ideal of such semigroups. Some of the basic topological-algebraic results will also be presented, including the fundamental fact that any compact right topological semigroup has an idempotent.
Day 2. I will introduce the Stone-Cech compactification $\beta S$ of a discrete space as a set of ultrafilters and show how, if S is a discrete semigroup, its operation extends to this compactification making it a right topological semigroup. Some of the algebra which is specific to $\beta S$ will be developed and some of the simple combinatorial applications will be presented.
Day 3. I will concentrate on the Ramsey Theoretic applications of the algebraic structure of $\beta S$. In particular, I will look at the commutative Central Sets Theorem and some of its consequences.
Distinguished Lecture: Partition Regularity of Matrices
I will discuss the notions of Kernel Partition Regularity (characterized by Rado in 1932) and Image Partition Regularity, which was used by Deuber in 1973 to prove a famous conjecture of Rado's, but which was only characterized in 1993. Included will be simple consequences such as van der Waerden's Theorem on arithmetic progressions and the finite versions of the Finite Sums Theorem.