ultrafilter – Mike Pawliuk

Ramsey and Ultrafilters 2 – Ramsey DocCourse Prague 2016

The following notes are from the Ramsey DocCourse in Prague 2016. The notes are taken by me and I have edited them. In the process I may have introduced some errors; email me or comment below and I will happily fix them.

Special thanks to Ivan Khatchatourian for some elaborations.

Title: Ramsey and Ultrafilters 2 (of 2)

Lecturer: Slawomir Solecki

Date: Friday October 21, 2016.

Main Topics: Abstraction of Gowers/Lupini/Furstenberg-Katznelson, worked example with Furstenberg-Katznelson, slides about tensors

Definitions: Forestation of a Poset, Semigroup generated by a poset, [Specific to these constructions], $\hat{Y^{\prime\prime}(M)}$

Lecture 1 – Lecture 2

These two lectures are based on the following 2016 paper of Solecki: “Monoid actions and ultrafilter methods in Ramsey theory”.
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Ramsey and Ultrafilters 1 – Ramsey DocCourse Prague 2016

Special thanks to Ivan Khatchatourian for clarifying some typos and adding some elaborations.

Title: Ramsey and Ultrafilter 1 (of 2)

Lecturer: Slawomir Solecki

Date: Wednesday October 19, 2016.

Main Topics: Gowers’ Theorem, Lupini’s Theorem, Furstenberg-Katznelson Theorem, Monoids, Semigroups.

Definitions: [many]

Lecture 1 – Lecture 2

These two lectures are based on the following 2016 paper of Solecki: “Monoid actions and ultrafilter methods in Ramsey theory“.
Continue reading Ramsey and Ultrafilters 1 – Ramsey DocCourse Prague 2016

Stevo’s Forcing Class Fall 2012 – Class 5

(This is the fifth lecture in Stevo Todorcevic’s Forcing class, held in the fall of 2012. You can find the fourth lecture here. Quotes by Stevo are in dark blue; some are deep, some are funny, some are paraphrased so use your judgement. As always I appreciate any type of feedback, including reporting typos, in the comments below.)

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Stevo’s Forcing Class Fall 2012 – Class 1

(In the fall of 2012 I will be taking Stevo Todorcevic’s class in Forcing at the University of Toronto. I will try to publish my notes here, although that won’t always be possible.)

Summary of class 1.

Discuss examples of ccc posets whose product is not ccc.
Prove a theorem of Baumgartner’s that relates the branches in a tree to its antichain structure.
Display the differences in chain conditions.

Continue reading Stevo’s Forcing Class Fall 2012 – Class 1

Hindman’s Theorem write-up

It came to my attention that Leo Goldmakher had written up notes for a lecture I gave in August 2011 on the proof of Hindman’s Theorem via ultrafilters. The notes are quite nice so I thought I would share them.

Here is a link to the notes (pdf) and here is Leo’s website.

The lecture I gave follows the papers:

“An Algebraic Proof of van der Waerden’s Theorem” by Vitaly Bergelson, Hillel Furstenburg, Neil Hindman and Yitzhak Katznelson. (L’enseignement Mathematique, t. 35, 1989, p. 209-215)
“Ultrafilters: Some Old and some New Results” (pdf) by W.W. Comfort. (Bulletin of the AMS, Volume 83, Number 4, July 1977)