Instructor: Professor Mohamed Omar, York University

Course Dates: September 4th - November 20th, 2024
Mid-Semester Break: October 14th - 18th, 2024
Lecture Times: Wednesday, 9:00 AM - 12:00 PM (ET)
Office Hours: By appointment

Registration Fee: PSU Students - Free | Other Students - CAD$500

Capacity Limit: 24 students

Format: Online via Zoom

Course Description

This graduate topics course is centered on the application of algebraic methods to problems in extremal combinatorics. The course will focus on methods that have been developed within the past two decades, with a highlight on methods that are state-of-the art; it will also feature some classical results whose techniques are ubiquitous in the literature. Central themes may include but are not limited to linear independence and rank methods applied to extremal problems in combinatorics, polynomial methods , the Combinatorial Nullstellensatz, and the slice and partition rank polynomial methods. Much of the course work will focus on analyzing themes in research articles that use these tools.

There is no mandatory textbook; the primary resources will be notes from class and research papers that we share in class. Specific resources will be shared during the course.

Prerequisites: (recommended) Linear Algebra, one of Combinatorics or Graph Theory

Course Grading: The course grade will be based on 3 assignments (60%) and a final project (30%) with a presentation (10%).

• Assignment 1 (20%): Due Sept 25
• Assignment 2 (20%): Due Oct 23
• Assignment 3 (20%): Due Nov 13
• Final Project (30%): Due Nov 27
• Final Presentation (10%): Nov 20

Assignments will be problems based on course material.

The final project is a 5-page latex written document (standard font) on an extremal combinatorics problem of your choice and a detailed account of algebraic approaches to resolve it. Your presentation will be in-class (via Zoom) and the amount of time for the presentation will depend on course enrollment.

The final grade is a letter grade, whereas throughout the term you will be getting numerical grades. The standard translation from numerical grades to letter grades will be used (or more generous one depending on the distribution of grades).

Tentative Course Schedule: This tentative schedule is subject to change but gives a guideline of what to expect for each lecture of the semester.

• Wed Sept 4: Course Intro & Odd Town Even Town Theorems
• Wed Sept 11: Mantel’s Theorem & Turan’s Theorem (algebraic approaches)
• Wed Sept 18: Turan’s Theorem Continued & Bipartite graphs (algebraic approaches)
• Wed Sept 25: Applications of Linear Independence of Polynomials
• Wed Oct 2: Combinatorial Nullstellensatz (theory and first applications)
• Wed Oct 9: Combinatorial Nullstellensatz II (more applications)
  
• Wed Oct 16: Reading Week
• Wed Oct 23: Slice Rank Polynomial Method (theorem proof & first applications)
• Wed Oct 30: Slice Rank Polynomial Method (more applications)
• Wed Nov 6: Slice Rank Polynomial Method & Intro to Partition Rank Polynomial Method
• Wed Nov 13: Applications of Partition Rank Polynomial Method
• Wed Nov 20: Final Presentations

Academic Integrity:

Discussion between students for general understanding of the material is strongly encouraged. All work to be submitted must reflect strictly individual effort, however. Plagiarism and cheating of any form will be dealt with according to university policy.

In addition, whenever a student submits work obtained through Course Hero, One Class, Chegg, or a similar website, the submitting student will be charged with plagiarism and the uploading student will be charged with aiding and abetting. Note also that exams, tests, and other assignments are the copyrighted works of the professor assigning them, whether copyright is overtly claimed or not (i.e. whether the © is used or not). Scanning these documents constitutes copying, which is a breach of Canadian copyright law, and the breach is aggravated when scans are shared or uploaded to third party repository sites.

Missed Work:

Students are required to inform their instructors of any situation which arises during the semester which may have an adverse effect upon their academic performance, and must request any considerations and accommodations according to the relevant policies and well in advance. Failure to do so will jeopardize any academic appeals.

Religious Accommodations:

If you require a religious accommodation for assessment, follow the procedures on this webpage:  https://w2prod.sis.yorku.ca/Apps/WebObjects/cdm.woa/wa/regobs.

Accessibility Accommodations:

York University has policies in place to ensure that all students have an equal opportunity to attain their educational goals. Accommodations related to diagnosed learning disabilities may be made through Student Accessibility Services. If you would like confidential support or academic accommodations, please visit https://accessibility.students.yorku.ca/.