Selected Topics in Combinatorics and Geometry
MATH-678 / 2 credits
Teacher: Invited lecturers (see below)
Language: English
Frequency
Only this year
Summary
We introduce the following topics through Problems and Puzzles: Probability Puzzles/ The Probabilistic Method/ Monsky's Theorem, Sperner's Lemma, and p adic numbers/ Problems on Vectors/ Extremal Problems on Graphs/ Projective Geometry. If time allows: The Polynomial Method/ Continued Fractions.
Content
In this 6 weeks course we introduce selected mathematical topics through puzzles and problems in Combinatorics, Discrete Geometry, and Number Theory. The course is suitable only for graduate students.
There will be NO OVERLAP of topics with the course 'The Math of Puzzles' from the spring semester 2026. All lectures will be given by using computer presentation.
The topics we plan to cover and the schedule are as follows:
[Week 1]: Puzzles about Probablity
[Week 1]: The probabilistic Method and Applications [Week 2]: Puzzles about distances
[Week 3]: Monsky's Theorem, Sperner's Lemma and $p$ adic numbers [Week 4]: Puzzles on Vectors
[Week 5]: Extremal problems on Graphs [Week 6]: Projective Geometry
If time allows, we will also present the following two topics:
The Polynomial Method
Continued Fractions and Diophantine Approximation
The first week will be a bit different with two lectures.
Starting from the second week, the students will use the first class of the week to present their solutions in class for the HW puzzles of the topic of previous week. Then in the second class of the week there will be a lecture on a new topic followed by HW puzzles to be presented in the following week. Some open problems will be discussed as well every week.
Keywords
combinatorics, discrete geometry, probability, number theory
Learning Prerequisites
Recommended courses
Introduction to probability, Introduction to combinatorics, linear algebra, elementary calculus
Learning Outcomes
By the end of the course, the student must be able to:
- Use first moment method in probability
- Use forbidden subgraph in giving nontrivial upper bounds to various mathematical problems
- Use central projections and inversion to solve classical problems in geometry
- Use Soerner's Lemma in several classical mathematical
Resources
Bibliography
Any canonical textbook about the Probabilistic Method, Projective Geometry, Monsky's theorem, Extreme's Graph Theory
We will not follow any specific textbook.
Moodle Link
In the programs
- Number of places: 30
- Exam form: Oral presentation (session free)
- Subject examined: Selected Topics in Combinatorics and Geometry
- Courses: 14 Hour(s)
- Exercises: 10 Hour(s)
- TP: 15 Hour(s)
- Type: optional