Jozsef Solymosi

Office: MATH 220

MATH 309 Section 201

Term II, 2016-17

Topics in Geometry

Prerequisite: One of MATH 152, MATH 221, MATH 223 and one of MATH 220, MATH 226, CPSC 121. Specifically, it will be assumed that students are familiar with basic techniques of mathematical proof and reasoning such as induction and proof by contradiction. Students will be expected to write logically correct and mathematically coherent proofs as part of homework and examinations.

The course syllabus will be as follows:

_ Euler’s formula. Convexity, Graphs, Trees, Convex Polyhedron, Planar Graphs.

_ Planar Graphs. Drawings, Colouring, Structure, Kuratowski’s Theorem.

_ Geometric Graphs. Drawings, Intersections, Crossings.

_ Crossing Number. Basic Probability, Upper and Lower Bounds.

_ Point-Line incidences. Szemeredi-Trotter Theorem (bounds on incidences)

_ Metric problems in Discrete Geometry. The unit distances problem, distinct distances.

_ Combinatorics of point-sets. Erdos-Szekeres Theorem, Halving lines.

_ Circle Packing. Planar circle packing, Lattice Packing, Sphere Packing. (Chapter 1. In Combinatorial geometry )

_ Geometry of Numbers. Pick’s Theorem, Minkowski’s Theorem, applications. (Chapter 1. In Combinatorial geometry )

Course notes (relevant notes from the web) will be provided after lectures. There are two advanced books which cover most of the topics of the course.

_ Lectures on discrete geometry / Jiri Matousek.

_ Combinatorial geometry [electronic resource via UBC Library] / János Pach, Pankaj K. Agarwal.

Evaluation: There will be two midterm exams and one final exam, as well as weekly homework assignments. Homework will be assigned on Thursdays, and due the following Thursday in class. Late homework will not be accepted.

Curve: 60-40% for term work/ final or final/term work.

The course mark will be computed as follows:

Final Exam: 50 percent

Midterm Exams: 20 percent x 2 = 40 percent

Homework: 10 percent

You are required to be present at all examinations. No makeup tests will be given. Non-attendance at an exam will result in a mark of zero being recorded.

Office hours: April 18 13:30-15:30 MATX 1102

HW #3

HW #4 Due date: February 14.

HW #5 Due date: March 14.

HW #6 Due date: April 6. (No solutions will be posted)

Practice questions (with hints) We are going to discuss the problems in class in February 14.

HW #1 solutions

HW #2 solutions

HW #3 solutions ver. 1

HW #3 solutions ver. 2

HW #3 solutions ver. 3

HW #4 solutions (for selected problems)

First midterm (with solutions)

Second midterm (with solutions)

HW #5 solutions

Practice questions for the final. Try to solve them on you own.

Practice solutions

Office hours: April 18 13:30-15:30 MATX 1102