Class time and place: MWF 2:00-2:50 pm, in Engineering Hall, room 106B6.
Office hours: Monday and Wednesday at 5:00-6:00 pm on Zoom.
The meeting number and the password was given in class.
Professor: Igor Mineyev. Email is not very efficient for discussions.
Please talk to me before/during/after classes and come to office hours.
For the purposes of this course, "constructive geometry" most of the time will mean "combinatorial geometry of graphs and its relations to group theory". The course will be a mixture of some algebra, some combinatorial geometry, and some generalized algorithms.
First we will review some standard group theory. Then I will ask you to forget it all. While it is helpful to have some knowledge of groups, we will start from scratch and define some geometric notions first. Only after that some notions of group theory will start appearing again, in a different, more natural, geometric form.
Usually a group is defined algebraically, as a set with a binary operation satisfying some properties. In this way, the structure of a general group is a black box, not so much visible to the eye. This 490 course will introduce groups as geometric objects and will emphasize studying groups as geometric objects. In this way, this course relates combinatorics, algebra and geometry, three different areas of mathematics. This course will be experimental, introductory, and fun. (!) It can be viewed as a first step before taking Math 503 Introduction to geometric group theory course that will hopefully run in Spring 2026. (Register for Math 503 as early as possible in Fall 2025 to make sure it will run.) For this 490 course I will take a novel approach that will be different from the introduction to geometric group theory usually given. Usually, groups are defined first and then their geometry is investigated, if ever. My plan is to first construct some specific geometric/combinatorial objects that I call grounds, and only then define groups as a consequence. Then we will see examples of grounds and groups, and various ways to construct them. Secretly, grounds are related to Cayley graphs, but they will be defined (a) algorithmically and constructively, (b) before any group is given, and (c) by combinatorial and geometric means.
In addition, there is an algorithmic component to studying groups. I am planing to interpret the combinatorial and geometric constructions in this 490 course as algorithms in a generalized sense.
Feel free to play the visual PathForms game. It is at a preliminary stage and is related to the course. The ColorTaiko! game might be more related to Math 503, but should be fun to play too. Click on the games' titles to see more information.
Prerequisites. There are no strict prerequisites for this course. The course should be accessible to an aspiring undergraduate student interested in mathematics who has experience with mathematical concepts and mathematical reasoning. As a fun preparation, read about groups and graphs, find some interesting examples of groups and graphs. It might be helpful to refresh some standard notions in group theory: a group, a group homomorphism, a subgroup, the quotient group, isomorphism theorems of groups, etc. If interested to go deeper, learn about concepts and interesting open problems in algebra, geometry, graph theory. Also, click on the links presented below and investigate.
Grading, etc. There will be some homework, posted here. There will be no exams. There will be a project: later in the course, the students will be randomly split into several groups and participate in a project on an exciting topic of their choosing related to the course.
The best way to contact. My email is mineyev at illinois edu, but I have a slight disability that makes it hard for me to type, and generally to deal with email. Vocal real-time communication is much more preferred and very much appreciated. Please talk to me before/during/after the class and during the office hours. Face-to-face communication (virtual or otherwise) is also much more efficient for discussing things. I would appreciate if you use it as much as possible rather than communicate by email.
I am happy to discuss any kind of math, or other things that you are interested in. Math is the only possible meaning of life. And fun too. Take initiative, talk to me about it. Any suggestions are also welcome. (In person, not by email. :)
Participation. Whether you participate during the class will not affect your grade, but your participation is helpful and very much appreciated. Ask questions, discuss concepts, come up with ideas. This helps exercising creativity, structuring the course, as gives even more fun to learning mathematics. And again, office hours are good for discussing any kinds of math.
Attendance. I expect you to be in class most of the time. It is OK if you need to skip one or two lectures for substantial reasons, but you still need to learn the material of those missed lectures. Skipping too many classes might negatively affect your grade for the class.
More information will appear here as we proceed.