And what mean all these mysteries to me,
Whose life is full of indices and surds?
x2 + 7x + 53
= 11 3
- Charles Lutwidge Dodgson
Memento mori Как резво ты скакал! А что в итоге?
От силы два абзаца в некрологе.
Но живо то, что сотворить ты смог.
Какой итог, дружок, какой итог?
- Игорь Минеев
If you are lonely when you are alone,
you are in bad company.
- Jean-Paul Sartre


How to pronounce this name. Igor Mineyev's Math Page
Thinker cv
talks
songs
UnderGround

Candidate A is your favorite, but you are told that you must vote for a worse candidate B because your vote for candidate A would help the worst candidate C. The current voting system punishes you for voting honestly.

Do you like two out of five candidates equally? The current ballot does not let you express your opinion. The result? Politicians with unpopular agendas are elected.

Do you want democracy? Demand a simple, purely mathematical solution: approval voting, or better the more general score voting aka range voting, or even better its variation called STAR voting.

Any one of these is better than ranked-choice voting aka IRV. Much better. Do not advocate for ranked-choice, it's a distraction.

Bees have a better democracy. We need to keep up.

Score voting is more important than the electoral college. It is more important than gerrymandering. The country is not polarized; it uses a corrupt voting system. Spread the word. Discuss. Demand that your politicians implement score voting.

Will the politicians allow score voting? This is not a mathematical question.

I am a Professor at UIUC Math Dept. My mathematical interests include subjects related to geometric group theory, in particular,
  • geometric grounds for group theory, open problems in group theory and topology/geometry,
  • the zero-divisor conjecture, the Kaplansky conjectures, the Atiyah problem; various types of (co)homology of groups,
  • hyperbolic groups, metric geometry, flows, CAT(0) and CAT(-1) spaces,
  • the Hanna Neumann conjecture and submultiplicativity,
  • geometric and topological rigidity, conformal structures, geometric analysis,
  • machine learning, artificial intelligence, image recognition, P vs. NP problem,
  • l2-Betti numbers, von Neumann algebras,
  • 3-manifolds and 4-manifolds, relative hyperbolicity,
  • quantum computing and information, entropy, error correction,
  • the geometry of the Riemann hypothesis,
  • the Novikov conjecture, the Baum-Connes conjecture,
  • and many other incoherent things...
  • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
My links and comments on things related to geometric group theory. (Hint: pretty much anything is related.) Grokipedia knows more about me than I do. Talk to me about any kind of math. In person.
Meditative math art – SpiroGraph
Three things can be watched endlessly: fire burning, water flowing, and SpiroGraph drawing an amazing variety of pretty-wild curves. This is an online visual exploration game inspired by classical spirograph curves: it uses ellipses rather than circles. Press the "random" button to see more curves. Download the pictures you like, there will be plenty. Explore various settings, create, or run this game as a screensaver.
Click to play Space Fish video Watch Space Fish on YouTube

Or watch it on YouTube. The video orange-and-blue SPACE FISH and SPACE CLOCK .
Make it full-screen and watch from the beginning to the end. When you finish your meditative journey, the highest mathematical truth might descend upon you. (If stuck, try different browsers.) Do you recognize the function in this video? (Hint: It was introduced in 1859. And no, it was not in Darwin's "The origin of species".)

ColorTaiko! is an IML project I was running in Spring 2024, Fall 2024 and Spring 2025. Play our fun "ColorTaiko!" game. Without any explanation, can you figure out how to play it? Here is more information Talk to me if you want to participate in this in Spring 2026.. The game is secretly related to several long-standing open problems about group algebras called Kaplansky conjectures. My article explains the relation between the Kaplansky conjectures and this game. We will keep improving the game from time to time, so come and play again.

My grant application to the Topology program at the National Science Foundation (NSF) in Fall 2023 asking for support for this game, for the research it is based on, and for the people involved in the project, was expressly NOT supported by NSF. This preliminary version of the game that you see here was created by our heroic efforts in spite of NSF's refusal to support that proposal. My grant application to the Topology program in Fall 2024 was not supported by NSF either. NSF did not value this project.

PathForms is another IML project that I am running in Spring 2025, Fall 2025, Spring 2026. Its goal is to engage the general public into learning, in a fun geometric way, Nielsen's algorithm coming from geometric group theory. Here is the information about this game.

IG4OR'S

Teaching in Spring 2026:


Paper work
linfty-cohomology and metabolicity of negatively curved complexes. Internat. J. Algebra Comput. Vol. 9, No. 1(1999), 51-77.
Higher dimensional isoperimetric functions in hyperbolic groups. Math. Z. 233 (2000), no. 2, 327-345.
l1-homology of combable groups and 3-manifold groups. International Conference on Geometric and Combinatorial Methods in Group Theory and Semigroup Theory (Lincoln, NE, 2000). Internat. J. Algebra Comput. 12 (2002), no. 1-2, 341-355.
Straightening and bounded cohomology of hyperbolic groups. GAFA, Geom. Funct. Anal. 11(2001), 807-839.
Bounded cohomology characterizes hyperbolic groups. Quart. J. Math. Oxford Ser., 53(2002), 59-73.
The Baum-Connes conjecture for hyperbolic groups. Joint with Guoliang Yu. Invent. Math. 149 (2002) 1, 97-122.
Ideal bicombings for hyperbolic groups and applications. Joint with N. Monod and Y. Shalom. Topology 43 (2004), no. 6, 1319-1344.
Non-microstates free entropy dimension for groups. Joint with D. Shlyakhtenko. GAFA, Geom. Funct. Anal., 15 (2005), 476-490.
Flows and joins of metric spaces. Geometry and Topology, Vol. 9 (2005), no. 13, 403-482. Here is how to type Latex symbols for this article. Also the file on GT web page.
Metric conformal structures and hyperbolic dimension. Conform. Geom. Dyn. 11 (2007), 137-163. Also the published version.
Relative hyperbolicity and bounded cohomology. Joint with A.Yaman. Preprint.
The topology and analysis of the Hanna Neumann Conjecture. J. Topol. Anal. (JTA), 3(2011), no. 3, 307-376. If you would like to learn about the Hanna Neumann conjecture and its generalizations, read my three papers in the reverse order. (Read this paper third).
Submultiplicativity and the Hanna Neumann Conjecture. Ann. of Math., 175 (2012), no. 1, 393-414. Also a latex leafage picture. Another leafage picture. Published version. (Read this paper second).
Groups, graphs, and the Hanna Neumann Conjecture. J. Topol. Anal. (JTA), 4(2012), no. 1, 1-12. Here is the pictures-only version of this article. Published version. (Read this paper first).
To Abstract via Algebraic Innateness: Hierarchical, Interpretable, and Task-Free Clustering. H. Yu, I. Mineyev, and L. R. Varshney. Proceedings of the 2019 Information Theory and its Applications Workshop (ITA), San Diego, California, 10-15 February 2019.
Orbit computation for atomically generated subgroups of isometries of Z^n. Joint with H. Yu, I. and L. R. Varshney. SIAM J. Appl. Algebra Geom. 5 (2021), no. 3, 479-505. Here is the preprint version of that paper.
A group-theoretic approach to computational abstraction: symmetry-driven hierarchical clustering. Joint with Haizi Yu and Lav R. Varshney. Journal of Machine Learning Research (JMLR), 24(47):1-61, 2023.
Learning from one and only one shot. Joint with Haizi Yu, Lav R. Varshney, James A. Evans. NPJ (Nature Partner Journals) Artificial Intelligence, volume 1, 13 (2025). This paper uses quasiconformal transformations and quasiconformal distortion to enable artificial intelligence to recognize letters/images with essentially no training.
The topology and geometry of units and zero-divisors: origami. Preprint 2024. How to look for zero-divisors and units in group algebras using geometry and graphs.
On zero-divisors and units in group rings of torsion-free CAT(0) groups. Joint with Manisha Garg. Preprint 2025.

Miscellaneous "Mathematics is a piece of cake: if you like it, you will get it."

"My theory is that no matter how hard people try to describe the real world, any theory would give only a rough approximation to the reality. Except for my theory."

"...a point has the non-zero zero homology and the zero non-zero homology..."

--I.V.Mineyev, hope-not-yet-complete works.

A Mathematician's Lament by Paul Lockhart. What you hated in school was not mathematics. A quote from teaching evaluations:
Instructor's weakness: high expectations.
Freedom.
Space curvature.
 Another quote from teaching evaluations: Do not teach proofs. They are useless. Teach what needs to be done on the homework and tests.
A very important statement. Funny math pictures from V.Troitsky.

G3
conference

 The mathematically correct name is "gee cube". As y'all know, G3 stands for the annual Geometric Group Theory on the Gulf Coast Conference, commonly abbreviated in various ways, like Geometric Groups on the Gulf.

G3 conference is currently not running.


Department of Mathematics,
University of Illinois at Urbana-Champaign,
250 Altgeld Hall,
Urbana, IL 61801, USA.
 		Email address: mineyev illinois edu
Please don't get discouraged if I don't reply. I have email anxiety and a mild form of keyboard incompatibility. I read email, but it takes time. Email also happens to be a distraction from deep thinking. I'm working on my email skills, it's a process. Meet me in person, I'm much nicer that way. :) If you want to write me something important, a regular mail letter might be a better option.