Some previous bachelor topics
Dedekind-finite rings
A ring is said to be a Dedekind-finite ring if $ab = 1$ implies $ba = 1$ for any two elements $a$ and $b$. Several classes of rings are known to be Dedekind-finite. There is a beautiful theorem of Kaplansky that states that if an element of a ring has more than one right inverse, then it in fact has infinitely many.
References:
- T. Y. Lam. A first course in noncommutative rings, volume 131 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 2001.
- T. Y. Lam. Exercises in modules and rings. Problem Books in Mathematics. Springer, New York, 2007.
- I. Rosenholtz. A pigeonhole proof of Kaplansky’s theorem. Amer. Math. Monthly, 99(2):132–133, 1992.
- M. Isaacs. Finite group theory. Graduate Studies in Mathematics, 92. American Mathematical Society, Providence, RI, 2008.
Ore’s conjecture
Is every element of a finite non-abelian simple group a commutator? The statement is Ore’s conjecture (1951). The conjecture is now a theorem; it was proved in 2010 using sophisticated mathematics. However, it is exciting to prove the conjecture for the alternating simple groups and with computers for sporadic simple groups.
References:
- O. Ore. Some remarks on commutators. Proc. Amer. Math. Soc. 2 (1951), 307-314.
Dearrangements
A derangement is a permutation that has no fixed points. Everything about dearrangements is intriguing, even counting them! They are intimately connected with many other topics in mathematics, including number theory, game theory, enumerative combinatorics, and more.
References:
- R. Stanley. Enumerative combinatorics. Volume 1. Second edition. Cambridge Studies in Advanced Mathematics, 49. Cambridge University Press, Cambridge, 2012. xiv+626 pp.
Commuting probability in finite (simple) groups
What is the probability that two randomly chosen elements of a finite group commute? A classic result states that this commuting probability is at most $5/8$. A nice theorem of Dixon states that, for finite non-abelian simple groups, the commuting probability is at most $1/12$.
References:
- W. Gustafson. What is the probability that two group elements commute? Amer. Math. Monthly 80 (1973), 1031-1034.
The Skolem—Noether theorem
Any automorphism of the full $n\times n$ matrix algebra is conjugated by some invertible $n\times n$ matrix. This is the celebrated Skolem—Noether theorem. Remarkably, there exists a friendly proof of this astonishing result!
References:
- M. Brešar. Introduction to noncommutative algebra. Universitext. Springer, Cham, 2014.
Combinatorial invariants of knots
An elementary method for distinguishing knots involves consistently coloring their arcs using algebraic structures. Conjugacy classes of groups naturally produce such colorings. More generally, quandles provide a robust framework for knot classification by coloring arcs. Furthermore, by assigning weights to each color, we can strengthen these invariants, leading to the discovery of quandle homology.
References:
- J. Carter, D. Jelsovsky, S. Kamada, L. Langford, M. Saito. Quandle cohomology and state-sum invariants of knotted curves and surfaces. Trans. Amer. Math. Soc. 355 (2003), no. 10, 3947-3989.
- M. Elhamdadi, Mohamed, S. Nelson. Quandles—an introduction to the algebra of knots. Student Mathematical Library, 74. American Mathematical Society, Providence, RI, 2015. x+245 pp.
Groups and lattices
Do you like algebra? Do you enjoy group theory? Chermak—Delgado lattice is what you need! The definition does not require advanced group theory. The beauty of this is that you will try to prove several open questions after doing some exercises and experiments.
References:
- M. Isaacs. Finite group theory. Graduate Studies in Mathematics, 92. American Mathematical Society, Providence, RI, 2008.
Concrete computer experiments with a famous equation in mathematical physics
This project will explore specific solutions to the famous Yang—Baxter equation in a natural experimental computer laboratory. We will study several related algebraic structures and implement different functions to play with specific properties of solutions.
References:
- Cedó, Ferran; Jespers, Eric; Okniński, Jan. Braces and the Yang-Baxter equation. Comm. Math. Phys. 327 (2014), no. 1, 101-116.
The Jones polynomial
In 1984 Jones discovered a new invariant of knots. The invariant assigns to each oriented knot (or link)
a Laurent polynomial with integer coefficients. This invariant is surprisingly simple and extremely
powerful. And Jones’ discovery was crucial in solving some old-and-famous
200-years-old conjectures.
References:
- C. C. Adams. The knot book. American Mathematical Society, Providence, RI, 2004. An elementary introduction to the mathematical theory of knots, Revised reprint of the 1994 original.
- W. B. R. Lickorish. An introduction to knot theory, volume 175 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1997.