Skew braces and the YBE
Below you will find some ideas.
The project concerns a crucial notion in modern mathematics: filtrations. This means decomposing an object (for us, a finite group) into small pieces that one can understand better. In this concrete project, we want to consider sequences of nested subgroups where each is normal in the next. So we naturally find the concept of subnormality. The idea first appeared in the work of Wielandt and is now an essential tool in proving several amusing theorems.
Let $p$ be a prime number. If $A$ and $B$ are non-empty subsets of the set $\mathbb{Z}/p$ of integers modulo $p$, then $|A+B|\geq\min\lbrace p,|A|+|B|-1\rbrace$. This sharp bound is known as the Cauchy—Davenport theorem. Remarkably, the theorem can be generalized to any finite group!
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.
The result is one of the most influential theorems in the theory of permutation groups. Briefly, the O’Nan-Scott theorem describes maximal subgroups of symmetric groups. Combined with the classification of finite simple groups, there are so many applications that one cannot even count them!
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 $1/12$.
Given a normal subgroup $N$ of $G$, can we reconstruct the structure of $G$ from that of $N$ and $G/N$? In general, no. However, there is a crucial case where this problem has a beautiful solution: If the orders of $N$ and $G/N$ are coprime, then $G$ is a semidirect product of $N$ and $G/N$. This is the celebrated Schur—Zassenhaus theorem. The proof is also enjoyable. It reduces the problem to the case where $N$ is abelian; in that case, one uses some basic group cohomology!
The idea is to study an elementary theorem (yet compelling) proved not so long ago by Deaconescu and Walls about divisibility relations among the set of orbits of actions by group automorphisms. The theorem is very elementary and has friendly and highly non-trivial applications.
In the study of some of Kaplansky’s conjectures, a mysterious property of groups pops up. The unique product property (UPP) is hard to check in practice. Are there torsion-free groups without the UPP? Yes, and a famous example is the one that Promislow found around 1988. Remarkably, only a few groups of this kind are known. The plan is to present examples of torsion-free groups without the UPP and show how this notion fits in geometric group theory.
Suppose that $n$ persons apply to $m$ jobs. Assume that each person applied to some jobs. When do we know that every person will get a job? Hall’s theorem answers the question. The result has several equivalent formulations and almost infinite applications!
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.
Let $K$ be a finite field (e.g., the field of integers modulo a prime number $p$).
The project is about “permutation polynomials”.
A permutation polynomial $f(X)\in K[X]$ such that the associated function $x\mapsto f(x)$
is bijective. In 1966, Carlitz presented a conjecture that motivated around 30 years of
intensive research in permutation polynomials. Although there was an immediate success in
some special cases, progress was made slowly
over the next three decades until Carlitz’s conjecture was finally resolved
in the affirmative by Fried, Guralnick, and Saxl in 1993.
An algebraic approach to combinatorial problems involves capturing some combinatorial structures using polynomials and arguing about their algebraic properties. This has led to simple solutions to several long-standing open problems. One of the main tools in this context is Alon’s combinatorial NullStellensatz. Examples of problems that can be solved with Alon’s theorem are: Cauchy—Davenport theorem, and Kakeya’s conjecture for finite fields.
In modern representation theory of finite groups, counting conjectures play a fundamental role. Among these conjectures, one stands out: McKay’s conjecture. John McKay first formulated it in 1972, and it is still open. However, the conjecture is known to be true in several cases. Different techniques are crucial for proving the conjecture in particular families of groups. For example, easy computer calculations prove McKay’s conjecture for huge sporadic simple groups!
This astonishing claim follows quickly from a theorem of Hurwitz about the possibility of writing products of a sum of squares as a sum of squares. A proof of Hurwitz’s theorem uses the representation theory of finite groups.
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.
In Ring and Module Theory, we studied the basic theory of character theory of finite groups.
However, because of the lack of time, only some applications are given.
Luckily, several applications fit perfectly with a bachelor’s thesis. For examples,
every finite group of order divisible by at most two primes is solvable (Burnside´s theorem).
There are (at most) finitely-many simple groups with a centralizer of involutions of order $n$. The theorem is the starting point for the classification of simple groups.
Can you describe groups of order 21? This is where Frobenius groups appear. Frobenius’ theorem goes far beyond this: it describes the structure of a finite group, assuming that one has a normal subgroup with particular properties. Frobenius groups have played a significant role in many areas of group theory.
Köthe’s conjecture is a problem in ring theory related to specific properties of ideals. The question was posed in 1930 and is still open. There are several easy-to-understand equivalent versions of the conjecture. The conjecture is true in several classes of rings.
Golod and Shafarevich proved this significant result in 1964. It results in non-commutative algebra, which solves several challenging problems (e.g., the class field tower problem). In combinatorial group theory, finding a counterexample to the generalized Burnside problem is crucial: For each prime $p$, there is an infinite group $G$ generated by three elements in which each element has order a power of $p$.
There are several open problems in ring theory known as Kaplansky’s conjectures. Recently, Giles Gardam found a two-pages counterexample to the celebrated conjecture on units of group algebras. This is just the story’s beginning: several other open problems exist!
Jacobson introduced this weird (yet intriguing) family of rings (without one) around 1940. In 2005 Rump found a surprising connection: radical rings produce solutions to the Yang—Baxter equation. Rump’s discovery is the starting point of several exciting developments where algebra, mathematical—physics, and combinatorics meet.
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!
Cayley—Hamilton states that every square matrix satisfies its characteristic equation. The Amitsur—Levitzki theorem deals with products of $2k$ matrices of size $k^2$. It is lovely and essential and when it comes to mathematics. The theorem is the starting point of a rich theory of rings with polynomial identities. (Interesting note: As a young man, Levitzki went to Göttingen to study chemistry, but attending a lecture by Emmy Noether converted him to mathematics.)
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.
FRACTRAN is a Turing-complete programming language invented by the mathematician John Conway. A FRACTRAN program is an ordered list of positive rational numbers and an initial positive integer. In this fantastic language, Conway learned how to write an astonishing prime number generator. Surprisingly, this FRACTRAN program is just a list of 14 rational numbers!
The recipe is as follows
After finishing, play with the database and find conjectures!
Is there a connection between algebraic logic, the famous Yang—Baxter equation in mathematical physics, and braid groups? Yes! The connection between these topics depends on a recently introduced new algebraic structure known as an L-algebra. There are several ways to get into this fascinating theory. One can start with algebraic logic (Hilbert and Heyting algebras are particular families of L-algebras), with topology, with the Garside theory of braid groups, or with combinatorial solutions to the Yang—Baxter equation. Surprisingly, these unrelated topics are now connected by the theory of L-algebras!
An elementary way of distinguishing knots is by reasonably coloring their arcs. Conjugacy classes of groups provide natural colorings. More generally, one can use quandles to determine knots by coloring arcs. These structures provide algebraic tools that serve as non-commutative colorings. Moreover, if we reasonably weigh each color, we make our invariants even more potent and discover the homology of quandles!
In 1984 Jones discovered a new invariant of knots. The Jones polynomial is surprisingly simple and extremely powerful. Moreover, Jones’ discovery was crucial in solving some old-and-famous 200-years-old conjectures.