Leandro Vendramin

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Some previous bachelor topics

Lagrange’s four-square theorem

Lagrange’s theorem states that every non-negative integer can be represented as a sum of four non-negative integer squares. A particularly elegant proof of this result can be found using ideas from the geometry of numbers, specifically Minkowski’s theorem. It can also be proved using quaternions and ideas similar to those we studied in the bachelor course Ring and Module Theory.

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A nice theorem of Jacobson

Here is an easy algebra exercise that we all know: if $x^2 = x$ for every element $x$ of a ring, then the ring is commutative. Fewer people know that a similar property holds if we assume $x^3 = x$ or $x^4 = x$. These cases are harder to prove. There is a very beautiful theorem of Jacobson stating that a ring is commutative if the condition $x^n = x$ holds for all $x$. This highly non-trivial beautiful result can be generalized in several different directions. And even better: one enters the theory of polynomial identity rings.

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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.

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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.

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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.

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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 $1/12$.

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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!

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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.

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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.

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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.

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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.

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