let $R$ be a commutative ring and $\mathcal{A}$ be a commutative $R$-algebra. in 2010, my thesis advisor Wenhua Zhao introduced the following notion: we say that a subspace $\mathcal{M}$ of $\mathcal{A}$ is a Mathieu subspace (or now frequently a Mathieu-Zhao subspace) of $\mathcal{A}$ if for any $a,b \in \mathcal{A}$ with $a^m \in \mathcal{M}$ for all $m \geq 1$, we have $a^mb \in \mathcal{M}$ when $m \gg 0$, i.e. there exists some $N \geq 1$ (potentially depending on $a,b$) such that $a^mb \in \mathcal{M}$ for all $m \geq N$. intuitively, my advisor always conceptualized Mathieu-Zhao subspaces as 'black holes' - they are a generalization of ideals where elements are absorbed after a point, as if gravity begins to pull the elements in. Mathieu-Zhao subspaces may be useful in a proof of the (famously open) Jacobian conjecture, as many equivalent conjectures can be restated in terms of Mathieu-Zhao subspaces.

let $R$ be a field, $G$ be a group, and $P$ the set of conjugacy classes of subgroups of $G$. denote by $\mathscr{B}_R(G)$ the free $R$-module generated by the set $\{[G/H_a] \mid a \in P\}$ where for any two basis elements $[G/H_a], [G/H_b]$ we have $$ [G/H_a] \cdot [G/H_b] = \sum[G/K_i]$$ where the sum is taken over all $G$-orbits in $G/H_a \times G/H_b$ and $K_i$ is the stabilizer of the $i$th $G$-orbit. extending this product via linearity makes $\mathscr{B}_R(G)$ into a commutative ring with identity $[G/G]$, which we call the Burnside algebra of $G$ over $R$. in this thesis, we characterize the Mathieu-Zhao subspaces of Burnside algebras of finite abelian groups and dihedral groups over fields of prime characteristic.

take a ring $R$ and consider its group of invertible elements $R^\times$. for a prime $p$, we say that $R$ satisfies the $\Delta_p$ condition if $u^p = 1$ for all $u \in R^\times$, i.e. $R^\times$ is an elementary abelian $p$-group. in this paper, we classify all finite commutative rings, finite-dimensional $k$-algebras, group rings, and path algebras over finite acyclic quivers satisfying the $\Delta_p$ condition and discuss some connections to Mersenne primes and Dedekind's problem, which remains open.

let $G$ be a graph (a tuple containing a set of vertices $V$ and a set of edges $E$). we say that $G$ is hamiltonian if there exists a path in $G$ that starts and ends at the same vertex while visiting every vertex in $G$ exactly once. let $c(G)$ denote the number of components (distinct portions of $G$ inaccessible by one another) of $G$. we define the subgraph induced by a subset $S$ of vertices of $G$ to be the subgraph of $G$ containing all vertices in $S$ and edges between vertices in $S$. for any subset of vertices $S$, we denote by $G - S$ the subgraph of $G$ with all vertices in $S$ and edges to vertices in $S$ removed (i.e. $G - S$ is the subgraph induced by the subset of vertices of G after removing all vertices in $S$). we say that $G$ is $H$-free if it contains no induced subgraph isomorphic to $H$. finally, we say that $G$ is $t$-tough if for every subset of vertices $S$ we have $t \cdot c(G-S) \leq |S|$, or more simply $t$ times the number of components in $G-S$ is less than or equal to the number of vertices removed. in this paper, we prove that every $3$-tough $(K_2 \cup 3K_1)$-free graph with at least 3 vertices is hamiltonian.