Abstract: Please come or contact the organizers if you are interested in giving a talk this semester or want to invite someone.

Abstract: A semigroup is a set $S$ equipped with an associative operation. The covering number of a semigroup $S$ is the minimum number of proper subsemigroups whose union is $S$. In this talk, I will introduce basic semigroup theory and some fundamental examples while proving the following theorem: If $S$ is a finite semigroup that is not a group nor generated by a single element, then the covering number of $S$ is 2. Similar questions have been studied for groups and loops. (Joint work with Luise-Charlotte Kappe and Marcin Mazur.)

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Abstract: A mode A is an algebra which is idempotent and in which every operation is a homomorphism from the appropriate power of A to A. We will explore some results on a particular class of modes which are constructed from semilattices. In particular, we will look at the question of when is a semilattice mode subdirectly irreducible, both in general and in the particular case of when the mode has a single binary operation.

Abstract: BCK-algebras are algebraic structures arising from non-classical logic. This talk will focus primarily on the classes of commutative BCK-algebras and commutative involutory BCK-algebras. Particularly, I will discuss some basic ideal theory and spectral properties of such algebras, looking at differences between the bounded and unbounded cases.

Abstract: In this talk we consider the centers of the centralizers of elements in finite groups. We will then obtain a lower bound on the order of a maximal abelian subgroup in terms of the indices of the centralizers of elements and the orders of the centers of the centralizers of elements. We will use this to obtain a lower bound for maximal abelian subgroups of semi-extraspecial groups.

Abstract: We explore the question of when does a polynomial with integer coefficients induce an automorphism of the infinite regular d-ary tree. This is well-known for d=2, and there are some partial results for d prime. We extend the results to the case when d is square-free.

Abstract: The Chermak-Delgado measure of a subgroup $H$ of a finite group $G$ is defined as the product of the order of $H$ with the order of the centralizer of $H$ in $G$, $|H||C_G(H)|$, and the set of all subgroups with maximal Chermak-Delgado measure forms a dual sublattice of the subgroup lattice of $G$. In this talk we step back from centralizers and consider four types of centralizer-like subgroups, defined using general words in the alphabet $\{x, y, x^{-1}, y^{-1} \}$ instead of the specific commutator word. We show that this generalization results in four sublattices of the subgroup lattice of a finite group, some of which may be equal to one another depending on the word. We consider which properties of the Chermak-Delgado lattice generalize to the new lattices, and which properties are specialized in the Chermak-Delgado lattice. (This work is joint with Elizabeth Wilcox.)

Abstract: This talk will consist of a (quick) introduction to universal algebra where we focus on three topics: the definability of principal congruences, classifying subdirectly irreducibles, and determining the clone of term operations. We will attempt to understand these topics by focusing on two examples: the two-element semilattice and the three-element non-transitive tournament (a.k.a. the triangle).

Abstract: The complex irreducible characters $\text{Irr}(G)$ of a finite group $G$ contain a lot of information about $G$ itself. Recently, it has been realized that some of this information can still be captured if, instead of considering the entire set $\text{Irr}(G)$, one only considers those irreducible characters taking values in some suitable subfield of $\mathbb{C}$. This motivates the following definitions: $\text{Irr}_\mathbb{F}(G)$ is the subset of irreducible characters taking values in the subfield $\mathbb{F}\subseteq \mathbb{C}$; and $\text{Cl}_\mathbb{F}(G)$ is the set of conjugacy classes of $G$ whose elements, when evaluated at every character of $G$, take values in $\mathbb{F}$. A result of Isaacs $\&$ Navarro says that $|\text{Irr}_\mathbb{F}(G)| = |\text{Irr}_\mathbb{F}(G/N)|$ and $|\text{Cl}_\mathbb{F}(G)| = |\text{Cl}_\mathbb{F}(G/N)|$ whenever $N\unlhd G$ contains no non-trivial $\mathbb{F}$-elements.

For a prime $p$, the $p$-Brauer characters of $G$ arise from representations of $G$ over $\overline{\mathbb{F}}_p$. They provide a link between the representation theory of $G$ in characteristic $0$ and in characteristic $p$. Whenever one has a relationship involving characters and conjugacy classes $G$, it is natural to wonder if there is an analogous relationship between the $p$-Brauer characters and $p$-regular conjugacy classes.

I will give some examples of the sorts of $\mathbb{F}$-generalizations alluded to in the first paragraph; introduce the basic notions of Brauer characters; and discuss a Brauer analogue of the Isaacs-Navarro result mentioned above.

Abstract: I will continue my exploration of covering numbers of semigroups by considering specific classes of semigroups. A monoid is a semigroup with an identity. An inverse semigroup $I$ is a semigroup such that for each element $a\in I$ there exists a unique element $a^{-1}\in I$ such that $aa^{-1}a=a$ and $a^{-1}aa^{-1}=a$. I will give a complete description of the covering number of monoids and inverse semigroups with respect to submonoids and inverse subsemigroups respectively (modulo the covering numbers of groups and semigroups). I will use Green's relations and other results to describe the structure of such semigroups.

Abstract: Often, we think of fractals as subsets of $\mathbb{R}^n$. Many definitions in fractal geometry can be generalized to any metric space, including groups equipped with a metric. In particular, the Hausdorff dimension and Box counting dimension can be defined on any metric space. Profinite groups can be equipped with a natural metric, under which we can discuss fractal properties. This will be an expository talk in which I define fractal dimensions and profinite groups. My goal is to set up the following question: Given two fractal subgroups of the automorphism group of the rooted infinite n-ary tree, what is the dimension of their intersection?

Abstract: The commuting probability $d(G)$ of a finite $G$ (introduced by Erdős and Turán in 1968), is defined to be the probability that two randomly chosen elements of $G$ commute. The commuting probability $d(G)$ is also called the commutativity degree of $G$. Erdős and Turán showed that $d(G)=k(G)/|G|$, where $k(G)$ is number of conjugacy classes of $G$. In 1973, W. H. Gustafson proved that $d(G)\leq 5/8$ for any non-abelian group $G$. Since then, there are numerous results concerning the structure of finite groups using various bounds on the commuting probability. In this talk, I will consider a $p$-local version of the commuting probability. Specifically, for a prime $p$, we define $d_p(G)$ to be the ratio $k_p(G)/|P|$, where $k_p(G)$ is the number of conjugacy classes of $p$-elements of $G$ and $P$ is a Sylow $p$-subgroup of $G$. Using the invariant $d_p(G)$, we obtain some new criteria for the existence of normal $p$-complements in finite groups.

Abstract: For a subgroup $H$ of a finite group $G,$ the Chermak-Delgado measure of $H$ is defined as $m_{G}(H) := |H||C_{G}(H)|$. The subgroups of $G$ with maximum Chermak-Delgado measure form a dual sublattice of the subgroup lattice of $G$. In this talk I will discuss some properties of such maximum-measure subgroups and calculate the Chermak-Delgado lattice for some classes of finite groups.

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