~~META:title=Algebra Seminar~~ [[http://www-history.mcs.st-and.ac.uk/Biographies/Galois.html|{{http://www.win.tue.nl/~aeb/at/mathematicians/galois1.jpg?110*135 |Evariste Galois}}]] [[ http://www-history.mcs.st-and.ac.uk/Mathematicians/Noether_Emmy.html|{{ http://seminars.math.binghamton.edu/AlgebraSem/emmy_noether.jpg?110*135|Emmy Noether}}]] \\ \\ **#####The Algebra Seminar#####** Unless stated otherwise, the seminar meets Tuesdays in room WH-100E at 2:50 p.m. There will be refreshments served at 4:00 in room WH-102. Organizers: [[:people:alex:start|Alex Feingold]] and [[:people:tongviet:start|Hung Tong-Viet]] To receive announcements of seminar talks by email, please join the seminar's [[http://www1.math.binghamton.edu/mailman/listinfo/algsem|mailing list]]. ---- =====Fall 2018===== * **August 28**\\ Organizational Meeting \\ **//Title of Talk//** \\ \\ **//Abstract//**: Please come or contact the organizers if you are interested in giving a talk this semester or want to invite someone. * **September 4**\\ Casey Donoven (Binghamton University)\\ **// Covering Number of Semigroups //** \\ \\ **//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.) * **September 11**\\ No Classes (University)\\ **// Title //** \\ \\ **//Abstract//**: Abstract text * **September 18**\\ Joe Cyr (Binghamton University)\\ **// Semilattice Modes //** \\ \\ **//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. * **September 25**\\ Matt Evans (Binghamton University)\\ **// Spectral properties of involutory BCK-algebras //** \\ \\ **//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. * **October 2**\\ Mark Lewis (Kent State University)\\ **// Centers of centralizers and maximal abelian subgroups //** \\ \\ **//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. * **October 9**\\ Fernando Guzman (Binghamton University)\\ **// Polynomials Automorphisms of the Regular d-ary Tree //** \\ \\ **//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. * **October 16**\\ Luise-Charlotte Kappe (Binghamton University)\\ **// A generalization of the Chermak-Delgado lattice to words in two variables //** \\ \\ **//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.) * **October 23**\\ Eran Crockett (Ripon College)\\ **// The variety generated by the triangle //** \\ \\ **//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). * **October 30**\\ Dan Rossi (Binghamton University)\\ **// Brauer characters and fields of values //** \\ \\ **//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. * **November 6**\\ Casey Donoven (Binghamton University)\\ **// Covering Number of Semigroups (cont.) //** \\ \\ **//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. * **November 13**\\ Casey Donoven (Binghamton University)\\ **// Fractal Subgroups of Profinite Groups //** \\ \\ **//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? * **November 20**\\ Hung Tong-Viet (Binghamton University)\\ **// Conjugacy classes of $p$-elements and normal $p$-complements //** \\ \\ **//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. * **November 27**\\ Nicholas Gardner (Binghamton University)\\ **// An Introduction to the Chermak-Delgado Lattice //** \\ \\ **//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. * **December 4**\\ Speaker (University)\\ **// Title //** \\ \\ **//Abstract//**: Abstract text ---- ---- * [[http://seminars.math.binghamton.edu/AlgebraSem/index.html|Pre-2014 semesters]]\\ * [[seminars:alge:fall2014]] * [[seminars:alge:spring2015]] * [[seminars:alge:alge_fall2015]] * [[seminars:alge:alge-spring2016]] * [[seminars:alge:alge-fall2016]] * [[seminars:alge:alge-Spring2017|Spring 2017]] * [[seminars:alge:alge-Fall2017|Fall 2017]] * [[seminars:alge:alge-Spring2018|Spring 2018]]