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seminars:alge

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: Alex Feingold and Hung Tong-Viet

To receive announcements of seminar talks by email, please join the seminar's mailing list.

## Spring 2019

• January 22
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.

• January 29
Ben Brewster (Binghamton University)
The values of the Chermak-Delgado measure

Abstract: Let $G$ be a finite group. For $H\leq G$, $m_G(H) = |H|\ |C_G(H)|$. Let $m^*(G) = max\{m_G(H)\mid H\leq G\}$ and $CD(G) = \{H\leq G\mid m_G(H)=m^*(G)\}$. Then $CD(G)$ is a self-dual modular sublattice of the subgroup lattice of $G$.

It is known that if $|G| > 1$, then not every subgroup of $G$ is a member of $CD(G)$, that is, $|\{m_G(H)\mid H\leq G\}| > 1$. Following some ideas of M. Tarnauceanu, we examine possibilities for $|\{m_G(H)\mid H\leq G\}|$, its form and the distribution of subgroups of same measure.

• February 5
Alex Feingold (Binghamton University)
An introduction to Lie algebras

Abstract: A Lie algebra is a vector space equipped with a bilinear product, denoted by $[\cdot,\cdot]$, such that $[x,x]=0$ and $[x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0$ (Jacobi Identity). I will give an introduction to the basic ideas and examples.

• February 12
Canceled due to inclement weather
• February 19
Daniel Rossi (Binghamton University)
The structure of finite groups with exactly three rational-valued irreducible characters

Abstract: Many results in the character theory of finite groups are motivated from the question: to what extent do the irreducible characters of a group $G$ control the structure of $G$ itself? Recently, it has been observed that certain results along these lines can be obtained when one looks not at the set of all irreducible characters of $G$, but only the subset of those characters taking values in some appropriate field. In this talk, I'll characterize the structure of finite groups which have exactly three rational-valued irreducible characters (for solvable groups, this characterization is due to J. Tent). I will attempt to give some of the flavor of the proof – which at one point includes a surprise cameo by the complex Lie algebra $sl(n)$.

• February 26
Casey Donoven (Binghamton University)
Thompson's Group $V$ and Finite Permutation Groups

Abstract: Thompson's group $V$ is group of homeomorphisms of Cantor space. It acts by exchanging finite prefixes in infinite strings over a two-letter alphabet. Generalizations of $V$ called $V_n$ act on n-letter alphabets. I will present more generalizations that add the action of finite permutation groups to the finite prefix exchanges. For a finite permutation group $G$ on $n$ points, the group $V_n(G)$ marries the finite prefix exchanges with iterated permutations from $G$. The primary theorem I will present states that $V_n$ is isomorphic to $V_n(G)$ if and only if $G$ is semiregular (i.e. $G$ acts freely). The proof involves the use of automata and orbit dynamics.

• March 5
Matt Evans (Binghamton University)
Spectra of cBCK-algebras

Abstract: BCK-algebras are algebraic structures that come from a non-classical logic. Mimicking a well-known construction for commutative rings, we can put a topology on the set of prime ideals of a commutative BCK-algebra; the resulting space is called the spectrum. I will discuss some results/properties of the spectrum of such algebras. A particularly interesting spectrum occurs when the underlying algebra is a so-called BCK-union of a specific algebra. In this case, the spectrum is a spectral space, meaning it is homeomorphic to the spectrum of a commutative ring.

• March 12
Hung Tong-Viet (Binghamton University)
Real conjugacy class sizes and orders of real elements

Abstract: In this talk, I will present some recent results concerning the structure of finite groups with restriction on the real conjugacy classes or on the orders of real elements.

• March 19
Spring Break
No Talk

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• March 26
Dikran Karagueuzian (Binghamton University)
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• April 2
John Brown (Binghamton University)
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• April 9
Jonathan Doane (Binghamton University)
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• April 16
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• April 23
Joseph Cyr (Binghamton University)
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• April 30
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• May 7
Joshua Carey
(Candidacy Exam Part 1)

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