I'll present a proof of the existence of this characteristic subgroup of a finite group. One of the interesting points is that this subgroup was first identified about 1989.
I’ll describe some of my recent research on extremal independent sets and graph colorings, intended for a general math audience. There will be a lot of pictures.
We show how to use a computational tool from abstract algebra, Gröbner bases, to solve a puzzle game called Circuit Scramble. In Circuit Scramble, the player has to figure out how to set the inputs to a logic circuit as True or False in order to make the final output True. (See here for a small example.) More generally (and, to be honest, more usefully) we'll see how Gröbner bases can also be used to formally verify the correctness of large circuits.
We will talk about embedding algebras into other algebras that have a transitive automorphism group.
We discuss a new notion of dependence in sequences of random variables, one that has structure that can be modeled nicely by digraphs. We prove some major results: that sequences of vertically dependent random variables under certain conditions remain identically distributed. We build on this idea to yield generalized forms of the binomial, negative binomial, geometric, and multinomial distributions. We also discuss generated such sequences, a notion of horizontal dependence, and applications of this work, including dependent random walks. The initial idea was built by Dr Andrzej Korzeniowski of University of Texas Arlington (the presenter’s PhD advisor), and has been expanded and generalized by the presenter for the last two years. We’ll also seek feedback and mention future directions for this work.
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