Here are my slides for my talk at the BLAST 2017 conference at Vanderbilt University in Nashville, Tennessee.

As a side note, I just noticed this other conference. All of the talks at that other conference on Laver tables are woefully outdated (i.e. 1995 or so). They only talk about the classical Laver tables. As an analogy, only talking about the classical Laver tables is like only talking about the cyclic groups of order $3^{n}$ and then claiming that they some how represent group theory as a whole. If you are going to give a talk about Laver tables or write a paper on the Laver tables, then please read the abridged version of my paper before you do so.

The classical Laver tables by themselves are a rather dead-end research area that have not been active within the last 20 years (one can probably try to analyze the fractal structure obtained from the classical Laver tables but such an analysis will probably be difficult and incremental). In order to advance further research in this area, one needs to consider the generalizations including Laver-like algebras, multigenic Laver tables, and functional endomorphic Laver tables. The classical Laver tables do not explain what the subalgebras of $\mathcal{E}_{\lambda}/\equiv^{\gamma}$ generated by multiple elements look like (one cannot even show that $\mathcal{E}_{\lambda}/\equiv^{\gamma}$ is locally finite without using the multigenic Laver tables). The classical Laver tables do not have any cryptographic applications. The classical Laver tables are just one sequence of structures, and it is hard to advance mathematics simply by looking at only one kind of structure with limited complexity. There is no reason at all to look at the classical Laver tables without looking at more general structures.

It is better to call the structures $A_{n}=(\{1,…,2^{n}\},*_{n})$ “classical Laver tables ” instead of simply “Laver tables.” There are other structures to consider.

How to give a classical Laver table talk.

The first step to giving a presentation on the classical Laver tables is to make sure you give your talk to the proper audience. The best audience to give a classical Laver table talk to is an audience of middle schoolers or maybe high schoolers (it is not that hard to fill out the multiplication table of a classical Laver table). Once you have your audience of middle schoolers present, you should get them to fill out an $8\times 8$ classical Laver table and then a $16\times 16$ classical Laver table. After they fill out the $16\times 16$ classical Laver table. And yes, middle schoolers are completely capable of filling out classical Laver tables. It is not that hard. After they are done filling out the tables, you can show them pictures that arise from the classical Laver tables on the projector and hint about how these objects come from infinity.