### How to Write a Convincing Mathematical Paper

Let ${X}$ be a Banach Space

Open any mathematical journal and read the first sentence of a paper chosen at random. You will probably find something along the following lines: “Let X be a Banach space”. That is fine if you know what a Banach space is, but meaningless if you don’t.

Picking a recent issue of the Bulletin of the American Mathematical Society from the shelf, I chose a few papers at random. The titles and opening sentences were as follows:

1. Arithmetic Hyperbolic Reflection Groups: Consider a finite volume polyhedron ${P}$ in the ${n}$-dimensional hyperbolic space ${\mathbb{H}^n}$.
2. Three Themes of Syzygies: Free resolutions are both central objects and fruitful tools in commutative algebra.
3. An Overview of Periodic Elliptic Operators: Elliptic PDEs with periodic coefficients, notably the stationary Schrödinger operator ${-\Delta + V(x)}$ with a periodic potential ${V}$, have been intensively studied.

Does anything here enlighten you in any way? Probably not. In reality, even professional mathematicians have great difficulty in understanding advanced mathematics in areas other than their own specialities and interests.

Of course, there is a great difficulty in explaining in simple terms the significance and content of a paper on advanced mathematics. For example, to elucidate the Banach space paper, one might begin by explaining that a Banach space is a complete, normed linear space. But that would immediately beg three questions:

• What does complete mean?
• What is a norm?
• What is a linear space?

The answers would involve further excursions into analysis and algebra, requiring more layers of explanation.

Baffling the Boffins

Mathgen is a program to randomly generate professional-looking mathematics papers, including theorems, proofs, equations, discussion, and references. The automatic generation process succeeds because of the limited and formalised language used in most mathematical papers. The Mathgen program uses only a handful of sentence templates, and yet it is capable of producing documents that are remarkably close to those found in mathematical journals.

Many mathematical publications use stilted writing styles and contain numerous stock phrases. Mathgen constructs papers starting from a basic template and replaces blanks by various textual elements. These in turn contain blanks, so the construction is recursive.

Randomly-generated mathematics books have also been produced by MathGen. A recent example is Galois Knot Theory: With Applications to Elementary Global K-Theory, by J. Maruyama. URL

MathGen papers accepted for publication

Stefan Friedl of the University of Cologne had a Mathgen paper accepted by the Journal of Algebra and Number Theory Academia. It was entitled On the uniqueness of prime, Jacobi functors. The anonymous referees report offered unqualified praise for the paper:

“I have gone through the paper. It is a good paper. In my view the results obtained are original, new and interesting. This paper may trigger further research in the direction of work. Recommendation: I recommend that this paper be accepted for publication in the Journal.”

Unsurprisingly, the journal requires that the author pay an article processing charge of US\$20 per page.

Mathgen was written by Nate Eldredge, Associate Professor at the University of Northern Colorado. His blog has a name reminiscent of the one you are reading now: \url{https://thatsmathematics.com}.

The software underlying Mathgen is free and released under the terms of the GNU General Public License, version 2.0.

Sample Papers generated using MathGen

To bulk out my CV, I decided to generate a few publications using Mathgen. Below are five papers “written” by me in collaboration with some very big names in mathematics. The authors, title and opening sentence of each paper are given. At a casual glance, the papers — each produced in under a second — look realistic. But, of course, they are all complete rubbish. Caveat lector.

“References”

${\bullet}$ P.~Lynch, D.~Selberg, U.~Peano and U.~Bernoulli, 2021: On the Convexity of Factors. The authors address the reducibility of totally complex random variables under the additional assumption that ${\mathrm{d}X \le \boldsymbol{g}}$.

${\bullet}$ P.~Lynch and H.~T.~Boole, 2021: Super-discretely finite homeomorphisms and commutative potential theory. The authors extend quasi-globally sub-orthogonal, co-p-adic, pseudo-universally co-dependent graphs.

${\bullet}$ P.~Lynch and W.~Cayley, 2021: Convergence Methods in Concrete K-Theory. Recent developments in dynamics have raised the question of whether there exists a completely ordered globally ultra-infinite subalgebra.

${\bullet}$ P.~Lynch and U. Napier, 2021: Unique Categories and Topological Number Theory. We wish to extend the results of [11] to partially co-associative random variables.

${\bullet}$ P.~Lynch and Q.~Bernoulli, 2021: Anti-discretely onto-triangles over ultra-admissible subrings. A central problem in stochastic graph theory is the derivation of characteristic, universal monodromies.

Try Mathgen for yourself at this URL.   Just add your name and click “Generate”.