Postcards from the edge, or Snapshots of the theory of generalised Moonshine

What would you do if you were asked to "add" knowledge? Would you say that "one plus one knowledge" is two "knowledges"? Less than that? More? Or something in between? Adding knowledge sounds strange, but it brings to the forefront questions that are as fundamental as they are eclectic. These are questions about the nature of knowledge and about the use of mathematics to model reality. In this chapter, I explore the mathematics of adding knowledge starting from what I believe...

While the Standard Model is in good shape, there are many reasons to believe it is incomplete. There are high expectations that the LHC will shed light on some well studied possibilities, like technicolor and supersymmetry. Emboldened by this optimism, we consider some non-mainstream ideas that if established would change dramatically the way we view the world.

Mathematical conception of infinite quantities forms a cornerstone of many disciplines of modern mathematics --- from differential calculus to set theory. In fact, it could be argued that the most significant revolutions in mathematics in the modern period were always triggered by a development in our understanding of infinity. From the pedagogical point of view, the students' comprehension of the concept of infinity is a competence of interdisciplinary value, helping them to...

In this article, we employ mathematical concepts as a tool to examine the phenomenon of consciousness experience and logical phenomena. Through our investigation, we aim to demonstrate that our experiences, while not confined to limitations, cannot be neatly encapsulated within a singular collection. Our conscious experience emerges as a result of the developmental and augmentative trajectory of our cognitive system. As our cognitive abilities undergo refinement and advanceme...

Motivated by the appearance of penumbral moonshine, and by evidence that penumbral moonshine enjoys an extensive relationship to generalized monstrous moonshine via infinite products, we establish a general construction in this work which uses singular theta lifts and a concrete construction at the level of modules for a finite group to translate between moonshine in weight one-half and moonshine in weight zero. This construction serves as a foundation for a companion paper i...

Computer science provides an in-depth understanding of technical aspects of programming concepts, but if we want to understand how programming concepts evolve, how programmers think and talk about them and how they are used in practice, we need to consider a broader perspective that includes historical, philosophical and cognitive aspects. In this paper, we develop such broader understanding of monads, a programming concept that has an infamous formal definition, syntactic su...

This is a non-standard paper, containing some problems in set theory I have in various degrees been interested in. Sometimes with a discussion on what I have to say; sometimes, of what makes them interesting to me, sometimes the problems are presented with a discussion of how I have tried to solve them, and sometimes with failed tries, anecdote and opinion. So the discussion is quite personal, in other words, egocentric and somewhat accidental. As we discuss many problems, hi...

Presented here are over one hundred conjectures ranging from easy to difficult, from many mathematical fields. I also summarize briefly methods and tools that have led to this collection.

We give an historical account, including recent progress, on some problems of Erd\H os in number theory.

Several examples are used to illustrate how we deal cavalierly with infinities and unphysical systems in physics. Upon examining these examples in the context of infinities from Cantor's theory of transfinite numbers, the only known mathematical theory of infinities, we conclude that apparent inconsistencies in physics are a result of unfamiliar and unusual rules obeyed by mathematical infinities. We show that a re-examination of some familiar limiting results in physics lead...