This paper seeks to present and justify 2 theses: that cryptography can be reduced to a purely categorical formulation, specifically that it can be reduced to the study of a single family of sets of functors; and that cryptography can be reduced to a purely 2-adic formulation, specifically it can be reduced to the study of functions Z m 2 → Z n 2 .

An isobaric polynomial is defined as a homogeneous polynomial of the form cijx i y j where the constant sum rule i + j = n is satisfied for some n. [Weis] This paper seeks to study such polynomials and provide new generalizations.

There are many different objects in mathematics that do not have innate geometric interpretations that are "direct" in some way-that is, realizations that can be seen, intuitively, as different "forms" of that object. This paper, building upon previous work and integrating some new ideas seeks to provide a general framework for doing so.

Dirichlet series, or, series of the form D(A, s) = ∞ i=1 a i i s are central objects in complex analysis and analytic number theory. This paper seeks to generalize them by introducing & studying new types of Dirichlet series-quaternionic Dirichlet series, and lattice Dirichlet series. That is, Dirichlet series over quaternions, and, various types of Dirichlet series over lattices.

In this paper, we will be doing an overview on a certain generalization of the Mandelbrot set-specifically, one to algebraic varieties defined as the zero sets of polynomials defined w.r.t. parameter spaces.

This paper studies some properties and generalizations of the "canonical structure" operator on strings, denoted C, which was originally introduced as part of the study of Papyrus Oxyrhynchus 90, in [Har25].

Analogously to the connection in number theory, in this paper, we establish and study the connection (or at least the most "direct" form of it) between overpacked lattices and Borcherds surfaces of index (k, 3).

In this paper, we will be discussing hyperbolas and other related algebraic varieties in n-dimensional space with group structures defined by projection multiplication, as well as detailing some of the theory that immediately emerges.

In this paper, we will extend the fundamental group π1 from its context in topological spaces to the context of graphs. We will also give some examples of fundamental groups of certain graphs, e.g. Kn.

In this paper, we will present a few new extensions of the theory of modular forms, a departure from the typical theory-i.e. the theory of modular forms over the matrix group SL2(Z) with inputs in H C .

In this paper, we will be giving some exposition on categorical products and of overpacked lattices, or, "overloaded lattices". We will also be detailing some immediate results of the constructions.

In this paper, we will be providing further generalizations for the Eisenstein series and ℘-function, which vastly generalize and enlarge the original notions. We will be focusing on results that pertain to specific kinds of subsets of C.