For a finite group \(G\), we introduce the complete suboperad \(Q_G\) of the categorical \(G\)-Barratt-Eccles operad \(P_G\). We prove that \(P_G\) is not finitely generated, but \(Q_G\) is finitely generated and is a genuine \(E_\infty\) \(G\)-operad (i.e., it is \(N_\infty\) and includes all norms). For \(G\) cyclic of order 2 or 3, we determine presentations of the object operad of \(Q_G\) and conclude with a discussion of algebras over \(Q_G\), which we call biased permutative equivariant categories.

In a seminal 2007 paper, Andrews introduced a class of combinatorial objects that generalize partitions called \(k\)-marked Durfee symbols. Multivariate rank generating functions for these objects have been shown by many to have interesting modularity properties at certain vectors of roots of unity. Motivated by recent studies of rank generating functions for strongly unimodal sequences, we apply methods of Andrews to define an analogous class of combinatorial objects called \(k\)-marked strongly unimodal symbols that generalize strongly unimodal sequences. We establish a multivariate rank generating function for these objects, which we study combinatorially. We conclude by discussing potential quantum modularity properties for this rank generating function at certain vectors of roots of unity.

We look at the minimal graded free resolution of the Rees algebra of an ideal \(I\) and take degree-\(d\) strands of this resolution to give a graded free resolution of \(I^d\). We give a bound on the regularity of \(I^d\) through this process. We also provide a detailed example going through the process of trimming the graded free resolution to obtain the value of the regularity from the degree-\(d\) strand.

We examine the concept of infinitesimals in the works of both Newton and Leibniz and expand on why their initial concept of the infinitesimal failed to address both mathematical and philosophical problems. Historically, this led to the development of real analysis and with theory of limits being rigorously introduced by Cauchy and Weierstrass in the 19th century, the use of infinitesimals in math was abandoned and made obsolete. Despite this new and rigorous foundation, much of the new theory was (and still is) rather unintuitive in the physical sense. However, we argue that it is not necessary to completely abandon infinitesimals, but rather under a different framework of mathematics, we can make the infinitesimal mathematically rigorous.