Various ring properties trivially satisfied by commutative rings have given place to various classes of rings. These include reflexive, semicommutative, abelian, and reversible rings. In this talk we present various examples of finite rings with the smallest possible cardinality satisfying those ring properties. In particular, special attention is devoted to a reflexive abelian non-semicommutative ring, whose cardinality has been shown to be 256.

The relationship between quantum link invariants and topological invariants of a link has been a motivating question in low-dimensional topology. For annular Khovanov homology, the wrapping conjecture of Hoste-Przytycki suggests that the maximal non-zero annular grading is related to embeddings of meridional disks. We provide further evidence for this conjecture by exhibiting infinite families of annular links for which the gap between the maximum non-zero annular Khovanov grading and the maximum non-zero annular Floer-theoretic gradings is arbitrarily large. We also show this gap exists at the decategorified level for some of the infinite families. Additionally, we show that certain satellite operations can not be used to construct counterexamples to the categorified wrapping conjecture.

A rational convex polytope, i.e., the convex hull of finitely many rational points in \mathbb{Q}^d, its lattice point count, and its associated combinatorial invariants provide information on quantities of geometric and algebraic interest. In algebraic geometry, a polytope $P$ corresponds to a projective toric variety $X_P$ and an ample line bundle $L$, whose Hilbert polynomial enumerates the lattice points in $P$. In commutative algebra, lattice point enumeration appears in the guise of Hilbert series of graded rings. In the representation theory of semisimple Lie algebras, the Kostant partition function enumerates the lattice points in flow polytopes. The lattice point enumerator results to be a quasipolynomial, known as the Ehrhart quasipolynomial of a rational polytope P, and encodes the number of integer lattice points in dilates of $P$. The $h^*$ -polynomial of $P$ is the numerator of the accompanying generating function. We provide two decomposition formulas for the $h^*$-polynomial of a rational polytope. The first decomposition generalizes a theorem of Betke and McMullen for lattice polytopes. We use our formula to provide a novel proof of Stanley's Monotonicity Theorem for the $h^*$-polynomial of a rational polytope. The second decomposition generalizes a result of Stapledon, which we use to provide rational extensions of the Stanley and Hibi inequalities satisfied by the coefficients of the $h^*$-polynomial for lattice polytopes. Lastly, we apply our results to rational polytopes containing the origin whose duals are lattice polytopes. This is joint work with Matthias Beck (SFSU & FU Berlin) and Ben Braun (Univ. of Kentucky).

Notable examples of periodic sphere packings include the hexagonal lattice in $\mathbb{R}^2$ which Gauss proved is the densest periodic planar circle packing, and the Leech lattice and $E_8$ in $\mathbb{R}^3$. In this talk, we will explore what a lattice is, and introduce a result by M. Faulhuber and S. Steinerberger which proves another extremal property of the hexagonal lattice, using the fact that it is a critical point in the space of lattices.

Using a standard embedding of $K[x]$, the algebra of polynomials with coefficients in a field $K$, into the algebra of row and column finite matrices over $K$, non-trivial factorizations of some irreducible polynomials in $K[x]$ are possible in the larger context. The row and column finite matrices involved in those factorizations resemble polynomials in several ways. We call these matrices \emph{entangled polynomials} as each one of them is induced by a finite number of polynomials. For every fixed $m \in \Z_+$, entangled polynomials induced by $m$ polynomials (the so-called $m$-nomials) form a ring, $K^{m}[x]$. The ring of $1$-nomials is precisely $K[x]$. When $n$ divides $m$, $K^{n}[x]$ is a subring of $K^{m}[x]$. In particular, all rings of $m$-nomials include the polynomials.
Given an algebra $A$ over a field $K$, a basis $\cB$ for $A$ is said to be amenable if one can naturally extend the $A$-module structure on the $K$-vector space $\bigoplus_{b \in \cB} Kb= K^{(B)}$ to the $K$-vector space $\prod_{b \in \cB} Kb = K^{\cB}$. A basis $\cB$ is congenial to another one $\cC$ if infinite linear combinations of elements of $\cB$ translate in a natural way to infinite linear combinations of elements of $\cC$. An amenable basis $\cB$ is simple if it is not properly congenial to any other amenable basis. We will present amenable bases in the ring of all $m$-nomials as well as some results about congeniality in the ring of all $2$-nomials. This will culminate to presenting a simple basis in the ring of all $2$-nomials.

Discrete Morse theory, a cell complex-analog to smooth Morse theory, has been developed over the past few decades since its original formulation by Forman in 1998. One of the main objects of concern in discrete Morse theory are discrete gradient vector fields on simplicial complexes. We prove that the characteristic polynomial of the Laplacian of a simplicial complex is a special generating function for gradients when the complex is either any graph or a triangulation of an orientable manifold of any dimension.

The relationship between nilpotent elements and commutators over rings has been studied in detail over the years. We will show that every nilpotent N in M_n(D), the ring of square matrices over a division ring, can be presented as a single commutator, that is, N = AB − BA for some matrices A, B in M_n(D). We will also construct an example illustrating that there exists a prime ring with unity over which some nilpotent matrices cannot be presented as commutators.

A numerical semigroup is a submonoid of the non-negative integers under addition. An important property of numerical semigroups is that their elements may have multiple factorizations. Given an element $n$ in a numerical semigroup $S$, graphs can be constructed using the multiple factorizations of $n$ as vertices. In this talk, we explore the minimal trade graphs of $n$ and the factorization support graph of $n$. The rank of the fundamental group of these graphs can be found by counting how many edges are present in each.