We study a twisted Alexander polynomial naturally associated to a hyperbolic knot in an integer homology 3-sphere via a lift of the holonomy representation to $\mathrm{SL}(2, \mathbb{C})$. It is an unambiguous symmetric Laurent polynomial whose coefficients lie in the trace field of the knot. It contains information about genus, fibering, and chirality, and moreover, is powerful enough to sometimes detect mutation.

We calculated this invariant numerically for all $313\, 209$ hyperbolic knots in $S^3$ with at most 15 crossings, and found that in all cases it gave a sharp bound on the genus of the knot and determined both fibering and chirality.

We also study how such twisted Alexander polynomials vary as one moves around in an irreducible component $X_0$ of the $\mathrm{SL}(2, \mathbb{C})$-character variety of the knot group. We show how to understand all of these polynomials at once in terms of a polynomial whose coefficients lie in the function field of $X_0$. We use this to help explain some of the patterns observed for knots in $S^3$, and explore a potential relationship between this universal polynomial and the Culler–Shalen theory of surfaces associated to ideal points.

In 1878, motivated by the requirements of the invariant theory of binary forms, J. J. Sylvester constructed, for every graph with possible multiple edges but without loops, its symmetrized graph monomial, which is a polynomial in the vertex labels of the original graph. We pose the question for which graphs this polynomial is nonnegative or a sum of squares. This problem is motivated by a recent conjecture of F. Sottile and E. Mukhin on the discriminant of the derivative of a univariate polynomial and by an interesting example of P. and A. Lax of a graph with four edges whose symmetrized graph monomial is nonnegative but not a sum of squares.We present detailed information about symmetrized graph monomials for graphs with four and six edges, obtained by computer calculations.

This paper presents empirical evidence supporting Goldfeld’s conjecture on the average analytic rank of a family of quadratic twists of a fixed elliptic curve in the function field setting. In particular, we consider representatives of the four classes of nonisogenous elliptic curves over $\mathbb{F}_q(t)$ with $(q, 6) = 1$ possessing two places of multiplicative reduction and one place of additive reduction. The case of $q = 5$ provides the largest data set as well as the most convincing evidence that the average analytic rank converges to 1/2, which we also show is a lower bound following an argument of Kowalski. The data were generated via explicit computation of the $L$-function of these elliptic curves, and we present the key results necessary to implement an algorithm to efficiently compute the $L$-function of nonisotrivial elliptic curves over $\mathbb{F}_q(t)$ by realizing such a curve as a quadratic twist of a pullback of a "versal" elliptic curve. We also provide a reference for our open-source library ELLFF, which provides all the necessary functionality to compute such $L$-functions, and additional data on analytic rank distributions as they pertain to the density conjecture.

Using the results of several extremely large recent computations, we tested positively the normality of a prefix of roughly four trillion hexadecimal digits of $\pi$. This result was used by a Poisson process model of normality of $\pi$: in this model, it is extraordinarily unlikely that $\pi$ is not asymptotically normal base 16, given the normality of its initial segment.

Let $A \supseteq B$ be cancellative abelian semigroups, and let $R$ be an integral domain. We show that the semigroup ring $R[B]$ can be decomposed, as an $R[A]$-module, into a direct sum of $R[A]$-submodules of the quotient ring of $R[A]$. In the case of a finite extension of positive affine semigroup rings, we obtain an algorithm computing the decomposition. When $R[A]$ is a polynomial ring over a field, we explain how to compute many ring-theoretic properties of $R[B]$ in terms of this decomposition. In particular, we obtain a fast algorithm to compute the Castelnuovo–Mumford regularity of homogeneous semigroup rings. As an application we confirm the Eisenbud–Goto conjecture in a range of new cases. Our algorithms are implemented in the MACAULAY2 package MONOMIAL ALGEBRAS.

Associated to any toric ideal are two special generating sets: the universal Gröbner basis and the Graver basis, which encode polyhedral and combinatorial properties of the ideal, or equivalently, its defining matrix. If the two sets coincide, then the complexity of the Graver bases of the higher Lawrence liftings of the toric matrices is bounded.

While a general classification of all matrices for which both sets agree is far from known, we identify all such matrices within two families of nonunimodular matrices, namely, those defining rational normal scrolls and those encoding homogeneous primitive colored partition identities. This also allows us to show that higher Lawrence liftings of matrices with fixed Gröbner and Graver complexities do not preserve equality of the two bases.

The proof of our classification combines computations with the theoretical tool of Graver complexity of a pair of matrices.

We investigate simple random walks on graphs generated by repeated barycentric subdivisions of a triangle. We use these random walks to study the diffusion on the self-similar fractal known as the Strichartz hexacarpet, which is generated as the limit space of these graphs. We make this connection rigorous by establishing a graph isomorphism between the hexacarpet approximations and graphs produced by repeated barycentric subdivisions of the triangle. This includes a discussion of various numerical calculations performed on these graphs and their implications to the diffusion on the limiting space. In particular, we prove that equilateral barycentric subdivisions—a metric space generated by replacing the metric on each 2-simplex of the subdivided triangle with that of a scaled Euclidean equilateral triangle—converge to a self-similar geodesic metric space of dimension log(6)/ log(2), or about 2.58. Our numerical experiments give evidence to a conjecture that the simple random walks on the equilateral barycentric subdivisions converge to a continuous diffusion process on the Strichartz hexacarpet corresponding to a different spectral dimension (estimated numerically to be about 1.74).