We present a theory that produces several examples in which the homotopy Lie algebra of a complex hyperplane arrangement is not finitely presented. We also present examples of hyperplane arrangements in which the enveloping algebra of this Lie algebra has an irrational Hilbert series. This answers two questions of Denham and Suciu.

The proof by Euclid that there exist infinitely many prime numbers is well known. The proof involves generating prime numbers that do not belong to a given finite set of primes, and one may ask whether all prime numbers can be obtained by this method. Daniel Shanks gave a heuristic argument that suggests that the answer is affirmative. Despite recent advances in computational number theory, numerical examples do not seem to make this conjecture convincing. We reformulate the problem in polynomial rings over finite fields and prove that in some explicitly characterized cases, Shanks’s argument does not hold. On the other hand, we have performed numerical computations that suggest that except for the above cases, Shanks’s conjecture is true.

We describe some theoretical results on triangulations of surfaces and we develop a theory on roots, decompositions, and genus surfaces. We apply this theory to describe an algorithm to list all triangulations of closed surfaces with at most a fixed number of vertices. We specialize the theory to the case that the number of vertices is at most 11, and we obtain theoretical restrictions on genus surfaces, allowing us to obtain a list of all triangulations of closed surfaces with at most 11 vertices.

We determine the rational homology group of $\mathrm{Out}(F_n)$ for $n\leq 6$. Combining this result with results of Conant and Vogtmann proves that the first two Morita classes are nontrivial. We conclude that these classes generate the nontrivial part of the rational homology in the range $n\leq 6$.

In this paper we gather experimental evidence related to the question of deciding whether a curve has a rational point. We consider all genus-$2$ curves over $\Bbb Q$ given by an equation $y^2 = f(x)$ with $f$ a square-free polynomial of degree 5 or 6, with integral coefficients of absolute value at most 3. For each of these roughly 200,000 isomorphism classes of curves, we decide whether there is a rational point on the curve by a combination of techniques that are applicable to hyperelliptic curves in general.

In order to carry out our project, we have improved and optimized some of these techniques. For 2 of the curves, our result is conditional on the Birch and Swinnerton-Dyer conjecture or on the generalized Riemann hypothesis.

The papers [Gálvez et al. 00, Kokubu et al. 03, Kokubu et al. 05] gave a method of constructing flat surfaces in threedimensional hyperbolic space. Generically, such surfaces have singularities, since any closed nonsingular flat surface is isometric to a horosphere or a hyperbolic cylinder. In [Sasaki et al. 08a], we defined a map, called the hyperbolic Schwarz map, from one-dimensional projective space to three-dimensional hyperbolic space using solutions of the Gauss hypergeometric differential equation. Its image is a flat front and its generic singularities are cuspidal edges and swallowtail singularities. In this paper we study the curves consisting of cuspidal edges and the creation and elimination of swallowtail singularities depending on the parameters of the hypergeometric equation.

We investigate, both numerically and mathematically, several questions about embedded minimal surfaces of finite total curvature in Euclidean space. We also describe how a theoretical construction can be implemented numerically to produce pictures of such surfaces.

We provide a proof of a conjecture in [Bailey et al. 07a] on the existence and form of linear recurrences for moments of powers of the Bessel function $K_0$.

Recently, Friedman proved Alon's conjecture for many families of $d$-regular graphs, namely that given any $\epsilon > 0$, ``most'' graphs have their largest nontrivial eigenvalue at most $2\sqrt{d-1}+\epsilon$ in absolute value; if the absolute value of the largest nontrivial eigenvalue is at most $2\sqrt{d-1}$, then the graph is said to be Ramanujan. These graphs have important applications in communication network theory, allowing the construction of superconcentrators and nonblocking networks, as well as in coding theory and cryptography. Since many of these applications depend on the size of the largest nontrivial positive and negative eigenvalues, it is natural to investigate their distributions. We show that these are well modeled by the $\beta=1$ Tracy-Widom distribution for several families. If the observed growth rates of the mean and standard deviation as a function of the number of vertices hold in the limit, then in the limit, approximately 52% of $d$-regular graphs from bipartite families should be Ramanujan, and about 27% from nonbipartite families (assuming that the largest positive and negative eigenvalues are independent).

We construct five new elements of degree 6 in the nucleus of the free alternative algebra. We use the representation theory of the symmetric group to locate the elements. We use the computer algebra system ALBERT and an extension of ALBERT to express the elements in compact form and to show that these new elements are not a consequence of the known degree-5 elements in the nucleus. We prove that these five new elements and four known elements form a basis for the subspace of nuclear elements of degree 6. Our calculations are done using modular arithmetic to save memory and time. The calculations can be done in characteristic zero or any prime greater than 6, and similar results are expected. We generated the nuclear elements using prime 103. We check our answer using five other primes.