In this paper we report on massive computer experiments aimed at finding spherical point configurations that minimize potential energy. We present experimental evidence for two new universal optima (consisting of 40 points in 10 dimensions and 64 points in 14 dimensions), as well as evidence that there are no others with at most 64 points. We also describe several other new polytopes, and we present new geometrical descriptions of some of the known universal optima.

We consider the problem of when an L-space homology sphere gives rise to lens spaces. We will show that when a knot in an L-space homology sphere $Y$ yields $L(p,q)$ by an integral Dehn surgery, then the slope $p$ is bounded by the genus of the knot and the correction term of $Y$, and we will demonstrate that many lens spaces are obtained from an L-space homology sphere whose correction term is equal to $2$.

Let $E: y^2=x^3+D$ be an elliptic curve, where $D$ is an integer that contains no primes $p$ with $6 \mid {\ord}_pD$. For a nontorsion rational point $P$ on $E$, write $x(nP)=A_n(P)/B_n^2(P)$ in lowest terms. We prove that for the sequence $\{B_{2^m}(P)\}_{m \ge 0}$, the term $B_{2^m}(P)$ has a primitive divisor for all $m \ge 3$. As an application, we give a new method for solving the Diophantine equation $y^2=x^3+d^n$ under certain conditions.

We shall investigate spacelike constant-mean-curvature surfaces with singularities in Minkowski 3-space. Recently, Kokubu, Rossman, Saji, Umehara, and Yamada gave useful criteria for cuspidal edges and swallowtails. Applying their criteria, we define a new class of generalized constant-mean-curvature surface and give some examples with cuspidal edges and swallowtails.

This paper studies $\EPDiff(S^1)$, the Euler-Poincaré equation for diffeomorphisms of $S^1$, with the Weil-Petersson metric on the coset space $\PSL_2(\R)\setminus\Diff(S^1)$. This coset space is known as the universal Teichmüller space. It has another realization as the space of smooth simple closed curves modulo translations and scalings. $\EPDiff(S^1)$ admits a class of solitonlike solutions (teichons) in which the ``momentum'' $m$ is a distribution. The solutions of this equation can also be thought of as paths in the space of simple closed plane curves that minimize a certain energy. In this paper we study the solution in the special case that $m$ is expressed as a sum of four delta functions. We prove the existence of the solution for infinite time and find bounds on its long-term behavior, showing that it is asymptotic to a one-parameter subgroup in $\Diff(S^1)$. We then present a series of numerical experiments on solitons with more delta functions and make some conjectures about these.

In this paper we present an algorithm for computing Hecke eigensystems of Hilbert--Siegel cusp forms over real quadratic fields of narrow class number one. We give some illustrative examples using the quadratic field $\Q(\sqrt{5})$. In those examples, we identify Hilbert--Siegel eigenforms that are possible lifts from Hilbert eigenforms.

The Bogomolov conjecture is a finiteness statement about algebraic points of small height on a smooth complete curve defined over a global field. We verify an effective form of the Bogomolov conjecture for all curves of genus at most $4$ over a function field of characteristic zero. We recover the known result for genus-$2$ curves and in many cases improve upon the known bound for genus-$3$ curves. For many curves of genus $4$ with bad reduction, the conjecture was previously unproved.

Let $X_{m,n}$ be the Segre-Veronese variety $\P^m \times \P^n$ embedded by the morphism given by $\cO(1,2)$. In this paper, we provide two functions $\underline{s}(m,n)\le \overline{s}(m,n)$ such that the $s$th secant variety of $X_{m,n}$ has the expected dimension if $s \leq \underline{s}(m,n)$ or $ \overline{s}(m,n) \leq s$. We also present a conjecturally complete list of defective secant varieties of such Segre-Veronese varieties.