The commutator $ab - ba$ turns any associative algebra into a Lie algebra. Summing the products of $n$ elements over all orderings, with signs, gives an $n$-ary bracket, and this bracket makes the associative algebra a homotopy Lie algebra. This project takes the associative algebra of holomorphic differential operators $\mathrm{Diff}_{*}(\mathcal{O}(\mathbb{C}),\mathcal{O}(\mathbb{C}))$, where $\mathcal{O}(\mathbb{C})$ denotes the algebra of Laurent series over $\mathbb{C}$ (see arXiv:math/0410185).
For $2p$ operators of order $p$, the bracket is again an operator of order $p$, and its coefficient is the Wronskian of the weights times an integer const(p) that depends only on $p$. The question became what that integer is. The full story (identity, conjectures, evidence) lives on the research page.
The backtracking search in the paper computes const(p) exactly up to p = 14. Replacing the enumeration by a dynamic programme over subsets, carried out modulo primes, reached p = 18: const(18) has 463 digits and was computed on a single node of the Hábrók cluster.
C++ Dynamic Programming Modular Arithmetic CombinatoricsThe solutions of the minimal surface equation, $$\left(1+u_y^2\right)u_{xx} - 2u_xu_yu_{xy} + \left(1+u_x^2\right)u_{yy} = 0,$$ describe two-dimensional minimal surfaces $\Sigma \subset \mathbb{E}^3$ in nonparametric form.
In his 2006 paper, Dr. Kiselev constructs two infinite families of such surfaces. I build further examples from these families with differential-geometric techniques; the first surface of one family is a self-intersecting minimal surface filling the trefoil knot - the same knot the homepage flies you through - and I believe these surfaces have something unexpected to say to algebraic topology.
Differential Geometry Algebraic Topology Jets Maple Python