To begin, we consider an n-ary bracket over any associative algebra, making it a homotopy Lie algebra. For this project, we considered 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 the Laurent series over $\mathbb{C}$ (see arXiv:math/0410185). This project resulted in the paper arXiv:2605.11137 with Prof. Arthemy V. Kiselev, an OEIS submission, an ANTS XVII poster and a Group36 talk. The full story - identity, conjectures, evidence - lives on the research page.
Differential Geometry Jets Maple CombinatoricsNaively expanding $\sum_{\sigma \in S_N} (-1)^\sigma\, w_{\sigma(1)} \cdots w_{\sigma(N)}^{(p)}$
becomes infeasible combinatorially fast, so I developed const_p_parallel.py (soon to be on GitHub I promise), which walks the tree instead:
a backtracking search over admissible weight distributions, parallelised across cores,
with which $\mathrm{const}(p)$ is calculated uptil $p = 14$.
The same runs feed the asymptotic fit $\log \mathrm{const}(p) \approx c \cdot p \log(p!) + d \cdot \log\bigl((2p)!\bigr)$, which matches with $R^2 \approx 0.9999$ - a very good fit.
Python Parallel Computing Combinatorics p-adicThe 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