Calculating const(p) from the Wronskian identity for the alternating compositions of weighted differential operators
now a paper
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.
Computing const(p): from backtracking to a subset dynamic programmesame paper
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.
The 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,
Arthemy 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.
From backtracking to a dynamic programme over subsets, modulo primes: const(18) has 463 digits.
Research
Questions worth chasing
The structure of late-growing permutations
Wronskian identity for iterated differential operators
Minimal surfaces from symmetries
Three lines of work, each with its own page to step into.
Teaching
Courses I've TA'ed
Quantum Mechanics 1 and Metric & Topological Spaces, with hints and recaps for each homework.
/now
Current state vector
Preparing for the Caltech Mathathon - super excited to be a part of it!
Back from the ASC School on Generalized Symmetries - Arnold Sommerfeld Center, LMU Munich, 5 - 9 October 2026 - school page ↗.
New preprint with A. V. Kiselev: The counts of parity-odd vs even late-growing permutations in the group S2p−1 differ by (−1)p - arXiv:2610.35779. The ornament from it is now on the site.
Contact
k.shah.8@student.rug.nl
Or arXiv, ORCID, Google Scholar and LinkedIn, all on the contact page.
arXiv:math/0410185 · math.RA · 2004
On the associative homotopy Lie algebras and the Wronskians
A. V. Kiselev
Fundam. Appl. Math. 11 (2005) n.1, 159-180
arXiv:math/0603424 · math.DG · 2006
Minimal surfaces associated with nonpolynomial contact symmetry flows
