δ(γ) = supx≠y dγ (x,y) / |x−y|Knot distortion: dγ (x,y) is the shorter arclength from x to y measured along the knot itself; |x−y| is the straight-line distance between the two points in 3-space.
This is a torus knot T(p,q).Click any two points on the knot : the orange straight line is the distance in space, the glowing cyan curve is the distance along the knot.
Knot parameter p 3
Knot parameter q 5
Tube thickness 0.110
Sampling resolution N=500
Find a high-distortion pair Clear selection
Straight-line distance |x−y| —
Arclength along the knot dγ —
Distortion ratio dγ /|x−y| —
Discrete-sampling lower bound (N=500 ） —
The values above are educational estimates from discrete sampling, not a rigorous computation of the continuous supremum.
Pardon settled one of Gromov's problems on knot distortion: he proved that certain increasingly complicated torus knotscannot keep their geometric distortion uniformly bounded . This does not mean that the optimal distortion of every knot has been fully classified, nor that distortion grows at any fixed rate as complexity increases — and the values measured in this animation are certainly not a rigorous proof.
The Hilbert–Smith conjecture (3-dimensional case)If a locally compact topological group acts faithfully on a connected 3-manifold, then that group must be a Lie group.
Glossary
Group action (group action): a family of symmetries that compose with one anotherFaithful action (faithful): distinct group elements really do give distinct transformationsLie group (Lie group): both a group and a smooth continuous geometric structure3-manifold (3-manifold): a space that looks locally like ordinary 3-dimensional space everywhere
Action mode
Regular action Pathological action
Regular action: the manifold undergoes continuous, smooth, invertible rotation and twisting; orbits trace smooth curves and nearby points move continuously — this is what a Lie group action looks like.
Red warning grid: here the motion is pushed to break up on ever smaller scales without limit. Pardon's theorem flags such behaviour as a candidate structure that cannot occur faithfully on a 3-manifold. This animation is illustrative, not a rigorous proof.
Pardon proved the Hilbert–Smith conjecture in the case of 3-manifolds, a major breakthrough on a problem roughly 70 years old.Note: the conjecture is not settled in all dimensions, and no such claim is made on this page.
The translucent body on screen is only a conceptual cross-section of the local structure of a 3-manifold (a schematic model); it does not treat the 2-sphere as a 3-manifold.
Left: a closed curve in the geometric space M (coloured along a pair of conjugate directions). Right, the disc:moduli space — each curve corresponds to one of these parameter points.
Perturbation ε 0.00
Naive count (illustrative data) 5 curves before perturbation
Counting points directly is not a stable invariant.
Three typical problems
① Extra symmetry Curves have automorphisms, so counting by naive cardinality double-counts or loses weights.
② Degenerate limits Several distinct curves degenerate to the same limiting object and their parameter points merge.
③ Anomalous dimension The actual dimension of the moduli space disagrees with the expected one; singularities and branching appear locally.
All figures here are illustrative and educational, not computations for any actual space.
Virtual fundamental cycle Fukaya category
[M]vir ∈ Ȟd (M)Virtual fundamental cycle: when the actual moduli space is non-smooth, of the wrong dimension, or singular, one builds an algebraic–topological surrogate carrying the correct expected dimension, so that curve counts remain invariant.
Four-step construction (geometric metaphor)
1 Break up the moduli space 2 Sweep singularities
3 Local patches 4 Virtual cycle
▷ Play all four Toggle: before / after perturbation
Moduli space appearance Before perturbation
Weighted count via the virtual cycle (schematic) +3
What is shown here is a geometric metaphor for the idea of the virtual fundamental cycle, not a step-by-step account of Pardon's construction.
Citation: Pardon brought methods from algebraic topology to bear on moduli spaces of pseudo-holomorphic curves, gave a new construction of virtual fundamental cycles, and applied it to several central problems in symplectic geometry. The notion of a virtual fundamental cycle rests on foundational work by a number of mathematicians; Pardon is not its sole originator.
Related applications
Applications related to the Arnold conjecture the rigorous definition of contact homology Counting pseudo-holomorphic curves
on a Liouville manifold (whose boundary extends to infinity) wrapped Fukaya category Conceptual scene.Drag the gold or purple Lagrangian changes the intersections.
Objects: Lagrangian submanifolds
Morphisms: their intersections / generators of the Floer complex
Combinatorial structure: encoded by counts of pseudo-holomorphic polygons (A∞ structure)
Wrapped Fukaya categories apply to certain non-compact symplectic manifolds
Show disc Wrapped mode Reset
Current intersections (generators) 0
Click a glowing intersection to inspect it locally.
With collaborators including Sheel Ganatra and Vivek Shende, Pardon studied and computed the structure of wrapped Fukaya categories of certain Liouville manifolds. The intersection counts shown here are illustrative, not research-level invariants.
Left: Gromov–Witten theory — counting curves mapped into the target space directly (with parametrisation and stable-map structure; different curves contribute different weights).Right: Donaldson–Thomas theory — the same target space discretised into points, sheaves and algebraic data (ideal sheaves / stable objects), not a count of ordinary mesh triangles.
ZGW (u,β) ↔ ZDT (q,β)，q = −eiu The correspondence of generating functions after the change of variables — schematic
▷ Count one by one Apply the substitution
Curves counted on the GW side 0 / 4
Objects encoded on the DT side 0 / 4
Align generating functions Not substituted
The MNOP conjecture predicts a deep correspondence between the generating functions of Gromov–Witten and Donaldson–Thomas invariants. Pardon introduced and developed strikingly original methods and obtained major breakthroughs for important complex threefolds (such as Calabi–Yau threefolds, and cases where the anticanonical bundle satisfies the relevant nef condition) — this does not amount to a proof of the full MNOP conjecture for all spaces.