01 / EXHIBITAlgebraic Geometry
Geometry of framed correspondences
Inspect an étale neighbourhood, its deleted loci, and the finite support of a framed homotopy.
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An interactive atlas of mathematical ideas
The Geometry
of Mathematics
Explore the structures behind advanced mathematical ideas. Research-level models in geometry, topology, motives, K-theory, and category theory.
11 exhibits · open to everyone · no account required
z³ − 1 = 0Three roots on the Riemann sphere · drag to rotateThe complete collection
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10 visualisations · all subjects
01 / EXHIBIT
02 / EXHIBITMotivic Homotopy Theory
A collision inside a framed homotopy
Watch two complex zeros meet without destroying finite flatness or the algebraic framing.
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03 / EXHIBITSpectral Sequences
The motivic Atiyah–Hirzebruch spectral sequence
Follow classical stable stems through a Postnikov tower into framed motivic cohomology.
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04 / EXHIBITNoncommutative Geometry
A genuine symmetric projective line
Keep the projective points and change the structure sheaf: the stalk at infinity becomes a field.
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05 / EXHIBITAlgebraic K-Theory
Semilocal Milnor K-theory
A divisor crosses a tetrahedron, yet its equation is a unit in the ring semilocalized at the vertices.
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06 / EXHIBITAlgebraic Topology
From framed points to the sphere spectrum
Connect a framed collision and a degree-d map of spheres to the Garkusha–Panin reconstruction theorem.
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07 / EXHIBITAlgebraic Topology
Reconstructing the sphere spectrum: Theorem 11.9
Follow twelve narrated scenes from Pontryagin–Thom intuition and polynomial framings to the Garkusha–Panin sphere reconstruction theorem.
Animation · Narrated · 12 scenesWatch
08 / EXHIBITCategory Theory
Framed correspondences and symmetric spectra
Explore polynomial framings, their composition and symmetric shuffles, stabilisation, and finite-flat homotopies.
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09 / EXHIBITMotivic Homotopy Theory
E-Framed Thom Laboratory
Explore normal framings, collisions of l.c.i. supports, stabilisation of embeddings, and explicit models of motivic Thom spectra.
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10 / EXHIBITAlgebraic K-Theory
The Hauptlemma in algebraic Kasparov K-theory
Move parametrised facets through a subdivided simplex, preserve the cube’s lateral boundary, and trace the construction of algebraic Kasparov spectra.
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11 / EXHIBITMotivic Homotopy Theory
Group schemes and motivic spectra
Explore Stiefel quotient towers, the untwisting of induced Thom spaces, and the comparison theorem for group-scheme spectra.
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Created by Grigory Garkusha, this collection develops interactive models around research in algebraic geometry, topology, motivic homotopy theory, and algebraic K-theory.
Every exhibit identifies its constructions, hypotheses, and limitations. The models invite exploration while keeping the mathematical statements explicit.
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Controls change meaningful parameters and reveal what the equations govern.
Read the mathematics
Definitions, references, and boundaries accompany the interactive model.
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Each standalone model has a permanent, shareable URL.
Sources
The mathematics behind the models.
Grigory Garkusha and Ivan Panin. Framed motives of algebraic varieties (after V. Voevodsky). J. Amer. Math. Soc. 34 (2021), 261–313.Read the reference ↗
Grigory Garkusha. Computing framed motives. Preprint; Section 3, Theorems 3.3, 3.5 and 3.8, as cited by the exhibit.Read the reference ↗
Grigory Garkusha. Symmetric quasi-coherent sheaves. Definitions underlying the example; the exhibit refers to the September manuscript.Read the reference ↗
Grigory Garkusha. Semilocal Milnor K-Theory. IMRN 2023, no. 24, 22069–22095; Definitions 2.3 and 3.1, Remark 3.2.Read the reference ↗
Grigory Garkusha. Correspondences and stable homotopy theory. Trans. London Math. Soc. 10(1) (2023), 124–155; Definition 2.3 and Theorems 2.4, 3.2, 5.2 and 5.5.Read the reference ↗
Grigory Garkusha. Algebraic Kasparov K-theory. I. Documenta Math. 19 (2014), 1207–1269; §7.2, the Hauptlemma and formula (8).Read the reference ↗
Grigory Garkusha. Group schemes and motivic spectra. Israel J. Math. 259 (2024), 727–758; Theorems 6.1 and 6.5.Read the reference ↗
Grigory Garkusha and Alexander Neshitov. Fibrant resolutions for motivic Thom spectra. Annals of K-Theory 8 (2023), no. 3, 421–488; Theorems 8.1, 9.13, 11.26, 12.15 and 12.16.Read the reference ↗