Every dualizable additive category is an almost-module category
The characterization links dualizable additive categories to idempotent ideals, nuclear modules, and algebraic K-theory.
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Algebraic Topology
Homotopy theory, homological algebra, algebraic treatments of manifolds
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Weil group fails to make number fields K(π,1) spaces
A new anima with the Weil group as fundamental group adds higher homotopy to produce better-behaved cohomology.
Koszul duality is one equivalence of 2-categories
A Morita category of operads and bimodules matches a dual category of cooperads and bicomodules.
Distinct primes: Adams operation ψ^p leaves the J-map unchanged
Spectrum-level proof: the ℓ-completed J-homomorphism ignores ψ^p, with oriented and ℓ-local variants.
KL transport map preserves topology while condensing point clouds
Condensed shapes stay homotopy equivalent to the data, fixing PCA artifacts on image sets.
AI agent wrote the Agda proof that π₃(S²) ≅ ℤ
In 22 days it completed a formal proof requiring the Hopf fibration and Blakers–Massey.
A 24-periodic formula counts rank-n bundles on CP^{n+2}
For odd n, existence is decided by the top two Chern classes; the count depends only on n mod 24 and c1,c2 mod 12.
· “Enumerating corank 2 complex vector bundles on odd complex projective spaces”
Orlik–Solomon algebras now cover locally geometric posets
For compact complex abelian groups, an explicit dga models the complement's rational cohomology.
This mathematics paper proves a sharp cutoff: if every edge of a finite graph has…
A finite graph with all edge curvatures above one half has vanishing first GLMY path homology, and one half is the best possible threshold.
Two-to-one bubble weights yield explicit topology at infinity
Interior bubbles cost twice boundary bubbles; homology of the resulting spaces is computed exactly.
Three roads lead to one canonical p-adic Mal'cev completion
Representations, Hopf algebras, and analytic coordinates are interchangeable toolkits for pro-p groups.
Relative hyperbolic curves are anabelian in families
Dominant maps into such a family are classified by π1-open maps of profinite étale homotopy types over the base.
4-manifold invariant distinguishes phantom homotopy classes
Extends the known fusion-2-category state sum and distinguishes homotopy classes invisible to homotopy groups.
· “Graded-fusion 2-categories and quantum homotopy invariants of 4-manifolds”
Viterbo isomorphism lifts to MU for every closed manifold
Symplectic cohomology of T*Q becomes a twisted Thom spectrum of the free loop space of Q.
· “Spectral Viterbo isomorphism: complex-oriented versus framed”
One matrix reduction proves the barcode isometry theorem
A single column reduction turns an interleaving into a matching between barcodes, uniting stability with persistent homology.
· “Matrix Algebra for Persistence Modules Yields a Proof of the Isometry Theorem”
Equivariant sphere-to-Stiefel degrees collapse to one residue class
When the Hurewicz index is 1, every equivariant map has degree congruent to a binomial coefficient mod m, or none exist.
· “On the degrees of equivariant maps from spheres to complex Stiefel manifolds”
Chaos makes the stochastic partition function grow exponentially
The paper separates the noise partition function from the dynamics, turning chaos into spontaneous supersymmetry breaking.
Contours govern persistence-module distances
A bijection ties simple norms to contours; continuous contours yield compact families.
All finite Coxeter groups have vanishing Bogomolov multiplier
A cut-and-paste topological proof shows every Schur multiplier generator is toral, linking rationality and bordism.
· “Topological perspectives on the vanishing of some Bogomolov multipliers”
One torsor unifies K-theory and TMF Galois actions
Subgroups of p-divisible groups yield Galois extensions on geometric fixed points, recovering KU and TMF actions.
Braid-kernel rule extends from surfaces to all complexes
Whether the kernel of the strand map has Goldberg's form is decided by a single tree condition on the complex.
· “Extending Goldberg's Exact Sequence to Braid Groups of Graphs and Simplicial Complexes”
Paclet computes simplicial homology inside the Wolfram Language
Torsion, Betti numbers, and Euler characteristics come straight from boundary matrices, no extra system needed.
· “SimplicialHomology: implementation of abstract simplicial complex in Mathematica”
Equivariant relative sectional category computes TC of G-maps
It gives a single framework that recovers LS category, topological complexity, and homotopic distance under symmetry.
· “Equivariant Relative Sectional Category and Induced Invariants”
Tits building is the unique minimal synthetic building for Lie rank 2
A new topological definition separates the Tits building from all exotic alternatives for groups of Lie type of rank 2.
A 'cut' structure makes Moore flow models left proper
Gluing weakly equivalent Moore flows along cofibrations preserves the weak homotopy type of executions.
One globular cell breaks left properness for multipointed d-spaces
Two execution paths separated before the cell attach merge afterward, so pushouts fail to preserve weak equivalences.
· “The q-model category of multipointed d-spaces is not left proper”
Two products on Γ-sets form a linear-logic model
The sphere spectrum becomes the common unit, and the assembly map supplies the mixor.
· “On the sphere spectrum from the viewpoint of linear logic”
Maximal real varieties need not stay maximal in Hilbert squares
Explicit Betti-number bounds imply Hilbert squares of real abelian varieties never reach maximality.
· “Lower bounds for the Smith-Thom deficiency of Hilbert squares”
Distance-matrix heatmaps encode every 1-cycle of a point cloud
A chain map shows that recurrence plots determine Cech homology, simplifying persistence computations for time series.
· “Distance Matrices of Ordered Point Clouds and Their Persistent Homology”
Fusion-stable Burnside units come from real representations
The result pins down when the Lefschetz map for endotrivial complexes is surjective: exactly over F2.
Boundary data alone fix a shape's radial extended persistence
Signed boundary intervals plus center-location rules recover interior homology; nested holes need a death cascade.
· “Computing extended persistent homology of radial distance filtrations of Euclidean shapes”
Virtual knot groups: left-orderable equals circularly orderable
For virtual knot groups, circularly orderable is the same as left-orderable — a dichotomy proved through one relative quotient.
Spin manifolds' largest obstruction grading found exactly
For every n=2^e+d with 4≤d≤7, the largest immersion-obstruction grading is 2^e−1, and explicit manifolds attain it.
· “Spin manifolds with nonzero dual Stiefel-Whitney classes of large grading”
Ball Mapper overlaps become a joint probability law
A new probabilistic reading lets Ball Mapper graphs of different sizes be compared by optimal transport.
For odd primes, equivariant THH of F_p is a divided-power ring
The C_p-fixed homotopy ring contains divided powers and an exterior class, reshaping equivariant periodicity and future trace-method…
· “Computations in Equivariant Topological Hochschild Homology”
One identity settles Sommen's bivector bills
The gap between cell counts and homology is boundary data; tracking it explains both the n=4 and n=5 overcounts.
Point-cloud cochains recover the manifold's Hodge geometry
At reconstruction scales, discrete Hodge energies on sampled complexes converge to the manifold's smooth continuum limits.
· “The Geometry of Cochains on Sampled Vietoris-Rips Complexes”
New invariant tells directed motion-planning structures apart
Sequential directed and parametrized topological complexity measure plans with ordered waypoints and varying parameters.
· “On the Sequential topological complexity of directed (parametrized) motion planning algorithms”
Localizing motives are never compactly generated over rational bases
Even over the sphere spectrum, the infinity-category of localizing motives has no compact generators.
A mathematics paper computes the rational homotopy effect of the isometry group of the…
The rational homotopy class of the F4-action on the Cayley plane is determined, showing F4 and F4/Spin(7) map isomorphically to the…
· “Homotopical Robustness of Isometries on the Cayley Plane”
Odd primes double SU(n) gauge-group homotopy types on spheres
Each qualifying odd prime p\le n+1 with n mod p in {0,\dots,p-j} adds one binary choice.
· “Lower bounds on the number of homotopy types of SU(n)-gauge groups over spheres”
Derived categories yield 3D TQFTs from modular tensor categories
State spaces are derived Homs into powers of the canonical coend; degree zero recovers the non-semisimple Reshetikhin–Turaev theory.
· “3-dimensional TQFTs from derived categories of quantum group representations”
One ratio sets the power-law tail of growing hypergraphs
New nodes vs. hyperedge size alone fixes the degree exponent, and stronger attachment builds simplicial structure up to gelation.
· “Preferential Attachment as a Simpliciality-Enforcing Mechanism in Hypergraphs”
Birational motivic homotopy is an ∞-topos
For finite-generic schemes, the birational category is a Postnikov-complete ∞-topos of dimension 0.
Tate filtration supports two-fold multiplication
The result fills a gap in the literature and settles the exact multiplicative strength of the filtration.
Persistence landscapes embed barcodes into L^p spaces
For every p from 1 to infinity, the landscape is a 1-Lipschitz embedding; for p=2 it lands in a Hilbert space.
· “A Hilbert space embedding of persistence diagrams and barcodes”
Exterior calculus brings de Rham cohomology to non-smooth spaces
Using the Lagrangian as a smoothing kernel, the support of a causal variational principle carries forms, Stokes and Gauß theorems.
Symmetrization yields lower bounds for distributional sectional category
Rational cup length of the base's kernel bounds dsecat; new product, fibration, and triangle inequalities follow.
Rack type t annihilates the classifying map in all degrees
For connected racks, every higher homology class lands in t-torsion; without connectedness, t^(n-1) kills torsion classes.
Medial random complexes get topological complexity at most 2
The value is exactly 2 for homogeneous complexes along a density-one set of vertex counts, improving the old bound of 4.
4-multicomplexes are Quillen equivalent to bicomplexes
For any ring and any page pair, truncating a multicomplex preserves its homotopy theory, answering an open question.
Defects in 2D reflection TQFTs form O(2)-dagger bicategories
If the theory is reflection positive, the defect bicategory is nearly a 3-Hilbert space with a spherical weight.
· “Topological defects in reflection positive topological field theories”
Concordant knots in homology spheres obey the Fox-Milnor rule
The classical slice-knot obstruction now applies to every oriented homology 3-sphere, not just the 3-sphere.
· “Fox-Milnor condition for concordant knots in homology 3-spheres”
Chamber gcds bound Betti numbers of cyclic covers
A non-resonance condition makes the integral cohomology of these covers torsion-free.
· “The cohomology groups of finite cyclic covers of complexified real arrangement complements”
Pointwise adjunctions lift to cofiltered limits of ∞-categories
A filtered-colimit formula rebuilds the missing Beck–Chevalley data, with applications to presentable categories and topos shape.
· “On Cofiltered Limits of infty-Categories and Adjunctions”
One matrix update tracks every bar in a Conley-Morse barcode
Split, merge, and transposition edits keep the barcode current, replacing repeated zigzag persistence runs.
· “Computing Conley-Morse Persistence Barcode Efficiently by Updating Matrix Decompositions”
Persistence diagrams become trainable stochastic processes
Learned edits give topological summaries a unique steady law and preserve structure during compression.
· “Stochastic Dynamics on Persistence Diagram Space via Reinforcement Learning”
Every reduced exterior cohomology theory is represented by a spectrum
Exterior spectra now represent proper cohomology theories, including compactly supported ones.
· “A stable framework for proper and exterior homotopy and cohomology theories”
Interaction of point clouds reduces to one forced integer curve
The overlap's Euler characteristic is the unique symmetric, stable interaction profile, computable in near-optimal time.
Minimality does not save Whitehead's theorem for finite spaces
For every n, two equally small minimal models of the same sphere wedge are weakly equivalent but not homotopy equivalent.
Every finite tensor-nilpotence height occurs in spectra
New spectra refute the retract conjecture and set the Bousfield lattice's size at $2^{2^{\aleph_0}}$.
Quaternionic torus lift problem collapses to two obstructions
New criterion: Borel-image check plus trivializing a gauge-valued descent defect decides liftability.
Cycles and currents combine into one unique homology theory
A differential refinement of singular homology pairs with cohomology, gives a cap product and Poincaré duality, and extends to…
Slice formula computes C-motivic spectra from Adams-Novikov pages
For tmf the formula yields the slices of motivic modular forms, and it reproves earlier slice conjectures.
Birational motivic homotopy splits by generic points
For a variety V, the birational motivic homotopy category of V equals that of its function field k(V).
· “Schematic Functorialities of Birational Motivic Homotopy Categories”
This paper computes the Batalin-Vilkovisky algebra structure of Rabinowitz loop homology…
For compact rank one symmetric spaces, string point invertibility over a field of characteristic p holds exactly when p equals the Euler…
· “Resonances and string point invertibility for compact rank one symmetric spaces”
Every fiber of multiparameter persistent homology is a polyhedral complex
For n-filters on a fixed simplicial complex, preimages of a persistence module are piecewise-linear, with dimension controlled by Betti…
· “The fiber of multiparameter persistent homology for simplicial complexes”
Local Milnor numbers bound first nonzero Betti numbers of fibers
General and atypical fibers obey vanishing windows set by singular strata, including at infinity.
This paper develops new spectral sequence tools for computing the mod two homology of…
Homological Real trace methods are developed, yielding spectral sequences that compute the continuous mod two Bredon homology of Real…
129 seconds: H4 of the Greene sphere is zero
A new algorithm skips 6.4 trillion cubes, making discrete homology faster than simplicial on noisy data.
· “Discrete homology computations by reduction to zero differentials”
Kernel combination keeps clustering faithful to diagram distances
Convex combination of sliced Wasserstein kernels matches 1-Wasserstein, and its weights reveal the separating homology.
Link barcode pairs deaths with births across dimensions
A small twist on the persistence reduction adds cross-dimensional links—stably and in near-linear time for graphs.
One extra form completes homotopy classification of Z/p⋊Z 4-manifolds
Quadratic 2-type plus a Z/p cap-product form decides homotopy equivalence in this family.
· “Homotopy classification of 4-manifolds with fundamental group mathbb{Z}/p rtimes mathbb{Z}”
Smaller predictive coding nets collapse topology earlier
New homology analysis ties early topological collapse to weaker reconstruction and a 3.6-layer delay versus MLPs.
· “Topological Simplification in Predictive Coding Networks”
Cover complexity exceeds Nielsen count by a characteristic radical
For torus-bundle maps, the extra cover cost is set by a degree-two characteristic class, not linear data.
· “Finite Cover Resolution Complexity of Nielsen Fixed Point Spectra”
Gluing n- and m-categories now gives an (n+m)-category
The Gray tensor product is the basic case, and a new Grothendieck construction makes it work for all diagrams.
Even non-stable layers act by η
Complex realization then classifies oriented rank-(n−2) bundles on split quadrics and yields corank-two splitting criteria.
· “Adjacent non-stable layers in the mathbb{A}¹-homotopy of the special linear tower”
Geometric and algebraic completion agree for tempered cohomology
One theorem covers K-theory, elliptic cohomology, and genuine equivariant TMF.
A new simplicial distance encodes LS category and discrete TC
One invariant yields both m-simplicial LS category and sequential discrete motion-planning complexity.
Pencil plane's cotangent space = σ-continuous odd functions
Derivatives along every line through the origin determine a germ's class, and the angular data obey a weak continuity condition.
· “The Zariski Cotangent Space at the Origin of the Pencil Diffeological Space”