67 papers
This paper constructs a presheaf of Koszul complexes on weighted finite subsets of Euclidean space, utilizing linearized coefficients and change-of-coordinates maps to form a Čech-Koszul bicomplex whose degree-0 cocycles capture homotopical relationships between locally defined least squares solutions on data overlaps.
This paper proves a conjecture by Brochier, Jordan, Safronov, and Snyder that characterizes fully-dualizable and invertible -algebras within higher Morita categories, thereby identifying the specific algebraic structures that generate -dimensional and invertible topological quantum field theories.
The paper demonstrates the existence of non-formal compact (almost) contact manifolds with specific first Betti numbers and, in certain dimensions, simple connectivity, thereby resolving a key problem in the geography of such manifolds.
This paper employs cellular sheaves and their cohomology to analyze the infinitesimal rigidity of graph-of-groups realizations, establishing algebraic conditions for Henneberg moves and proving that the Maxwell-count provides a necessary and sufficient condition for minimal rigidity in problems defined over real algebraic groups with specific subgroup constraints.
This paper establishes that the moduli of -valued constructible sheaves and perverse sheaves on a compact oriented manifold with a conically smooth stratification are -shifted Lagrangian, a result derived from constructing a relative left -Calabi--Yau structure via lax gluing of categorical cubes and identifying symplectic leaves for perverse sheaves with prescribed monodromy.
This paper introduces dynamic curvature-set persistent homology, a computationally tractable and stable method for distinguishing dynamic data by extending Kim and Mémoli's spatiotemporal framework to curvature sets, proving the resulting modules are antichain-decomposable to enable efficient erosion distance computation and successfully demonstrating its ability to detect parameter changes in the Boids model.
This paper proposes a fast and provably correct method for approximating the persistence diagrams of sliding window embeddings of quasiperiodic functions by integrating the Three Gap Theorem from number theory with the Persistent Künneth formula from topological data analysis.
This paper establishes a stability theorem for minimal projective resolutions of modules over finite metric posets by proving that a newly defined bottleneck distance between resolutions is bounded above by the Galois transport distance, thereby generalizing classical bottleneck stability to multiparameter persistence and providing a stability framework for Möbius homology.
The paper demonstrates that various categories of trees, including the dendroidal category , the category of trees with a -action, and the category of genuine equivariant trees , can be modeled as Grothendieck constructions on categories of trees with a fixed set of leaves.
This paper establishes homological stability for automorphisms of symmetric bilinear forms over a broad class of principal ideal domains, thereby determining a significant portion of the stable cohomology of odd orthogonal groups over the integers in low degrees.
This paper provides a complete characterization of all model category structures on finite lattices by utilizing transfer systems, thereby establishing new connections between abstract homotopy theory and equivariant methods.
This paper establishes the relationship between pairings of May operads and compatible pairs of indexing systems, demonstrating that operad pairings induce system pairings and that, in many cases, compatible indexing systems can be realized by pairings of -operads.
This paper investigates separable commutative algebras in equivariant homotopy theory, establishing conditions under which they are "standard" (arising from finite -sets) and demonstrating that while all such algebras are standard for -groups, non-standard examples exist for general finite groups, with the classification further refined by the presence of multiplicative norms and the solvability of the group.
This paper generalizes the classical Morlet-Burghelea-Lashof-Kirby-Siebenmann smoothing theory delooping of disc diffeomorphism groups to various disc embedding spaces, establishing their equivalence to specific loop spaces of quotient classifying spaces and demonstrating how these deloopings unify Hatcher and Budney group actions into a framed little discs operad action.
This thesis extends the relative -contractibility of Koras-Russell threefolds and their higher-dimensional prototypes to arbitrary Noetherian base schemes, thereby establishing the existence of the first known family of smooth "exotic" motivic spheres in dimensions that are -homotopic to, but not isomorphic to, the punctured affine space .
This paper establishes the compactness of a character variety classifying continuous group actions on algebraic real hyperbolic spaces of arbitrary dimension, unifying previous compactifications and proving rigidity results for irreducible representations via the introduction of algebraic and abstract cross-ratios.
This paper refines the string equation on from the Chow ring to algebraic cobordism to derive inductive formulas for cobordism-valued psi-class intersections and explicit classes up to , including their images in -theory.
This paper proposes a new formulation of 3-crossed modules equipped with a novel lifting mechanism, proving that they induce quasi-categories and arise naturally from the Moore complex of length 3, thereby establishing a robust foundation for extending the equivalence between higher algebraic structures and Gray 3-groups.
This paper introduces a homological stratification theory for rigidly-compactly generated tensor-triangulated categories, demonstrating its robust descent properties to provide a unified framework for answering when stratification descends and extending equivariant module spectrum results from finite groups to compact Lie groups.
This paper establishes a cohomology theory for the absolute semi-topological Galois group associated with Weierstrass polynomials, utilizing spectral sequences to derive obstruction theories and vanishing results that ultimately prove the -detectable Weierstrass realizability conjecture for abelian varieties, smooth complex projective curves, and ruled surfaces.