Model for assembly map of bordism-invariant functors.
problem Understanding assembly maps of bordism-invariant functors.
method Categorical model using oplax colimits of stable, hermitian, and Poincaré categories.
result Explicit description of the kernel of the assembly map.
We prove the existence of Verdier stratifications for sets definable in any o-minimal structure on (R, +, .). It is also shown that the Verdier condition (w) implies the Whitney condition (b) in o-minimal structures on (R, +, .). As a consequence the Whitney Stratification Theorem holds. The existence of (wf)-stratific…
Proves Verdier duality for sheaves on stratified spaces.
problem Verdier duality for constructible sheaves on stratified spaces.
method Uses conically smooth stratified spaces and Lurie's Verdier duality.
result Shows equivalence between constructible sheaves and cosheaves.
Three counterexamples show higher eigenvalue multiplicities than conjectured.
problem Determining the maximum eigenvalue multiplicity for closed hyperbolic surfaces.
method Applying the twisted Selberg trace formula to induced representations of triangle groups.
result Found counterexamples with higher eigenvalue multiplicities than previously conjectured.
The paper bounds eigenvalue multiplicities for hyperbolic surfaces using short geodesics.
problem Bounding the multiplicity of Laplacian eigenvalues for hyperbolic surfaces.
method Using the number of short closed geodesics and surface genus.
result Upper bounds on eigenvalue multiplicities, showing sublinear behavior under certain conditions.
3028 obstructions found for embedding without knots.
problem Finding obstructions for knotless embedding.
method Surveying recent work, updating obstructions, and proposing new questions.
result New obstructions with μ=6 and insights into connectivity.
We show that Verdier duality for certain sheaves on the moduli spaces of graphs associated to Koszul operads corresponds to Koszul duality of operads. This in particular gives a conceptual explanation of the appearance of graph cohomology of both the commutative and Lie types in computations of the cohomology of the ou…
New constructions from non-separating planar graphs improve understanding of graph linkability and knotability.
problem Understanding linkability and knotability of graph complements.
method Using maximal non-separating planar graphs to construct examples of maximal linkless and knotless graphs, and analyzing their Colin de Verdière invariant.
result The Colin de Verdière invariant of the complement of a maximal non-separating planar graph satisfies μ(cG) ≤ n-4, and equality holds.
In this paper we describe the notion of a weak lipschitzianity of a mapping on a Cq stratification. We also distinguish a class of regularity conditions that are in some sense invariant under definable, locally Lipschitz and weakly bi-Lipschitz homeomorphisms. This class includes the Whitney (B) condition and the …
In this paper we prove that every definable set has a definable triangulation which is locally Lipschitz and weakly bi-Lipschitz on the natural simplicial stratification of the simplicial complex. We also distinguish a class T of regularity conditions and give a universal construction of a definable triangulation with …
The paper proves that most metrics satisfy a strong version of Arnold's conjecture for Laplace eigenvalues.
problem Understanding metrics that satisfy a strong version of Arnold's conjecture for Laplace eigenvalues.
method Using geometric characterizations and perturbation theory, the paper proves the conjecture for most metrics.
result The Strong Arnold Hypothesis is satisfied for all metrics except for a set of infinite codimension.
Extended bounds on small eigenvalues for pseudo-Laplacians on hyperbolic surfaces.
problem Bounding small eigenvalues of pseudo-Laplacians on hyperbolic surfaces.
method Extended Otal-Rosas bound and Colin de Verdière's spectral theory to hyperbolic surfaces with multiple cusps.
result Extended bounds on small eigenvalues for pseudo-Laplacians.
We extend to orbifolds classical results on quantum ergodicity due to Shnirelman, Colin de Verdière and Zelditch, proving that, for any positive, first-order self-adjoint elliptic pseudodifferential operator P on a compact orbifold X with positive principal symbol p, ergodicity of the Hamiltonian flow of p implies quan…
SGD avoids critical points on weakly convex functions.
problem Non-convergence of SGD to critical points on specific manifolds.
method Stochastic subgradient descent, Verdier stratification, angle condition.
result SGD converges to local minimizers on weakly convex functions.
We present some applications of ideas from partial differential equations and differential geometry to the study of difference equations on infinite graphs. All operators that we consider are examples of "elliptic operators" as defined by Y. Colin de Verdiere. For such operators, we discuss analogs of inequalities of C…
For germs of subanalytic sets, we define two finite sequences of new numerical invariants. The first one is obtained by localizing the classical Lipschitz-Killing curvatures, the second one is the real analogue of the evanescent characteristics introduced by M. Kashiwara. We show that each invariant of one sequence is …
The monodromy conjecture states that every pole of the topological (or related) zeta function induces an eigenvalue of monodromy. This conjecture has already been studied a lot; however, in full generality it is proven only for zeta functions associated to a polynomial in two variables. In this article we consider zeta…
New stability conditions identified from quadratic differentials on surfaces.
problem Identifying stability conditions from quadratic differentials.
method Comparison of exchange graphs from tilting hearts and flipping mixed angulations.
result Spaces of stability conditions identified with moduli spaces of quadratic differentials.
A Delaunay cell decomposition of a surface with constant curvature gives rise to a circle pattern, consisting of the circles which are circumscribed to the facets. We treat the problem whether there exists a Delaunay cell decomposition for a given (topological) cell decomposition and given intersection angles of the ci…
We study rigidity of polyhedral surfaces and the moduli space of polyhedral surfaces using variational principles. Curvature like quantities for polyhedral surfaces are introduced. Many of them are shown to determine the polyhedral metric up to isometry. The action functionals in the variational approaches are derived …
Study finds maximal linklessly embeddable graphs up to 11 vertices and their complements.
problem Characterizing linklessly embeddable graphs and their complements.
method Comprehensive search and verification of graphs up to 11 vertices.
result For graphs of order 11, either the graph or its complement is intrinsically linked.
Paper proves existence and uniqueness of circle patterns on surfaces with assigned geodesic curvatures.
problem Existence and uniqueness of circle patterns on surfaces with prescribed geodesic curvatures.
method Applied Perron's method and Thurston's algorithm to prove existence and convergence.
result Existence and uniqueness of circle patterns on surfaces with prescribed geodesic curvatures.
In 2004, Sormani and Wei introduced the covering spectrum: a geometric invariant that isolates part of the length spectrum of a Riemannian manifold. In their paper they observed that certain Sunada isospectral manifolds share the same covering spectrum, thus raising the question of whether the covering spectrum is a sp…
The paper classifies K-contact forms on 3-manifolds and connects their orbits to spectral invariants.
problem Classifying K-contact forms with specific properties on 3-manifolds.
method Analyzing the Reeb vector field and its orbits, proving diffeomorphism results, and relating to spectral invariants.
result Compact 3-manifolds carrying such K-contact forms are diffeomorphic to lens spaces with specific orbit properties.
The paper studies circle packings on surfaces with boundary and their total geodesic curvatures.
problem Existence and rigidity of circle packings with conical singularities.
method Variational principle and combinatorial Ricci flow.
result Existence and rigidity of circle packings with prescribed total geodesic curvature.
Develops a new theory of localization in algebraic geometry.
problem Localization of cohomological theories on closed subsets.
method Categorical and algebro-geometric approach, focusing on torsors and translation groupoids.
result Found that localization often results in a torsor of supported refinements rather than a localized class.
The tame flows are ``nice'' flows on ``nice'' spaces. The nice (tame) sets are the pfaffian sets introduced by Khovanski, and a flow Φ:R×X→X on pfaffian set X is tame if the graph of Φ is a pfaffian subset of R×X×X. Any compact tame set admits plenty tame flows. We prove …
The paper applies Poincaré duality to supergravity, proving its equivalence to other formulations.
problem Relating different formulations of supergravity on supermanifolds.
method Proving relative Poincaré duality and using it to connect differential and integral forms.
result Relative Poincaré duality provides a rigorous definition of picture changing operators in supergravity.
Stochastic subgradient descent avoids critical points in definable functions.
problem Finding local minima in definable functions.
method Stochastic subgradient descent with density-like perturbation.
result SGD converges to a local minimum in definable functions.
Researchers approximate spectral targets on manifolds with constant negative curvature.
problem Prescribing an arbitrary finite portion of the Laplace-Beltrami spectrum on manifolds of constant negative curvature.
method Constructing macroscopically heterogeneous hyperbolic covering manifolds in d≥3 and using discrete spectral limit theorems in d=2. result Any finite strictly increasing target list can be approximated to arbitrary precision by a closed manifold of constant negative curvature.
The paper defines projective structures for Lie bialgebras and Poisson-Lie groups.
problem Defining projective analogues of Lie bialgebras and Poisson-Lie groups.
method Introducing projective tensor products and adapting classical notions to these structures.
result Every quasi-triangular projective r-matrix gives rise to a projective Banach Lie bialgebra.
Characterizes minor-minimal separating projective planar graphs and their generalizations.
problem Understanding projective planar graphs and their properties.
method Analyzing minors, embeddings, and specific link types.
result Partial characterization of minor-minimal separating projective planar graphs and their generalizations.
Legendrian knots can be represented by projections with multi-crossings.
problem Representing Legendrian knots with multi-crossings.
method Investigating übercrossing and petal projections in front and Lagrangian projections.
result Legendrian knots with übercrossing projections in front are smoothly isotopic to the unknot.
The study classifies holomorphic projective connections on complex threefolds.
problem Characterizing holomorphic projective connections on complex threefolds.
method Analyzing properties of holomorphic projective connections on complex projective threefolds.
result Holomorphic projective connections on complex threefolds are either flat or translation invariant on abelian threefolds.
Defines projective Ricci curvature and proves rigidity for sprays.
problem Defining and studying projective Ricci curvature.
method Introduced projective Ricci-flat sprays and studied Randers metrics.
result Global rigidity result for projectively Ricci-flat sprays with nonnegative Ricci curvature.
The paper calculates delta invariants for specific geometric structures.
problem Computing delta invariants for projective bundles and cones of Fano type.
method Provides a precise formula for delta invariants.
result A formula to compute delta invariants for projective bundles and cones of Fano type.
This paper proposes a method to select project schedules with the lowest risk.
problem Selecting schedules that meet project deadlines while minimizing risk.
method Integrating aleatory uncertainty into project scheduling to quantify and compare risks.
result Proposes a method to select schedules with the lowest risk.
Projective rigidity of circle packings on complex surfaces proved.
problem Proving rigidity of circle packings on complex projective surfaces.
method Proved projective rigidity through triangulations and complex projective structures.
result Space of circle packings is projectively rigid on complex projective surfaces.
A small projective 4-manifold created via Dehn filling.
problem Creating a small positive Euler characteristic closed convex projective 4-manifold.
method Explicit construction through continuous path of projective cone-manifolds and Dehn filling of a cusped hyperbolic 4-manifold.
result Obtained a closed orientable convex projective four-manifold with small positive Euler characteristic.
Classifies flat projective structures with specific symmetries.
problem Classifying local projective structures with non-trivial Lie symmetries.
method Analyzes flat projective structures with positive-dimensional Lie algebra of projective vector fields.
result Obtained a classification of flat projective structures.
The authors give a complete classification of projective threefolds admitting a holomorphic normal projective connection. Moreover, they prove a general structure theorem on complex projective manifolds admitting a holomorphic normal projective connection, saying in particular, that any such manifold is either the proj…
In the asymmetric setting, Hilbert's fourth problem asks to construct and study all (non-reversible) projective Finsler metrics: Finsler metrics defined on open, convex subsets of real projective n-space for which geodesics lie on projective lines. While asymmetric norms and Funk metrics provide many examples of esse…
Paper studies pseudo-projective tensors on warped products.
problem Characterizing pseudo-projectively flat warped products.
method Analyzes sequential warped products and pseudo-projective tensors.
result Necessary and sufficient conditions for pseudo-projectively flat sequential warped products.
Study projective KLT varieties with projectively flat cotangent sheaves.
problem Uniformisation problems on projective varieties with klt singularities.
method Generalising Jahnke-Radloff's work, study torus quotients and varieties with semistable cotangent sheaves and extremal Chern classes.
result Torus quotients are the only klt varieties with semistable cotangent sheaves and extremal Chern classes.
In this paper, we introduce the weighted projective Ricci curvature as an extension of projective Ricci curvature introduced by Z. Shen. We characterize the class of Randers metrics of weighted projective Ricci flat curvature. We find the necessary and sufficient condition under which a Kropina metric has weighted proj…
New framework uses elliptic operators to study projective maps.
problem Understanding projective structures on Riemannian manifolds.
method Develops two elliptic operators of second and fourth order.
result Establishes a natural correspondence between analytical and geometric properties.
Projective manifolds with specific bundles are isomorphic to simpler spaces.
problem Characterizing projective manifolds with tangent bundles containing strictly nef subsheaves.
method Analyzing the structure of the tangent bundle and using properties of strictly nef subsheaves.
result Projective manifolds with the described bundles are isomorphic to projective bundles over hyperbolic manifolds or projective spaces.
An axis of a link projection is a closed curve which lies symmetrically on each region of the link projection. In this paper we define axis systems of link projections and characterize axis systems of the standard projections of twist knots.