Study cohomological equation for robotic screw motions on SE(3).
problem Understanding obstruction phenomena in robotic rigid-body motion.
method Combining Fourier analysis and Peter-Weyl theory, reduce to finite-dimensional linear transport systems.
result Explicit screw motion illustrates resonance conditions and finite-dimensional obstructions.
Analyzes complex structure deformations using cohomology contraction methods.
problem Deforming complex structures and identifying obstructions.
method Refined power series method for (p,q)-forms and complex structures, using Frölicher spectral sequence. result All obstruction classes lie in the kernel of contraction maps under natural vanishing conditions.
This paper extends some results of Hatcher and Quinn beyond the metastable range. We give a bordism theoretic obstruction to deforming a map between manifolds simultaneously off of a collection of pairwise disjoint submanifolds under the assumption that it can be deformed off of any proper subcollection in a homotopy c…
Invariant obstructs separating coassociative 4-folds.
problem Obstructing the separation of coassociative 4-folds.
method Defining a Z2-valued invariant for coassociative 4-folds with spin structures. result Invariant provides an obstruction to separating coassociative 4-folds.
Novel theory combines combinatorial and topological elements.
problem Understanding combinatorial phenomena at the intersection of topology.
method Synthesizes combinatorial and topological approaches with a new framing concept.
result Framed combinatorial spaces exhibit better behavior than classical spaces.
Study on Legendrian knots and their non-orientable Lagrangian fillings.
problem Conditions for Legendrian knots to have non-orientable exact Lagrangian fillings.
method Developed combinatorial obstructions and classified fillability for various knot families.
result Completely determined decomposably non-orientable fillability for alternating and plus-adequate knots.
Paper proves index theorem for self-adjoint elliptic boundary problems.
problem Proving index theorem for self-adjoint elliptic boundary problems.
method Topological and pseudo-differential methods, generalized Atiyah-Singer approach.
result Removed technical assumption to prove index theorem.
Hypothesis testing in singular models is fundamentally about identifiable vs. non-identifiable parameters.
problem Testing in singular models is inherently problematic due to non-identifiability and degeneracy of Fisher information.
method Formalized the overlap obstruction and showed that hypotheses over non-identifiable parameters are untestable, while those over identifiable parameters reduce to classical testing.
result Hypotheses over non-identifiable parameters are untestable, while those over identifiable parameters reduce to classical testing.
On the one hand, we prove that the Clifford torus in C2 is unstable for Lagrangian mean curvature flow under arbitrarily small Hamiltonian perturbations, even though it is Hamiltonian F-stable and locally area minimising under Hamiltonian variations. On the other hand, we show that the Clifford torus is r…
Link concordance equals homotopy for high-dimensional spheres.
problem Understanding when immersions of high-dimensional spheres are homotopically trivial.
method Developed stratified Morse theory for generic immersions, using gradient-like vector fields and Cerf theory.
result Every link of high-dimensional spheres is homotopically trivial, resolving a long-standing conjecture.
For gauge groups U(1) and SO(3) we classify invariant G2-instantons for homogeneous coclosed G2-structures on Aloff-Wallach spaces Xk,l. As a consequence, we give examples where G2-instantons can be used to distinguish between different strictly nearly parallel G2-structures on the same Aloff-Walla…
This paper studies torsion obstructions to complex sections on manifolds.
problem Torsion obstructions to finding complex sections on almost complex manifolds.
method Calculations using the Adams-Novikov spectral sequence for Thom spectra.
result Torsion obstructions for finding r complex sections of order p vanish for r<p2−p. The paper presents a new method to create exotic 4-manifolds and surfaces that remain exotic after stabilization.
problem Stabilization of exotic 4-dimensional phenomena and knotted surfaces.
method Elementary approach to constructing exotic 4-manifolds and surfaces, including examples in closed, simply connected 4-manifolds.
result The construction yields exotic surfaces in the 4-ball that remain exotic after stabilization, detected by Khovanov homology.
The paper argues against the inefficiency of explaining deep learning phenomena.
problem Explaining deep learning phenomena often leads to ad hoc hypotheses.
method Analyzing recent literature to assess the relevance of these phenomena.
result Many phenomena do not appear in real-world applications and may be inefficient.
In this paper, we consider a concentration of measure problem on Riemannian manifolds with boundary. We study concentration phenomena of non-negative 1-Lipschitz functions with Dirichlet boundary condition around zero, which is called boundary concentration phenomena. We first examine relation between boundary concen…
Fewer obstructions for small graphs in knotless embedding.
problem Finite obstructions for knotless embedding of small graphs.
method Analysis of graph minor obstructions and knotless embedding.
result There are only three obstructions for size 23 graphs, significantly fewer than previously known.
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.
Sharp changes in time series representing market dynamics are studied by means of the self--similar analysis suggested earlier by the authors. These sharp changes are market booms and crashes. Such crises phenomena in markets are analogous to critical phenomena in physics. A simple classification of the market crisis p…
The paper develops obstructions for embedding 2D complexes into 4D space.
problem Embedding 2D complexes into 4D space and understanding obstructions.
method Uses Goodwillie-Weiss calculus and intersections of Whitney disks.
result Two approaches to obstructions lead to the same result.
Global obstructions found for conformally Einstein metrics in 6D.
problem Obstructing the existence of conformally Einstein metrics in six dimensions.
method Presentation of global conformal invariants.
result Nontrivial global conformal invariant obstructing conformally Einstein metrics.
Complete surgery obstructions for manifolds with finite fundamental group, disproving a conjecture.
problem Disproving the Oozing Conjecture for manifolds with finite fundamental group.
method Description and calculation of surgery obstructions up to homotopy equivalence.
result New obstructions found for Arf invariant product formulas in codimensions ≥ 4, disproving the Oozing Conjecture.
Study on Lane-Emden equation on curved spaces, revealing new existence and non-existence phenomena.
problem Existence and non-existence of positive solutions for the Lane-Emden equation on Riemannian models.
method Analysis of the subcritical Lane-Emden equation on various Riemannian manifolds with polynomial volume growth.
result Subcritical regime divides into three ranges with distinct existence and non-existence phenomena.
Given a polarized manifold there are obstructions for asymptotic Chow semistability described as integral invariants. One of them is an obstruction to the existence for the first Chern class of the polarization to admit a constant scalar curvature Kähler (cscK) metric. A natural question is whether or not the other obs…
The article classifies cubiquitous sublattices and applies them to branched covers.
problem Understanding cubiquitous sublattices as obstructions to rational homology 4-balls.
method Developed a geometric Wu obstruction to classify cubiquitous sublattices and applied it to branched covers.
result Completely classified which sublattices with orthogonal bases are cubiquitous.
The paper explores higher property T in lattices and its connections to geometric phenomena.
problem Understanding higher property T in lattices and related geometric phenomena.
method Operator-algebraic characterizations of higher property T and connections to lattice geometry.
result Unified framework for understanding higher property T and related geometric phenomena.
For a 3-manifold with torus boundary admitting an appropriate involution, we show that Khovanov homology provides obstructions to certain exceptional Dehn fillings. For example, given a strongly invertible knot in S^3, we give obstructions to lens space surgeries, as well as obstructions to surgeries with finite fundam…
New obstructions found for smooth desingularization of compact Einstein orbifolds.
problem Finding obstructions to desingularizing compact Einstein orbifolds.
method Identifying new obstructions specific to compact Einstein 4-orbifolds. result Almost all flat orbifold metrics on T4/Z2 are not limits of Ricci-flat metrics. For ℓ>1, we develop L(2)-signature obstructions for (4ℓ−3)-dimensional knots with metabelian knot groups to be doubly slice. For each ℓ>1, we construct an infinite family of knots on which our obstructions are non-zero, but for which double sliceness is not obstructed by any previously known invari…
Study identifies obstructions for solving a 4th-order boundary problem.
problem Solving a 4th-order boundary problem with specific curvature conditions.
method Derived Kazdan-Warner type identities using variational formulation and conformal variations.
result Obtained nontrivial integral obstructions to solvability.
Developing a singular dimension descent method for positive scalar curvature obstructions
problem Positive scalar curvature obstructions in arbitrary dimensions
method Schoen--Yau type singular dimension descent method
result Proving obstructions to positive scalar curvature on enlargeable manifolds
Study Euler obstruction of 1-forms on determinantal singularities.
problem Understanding the Euler obstruction of 1-forms on determinantal singularities.
method Investigation of connections between local Euler obstruction and PHN index.
result Explicit computations of Euler obstruction for specific singularities.
Proves Massey's theorems on complex structure obstructions.
problem Finding complex structures on real vector bundles.
method Fractional Stiefel-Whitney classes and combinations of Pontryagin, Chern, and Euler classes.
result Determines the second obstruction for rank six bundles.
The paper explores properties of CR hypersurfaces and their flatness.
problem Analyzing the flatness properties of CR hypersurfaces.
method Investigation of local and global properties of strongly pseudoconvex CR hypersurfaces.
result Existence of obstruction flat points on compact CR hypersurfaces.
Paper connects Stokes phenomena to quantum groups and Poisson-Lie groups.
problem Analyse Stokes phenomena in Poisson-Lie groups and quantum groups.
method Use Ug-valued Stokes phenomena to construct quantum group U_hg and relate it to Poisson-Lie group G*.
result Show that Ug-valued Stokes phenomena can be obtained as a semiclassical limit of the KZ associator.
In this paper we observe that 2-dimensional 0-surgery occurs in natural processes, such as tornado formation and other phenomena reminiscent of hole drilling. Inspired by such phenomena, we introduce new theoretical concepts which enhance the formal definition of 2-dimensional 0-surgery with the observed dynamics. To d…
Study on a specific obstruction in four-dimensional geometry.
problem Singular Yamabe obstruction of a hypersurface.
method Derived various explicit formulas from the original definition.
result Relate formulas to literature and prove elementary.
Introduce generalized Ueda obstruction classes for line bundles and apply them to non-semi-positivity.
problem Non-semi-positivity of line bundles
method Introduce generalized Ueda obstruction classes and use Dolbeault resolution
result Recover classical examples and provide new examples of nef but not semi-positive line bundles
Surgery obstructions extended to integer homology spheres using Heegaard Floer homology.
problem Obstructing knots in integer homology spheres using surgery.
method Extending Heegaard Floer homology obstructions to all integer homology spheres for both positive and negative surgeries.
result Deduced a lower bound on b2(W) for smooth cobordism between integer homology spheres. Extended solitons show constant curvature on compact manifolds.
problem Understanding self-similar solutions to geometric flows.
method Analyzing extended solitons and applying integral inequalities.
result Closed ambient obstruction solitons are ambient obstruction flat and have constant scalar curvature.
Study the topological information of map germs using Euler obstruction.
problem Capturing topological information from map germs with complex singular spaces.
method Investigates the Euler obstruction and its relation to local Euler obstruction and Brasselet number.
result Relates Chern number to the number of cusps in a perturbed map-germ.
We give a survey of some known results and of the many open questions in the study of generic phenomena in geometrically interesting groups.
New obstructions show some 4-manifold homeomorphisms are pseudo-isotopic but not isotopic.
problem Tackling the difference between pseudo-isotopy and isotopy for 4-manifolds.
method Defining and showing realizable obstructions in Whitehead groups.
result Found homeomorphisms that are pseudo-isotopic but not isotopic.
Paper investigates rigidity phenomena for weighted Ricci curvature bounds with Laplacian comparison theorem.
problem Investigating rigidity phenomena for weighted Ricci curvature bounds.
method Derived comparison geometric estimates and generalized for non-symmetric Laplacian.
result Obtained rigidity results for Laplacian comparison theorem, diameter comparisons, and volume comparisons.
Geometrically, a new obstruction is found for 4-manifold realizations.
problem Realizing normal 1-types of 4-manifolds with a given boundary.
method Three-stage obstruction theory, with a geometric tertiary obstruction.
result A new geometric tertiary obstruction for 4-manifold realizations.
Study torsion obstructions to positive scalar curvature on manifolds.
problem Obstructions to positive scalar curvature metrics on manifolds.
method Analysis of torsion classes in integral homology.
result Construction of manifolds without positive scalar curvature metrics but whose Cartesian products do.
New curvature obstruction for Killing vector fields on Lorentzian manifolds.
problem Existence of timelike or causal Killing vector fields on Lorentzian manifolds.
method New curvature obstruction in terms of timelike or null sectional curvature.
result Extension of Gauss-Bonnet-Chern obstruction to non-zero timelike sectional curvature.
BSFP method reveals latent patterns in multi-omic data for predicting lung function in HIV-associated OLD.
problem Limited understanding of multi-omic molecular phenomena and clinical outcomes in obstructive lung disease.
method Bayesian Simultaneous Factorization and Prediction (BSFP) method for multi-omic data, accommodating imputation and full posterior inference.
result BSFP reveals distinct clusters of patients with OLD and multi-omic patterns related to lung function decline.
Develops formal moduli theory for splitting complex supermanifolds.
problem Tackles the splitting problem of complex supermanifolds.
method Constructs a filtered dg Lie algebra to control splittings and transfers the theory to a minimal filtered L∞-model. result Recover classical obstruction classes as leading terms of Maurer-Cartan representatives and proves the existence of higher obstructions.