Generalizes embedding complex Grassmannians into quadrics.
problem Holomorphic isometric embeddings of complex Grassmannians into quadrics.
method Generalization of do Carmo-Wallach theory for moduli spaces.
result Moduli spaces of embeddings discussed.
Study holomorphic isometric embeddings of a Grassmannian into quadrics.
problem Holomorphic isometric embeddings of complex Grassmannian into quadrics.
method Generalization of do Carmo--Wallach theory to study moduli space.
result Moduli space of embeddings up to equivalence discussed.
We introduce (k,l)-regular maps, which generalize two previously studied classes of maps: affinely k-regular maps and totally skew embeddings. We exhibit some explicit examples and obtain bounds on the least dimension of a Euclidean space into which a manifold can be embedded by a (k,l)-regular map. The problem c…
Study of embedding spaces using homotopy theory and operads.
problem Understanding the stable homotopy type of embedding spaces.
method Analysis of cubes of framed configuration spaces, homotopy theory of presheaves, operadic structures.
result Induced action of the Poisson operad on the homology of configuration spaces is a homotopy invariant.
Explains Hodge theory and Kodaira embedding theorem for complex manifolds.
problem Understanding complex manifold properties and their geometric implications.
method Expository review of harmonic forms, Hodge theory, and Kodaira embedding theorem.
result Establishes connections between de Rham cohomology, Dolbeault cohomology, and projective varieties.
Solves Skopenkov's problem on graph embedding criteria.
problem Criteria for toroidal embedding of one-vertex ribbon graphs.
method Analyzes one-vertex ribbon graphs with additional disc structure.
result Provides solutions to Skopenkov's problem.
This is the next step of uncovering the relation between string theory and exotic smooth R^4. Exotic smoothness of R^4 is correlated with D6 brane charges in IIA string theory. We construct wild embeddings of spheres and relate them to a class of topological quantum Dp-branes as well to KK theory. These branes emerge w…
Develops a new theory of width for embedded circles in Riemannian manifolds.
problem Defining and understanding the width of embedded circles in Riemannian manifolds.
method Morse-Lusternik-Schnirelmann theory applied to geodesics and minimising configurations.
result Classifies configurations of minimising geodesics intersecting embedded circles.
Survey simplifies embedding theorems for manifolds.
problem Embedding manifolds into Euclidean space.
method Combining complement and neighborhood ideas to reduce embeddability to algebraic problems.
result Clarified exposition of Browder-Levine theorem on realization of normal systems.
Well-quasi-orders proved on embedded planar graphs.
problem Proving well-quasi-orders on embedded planar graphs.
method Careful analysis and extensions of classical methods for embedded minor relations.
result Embedded minor relations are well-quasi-orders on various classes of embedded planar graphs.
New constraints on embedded spheres and projective planes in 4-manifolds from Seiberg-Witten theory.
problem Constraints on configurations of embedded spheres and real projective planes in 4-manifolds.
method Equivariant Seiberg-Witten invariants and gluing formula for relative Seiberg-Witten invariants.
result Existence of certain configurations of surfaces leads to 4-manifolds of non-simple type.
Solenoids are ``inverse limits'' of the circle, and the classical knot theory is the theory of tame embeddings of the circle into the 3-space. We give some general study, including certain classification results, of tame embeddings of solenoids into the 3-space as the ``inverse limits'' of the tame embeddings of the ci…
New gauge theory invariant detects non-smooth isotopy of P2-knots.
problem Detecting non-smooth isotopy of P2-knots. method Real Seiberg-Witten theory to construct a gauge theoretic invariant.
result Found a family of P2-knots that are topologically isotopic but not smoothly isotopic. Study non-Abelian gauge theories using Poisson bracket structures.
problem Defining a Poisson bracket structure on solution spaces.
method Using coisotropic embedding theorem.
result Defined Poisson bracket structure for non-Abelian gauge theories.
This paper builds on the theory of generalised functions begun in [1]. The Colombeau theory of generalised scalar fields on manifolds is extended to a nonlinear theory of generalised tensor fields which is diffeomorphism invariant and has the sheaf property. The generalised Lie derivative for generalised tensor fields …
Proves h-principle for loose Legendrian embeddings in contact topology.
problem Existence of non-loose Legendrian embeddings.
method h-principle from microlocal sheaf theory.
result Existence of non-loose Legendrian embeddings.
MCE reduces embedding instability in nonlinear dimensionality reduction.
problem Embedding instability caused by random initialization.
method Median of multiple embeddings (MCE) based on large deviation theory.
result MCE achieves consistency at an exponential rate and effectively mitigates instability.
The paper explores theories behind graph and relational data vector embeddings.
problem Understanding the foundations of vector embeddings for graphs and relational structures.
method Proposes two theoretical approaches to understand vector embeddings.
result Draws connections between various embedding techniques and suggests future research directions.
We develop a complete local theory for CR embedded submanifolds of CR manifolds in a way which parallels the Ricci calculus for Riemannian submanifold theory. In particular, we establish the subtle relationship between the submanifold and ambient standard tractor bundles, allowing us to relate the respective normal Car…
Given Poincare spaces M and X, we study the possibility of compressing embeddings of M x I in X x I down to embeddings of M in X. This results in a new approach to embedding in the metastable range both in the smooth and Poincare duality categories.
The paper proves the existence of at least 4 embedded minimal tori in a three-sphere with positive Ricci curvature.
problem Proving the existence of embedded minimal tori in three-spheres with positive Ricci curvature.
method The proof relies on a multiplicity one theorem for the Simon-Smith min-max theory.
result There exist at least 4 distinct embedded minimal tori in the three-sphere with positive Ricci curvature.
Notes a flaw in a proof about embedding graphs.
problem A flaw in proving Sachs' conjecture about graph embeddings.
method Analyzing Stanfield's proof for gaps.
result Identifies a significant error in the proof.
Study General Relativity using field theories and Poisson brackets.
problem Defining a Poisson bracket structure on solution spaces of field theories.
method Applying Poisson bracket structure to first order Hamiltonian field theories, focusing on General Relativity as a gauge theory.
result Established a Poisson bracket structure for General Relativity.
The paper establishes a new pseudoisotopy result for embedding spaces, leading to computations of homotopy groups of long knots.
problem Computing homotopy groups of spaces of long knots in high codimension.
method Using pseudoisotopy results and algebraic K-theory, the paper describes the difference in homotopy types of block and ordinary embeddings of a codimension at least three embedding.
result The homotopy type of spaces of long knots of codimension at least 3 is determined explicitly, including torsion information.
This text is the extended version of a talk given at 6th Meeting of Integrable Systems and Quantum Filed Theory at Peyresq hold from June 10 2006 to June 17, 2006 at Peyresq, France. The goal of this lecture is to give a brief introduction to Cartan-Kähler's theory. As examples to the application of this theory, we cho…
Improved learning theory for kernel distribution regression with two-stage sampling.
problem Distribution regression problem and two-stage sampling setting.
method Kernel methods, near-unbiased condition, new error bounds, convergence rates.
result Strictly improved convergence rates for three important classes of kernels.
New learning rates for embeddings in RKHSs, even when the target is not Hilbert-Schmidt.
problem Applying conditional mean embeddings to complex ML/RL settings with infinite-dimensional RKHSs.
method Developed novel learning rates using interpolation theory for RKHSs, derived explicit adaptive rates for sample estimator.
result Achieved uniform convergence rates in the output RKHS for certain parameter regimes.
Paper extends Miyazawa's construction to more 4-manifolds, finding exotic involutions and embeddings.
problem Finding exotic involutions and embeddings in 4-manifolds.
method Real Seiberg-Witten theory.
result Many infinite families of exotic involutions and embeddings.
We construct an infinite family of homology theories of framed links in thickened surfaces, as well as a homology theory whose graded Euler characteristic is exactly the Kauffman bracket of the link in the surface. Both theories are based on ideas coming from Asaeda, Przytycki and Sikora's categorification of the Kauff…
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.
Notes on embedding criteria for smooth manifolds.
problem Conditions for embedding smooth manifolds into Euclidean space.
method Linking recent results to classical criteria and K-theory.
result Recent results connect to classical embedding criteria.
Strong parallels can be drawn between the theory of minimal hypersurfaces and the theory of phase transitions. Borrowing ideas from the former we extend recent results on the regularity of stable phase transition interfaces to the finite Morse index case. As an application we present a PDE-based proof of the celebrated…
Segre embedding was introduced by C. Segre (1863--1924) in his famous 1891 article \cite{segre}. The Segre embedding plays an important roles in algebraic geometry as well as in differential geometry, mathematical physics, and coding theory. In this article, we survey main results on Segre embedding in differential geo…
Smoothly knotted 5RP^2 found in 4-sphere.
problem Finding knotted embeddings in higher dimensions.
method Topological and smooth knotting analysis in 4-sphere.
result Smoothly knotted 5RP^2 found in 4-sphere.
In this paper, we derive new adjunction inequalities for embedded surfaces with non-negative self-intersection number in four-manifolds. These formulas are proved by using relations between Seiberg-Witten invariants which are induced from embedded surfaces. To prove these relations, we develop the relevant parts of a F…
Graphs embeddable on torus and linklessly in 3D can be embedded linklessly in standard torus.
problem Embedding linklessly in a standard torus for graphs embeddable on torus and in 3D.
method Analyzing graphs of order 9 and below, showing linkless embedding in standard torus.
result For graphs of order 9 and below, linkless embedding in standard torus is possible.
Exposes two methods for constructing flat surfaces in 4D spaces.
problem Building locally flat embedded surfaces in 4-manifolds.
method Direct methods and surgery theory.
result Every primitive second homology class in a closed, simply connected 4-manifold is represented by a locally flat embedded torus.
The smoothing theory is revised to generalize to different disc embedding spaces.
problem Generalizing the smoothing theory to various disc embedding spaces.
method Revising the Morlet-Burghelea-Lashof-Kirby-Siebenmann theorem to apply to different versions of disc smooth embedding spaces.
result The delooping of disc embedding spaces is shown to be compatible with the Hatcher and Budney actions.
Paper connects algebraic K-theory to foam geometry.
problem Identifying K-theory groups with foam cobordisms.
method Using stratified manifolds with flat bundles over facets, and gluing conditions.
result First K-theory group identified with 1-foam cobordism group.
We use the theory of oriented matroids to show that any linear embedding of K9, the complete graph on nine vertices, contains a non-split link with three components.
We prove that an embedded cobordism between manifolds with boundary can be split into a sequence of right product and left product cobordisms, if the codimension of the embedding is at least two. This is a topological counterpart of the algebraic splitting theorem for embedded cobordisms of the first author, A. Nemethi…
Let X be any rational ruled symplectic four-manifold. Given a symplectic embedding $ι:B_{c}\into X$ of the standard ball of capacity c into X, consider the corresponding symplectic blow-up $\tX_ι$. In this paper, we study the homotopy type of the symplectomorphism group $\Symp(\tX_ι)$, simplifying and extending t…
This paper solves PDEs for embedding discrete lattices into smooth manifolds.
problem Embedding discrete lattices into smooth manifolds while preserving geometric and topological properties.
method Rigorous mathematical framework and analysis of partial differential equations (PDEs).
result Existence and regularity of solutions to PDEs under initial boundary conditions.
The paper generalizes free boundary min-max theory to equivariant settings.
problem Existence of minimal hypersurfaces with free boundary in manifolds with boundary.
method Free boundary min-max theory generalized to equivariant settings.
result Existence of nontrivial smooth almost properly embedded G-invariant minimal hypersurfaces with free boundary. New embeddings show answer to Baker-Laidacker question can be yes or no.
problem Answer to Baker-Laidacker question about disjoint compacta in R^N.
method Use of specific wild Cantor sets and Antoine's methods.
result Answer to Baker-Laidacker question can be twofold.
Szűcs introduced cobordism of singular maps to compute groups of immersions and embeddings.
problem Computing cobordism groups of immersions and embeddings in dimensions where classical theory fails.
method Investigation of classifying spaces constructed by Szűcs and Rimányi.
result Collection and organization of results towards computation of cobordism groups of singular maps.
Measures time-delay embedding for noisy, sparse data.
problem Applying Takens' embedding theorem to real-world, noisy data.
method Formulated a measure-theoretic generalization of the embedding theorem, using optimal transport.
result Reconstructed full state of dynamical systems from time-lagged partial observations robust to noise and sparsity.
New theory for eigenvectors of generalized Laplacian matrices, addressing dependency issues.
problem Dependency in random matrix theory hinders eigenvector analysis for latent embeddings.
method Introduces generalized Laplacian matrices and a new asymptotic theory framework.
result Established asymptotic normalities for spiked eigenvectors and eigenvalues.