Study confirms conjectures for topologically slice knots' concordance group.
problem Primary decomposition conjectures for knot concordance groups.
method Use of amenable L2-signatures, Ozsváth-Szabó d-invariants, and Némethi's Heegaard Floer homology results. result Smooth concordance group of topologically slice knots has large subgroup with true primary decomposition.
Study shows knots with coprime polynomials can't be concordant.
problem Primary decomposition of knot concordance.
method Using von Neumann rho-invariants and solvable filtration.
result Nonnegative integer n-solvable knots with coprime polynomials are not concordant.
New tool: relative Hopf invariant for Poincaré surgery.
problem Non-simply connected Poincaré surgery.
method Relative Hopf invariant in equivariant setting.
result Established Poincaré embedding results in relative setting.
Simplified combinatorial descriptions of branched spines for 3-manifolds using primary MP move and sliding moves.
problem Combinatorial descriptions of branched spines for 3-manifolds and their equivalence relations.
method Demonstrated that 16 MP moves on branched spines are derived from a primary MP move, pure sliding moves, and their inverses.
result Simpler combinatorial descriptions for closed 3-manifolds and combed 3-manifolds.
A new method for CT using graph-based regularization.
problem Transfer calibrations between instruments without suitable transfer standards.
method Employing manifold regularization of PLS objective to enforce invariant projections in latent variable space.
result Implicit removal of inter-device variation in predictive directions.
Study delta invariant of curves on rational surfaces using topological methods.
problem Calculate delta invariant for curves embedded in rational singularities.
method Use topological techniques and Poincaré series.
result Develop formulae for delta invariant in terms of embedded data.
Classifies stable diffeomorphism of spin 4-manifolds with specific fundamental groups.
problem Classifying stable diffeomorphism of spin 4-manifolds with given fundamental groups.
method Formulated conjectural relationships between algebraic invariants and obstructions, proved for specific groups.
result Proved conjectures for specific fundamental groups, providing complete algebraic stable classification.
A new method uses physics-informed neural networks to solve reliability analysis problems without simulations.
problem Solving reliability analysis problems without the need for expensive simulations.
method Physics-informed neural networks to learn directly from problem physics.
result Eliminates the need for expensive simulations and achieves highly accurate results.
We show that for each Seifert form of an algebraically slice knot with nontrivial Alexander polynomial, there exists an infinite family of knots having the Seifert form such that the knots are linearly independent in the knot concordance group and not concordant to any knot with coprime Alexander polynomial. Key ingred…
The geometric Hopf invariant of a stable map F is a stable Z_2-equivariant map h(F) such that the stable Z_2-equivariant homotopy class of h(F) is the primary obstruction to F being homotopic to an unstable map. In this paper we express the geometric Hopf invariant of the Umkehr map F of an immersion f:M^m \to N^n in t…
We present a sparse and invariant representation with low asymptotic complexity for robust unsupervised transient and onset zone detection in noisy environments. This unsupervised approach is based on wavelet transforms and leverages the scattering network from Mallat et al. by deriving frequency invariance. This frequ…
Rotation-equivariant CNN reveals common features in V1 neurons.
problem V1 models fail to predict natural stimuli responses accurately.
method Rotation-equivariant convolutional neural network model.
result Rotation-equivariant network outperforms regular CNN and reveals common features.
For a link L in the 3-sphere and for a prime p, we express the p-primary information on the first homology group of pm-fold branched covers of L in terms of its p-adic Milnor higher linking invariants, using the completed Alexander module of the pro-p completion of the link group of L.
For each integer N≥2, Mariño and Moore defined generalized Donaldson invariants by the methods of quantum field theory, and made predictions about the values of these invariants. Subsequently, Kronheimer gave a rigorous definition of generalized Donaldson invariants using the moduli spaces of anti-self-dual conne…
We show that some ternary quasigroups appear naturally as invariants of classical links and links on surfaces. We also note how to obtain from them invariants of Yoshikawa moves. In our previous paper, we defined homology theory for algebras satisfying two axioms derived from the third Reidemeister move. In this paper,…
Survey recent constructions of cyclic cocycles for Lie groups.
problem Higher index theory for proper cocompact G-actions. method Constructions of cyclic cocycles on Harish-Chandra Schwartz algebra.
result Applications to higher index theory.
We study primary and secondary invariants of leafwise Dirac operators on foliated bundles. Given such an operator, we begin by considering the associated regular self-adjoint operator Dm on the maximal Connes-Skandalis Hilbert module and explain how the functional calculus of Dm encodes both the leafwise calculus…
Study controls bifurcations in Eulerian flows with multiple Hopf singularities.
problem Bifurcation analysis and control of nonlinear Eulerian flows with non-resonant n-tuple Hopf singularities.
method Analysis of CW complex bifurcations of flow-invariant Clifford hypertori, using leaf-bifurcation varieties.
result Tertiary toral CW complex bifurcates from and persists outside a secondary toral CW complex.
Proposes a criterion for selecting relevant auxiliary variables in incomplete data analysis.
problem Selecting useful auxiliary variables for incomplete data analysis.
method Formulates model selection problem, proposes an information criterion based on Kullback-Leibler divergence.
result Proposed information criterion is an asymptotically unbiased estimator of Kullback-Leibler divergence.
The paper finds new constant p-mean curvature surfaces in the Heisenberg group.
problem Discovering new examples of constant p-mean curvature surfaces. method Utilizing the theory and approach for constructing such surfaces.
result Complete description of rotationally invariant surfaces of constant p-mean curvature. Study topological Iwasawa invariants for 3-sphere links, proving density results.
problem Detect and analyze Iwasawa invariants for 3-sphere links.
method Explicit criteria for detecting Iwasawa invariants, statistical analysis of 2-bridge links.
result Density of 2-bridge links with specific Iwasawa invariants.
The paper investigates exotic smooth structures on manifolds with group actions.
problem Existence of homeomorphic but not diffeomorphic smooth manifolds with shared basic spectra.
method Investigates Riemannian Laplacian eigenvalues and eigenfunctions on manifolds with compact Lie group actions.
result Establishes the existence of homeomorphic yet not diffeomorphic manifolds with shared basic spectra.
Motivated by families of formal moduli problems, in this note we generalize the notion of L-infinity space by allowing sheaves of L-infinity algebras over any (reasonable) nilpotent dg manifold. We discuss various examples including those coming from Lie algebroids. Given a Lie algebroid, we show that there is an L-inf…
Study blockchain's impact on primary financial market challenges.
problem Challenges of blockchain in securities issuance and trading.
method Hybrid method combining interviews and surveys.
result Complex due diligence, mismatch, and difficult monitoring are significant challenges.
Engel manifolds show transverse tori can be made to have various formal invariants.
problem Understanding transverse tori in Engel manifolds.
method Analogous to transverse knots, classify formal invariants and show their uniqueness.
result Engel manifolds can have infinitely many transverse isotopy classes of tori with specific invariants.
Reduces a complex hypersurface to a simplified equation with primary invariants.
problem Analyzing Levi degenerate CR manifolds in 5 dimensions.
method Applying Lie's theory, integrating and straightening chains, and using Poincaré-Moser reduction.
result Shows a convergent change of coordinates that simplifies the equation of the manifold.
Proves cohomology of elliptic structures on Lie groups can be algebraic.
problem Computing cohomology of elliptic structures on compact semisimple Lie groups.
method Used spectral sequences to construct an isomorphism between left-invariant and usual differential complexes.
result Reduced analytical problem to algebraic computation.
Study local equivalence of rigid 5D hypersurfaces in C^3, finding necessary and sufficient conditions for rigid biholomorphism.
problem Local equivalence problem for real-analytic rigid hypersurfaces in C^3.
method Cartan-type reduction to an appropriate {e}-structure, finding primary invariants.
result Identify necessary and sufficient condition for rigid biholomorphism using invariants.
A new geometric model for V1 hypercolumns combines symplectic and spherical models.
problem Understanding the structure of V1 hypercolumns in the visual cortex.
method A differential geometric model based on conformal geometry.
result Combines features of symplectic and spherical models of hypercolumns.
Aux-NAS uses auxiliary labels to improve primary task performance without extra inference cost.
problem Improving primary task performance using auxiliary labels without increasing inference cost.
method Architecture-based approach with a flexible asymmetric structure for primary and auxiliary tasks, using Neural Architecture Search (NAS) to evolve networks with only primary-to-auxiliary connections.
result Achieves improved performance on multiple tasks without increasing inference cost.
AFLAC improves domain generalization by balancing invariance and accuracy.
problem Balancing domain invariance and classification accuracy for domain generalization.
method Adversarial feature learning with accuracy constraint (AFLAC).
result AFLAC outperforms domain-invariance-based methods on synthetic and real-world datasets.
The theory of acceptance sets and their associated risk measures plays a key role in the design of capital adequacy tests. The objective of this paper is to investigate, in the context of bounded financial positions, the class of surplus-invariant acceptance sets. These are characterized by the fact that acceptability …
Classifies Real primary Hopf surfaces and their associated groups.
problem Classifying Real primary Hopf surfaces and their associated groups.
method Complete classification up to Real biholomorphisms and equivariant diffeomorphisms.
result Detailed description of groups associated with Real primary Hopf surfaces.
We study smooth maps between smooth manifolds with only fold points as their singularities, and clarify the obstructions to the existence of such a map in a given homotopy class for certain dimensions. The obstructions are described in terms of characteristic classes, which arise as Postnikov invariants, and can be int…
The paper shows infinite rank in T_1/T_2 and new insights into Alexander polynomials and concordance.
problem Understanding the structure of topologically slice knots and their concordance.
method Analyzing surgery manifolds of satellite links and using Ozsváth-Szabó d-invariants.
result There exist infinitely many knots with specific Alexander polynomials that are not concordant to any knot with coprime Alexander polynomial.
Quantum cocycle invariants derived from Yang-Baxter cohomology.
problem Constructing stronger quantum knot invariants.
method Developing quantum cocycle invariants using Yang-Baxter cohomology and deformation theory.
result Quantum cocycle invariants yield stronger invariants in certain examples.
New findings on knot concordance show limitations to primary decompositions.
problem Understanding the structure of knot concordance groups.
method Analyzing homomorphisms and polynomial factorizations of Alexander polynomials.
result Primary decompositions of topologically slice knots cannot exist.
Develops an oblique projection technique to approximate a foliation for non-normal dynamics.
problem Modeling dynamics far from a primary Spectral Submanifold (SSM) in non-normal systems.
method Oblique projection technique based on experimental data.
result Approximates a stable invariant foliation for non-normal dynamics efficiently.
Paper constructs a transfer map for codimension 2 submanifolds in higher index theory.
problem Higher index theory of codimension 2 submanifolds.
method Construction of codimension 2 transfer map and adjoint relationship with cyclic cohomology.
result Established adjoint relationship between codimension 2 transfer map and co-transfer map in cyclic cohomology.
The paper tackles online learning with two types of losses and shows it's impossible without certain assumptions.
problem Online learning with primary and secondary losses where the secondary loss is bounded by a linear threshold.
method Analyzes the feasibility of achieving low regret with respect to the primary loss while keeping the secondary loss within a linear threshold.
result Achieving the goal is impossible without bounded variance assumption on the secondary loss.
Study Ricci flow on torus bundles and related manifolds.
problem Compute Ricci flow formulas for torus bundles.
method Use invariant metrics compatible with connections on principal G-bundles.
result Solutions to Ricci flows on Heisenberg groups and Berger 3-spheres.
Programs compute Heegaard Floer invariants from open books.
problem Computing Heegaard Floer invariants of 3-manifolds.
method SageMath programs that analyze Heegaard diagrams and compute HF-hat.
result Computes Heegaard Floer homology and contact invariants.
Some connected components of a moduli space are mundane in the sense that they are distinguished only by obvious topological invariants or have no special characteristics. Others are more alluring and unusual either because they are not detected by primary invariants, or because they have special geometric significance…
The following are expanded lecture notes for the course of eight one hour lectures given by the second author at the 2014 summer school Asymptotic Analysis in General Relativity held in Grenoble by the Institut Fourier. The first four lectures deal with conformal geometry and the conformal tractor calculus, taking as p…
A method improves capsule networks by reducing information dilution and enhancing performance.
problem Capsule networks struggle with datasets containing background and complex objects.
method Restrict the activation values of primary capsule layers to highlight discriminative capsules.
result The method achieves better performance on various datasets.
We study holomorphic locally homogeneous geometric structures modelled on line bundles over the projective line. We classify these structures on primary Hopf surfaces. We write out the developing map and holonomy morphism of each of these structures explicitly on each primary Hopf surface.
A new method for optimizing hierarchical multi-objective problems.
problem Symmetry and neglect of objective hierarchy in existing multi-objective methods.
method Priority-Constrained Descent (PCD) framework exploiting hierarchical objective structures.
result Pareto dominance and better per-objective performance with secondary progress guarantees.
New training method improves model performance across various tasks.
problem Training deep neural networks using only the ground-truth class information.
method Maximizes likelihood of ground-truth classes while neutralizing complement classes.
result Training with both primary and complement objectives improves model performance across multiple tasks.