New framework predicts choice with set-related invariances.
problem Accurately predicting choice behavior in large-scale data.
method A learning framework that captures set-related invariances, derived from economics.
result Demonstrated utility on three large choice datasets.
Curved metrics on Wallach spaces bounded by curves under flow.
problem Properties of positively curved Riemannian metrics on Wallach spaces.
method Normalized Ricci flow analysis on specific Wallach spaces.
result Set of metrics forms bounded curves asymptotically.
Rigidity results for initial data sets related to the positive mass theorem.
problem Rigidity of initial data sets in general relativity.
method Establishing conditions for weak outermost marginally outer trapped surfaces and rigidity results for Riemannian manifolds.
result Marginally outer trapped surfaces are weakly outermost under certain conditions.
We formulate a notion of stability for maps between polarised varieties which generalises Kontsevich's definition when the domain is a curve and Tian-Donaldson's definition of K-stability when the target is a point. We give some examples, such as Kodaira embeddings and fibrations. We prove the existence of a projective…
Study proves estimate for Hessian quotient equations on 2D Riemannian manifolds.
problem Problems posed by Delanoë and Urbas related to Hessian quotient equations.
method Maximum principle argument and new test function introduced to prove estimate.
result Unobstructed second order a priori estimate for real Hessian quotient equation in 2D.
We consider learning high-dimensional multi-response linear models with structured parameters. By exploiting the noise correlations among responses, we propose an alternating estimation (AltEst) procedure to estimate the model parameters based on the generalized Dantzig selector. Under suitable sample size and resampli…
We study topological structures of the sets (0,1/2)3∩Ω and (0,1/2)3∖Ω, where~Ω is one special algebraic surface defined by a symmetric polynomial in variables a1,a2,a3 of degree~12. These problems arise in studying of general properties of degenerate singular points of dynamical systems ob…
Researchers prove an index formula for spinors on 3-manifolds branching along graphs.
problem Index formula for Dirac operators on 3-manifolds with branch points.
method Analyzes Dirac operator on two-valued spinors on a 3-manifold with a graph branch, with boundary conditions.
result Index formula vanishes when the branch is a smooth curve, extends to graphs with vertices.
Distributed learning of probabilistic models from multiple data repositories with minimum communication is increasingly important. We study a simple communication-efficient learning framework that first calculates the local maximum likelihood estimates (MLE) based on the data subsets, and then combines the local MLEs t…
Minimizing a function over an intersection of convex sets is an important task in optimization that is often much more challenging than minimizing it over each individual constraint set. While traditional methods such as Frank-Wolfe (FW) or proximal gradient descent assume access to a linear or quadratic oracle on the …
The paper calculates critical points of systole function on Teichmüller space.
problem Critical points with pathological feature in surfaces of large genus.
method Integer linear programming with symmetry breaking technique.
result Found minimal filling sets of systoles in genus 5 with 8 geodesics.
New algebraic geometry and statistical manifold connections proven.
problem Understanding the structure of statistical manifolds and their algebraic properties.
method Developed relations between algebraic geometry, information theory, and Topological Field Theory.
result Statistical pre-Frobenius manifolds form algebraic varieties and have hexagonal, isoclinic webs.
Background: In cognitive neuroscience the potential of Deep Neural Networks (DNNs) for solving complex classification tasks is yet to be fully exploited. The most limiting factor is that DNNs as notorious 'black boxes' do not provide insight into neurophysiological phenomena underlying a decision. Layer-wise Relevance …
Abstract invariant cannot be expressed using various slice-torus invariants.
problem Cannot express Iida-Taniguchi's slice-torus invariant using other known invariants.
method Analysis of various known invariants and their properties.
result Iida-Taniguchi's slice-torus invariant cannot be realized as a linear combination of other invariants.
The θ invariant encompasses the Rozansky-Overbay invariant.
problem None explicitly stated in the abstract.
method Generalization of the Rozansky-Overbay invariant using the θ invariant. result The θ invariant recovers the Rozansky-Overbay invariant. Non-invariant complex structures on Lie groups are not biholomorphic to invariant ones.
problem Understanding non-invariant deformations of complex structures on Lie groups.
method Computed cohomologies to show non-biholomorphicity.
result Non-invariant complex structures are not biholomorphic to invariant ones.
Study of Bauer-Furuta invariants under Lie group actions and Galois coverings.
problem Investigating invariants of 4-manifolds under group actions and Galois coverings.
method Functorial approach to equivariant invariants and study in Galois covering situations.
result Ordinary invariants of quotients are determined by equivariant invariants of the covering manifold.
The Kuperberg invariant is shown to be gauge invariant for certain framed 3-manifolds.
problem Exploring gauge invariance of the Kuperberg invariant for specific 3-manifolds.
method Using hyperbolic 3-manifolds and finite-dimensional Hopf algebras.
result First examples of gauge invariants of general finite-dimensional Hopf algebras via topological methods.
Paper introduces new invariant for pairs of immersions.
problem Understanding behavior of immersions through tangencies and triple points.
method Introduces J2+-invariant for oriented pairs of immersions, invariant under inverse tangencies and triple points. result Invariant changes under direct tangencies but remains invariant under orientation change and inverse tangencies.
New polynomial invariant distinguishes singular links.
problem Distinguishing singular links using existing invariants.
method Generalized quandle polynomial to singquandles and constructed a singular link invariant.
result New polynomial invariant distinguishes singular links with same counting invariant.
We show that the perturbative g invariant of rational homology 3-spheres can be recovered from the LMO invariant for any simple Lie algebra g, i.e, the LMO invariant is universal among the perturbative invariants. This universality was conjectured in [25]. Since the perturbative invariants dominate …
Constructs universal link invariants from intersections in configuration spaces.
problem Globalise topologically all coloured Jones polynomials and ADO polynomials.
method Defines new link invariants from graded intersections in configuration spaces.
result Recover all coloured Jones polynomials and ADO polynomials for links.
Grid homology confirms the Upsilon invariant in knot theory.
problem Verifying the equivalence of Upsilon invariants in knot theory.
method Reconstructed Upsilon invariant using grid homology and proved equivalence.
result Upsilon invariants in knot Floer and grid homology are equivalent.
New invariant CWR for alternating links is stronger than existing invariants.
problem Developing a stronger invariant for alternating links.
method Introducing CWR invariant as an array of two-variable polynomials. result The CWR invariant is stronger than classical invariants like HOMFLYPT and Kauffman polynomials. Defines knot concordance invariant using instanton homology and Donaldson invariants.
problem Knot concordance and its classification.
method Defines an invariant φ for knots in the 3-sphere using Donaldson invariants and Floer's instanton homology. result The invariant φ coincides with a special case of an invariant defined by Froyshov. Combines combinatorial method to extend Milnor invariants to welded links.
problem Extending Milnor invariants to welded links.
method Combinatorial approach.
result Invariance of extended Milnor invariants for welded links.
New family of knots with epsilon invariant nonzero despite Upsilon and phi being zero.
problem Comparing smooth concordance invariants.
method Building an infinite family of knots.
result Found knots with epsilon invariant nonzero but Upsilon and phi zero.
Paper introduces a new invariant for virtual knotoids and proves it's a Vassiliev invariant of order one.
problem Tackles the problem of understanding invariants for virtual knotoids.
method Uses a 0-smoothing invariant constructed from local modifications at classical crossings.
result Demonstrates that the 0-smoothing invariant provides less information than the gluing invariant.
We construct two knot invariants. The first knot invariant is a sum constructed using linking numbers. The second is an invariant of flat knots and is a formal sum of flat knots obtained by smoothing pairs of crossings. This invariant can be used in conjunction with other flat invariants, forming a family of invariants…
As nilpotent studies in knot theory, we focus on invariants of Milnor, Orr, and Kontsevich. We show that the Orr invariant of degree k is equivalent to the tree reduction of the Kontsevich invariant of degree <2k. Furthermore, we will see a close relation between the Orr invariant and the Milnor invariant, and …
Formula connects surface and curve invariants via slice transitions.
problem Computing surface invariants from curve invariants.
method Introducing differential measures for local changes across singular slice transitions.
result Explicit formula for surface invariant change during quadruple-point events.
In this article we introduce a family of transverse invariants arising from the deformations of Khovanov homology. This family includes the invariants introduced by Plamenevskaya and by Lipshitz, Ng, and Sarkar. Then, we investigate the invariants arising from Bar-Natan's deformation. These invariants, called β-invar…
New concordance invariants phi and phi_j are defined and studied.
problem Understanding the relationships between different concordance invariants.
method Defined and analyzed new invariants phi and phi_j, and provided recursive formulas.
result Found infinitely many knots with specific combinations of zero and nonzero phi invariant.
We recall the definition of the quadratic helicity invariant and of the higher asymptotic ergodic M-invariant. We present a simpler new proof (in part) that the M-invariant is ergodic. The M-invariant is a higher invariant, this means that for the magnetic field with closed magnetic lines the invariant is not a f…
Defines new link-homotopy invariants using Milnor's higher order link invariants.
problem Link-homotopy invariants for link maps of multiple components.
method Uses Milnor's higher order link invariants and combinatorial theory of cut-diagrams.
result Provides practical algorithms to compute these invariants and detects families of examples.
New invariants for singular knots and links defined using shadow structures.
problem Defining invariants for singular knots and links.
method Introducing action of singquandles on sets and defining shadow counting and polynomial invariants.
result Enhanced shadow counting invariant for singular knots and links.
Paper calculates L-invariant and L*-invariant for complex surface sums.
problem Calculating invariants for complex surface sums.
method Using pants complexes and dual curve complexes.
result First example of arbitrary large invariants for bridge numbers.
We discuss an universal bordism invariant obtained from the Atiyah-Patodi-Singer eta-invariant from the analytic and homotopy theoretic point of view. Classical invariants like the Adams e-invariant, ρ-invariants and String-bordism invariants are derived as special cases. The main results are a secondary index theo…
Paper proves Rohlin invariant's uniqueness and extends homology sphere invariants.
problem Proving the uniqueness of Rohlin invariant and extending homology sphere invariants.
method Using the Rohlin invariant's uniqueness, the paper extends invariants from trivial 2-cocycles to those with 2-torsion.
result Generalized invariants of homology spheres with 2-torsion values.
Study annular concordance invariants and refine transverse link invariants.
problem Understanding annular concordance and transverse links.
method Defined and studied annular concordance invariant, introduced modified chain complex, and refined transverse invariant.
result Refined transverse invariant and studied its relationship with transverse and braid monodromy properties.
The universal sl_2 invariant of string links has a universality property for the colored Jones polynomial of links, and takes values in the h-adic completed tensor powers of the quantized enveloping algebra of sl_2. In this paper, we exhibit explicit relationships between the universal sl_2 invariant and Milnor invaria…
We consider Milnor invariants for certain covering links as a generalization of covering linkage invariants formulated by R. Hartley and K. Murasugi. A set of Milnor invariants for covering links is a cobordism invariant of a link, and that this invariant can distinguish some links for which the ordinary Milnor invaria…
The study explores knot invariants using roots of unity.
problem Understanding knot invariants at roots of unity.
method Using Reshetikhin-Turaev method and generalizing ADO invariants.
result Clarified definitions and connections between different invariants.
Enhances quantum invariants using tribracket brackets.
problem Quantum invariants of tribracket-colored knots and links.
method Introduces tribracket brackets as skein invariants.
result Provides new quantum invariants and examples.
Researchers prove birational invariance of BCOV invariant using motivic integration.
problem Proving the birational invariance of BCOV invariant for Calabi-Yau manifolds and varieties.
method Motivic integration theory applied to Calabi-Yau varieties with Kawamata log terminal singularities.
result Birational Calabi-Yau manifolds have the same BCOV invariant.
Power series invariant of hyperbolic 3-manifolds matches knot invariants.
problem Understanding topological invariants of hyperbolic 3-manifolds.
method Perturbative power series associated with ideally triangulated cusped hyperbolic 3-manifolds.
result The power series agrees with Kashaev and Andersen-Kashaev invariants to all orders.
New method to calculate 3-manifold invariants via skew-racks.
problem Calculating invariants of 3-manifolds.
method Introducing skew-racks with good involution and Property FR, defining cocycle invariants.
result Established new approach to obtain 3-manifold invariants via Dehn surgery.
New invariant fully describes finite type invariants of knots in homology 3-spheres.
problem Constructing a universal finite type invariant for knots in homology 3-spheres.
method Refined construction of a new invariant that is strictly stronger and universal.
result New invariant fully describes the graded space of finite type invariants of knots in homology 3-spheres.