A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
We show that the perturbative invariant of rational homology 3-spheres can be recovered from the LMO invariant for any simple Lie algebra , i.e, the LMO invariant is universal among the perturbative invariants. This universality was conjectured in [25]. Since the perturbative invariants dominate …
New invariant from non-acyclic flat connections.
Power series invariant of hyperbolic 3-manifolds matches knot invariants.
The paper constructs new bimetric conformal invariants using metric perturbations.
The universal perturbative invariants of rational homology spheres can be extracted from the Chern-Simons partition function by combining perturbative and nonperturbative results. We spell out the general procedure to compute these invariants, and we work out in detail the case of Seifert spaces. By extending some prev…
New approach to quantum knot invariants using perturbed Gaussian generating functions.
While several feature scoring methods are proposed to explain the output of complex machine learning models, most of them lack formal mathematical definitions. In this study, we propose a novel definition of the feature score using the maximally invariant data perturbation, which is inspired from the idea of adversaria…
New quantum invariant is asymptotically multiplicative under cyclic covers.
New invariant counts graph configurations in 3D manifolds.
We derive a gauge theoretic invariant of integral homology 3-spheres which counts gauge orbits of irreducible, perturbed flat SU(3) connections with sign given by spectral flow. To compensate for the dependence of this sum on perturbations, the invariant includes contributions from the reducible, perturbed flat orbits.…
New knot invariants derived using quantum cluster algebras.
A perturbative SU(3) Casson invariant for integral homology 3-spheres is defined. Besides being fully perturbative, it has nice properties: (1) is an integer. (2) It is preseved under orientation change. (3) A connected sum formula holds. Explicit calculations of the invariant for $1/k…
The integrality of the Kontsevich integral and perturbative invariants is discussed. We show that the denominator of the degree part of the Kontsevich integral of any knot or link is a divisor of . We also show that the denominator of of the degree part of the universal perturbative invari…
Study of knot invariant growth for twisted knots.
We study the relationship between Bar-Natan's perturbation in Khovanov homology and Szabo's geometric spectral sequence, and construct a link invariant that generalizes both into a common theory. We study a few properties of the new invariant, and introduce a family of s-invariants from the new theory in the same spiri…
New invariants for 3-manifolds derived from equivariant Cerf theory.
We construct power series invariants of rational homology 3-spheres from quantum PSU(n)-invariants. The power series can be regarded as perturbative invariants corresponding to the contribution of the trivial connection in the hypothetical Witten's integral. This generalizes a result of Ohtsuki (the case) which l…
We introduce a new class of perturbations of the Seiberg-Witten equations. Our perturbations offer flexibility in the way the Seiberg-Witten invariants are constructed and also shed a new light to LeBrun's curvature inequalities.
The paper studies invariant complex manifolds in holomorphic slow-fast systems.
Prove that collapsing CSC metrics can be perturbed to invariant collapsing CSC metrics.
In this paper we prescribe a fourth order conformal invariant on the standard sphere, with , and study the related fourth order elliptic equation. We first find some existence results in the perturbative case. After some blow up analysis we build a homotopy to pass from the perturbative case to the non-pert…
Formula calculates invariant for 3-manifolds with torus boundaries.
We develop several methods that allow us to compute all-loop partition functions in perturbative Chern-Simons theory with complex gauge group G_C, sometimes in multiple ways. In the background of a non-abelian irreducible flat connection, perturbative G_C invariants turn out to be interesting topological invariants, wh…
We consider quantum invariants of 3-manifolds associated with arbitrary simple Lie algebras. Using the symmetry principle we show how to decompose the quantum invariant as the product of two invariants, one of them is the invariant corresponding to the projective group. We then show that the projective quantum invarian…
We propose an extension of the recently-proposed volume conjecture for closed hyperbolic 3-manifolds, to all orders in perturbative expansion. We first derive formulas for the perturbative expansion of the partition function of complex Chern-Simons theory around a hyperbolic flat connection, which produces infinitely-m…
We develop techniques for computing the integer valued SU(3) Casson invariant. Our method involves resolving the singularities in the flat moduli space using a twisting perturbation and analyzing its effect on the topology of the perturbed flat moduli space. These techniques, together with Bott-Morse theory and the spl…
We extend the Chern-Simons perturbative invariant of Axelrod and Singer to non-acyclic connections. We construct a solution of the quantum master equation on the space of functions on the cohomology of the connection. We prove that this solution is well defined up to master homotopy. We discuss also invariants of links…
Counts minimal tori in Riemannian manifolds with 6 or more dimensions.
The celebrated KAM Theory says that if one makes a small perturbation of a non-degenerate completely integrable system, we still see a huge measure of invariant tori with quasi-periodic dynamics in the perturbed system. These invariant tori are known as KAM tori. What happens outside KAM tori draws a lot of attention. …
Study linear perturbations of Spin(7) metrics, finding only rank one nilpotent matrices.
Based on the Hamiltonian dimensional reduction of axially symmetric, Ricci-flat Lorentzian spacetimes to a Einstein-wave map system with the (negatively curved) hyperbolic 2-plane target, we construct a positive-definite, (spacetime) gauge-invariant energy functional for linear axially symmetric perturbatio…
We give a purely topological definition of the perturbative quantum invariants of links and 3-manifolds associated with Chern-Simons field theory. Our definition is as close as possible to one given by Kontsevich. We will also establish some basic properties of these invariants, in particular that they are universally …
GCNNs gain rotation invariance with more training augmentation, making SVD-Universal more effective.
Vogel's construction links knot invariants to Lie algebras, revealing new insights.
Study analyzes perturbations in singular subspaces under random noise.
This paper analyzes invariants in non-perturbative complex Chern-Simons theory.
Adversarial examples are malicious inputs crafted to cause a model to misclassify them. Their most common instantiation, "perturbation-based" adversarial examples introduce changes to the input that leave its true label unchanged, yet result in a different model prediction. Conversely, "invariance-based" adversarial ex…
The Yamabe Invariant of a smooth compact manifold is by definition the supremum of the scalar curvatures of unit-volume Yamabe metrics on the manifold. For an explicit infinite class of 4-manifolds, we show that this invariant is positive but strictly less than that of the 4-sphere. This is done by using spin^c Dirac o…
We introduce a gauge-theoretic integer lift of the Rohlin invariant of a smooth 4-manifold X with the homology of . The invariant has two terms; one is a count of solutions to the Seiberg-Witten equations on X, and the other is essentially the index of the Dirac operator on a non-compact manifold with e…
Generic smooth minimal hypersurfaces exist in 8D manifolds.
Local index density of perturbed de Rham complex is invariant under certain conditions.
Novel geometry-informed irreversible perturbation accelerates Langevin dynamics convergence.
We study the general structure of formal perturbative solutions to the Hamiltonian perturbations of spatially one-dimensional systems of hyperbolic PDEs. Under certain genericity assumptions it is proved that any bihamiltonian perturbation can be eliminated in all orders of the perturbative expansion by a change of coo…
As one of the problems in his list [20], T. Ohtsuki proposed to study relations between quandle cocycle invariants and quantum invariants. The aim of this paper is to answer one of those questions. We prove that the coefficient of the finite perturbative expansion of the quandle shadow cocycle invariant defined by $(\Z…
Study of mean curvature flows with conical singularities using mathematical techniques.
Deep neural networks (DNNs) have achieved superior performance in various prediction tasks, but can be very vulnerable to adversarial examples or perturbations. Therefore, it is crucial to measure the sensitivity of DNNs to various forms of perturbations in real applications. We introduce a novel perturbation manifold …
Study delocalized eta invariants for signature operators on proper manifolds.
We propose a non-perturbative formulation of the Atiyah-Patodi-Singer(APS) index in lattice gauge theory, in which the index is given by the invariant of the domain-wall Dirac operator. Our definition of the index is always an integer with a finite lattice spacing. To verify this proposal, using the eigenmode set o…