Witt algebra acts on categorified quantum groups in type A.
problem Action of Witt algebra on categorified quantum groups.
method Construction of action on categorified quantum group and foams.
result Action of Witt algebra on foams recovers previous results.
New polynomial invariants for knots and links from quandle coloring quiver decategorification.
problem Defining new polynomial invariants for knots and links.
method Decategorification of the quandle coloring quiver to create polynomial invariants.
result The invariants are not determined by the quandle counting invariant.
Decategorifies higher actions in Heegaard Floer homology.
problem Understanding higher actions in Heegaard Floer homology.
method Decategorification of bordered Heegaard Floer strands algebras and identification with actions on exterior powers of homology groups.
result Identifies decategorifications with certain actions on exterior powers of homology groups of surfaces.
In two previous papers, the author showed how to decompose the Khovanov homology of a link L into the algebraic pairing of a type D structure and a type A structure (as defined in bordered Floer homology), whenever a diagram for L is decomposed into the union of two tangles. Since Khovanov homol…
We relate decategorifications of Ozsváth-Szabó's new bordered theory for knot Floer homology to representations of Uq(gl(1∣1)). Specifically, we consider two subalgebras Cr(n,S) and Cl(n,S) of Ozsváth- Szabó's algebra B(n,S), an…
We introduce an sl(n) homology theory for knots and links in the thickened annulus. To do so, we first give a fresh perspective on sutured annular Khovanov homology, showing that its definition follows naturally from trace decategorifications of enhanced sl(2) foams and categorified quantum gl(m), via classical skew Ho…
We give a simple, combinatorial construction of a unital, spherical, non-degenerate ∗-planar algebra over the ring Z[q1/2,q−1/2]. This planar algebra is similar in spirit to the Temperley-Lieb planar algebra, but computations show that they are different. The construction comes from the combinator…
New algebraic structures help categorify link invariants.
problem Classifying and distinguishing links and virtual links.
method Introducing mc-biquandles and categorifying homsets.
result New link invariants defined via mc-biquandle coloring quivers.
New invariants for virtual knots and links defined via quiver representations.
problem Defining new invariants for virtual knots and links.
method Quiver representations associated to virtual biquandles and rings.
result New polynomial invariants for virtual knots and links.
New polynomial invariants from quandle action quivers.
problem Classical and virtual knot and link invariants.
method Categorification of quandle counting invariant using quandle action quivers.
result Quandle action polynomials as decategorifications.
New polynomial invariants derived from birack and switch structures.
problem Polynomial invariants of braids.
method Switch structures, birack colorings, quiver-valued invariants.
result New polynomial invariants of braids.
Define quiver representation-valued invariants for classical and virtual knots
problem Define quiver representation-valued invariants for classical and virtual knots
method Define an infinite family of quiver representation-valued invariants of classical and virtual knots associated to a choice of data vector consisting of a biquandle, abelian group, set of biquandle arrows weights with values in the abelian group, coefficient ring and set of biquandle endomorphisms.
result Extract four new polynomial invariants as decategorifications
Theory for algebraic data on categories via concentration structures.
problem Defining algebraic structures on categories.
method Introducing concentration structures and concentration monoids.
result Every group can be represented as a concentration monoid of a trivial category.
Abstract TQFT for sutured manifolds using Floer homology.
problem Classical Frohman-Nicas TQFT for Alexander polynomial.
method Decategorification of bordered sutured Heegaard Floer homology.
result Generalization to arbitrary cobordisms between surfaces.
We define a homology HN for closed braids by applying Khovanov and Rozansky's matrix factorization construction with potential axN+1. Up to a grading shift, H0 is the HOMFLYPT homology defined in arXiv:math/0505056. We demonstrate that, for N≥1, HN is a $\mathbb{Z}_2\o…
Bordered Heegaard Floer homology is an invariant for 3-manifolds, which associates to a surface F an algebra A(Z), and to a 3-manifold Y with boundary, together with an orientation-preserving diffeomorphism φfrom F to \bdy Y, a module over A(Z). We study the Grothendieck group of modules over A(Z), and define an invari…
Khovanov and Rozansky's categorification of the HOMFLY-PT polynomial is invariant under braidlike isotopies for any link diagram and Markov moves for braid closures. To define HOMFLY-PT homology, they required a link to be presented as a braid closure, because they did not prove invariance under the other oriented Reid…
This is a continuation of our previous work arXiv:1601.05617 on trace and inverse trace of Steklov eigenvalues. More new inequalities for the trace and inverse trace of Steklov eigenvalues are obtained.
Study of torus surgeries on knot traces, finding exotic surfaces and traces.
problem Understanding exotic surfaces and traces through torus surgeries.
method Realizing annulus twisting as torus surgery, using key technical insight.
result Exotic elliptic surfaces and traces discovered, improving known geography.
Classifies knot traces with specific trisection genus limits.
problem Classifying knot traces with specific trisection genus limits.
method Classifying knot traces with specific trisection genus limits.
result Infinitely many knots have traces with trisection genus 3 and 4, and arbitrarily large trisection genus.
We reconsider the su(3) link homology theory defined by Khovanov in math.QA/0304375 and generalized by Mackaay and Vaz in math.GT/0603307. With some slight modifications, we describe the theory as a map from the planar algebra of tangles to a planar algebra of (complexes of) `cobordisms with seams' (actually, a `canopo…
Guillemin trace formula adapted for group actions.
problem Distributional trace for proper, cocompact group actions.
method Developing an equivariant version of the distributional trace.
result Equivariant Guillemin trace formula for group actions.
Paper derives trace formula for magnetic Laplacian at zero energy.
problem Trace formula for magnetic Laplacian at zero energy.
method Generalizes Gutzwiller trace formula, focuses on zero energy level.
result Derives trace formula at zero energy level.
In this paper, we obtain some new estimates for the trace and inverse trace of Steklov eigenvalues. The estimates generalize some previous results of Hersch-Payne-Schiffer , Brock}, Raulot-Savo and Dittmar.
New methods derive a generalized Frenkel trace formula for Lie groups.
problem Deriving a generalized Frenkel trace formula for Lie groups.
method Applying supersymmetric localization to quantum mechanical and gauged sigma models.
result Presented two complementary approaches for the derivation of the trace formula.
Examines a new type of analytic torsion on Riemannian manifolds.
problem Analyzing a new trace formula for Riemannian manifolds.
method Uses residue-trace instead of spectral zeta function quasi-trace.
result Defines and examines the residue analytic torsion.
Derives Selberg trace formula on Riemann surfaces and generalizes to other spaces.
problem Deriving and generalizing the Selberg trace formula.
method Supersymmetric localization principle and path integral derivation.
result Derives Selberg trace formula on arbitrary compact Riemann surfaces and generic compact locally symmetric spaces.
Introduces a new model for mapping matrices to matrices, subsuming linear regression.
problem Learning matrix-to-matrix mappings from data.
method Partial trace regression model, leveraging quantum information theory.
result Relevance demonstrated in matrix-to-matrix regression and positive semidefinite matrix completion.
CausalSim corrects bias in trace-driven simulations for more accurate results.
problem Bias in trace-driven simulations due to system conditions during trace collection.
method CausalSim learns a causal model of system dynamics and latent factors from an RCT to remove bias from trace data.
result CausalSim reduces simulation errors by 53% and 61% compared to baselines, providing more accurate insights.
New findings on knots and their traces, distinguishing L-space knots by their 0-trace.
problem Characterizing knots by their traces and Dehn surgeries.
method Combining Heegaard Floer homology with surface homeomorphisms and dynamics.
result The 0-trace detects every L-space knot, contrasting with characterizing surgeries.
This paper investigates the strength of the trace field as a commensurability invariant of hyperbolic 3-manifolds. We construct an infinite family of two-component hyperbolic link complements which are pairwise incommensurable and have the same trace field, and infinitely many 1-cusped finite volume hyperbolic 3-manifo…
We give axioms which characterize the local Reidemeister trace for orientable differentiable manifolds. The local Reidemeister trace in fixed point theory is already known, and we provide both uniqueness and existence results for the local Reidemeister trace in coincidence theory.
Clarifies a trace for Heisenberg operators on contact manifolds.
problem Calculating the index of Heisenberg elliptic operators on contact manifolds.
method Introduced a new trace on Heisenberg pseudodifferential operators and constructed a cocycle in periodic cyclic cohomology.
result Simplified the construction of the trace on Heisenberg pseudodifferential operators.
Researchers compute trace formula for magnetic Laplacian on hyperbolic surfaces.
problem Analyzing the magnetic Laplacian on compact hyperbolic surfaces.
method Computed the trace formula for magnetic Laplacian energies above the Mane critical level.
result Asymptotic behavior of trace formula coefficients near the Mane critical level.
Sharp inequality found on three-balls for fourth order Sobolev traces.
problem Fourth order Sobolev trace inequality on three-balls.
method Established through equivalence to a third order Sobolev inequality on two-spheres.
result Sharp fourth order Sobolev trace inequality on three-balls.
Infinite knots have non-integer trace values.
problem Finding knots with non-integer trace values.
method Proved existence of infinitely many non-homeomorphic hyperbolic knot complements with specific trace properties.
result Infinitely many non-homeomorphic hyperbolic knot complements with non-integer trace values.
Introduces expected eligibility traces for more efficient credit assignment in reinforcement learning.
problem Efficiently assigning credit to states and actions in reinforcement learning.
method Introduces expected eligibility traces, allowing updates to counterfactual sequences.
result Substantial improvements in temporal-difference learning can be achieved with expected traces.
We define a torsion invariant T for every balanced sutured manifold (M,g), and show that it agrees with the Euler characteristic of sutured Floer homology SFH. The invariant T is easily computed using Fox calculus. With the help of T, we prove that if (M,g) is complementary to a Seifert surface of an alternating knot, …
Quantum trace map defined for 3-manifolds with torus boundaries.
problem Quantifying topological structures of 3-manifolds with torus boundaries.
method Defining a quantum trace map from skein module to a quantum torus module.
result Established a 3D quantum trace map for 3-manifolds with torus boundaries.
New knots found that resist trace detection.
problem Knots not detected by any trace.
method Extended existing results to new knots.
result Knots exist that share traces with infinitely many distinct knots and are not detected by any trace.
We study new invariants of elliptic partial differential operators acting on sections of a vector bundle over a closed Riemannian manifold that we call the relativistic heat trace and the quantum heat traces. We obtain some reduction formulas expressing these new invariants in terms of some integral transforms of the u…
New hyperbolic manifolds found with same trace ring.
problem Finding non-commensurable hyperbolic manifolds with identical trace rings.
method Proved existence of infinitely many non-commensurable manifolds with same ambient group and trace ring.
result Infinitely many non-commensurable hyperbolic manifolds with the same ambient group and trace ring.
Sub-Riemannian Selberg trace formulae for compact quotients of SL(2, R)
problem Computing zeta-regularized determinants of sub-Laplacians
method Using Fourier decomposition and Selberg trace formulae
result Compact determinant formula expressed in terms of base hyperbolic surface and relative Selberg product
The abstract discusses detecting knotted spheres through their traces in high dimensions.
problem Detecting knotted spheres in high-dimensional spaces.
method Generalizing the RBG link construction to all dimensions and using surgery.
result Existence of non-isotopic smooth (n−2)-knots with diffeomorphic traces. Study categorifies link invariants using Soergel bimodules.
problem Categorification of link invariants.
method Explicit computation of derived traces of Soergel bimodules.
result Derived annular Khovanov-Rozansky link invariant.
The study computes trace fields and minimal polynomials for specific knots and links.
problem Computing trace fields and minimal polynomials for specific knots and links.
method Using factorization theorems for sparse polynomials.
result Results depend on the degrees of the trace fields over Q being sufficiently large.
Proposes GTTN for discovering all low-rank structures in deep multi-task learning.
problem Discovering all low-rank structures among tasks in deep multi-task models.
method Introduces GTTN, a convex combination of matrix trace norms of all tensor flattenings, to automatically determine the importance of components.
result Demonstrates the effectiveness of GTTN on real-world datasets.
A novel method for parallel transport and geodesics on submanifolds.
problem Understanding parallel transport and geodesics on submanifolds.
method Rolling tangent space to visualize and analyze parallel transport and geodesics.
result Conditions for parallel transport and geodesics are simplified and visualized in the tangent space.