We find the degree of the Hilbert polynomial for HOMFLYPT homology.
problem Understanding the growth and structure of HOMFLYPT homology.
method Analyzing the degree of the Hilbert polynomial for closed braids.
result The degree of the Hilbert polynomial is l−1 for a braid with l components. The paper calculates the Hilbert polynomials for configuration spaces over graphs with a short circumference.
problem Calculating the Betti numbers of configuration spaces over graphs with a short circumference.
method Using a combinatorial approach based on the canonical 1-bridge decomposition of the graph.
result An expression for the Hilbert polynomial of a graph in terms of its canonical 1-bridge decomposition.
We prove that the metric balls of a Hilbert geometry admit a volume growth at least polynomial of degree their dimension. We also characterise the convex polytopes as those having exactly polynomial volume growth of degree their dimension.
Paper solves a central question about nonnegative polynomials related to isoparametric polynomials.
problem Whether a given nonnegative polynomial is a sum of squares of polynomials.
method Solves the problem completely for nonnegative polynomials associated with isoparametric polynomials.
result The paper provides a complete solution for the specific case of isoparametric polynomials.
Computes differential invariants for conformal metrics.
problem Local recognition of conformal structures.
method Computes Hilbert polynomial and Poincare function.
result Resolves the local recognition problem for conformal structures.
In this article we prove an upper bound for a Hilbert polynomial on quaternionic Kaehler manifolds of positive scalar curvature. As corollaries we obtain bounds on the quaternionic volume and the degree of the associated twistor space. Moreover the article contains some details on differential equations of finite type.…
This work learns kernels for structured prediction using polynomial transformations.
problem Learning effective kernel functions for structured prediction.
method Polynomial kernel transformations (Schoenberg transforms and Gegenbaur transforms) learned using HSIC and matrix decomposition.
result State-of-the-art results on real-world datasets.
Acceleration in Hilbert spaces reduces computations but not accuracy.
problem Improving learning accuracy with fewer computations.
method Analysis of Nesterov acceleration and heavy-ball methods in Hilbert spaces.
result Acceleration can reduce computations but not improve accuracy with respect to gradient descent.
We solve for functions from their truncated Hilbert transforms using Chebyshev series.
problem Finding functions from their truncated Hilbert transforms.
method Express functions in Chebyshev series and numerically estimate coefficients.
result Numerical methods work well for extrapolating functions from truncated Hilbert transforms.
The intersection of a complex plane curve with a small three-sphere surrounding one of its singularities is a non-trivial link. The refined punctual Hilbert schemes of the singularity parameterize subschemes supported at the singular point of fixed length and whose defining ideals have a fixed number of generators. We …
New representation of curves helps prove complex geometry result.
problem Understanding cohomologically stable curves in projective space.
method Using commuting matrix polynomials to represent curves and show isomorphism to hyperkähler quotient.
result Hilbert scheme isomorphic to a hyperkähler quotient.
Researchers compute and describe differential invariants for self-dual conformal structures.
problem Local recognition problem for self-dual conformal structures.
method Quotient of self-duality equation, Hilbert polynomial, differential invariants, Poincaré function.
result Resolved the local recognition problem for self-dual conformal structures.
Study on gradient descent in Hilbert spaces with Markov chains, focusing on mixing coefficients.
problem Analyzing convergence of gradient descent in Hilbert spaces with stationary Markov chains.
method Examined strictly stationary Markov chains with φ- and β-mixing coefficients, derived probabilistic upper bounds. result Probabilistic upper bounds on convergence behavior of gradient descent algorithm based on mixing coefficients.
We conjecture an expression for the dimensions of the Khovanov-Rozansky HOMFLY homology groups of the link of a plane curve singularity in terms of the weight polynomials of Hilbert schemes of points scheme-theoretically supported on the singularity. The conjecture specializes to our previous conjecture relating the HO…
Seidel and Smith have constructed an invariant of links as the Floer cohomology for two Lagrangians inside a complex affine variety Y. This variety is the intersection of a semisimple orbit with a transverse slice at a nilpotent in the Lie algebra sl2m. We exhibit bijections between a set of generators for the Sei…
A method for interpreting SVMs using polynomial kernels, revealing model complexity.
problem Interpreting SVMs built with truncated orthogonal polynomial kernels.
method Orthogonal Representation Contribution Analysis (ORCA) with normalized Orthogonal Kernel Contribution (OKC) indices.
result The method reveals structural aspects of model complexity not captured by predictive accuracy.
Study analyzes learnability of RKHS under L∞ norm for kernel methods.
problem Understand performance of kernel methods and random feature models.
method Relate L∞ learnability to kernel spectrum decay and establish sample complexity bounds.
result Conditions for efficient L∞ learning of RKHS identified.
A representation of the Jacobi algebra h1⋊su(1,1) by first order differential operators with polynomial coefficients on the manifold C×D1 is presented. The Hilbert space of holomorphic functions on which the holomorphic first order differential operators with …
HR in 8D encodes unique conformal gravity with negative curvature.
problem Holographic Renormalisation in 8D Einstein Gravity.
method Relating HR to Topological Regularisation and adding the Euler term.
result The unique conformal gravity theory reproduces the polynomial and cancels divergent terms.
Develops resolvent degree theory for algebraic geometry problems.
problem Hilbert's 13th Problem and related conjectures.
method Extends Brauer's resolvent degree theory to algebraic geometry.
result Hilbert's 13th Problem and related conjectures are equivalent to enumerative geometry problems.
Given a planar curve singularity, we prove a conjecture of Oblomkov-Shende, relating the geometry of its Hilbert scheme of points to the HOMFLY polynomial of the associated algebraic link. More generally, we prove an extension of this conjecture, due to Diaconescu-Hua-Soibelman, relating stable pair invariants on the c…
We emphasize some properties of coherent state groups, i.e. groups whose quotient with the stationary groups, are manifolds which admit a holomorphic embedding in a projective Hilbert space. We determine the differential action of the generators of the representation of coherent state groups on the symmetric Fock space…
The paper defines functions that induce bounded composition operators on RKHSs with analytic positive definite functions.
problem Characterizing functions that induce bounded composition operators on RKHSs.
method Intrinsic properties of RKHSs and asymptotic properties of orthogonal polynomials.
result Only affine transforms can induce bounded composition operators in a large class of RKHSs.
Polynomial invariant of quandles counts random link colorings.
problem Counting Q-colorings of random braids in quandles. method Average number of Q-colorings for large n. result The average number of Q-colorings coincides with a polynomial PQ. This paper provides mathematical foundations for regression methods used in forward initial margin approximation.
problem Developing robust methods for approximating forward initial margin.
method Introduces mathematical rigor to show that regression methods are variations of approximating the conditional expectation function.
result Each regression method is a numerical estimation of the conditional expectation with a different functional form.
Groups with hyperbolic properties don't have strong Property (T).
problem Proving groups with hyperbolic properties don't have strong Property (T).
method Constructing an unbounded affine representation with polynomial growth.
result Groups with hyperbolic properties do not have strong Property (T).
Spectral filters enhance option pricing methods using Hilbert transforms.
problem Improving convergence rates of option pricing methods.
method Using spectral filters to improve convergence of numerical schemes based on discrete Hilbert transforms.
result Improved convergence rates, especially polynomial convergence, achieved with spectral filtering.
In this paper we give a quantum statistical interpretation for the bracket polynomial state sum <K> and for the Jones polynomial. We use this quantum mechanical interpretation to give a new quantum algorithm for computing the Jones polynomial. This algorithm is useful for its conceptual simplicity, and it applies to al…
Paper analyzes online learning without regularization, proving strong convergence.
problem Online learning without explicit regularization terms.
method Stochastic gradient descent in RKHS with polynomially decaying step sizes.
result Strong convergence of the last iterate in RKHS norm with polynomial step sizes.
The paper extends invariant theory to non-compact and non-reductive actions, classifying four regimes.
problem Extending invariant theory to non-compact and non-reductive actions.
method Examined two specific settings: discrete subgroups of Lorentz group acting on Rn,1 and cocompact actions on smooth manifolds. result Classification of invariant-theoretic regimes into four categories, identifying boundaries of Hilbert--Weyl and Schwarz theorems.
This paper formulates a generalization of our work on quantum knots to explain how to make quantum versions of algebraic, combinatorial and topological structures. We include a description of previous work on the construction of Hilbert spaces from the states of the bracket polynomial with applications to algorithms fo…
The paper constructs instanton complexes on stratified pseudomanifolds.
problem Analyzing functions with non-isolated critical points on singular spaces.
method Constructing Witten instanton complexes and Hilbert complexes.
result Proves Morse inequalities for stratified pseudomanifolds.
Ridge regression performs optimally in noisy environments with heavy-tailed distributions.
problem Performance of ridge regression in noisy environments with heavy-tailed noise.
method Established excess risk bounds using integral operator framework and Fuk-Nagaev inequality.
result Ridge regression achieves optimal convergence rates under heavy-tailed noise, demonstrating robustness.
We present a holomorphic representation of the Jacobi algebra hn⋊sp(n,R) by first order differential operators with polynomial coefficients on the manifold Cn×Dn. We construct the Hilbert space of holomorphic functions on which these differential operators a…
Develops analytic methods for Lefschetz and Morse theories on stratified pseudomanifolds.
problem Analytic framework for Lefschetz and Morse theories on stratified pseudomanifolds.
method Heat kernel and Witten deformation based techniques for global and local Lefschetz numbers and Morse polynomials.
result Formulas for Lefschetz numbers and Morse polynomials as supertraces over cohomology groups of Hilbert complexes.
New kernel improves graph learning with fewer labeled data.
problem Limited kernels for node-level problems on graphs.
method Derived from a regularization framework, transductive kernel for graphs with node features.
result Improved learning on fewer training points and non-Euclidean data.
We prove that the ring $\Aff{\R}{M}$ of all polynomials defined on a real algebraic variety M⊂Rn is dense in the Hilbert space $L^2(M,e^{-|x|^2}\deμ)$, where $\deμ$ denotes the volume form of M and $\deν=e^{-|x|^2}\deμ$ the Gaussian measure on M.
Quantum BPS invariants linked to combinatorics of Lyndon words.
problem Relating quantum BPS invariants to combinatorics on words.
method Constructing combinatorial models and using difference equations.
result BPS invariants expressed in terms of Lyndon words.
Stochastic gradient descent achieves polynomial convergence rates for noiseless linear models.
problem Convergence analysis of stochastic gradient descent in noiseless linear models.
method Fixed step-size stochastic gradient descent on least-square risk.
result Polynomial convergence rates depend on the regularities of the optimum and feature vectors.
Paper develops an online learning algorithm for functional data models.
problem Recovering slope functions or predictors in functional data models.
method Online regularized learning algorithm in reproducing kernel Hilbert spaces with polynomially decaying step-size.
result Established fast convergence rates for estimation error without capacity assumption.
We compute the Heegaard-Floer link homology of algebraic links in terms of the multivariate Hilbert function of the corresponding plane curve singularities. The main result of the paper identifies four homologies: (a) the Heegaard-Floer link homology of the local embedded link of the germ, (b) the lattice homology asso…
Extends neighborhood regression to algebraic structures for encoding conditional independence.
problem Encoding conditional independence statements in Gaussian distributions.
method Defining a neighborhood lattice based on generalized neighborhood regression.
result Algebraic structure provides an economic encoding of all conditional independence statements.
The paper analyzes deep neural networks' expressivity and training, revealing critical expressivity issues.
problem Critical expressivity issues in deep neural networks.
method Quantitative analysis using Hilbert space and Hermite polynomials for feature mapping and activation function design.
result Deep neural networks evolve to the edge of chaos, but expressivity depends on overcoming convergence.
This paper is an introduction to Khovanov homology, starting with the Kauffman bracket state summation, emphasizing the Bar-Natan Canopoloy and tangle cobordism approach. The paper discusses a simplicial approach to Khovanov homology and a quantum model for it so that the graded Euler characteristic that produces the J…
The paper develops SGD for estimating operators from data.
problem Estimating operators from data in infinite-dimensional spaces.
method Regularized SGD with operator-valued kernels.
result Near-optimal convergence rates for prediction and estimation.
Abstract: Review and properties of Gieseker stability for Higgs sheaves.
problem Defining and studying Gieseker stability for Higgs sheaves.
method Review and prove properties similar to classical Gieseker stability for coherent sheaves.
result Properties of Gieseker stability for Higgs sheaves extend to Higgs case, including direct sums and morphisms.
In the framework of finite order variational sequences a new class of Lagrangians arises, namely, \emph{special} Lagrangians. These Lagrangians are the horizontalization of forms on a jet space of lower order. We describe their properties together with properties of related objects, such as Poincaré--Cartan and Euler--…
New method finds global minima using function evaluations and kernel approximations.
problem Finding global minima of smooth functions with limited evaluations.
method Approximates the function using infinite sums of square smooth functions and solves the optimization problem with polynomial time complexity.
result Achieves optimal number of function evaluations with theoretical guarantees and nearly optimal convergence rate.