Solves Tian's stabilization problem for toric Fano manifolds.
problem Tian's stabilization problem for equivariant global log canonical thresholds.
method Expressed complex singularity exponents in terms of support and gauge functions from convex geometry.
result First general result on Tian's problem.
Study proves stability of big bang singularity in complex system.
problem Stability of Kasner solutions in Einstein-Maxwell-scalar field-Vlasov system.
method Detailed mathematical structures and new delicate arguments.
result Nonlinear stability with Kasner exponents in full strong sub-critical regime.
Researchers find explicit solutions to complex Monge-Ampère equation.
problem Solving complex Monge-Ampère equation with constant right-hand side.
method Explicit pluripotential and viscosity solutions.
result Presented solutions lie in Wloc1,2∩Wloc2,1 and are not Dini continuous. Formula for harmonic current dimension on foliated surfaces, extending Brunella's inequality.
problem Calculating the dimension of harmonic currents on foliated complex surfaces.
method Proving a formula involving Furstenberg entropy and Lyapunov exponent.
result Hausdorff dimension of harmonic current is bounded and can be calculated precisely.
Researchers find Kähler-Einstein metrics near isolated log terminal singularities.
problem Existence of Kähler-Einstein metrics with positive curvature near isolated log terminal singularities.
method Solving complex Monge-Ampère equations to analyze the existence of metrics.
result Existence of smooth solutions in subcritical regimes, with critical exponent expressed in terms of normalized volume.
Study asymptotic behavior of second Chern forms on degenerating Kähler-Einstein surfaces.
problem Asymptotic behavior of second Chern forms on degenerating Kähler-Einstein surfaces with ADE singularities.
method Investigates a function on the unit disc defined by fiber integrals of the forms with a smooth test function, showing a lower bound of Hölder exponent at the origin for both cscK-metrics and Ricci-flat metrics.
result Shows bounds of Hölder exponent for both cscK-metrics and Ricci-flat metrics.
New groups found with critical exponents close to but less than max.
problem Finding discrete isometry groups with critical exponents near maximum.
method Analyzing complex hyperbolic spaces to construct groups.
result Discrete isometry groups with critical exponents arbitrarily close to max but less.
We give sharp C2,α estimates for solutions of some fully nonlinear elliptic and parabolic equations in complex geometry and almost complex geometry, assuming a bound on the Laplacian of the solution. We also prove the analogous results to complex Monge-Ampère equations with conical singularities. As an application…
We prove the existence of Veech groups having a critical exponent strictly greater than any elementary Fuchsian group (i.e. >21) but strictly smaller than any lattice (i.e. <1). More precisely, every affine covering of a primitive L-shaped Veech surface X ramified over the singularity and a non-periodic …
Wavelet analysis reveals limitations in detecting multifractality in signals with isolated singularities.
problem Detecting multifractality in signals with isolated singularities using detrended fluctuation analysis and wavelet leaders.
method Comparison of detrended fluctuation analysis and wavelet leaders on signals with isolated singularities.
result Signals with isolated singularities can artefactually give rise to broad multifractal spectra, leading to incorrect inference of multifractality.
We study the holonomy cocycle H of a holomorphic foliation \Fc by Riemann surfaces defined on a compact complex projective surface X satisfying the following two conditions: 1) its singularities E are all hyperbolic; 2) there is no holomorphic non-constant map \C\to X such that out of E the image of \C is locally conta…
Study on solutions to conformally invariant fourth order equations, classifying their properties.
problem Classify qualitative properties of solutions to conformally invariant fourth order equations.
method Analyze two cases: removable and non-removable singularities, using Pohozaev-type invariant.
result Non-existence of semi-singular solutions, classifying them as multiples of the Emden--Fowler solution.
In this note, we prove the existence of a closed geodesic of positive length on any compact developable orbifold of dimension 3, 5, or 7. The argument uses the stratification of the singular locus, and reduces the problem of existence of a closed geodesic on a compact developable orbifold to the case of even dimensiona…
The miltifractal properties and scaling behaviour of the exchange rate variations of the Iranian rial against the US dollar from a daily perspective is numerically investigated. For this purpose the multifractal detrended fluctuation analysis (MF-DFA) is used. Through multifractal analysis, the scaling exponents, gener…
Study the relationship between canonical polynomials and elliptic sequences for elliptic singularities.
problem Understanding the relationship between canonical polynomials and elliptic sequences for elliptic singularities.
method An inductive setup of elliptic germs and comparison of their canonical polynomials.
result The exponents of the canonical polynomial determine the elliptic sequence and vice versa under certain conditions.
In this article, we solve the strong openness conjecture on the multiplier ideal sheaves for the plurisubharmonic functions posed by Demailly. We prove two conjectures about the growth of the volumes of the sublevel sets of plurisubharmonic functions related to the complex singularity exponents and quasi-plurisubharmon…
Recent results on ergodic theory for Riemann surface laminations and foliations.
problem Ergodic theorems for laminations and foliations on Riemann surfaces.
method Leafwise Poincaré metric, directed positive harmonic currents, multiplicative cocycles, Lyapunov exponents.
result Definition and study of canonical Lyapunov exponents for singular holomorphic foliations.
The paper proves optimal smoothness for certain Lagrangian graphs with specific Hölder continuity.
problem Optimal regularity for Hölder continuous Hamiltonian stationary Lagrangian graphs.
method Establishing smoothness conditions based on Hölder exponent and Lagrangian phase properties.
result Smoothness of graphs is achieved when Hölder exponent is strictly greater than 1/3 and Lagrangian phase is supercritical.
P2P lending activities have grown rapidly and have caused the huge and complex networks of debtor-creditor relationships. The aim of this study was to study the underlying structural characteristics of networks formed by debtor-creditor relationships. According attributes of P2P lending, this paper model the networks o…
The paper proves rigidity for complex Kleinian groups.
problem Characterizing hyperconvex subgroups of complex Kleinian groups.
method Analyzing critical exponents and using representations of PSL(2, C).
result Uniform lattices in PSL(2, C) are the only (d-k)-hyperconvex subgroups with a specific critical exponent.
Study on virtual singular braid groups with algebraic properties and homomorphisms.
problem Algebraic properties and homomorphisms of virtual singular braid groups.
method Numerical invariants, homomorphisms, semi-direct product decompositions, presentations, and quotients.
result Determined all group homomorphisms from VSGn to Sn and obtained corresponding semi-direct product decompositions. Improved loss scaling for stochastic momentum algorithms in high dimensions.
problem Improving loss scaling for stochastic momentum algorithms in high dimensions.
method Dimension-adapted Nesterov acceleration (DANA) scales momentum hyperparameters based on model size and data complexity.
result DANA improves loss scaling exponents across various data and target complexities.
For n≥5, we consider positive solutions u of the biharmonic equation \[ Δ^2 u = u^\frac{n+4}{n-4} \qquad \text{on}\ \mathbb R^n \setminus \{0\} \] with a non-removable singularity at the origin. We show that ∣x∣2n−4u is a periodic function of ln∣x∣ and we classify all periodic functions obta…
We started from computer experiments with simple one-dimensional ergodic dynamical systems called interval exchange transformations. Correlators in these systems decay as a power of time. In the simplest non-trivial case the exponent is equal to 1/3. We found a formula connecting characteristic exponents with explicit …
Study on Δ invariants of plumbed 3-manifolds, comparing with Heegaard-Floer homology.
problem Analyzing Δ invariants of plumbed 3-manifolds. method Expressing Δ of Seifert manifolds in terms of singularity theory invariants, providing examples for non-Seifert manifolds, comparing with Heegaard-Floer homology. result Interesting behavior of Δ invariants for non-Seifert manifolds, comparison with Heegaard-Floer homology. Optimal batch size minimizes training time for neural networks.
problem Minimizing training time for two-layer neural networks with SGD.
method Characterized optimal batch size as a function of target hardness (information exponents). Used Correlation loss SGD to overcome limitations.
result Optimal batch size minimizes training time without changing total sample complexity.
Reverse Hölder inequalities on Fano metrics with applications to geodesics and singularities.
problem Establishing reverse Hölder inequalities on Kähler metrics of Fano varieties.
method Using log-concavity and properties of Ricci potentials, the inequality is proven for Fano manifolds with log terminal singularities.
result The inequality holds for Fano varieties with log terminal singularities and the constant depends only on p and the dimension of X.
Paper analyzes error exponent in agnostic PAC learning.
problem Analyzing performance of agnostic PAC learning.
method Using error exponent from Information Theory to analyze PAC learning.
result Improved distribution-dependent error exponent for agnostic learning.
We study several new invariants associated to a holomorphic projective structure on a Riemann surface of finite analytic type: the Lyapunov exponent of its holonomy which is of probabilistic/dynamical nature and was introduced in our previous work; the degree which measures the asymptotic covering rate of the developin…
A new flow connects manifold invariants with critical exponents.
problem Understanding invariants of non-positively curved manifolds.
method Constructing the natural flow and relating it to the critical exponent.
result Established connections between manifold invariants and critical exponents.
Consider a flat bundle over a complex curve. We prove a conjecture of Fei Yu that the sum of the top k Lyapunov exponents of the flat bundle is always greater or equal to the degree of any rank k holomorphic subbundle. We generalize the original context from Teichmueller curves to any local system over a curve with non…
New learning rate approach reveals phase transitions in SGD performance.
problem Understanding feature learning dynamics in neural networks.
method Characterizing the relationship between learning rate(s) and sample complexity for gradient-based algorithms.
result Phase transition from information exponent to generative exponent regime with different learning rates.
Paper relaxes symmetry conditions for universal feature selection in noisy data.
problem Feature selection in noisy data with weak symmetry.
method Developed a universal feature selection framework using singular value decomposition of canonical dependence matrix.
result Selected features achieve asymptotically optimal error exponents up to a residual term.
Study critical exponents for L^p-cohomology of higher rank Lie groups and manifolds.
problem Investigate critical exponents for vanishing L^p-cohomology in higher rank Lie groups and manifolds.
method Examine SL3(R) and 5-dimensional solvable Lie groups, use spectral sequence arguments. result Discover a continuum of quasi-isometry classes of rank 2 solvable Lie groups.
Neural network learns low-dimensional polynomials with SGD near information-theoretic limit.
problem Learning a single-index target function with gradient descent.
method Two-layer neural network optimized by SGD on squared loss.
result Sample and runtime complexity of n≃T=Θ(d⋅polylogd) for polynomial single-index models, matching information theoretic limit up to polylogarithmic factors. Study of spacelike singularities in spherical spacetimes with scalar matter.
problem Characterize spacelike singularities in spherically symmetric spacetimes with scalar matter.
method Analyzes the properties of spacelike singularities in spherically symmetric spacetimes with scalar matter, proving inverse polynomial blow-up rates and providing a BKL-type expansion.
result Provides a rigorous description of Kasner-like singularities in spherically symmetric gravitational collapse.
We present an explicit formula for the discrete power function introduced by Bobenko, which is expressed in terms of the hypergeometric τfunctions for the sixth Painlevé equation. The original definition of the discrete power function imposes strict conditions on the domain and the value of the exponent. However, we sh…
The aim of this paper is to prove the existence of weak solutions to the equation Δu+up=0 which are positive in a domain Ω⊂RN, vanish at the boundary, and have prescribed isolated singularities. The exponent p is required to lie in the interval (N/(N−2),(N+2)/(N−2)). We also prove the exist…
Keeping a basic tenet of economic theory, rational expectations, we model the nonlinear positive feedback between agents in the stock market as an interplay between nonlinearity and multiplicative noise. The derived hyperbolic stochastic finite-time singularity formula transforms a Gaussian white noise into a rich time…
This work improves online SGD's sample complexity for multi-index models by considering higher-order terms.
problem Suboptimal sample complexity for learning multi-index models using online SGD.
method Focus on both second- and higher-order terms to improve sample complexity.
result Online SGD achieves ildeO(dPL−1) samples for multi-index models. We prove a generalization of the fundamental inequality of Guivarc'h relating entropy, drift and critical exponent to Gibbs measures on geometrically finite quotients of CAT(-1) metric spaces. For random walks with finite superexponential moment, we show that the equality is achieved if and only if the Gibbs density is…
Motivated by the equation satisfied by the extremals of certain Hardy-Sobolev type inequalities, we show sharp Lq regularity for finite energy solutions of p-laplace equations involving critical exponents and possible singularity on a sub-space of Rn, which imply asymptotic behavior of the solutions at i…
Stock market comovements are examined using cointegration, Granger causality tests and nonlinear approaches in context of mutual information and correlations. Underlying data sets are affected by non-stationarities and trends, we also apply AMF-DFA and AMF-DXA. We find only 170 pair of Stock markets cointegrated, and a…
Study on sample complexity of policy gradient for stabilizing linear systems under multiplicative noise.
problem Learning optimal feedback gain for stabilizing linear systems with multiplicative noise.
method Analyzes the sample complexity of policy gradient methods, addressing the cusp obstruction and using symmetry to control divergent parts of the gradient.
result Proves that projected mini-batch policy gradient attains total sample complexity of O(1/η) when noise density is known and O(η^(-(2s+1)/(2s))) when estimated, for C^s noise densities with s ≥ 2.
We present new minimax results that concisely capture the relative benefits of source and target labeled data, under covariate-shift. Namely, we show that the benefits of target labels are controlled by a transfer-exponent γ that encodes how singular Q is locally w.r.t. P, and interestingly allows situations where tr…
Regularization is used to find a solution that both fits the data and is sufficiently smooth, and thereby is very effective for designing and refining learning algorithms. But the influence of its exponent remains poorly understood. In particular, it is unclear how the exponent of the reproducing kernel Hilbert space~(…
Study efficient estimation of hidden subspaces in Gaussian Multi-index models.
problem Estimating hidden subspaces in Gaussian Multi-index models with low-dimensional projections.
method Introduced the generative leap exponent and developed an agnostic sequential estimation procedure using spectral U-statistics.
result Achieved optimal sample complexity of $n=Θ(d^{1 \vee \k/2})$ for efficient estimation.
Gradient flossing stabilizes RNN training by controlling Lyapunov exponents.
problem Gradient instability in RNNs leading to exploding and vanishing gradients.
method Regularizing Lyapunov exponents through backpropagation using differentiable linear algebra.
result Gradient flossing improves RNN training success rate and convergence speed.