The paper shows conditions for vanishing of certain topological invariants on specific types of manifolds.
problem Conditions for vanishing of topological invariants on compact manifolds.
method Refining Farrell's idea, using Jiang subgroup and non-positive curvature conditions.
result The χy-genus of a non-positively curved compact Kähler manifold vanishes under certain conditions. The concordance group of algebraically slice knots is the subgroup of the classical knot concordance group formed by algebraically slice knots. Results of Casson and Gordon and of Jiang showed that this group contains in infinitely generated free (abelian) subgroup. Here it is shown that the concordance group of algebr…
The paper classifies fixed subgroups in a specific group product.
problem Classifying fixed subgroups in a specific group product.
method Complete classification through direct product analysis.
result Infinitely many fixed subgroups if and only if k ≥ 2.
This paper studies fixed points of graph selfmaps and iwip endomorphisms of free groups.
problem Fixed points and fixed subgroups of selfmaps on graphs and surfaces.
method Analyzes iwip outer endomorphisms of free groups acting on stable trees.
result A sufficient condition for equality in a bound on a fixed point quantity is found.
Study controls curvature in Ricci flows using necks.
problem Controlling curvature in Ricci flows.
method Introducing necks of maximal symmetry and decomposing curvature into uniform bounds.
result Established L1-bounds on Riemann curvature tensor. Clarifies the theory of the deconfounder by Imai and Jiang.
problem Theoretical requirements for the deconfounder algorithm.
method Clarifies the assumption of 'no unobserved single-cause confounders' using empirical studies.
result Imai and Jiang's clarification of the assumption does not hold for counterexamples proposed by Ogburn et al. (2020).
Paper extends BIP to nilmanifold products and characterizes fixed points.
problem Lack of Bounded Index Property for fixed points in aspherical manifolds.
method Extended BIP to iterates and proved BIP_k for nilmanifold products.
result Proved BIP_k for certain nilmanifold products.
3D Ricci flows have bounded diameter before Type I singularities.
problem Bounding the diameter of 3D Ricci flows before Type I singularities.
method Introduced a neck-region concept and proved packing measure Ahlfors regularity.
result Uniformly bounded diameter up to Type I singular time.
Study shows some complex shapes don't fit a certain property.
problem Some aspherical manifolds lack the Bounded Index Property (BIP).
method Analyzes specific aspherical manifolds to prove they don't have BIP.
result Proves existence of aspherical manifolds without BIP.
We show that the infinite cyclic cover of the exterior of the untwisted Whitehead double of a non-trivial knot does not embed in any compact 3-manifold, answering a question of Jiang, Ni, Wang and Zhou.
Unified proof of smooth fibration theorems for collapsed manifolds.
problem Smooth fibration theorems for collapsed manifolds with Ricci curvature bounded below.
method Generalized Reifenberg condition and transformation technique for almost splitting maps.
result Unified proof of smooth fibration theorems in many previous works.
We provide additional statistical background for the methodology developed in the clinical analysis of knee osteoarthritis in "A Precision Medicine Approach to Develop and Internally Validate Optimal Exercise and Weight Loss Treatments for Overweight and Obese Adults with Knee Osteoarthritis" (Jiang et al. 2020). Jiang…
A relation between the dilatation of pseudo-Anosov braids and fixed point theory was studied by Ivanov. In this paper we reveal a new relationship between the above two subjects by showing a formula for the dilatation of pseudo-Anosov braids by means of the representations of braid groups due to B. Jiang and H. Zheng.
Unified routing and arbitrage with concave continuation.
problem Combining routing and arbitrage in financial markets.
method Extending AMM trade functions to negative inputs via concave continuation.
result Unified approach unifies routing and arbitrage.
Global regularity proved for 4D Ricci flow with scalar curvature integral bound.
problem Global regularity of 4D Ricci flow with integral scalar curvature bound.
method Extended Ge-Jiang's result to include integral bound on scalar curvature.
result Global ε-regularity for 4D Ricci flow with integral scalar curvature bound. Study shows offline RL under Q⋆-approximation and partial coverage is harder than previously thought.
problem Theoretical limits of offline reinforcement learning under Q⋆-approximation and partial coverage. method Introduced a decision-estimation framework to decompose offline RL complexity into decision and value estimation errors.
result Answered the open question by proving sample inefficiency under partial coverage is not guaranteed by Q⋆-realizability and Bellman completeness. Survey on biharmonic Riemannian submersions, a dual concept of biharmonic submanifolds.
problem Understanding biharmonic Riemannian submersions as a dual concept.
method Short survey focusing on biharmonic Riemannian submersions.
result Survey provides insights into the dual concept of biharmonic submanifolds.
New criterion for branched covers between 2-spheres.
problem Existence of specific types of branched covers between 2-spheres.
method Combining complex analysis and topology techniques.
result Established a complete criterion for the existence of branched covers.
Poisson actions of Poisson Lie groups have an interesting and rich geometric structure. We will generalize some of this structure to Dirac actions of Dirac Lie groups. Among other things, we extend a result of Jiang-Hua-Lu, which states that the cotangent Lie algebroid and the action algebroid for a Poisson action form…
For the product S1×S2 of any two connected compact hyperbolic surfaces S1 and S2, we give a finite bound B such that for any self-homeomorphism f of S1×S2 and any fixed point class F of f, the index ∣ind(f,F)∣≤B, which is an affirmative answer for a special c…
Jiang et al. (2020) found no uniformly tight generalization bounds for neural networks in the overparameterized setting.
problem Finding uniformly tight generalization bounds for neural networks in the overparameterized setting.
method Examined more than a dozen generalization bounds, proving that no bounds can be uniformly tight in the overparameterized setting.
result No generalization bounds can be uniformly tight in the overparameterized setting.
An important theorem about biharmonic submanifolds proved independently by Chen-Ishikawa [CI] and Jiang [Ji] states that an isometric immersion of a surface into 3-dimensional Euclidean space is biharmonic if and only if it is harmonic (i.e, minimal). In a later paper [CMO2], Cadeo-Monttaldo-Oniciuc shown that the theo…
New insights on offline RL with state aggregation and trajectory data.
problem Understanding sample complexity in offline policy evaluation.
method Analyzing concentrability coefficient in aggregated Markov Transition Model.
result Sample complexity depends on concentrability coefficient in aggregated model.
The paper discusses convergence of Bergman kernels on complex manifolds.
problem Convergence of Bergman kernels on complex manifolds with given conditions.
method Analysis of Gromov-Hausdorff limits, Hermitian line bundles, and positive currents.
result Uniform asymptotic expansion and estimates of Bergman kernels.
Improved SGD methods converge faster for nonconvex optimization.
problem Nonconvex optimization challenges in machine learning.
method Adaptive SGD with line-search and Polyak stepsizes.
result Unified convergence rates for various nonconvex functions.
A compact polyhedron X is said to have the Bounded Index Property for Homotopy Equivalence (BIPHE) if there is a finite bound B such that for any homotopy equivalence f:X→X and any fixed point class F of f, the index ∣ind(f,F)∣≤B. In this note, w…
In this paper, we prove sharp gradient estimates for a positive solution to the heat equation ut=Δu+aulogu in complete noncompact Riemannian manifolds. As its application, we show that if u is a positive solution of the equation ut=Δu and logu is of sublinear growth in both spatial and time directions the…
There are certain families of words and word sequences (words in the generators of a two-generator group) that arise frequently in the Teichm{ü}ller theory of hyperbolic three-manifolds and Kleinian and Fuchsian groups and in the discreteness problem for two generator matrix groups. We survey some of the families of su…
Study strong Nielsen equivalence on punctured disc homeomorphisms.
problem Characterize strong Nielsen equivalence of periodic points in punctured disc homeomorphisms.
method Braid theory and trace formula application.
result Necessary and sufficient conditions for strong Nielsen equivalence of periodic points.
Can we use deep learning to predict when deep learning works? Our results suggest the affirmative. We created a dataset by training 13,500 neural networks with different architectures, on different variations of spiral datasets, and using different optimization parameters. We used this dataset to train task-independent…
Study short-time existence of Ricci-DeTurck flow from rough metrics with Morrey-type integrability.
problem Short-time existence of Ricci-DeTurck flow from rough metrics with specific integrability condition.
method Rough existence theory, preservation and improvement of scalar curvature bounds.
result Preservation and improvement of distributional scalar curvature lower bounds under certain conditions.
In this paper, we obtain analogues of Jorgensen's inequality for non-elementary groups of isometries of quaternionic hyperbolic n-space generated by two elements, one of which is loxodromic. Our result gives some improvement over earlier results of Kim [10] and Markham [15]}. These results also apply to complex hyper…
We show that on-policy policy gradient (PG) and its variance reduction variants can be derived by taking finite difference of function evaluations supplied by estimators from the importance sampling (IS) family for off-policy evaluation (OPE). Starting from the doubly robust (DR) estimator (Jiang & Li, 2016), we provid…
Paper confirms Chen's biharmonic conjecture for hypersurfaces in 5D.
problem Chen's conjecture on biharmonic submanifolds in Euclidean spaces.
method Analyzes hypersurfaces in \(\mathbb{R}^5\) for \(n=4\).
result Chen's conjecture is confirmed for hypersurfaces in \(\mathbb{R}^5\) when \(n=4\).
Direct proof shows adaptive gradient descent converges near-linearly for convex functions.
problem Proving near-linear convergence of adaptive gradient descent for convex functions.
method Direct Lyapunov-based argument for convex functions with unique minimizer.
result Direct proof of near-linear convergence for convex functions.
Graphs with maximum degree Δ have at most O(1) equiangular lines for λ < 3/sqrt(2).
problem Finding the maximum number of equiangular lines in graphs with a given maximum degree.
method Using eigenfunctions and nodal domains to estimate the multiplicity of eigenvalues.
result The maximum multiplicity of λ as the second largest eigenvalue is O(1) for graphs with maximum degree Δ and cyclomatic number.
Breaks the hardness conjecture for batch RL with a novel tournament-based approach.
problem Sample-efficient reinforcement learning from exploratory data.
method BVFT algorithm using pairwise comparison and state-action partition.
result Solves the learning problem in a setting previously thought impossible.
New methods resolve conflicting treatment effect estimates in health tech assessments.
problem Conflicting conclusions from different sponsors analyzing the same data.
method Arbitrated indirect treatment comparisons (ArMAIC) targeting a common target population.
result Estimates treatment effects in a common target population, resolving the MAIC paradox.
Given a closed symplectic 4-manifold (X,ω), we define a twisted version of the Gromov-Taubes invariants for (X,ω), where the twisting coefficients are induced by the choice of a surface bundle over X. Given a fibered 3-manifold Y, we similarly construct twisted Lefschetz zeta functions associated with surface b…
In this paper, we compute the subgroup distortion of all finitely generated subgroups of all finitely generated 3-manifold groups, and the subgroup distortion in this case can only be linear, quadratic, exponential and double exponential. It turns out that the subgroup distortion of a subgroup of a 3-manifold group is …
Given a standard complex semisimple Poisson Lie group (G,πst), generalised double Bruhat cells Gu,v and generalised Bruhat cells Ou equipped with naturally defined holomorphic Poisson structures, where u, v are finite sequences of Weyl group elements, were defined and studied by Jiang Hua Lu and the auth…
Regular subgroups of SL3(R) are identified and ruled out.
problem Identifying and characterizing regular subgroups of SL3(R).
method Using Kapovich–Leeb–Porti and Guichard–Wienhard divergent subgroups criteria, and Oh's results.
result Regular subgroups of SL3(R) are precisely lattices in minimal horospherical subgroups.
Study on braid group quotients by congruence subgroups.
problem Understanding the image of congruence subgroups in GL(n,Z).
method Characterization through symplectic congruence subgroups.
result Open problem solved: image of congruence subgroups in GL(n,Z).
The paper explores geometric finiteness in mapping class groups and constructs new examples of these subgroups.
problem Understanding geometric finiteness in mapping class groups and constructing new examples.
method Examined several constructions of subgroups and determined conditions for geometric finiteness.
result Provides new examples of parabolically geometrically finite and reducibly geometrically finite subgroups.
Proposes a new method for finding non-redundant, standout subgroups in numeric datasets.
problem Mining large numbers of redundant subgroups in numeric datasets.
method Dispersion-aware problem formulation based on MDL principle for subgroup set discovery.
result Empirically demonstrates SSD++ returns outstanding subgroup lists.
Proves Congruence Subgroup Property for two types of groups.
problem Proving Congruence Subgroup Property for specific groups.
method Elementary proof of Johnson filtration and geometric subsurface inclusions.
result Proves Congruence Subgroup Property for nilpotent quotients and subsurface subgroups.
New method constructs non-quasiconvex subgroups in hyperbolic groups.
problem Creating non-quasiconvex subgroups in hyperbolic groups.
method Using Stallings-like techniques on right-angled Coxeter groups (RACGs).
result Explicit examples of non-quasiconvex subgroups constructed.
Characterizes knotted subgroups of Lie groups and provides examples.
problem Defining and understanding knotted subgroups of Lie groups.
method Geometric equivalence, one-parameter subgroups, infinitesimal elements, canonical forms, spectrum analysis.
result Completely classified knotted subgroups of SL(2,R) and SL(3,R).