According to seminal work of Kontsevich, the unstable homology of the mapping class group of a surface can be computed via the homology of a certain lie algebra. In a recent paper, S. Morita analyzed the abelianization of this lie algebra, thereby constructing a series of candidates for unstable classes in the homology…
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Study of CB generating sets for infinite-type surfaces.
problem Understanding CB generating sets for infinite-type surfaces.
method Constructing CB generating sets for specific infinite-type surfaces.
result Examples of surfaces with and without CB generating sets.
Automorphisms of fine curve graph match surface homeomorphisms.
problem Understanding automorphisms of curve graphs for surfaces.
method Building on previous work, proving isomorphism to surface homeomorphisms.
result The group of automorphisms of the fine curve graph is isomorphic to the extended mapping class group of the surface.
New Orlicz Brunn-Minkowski inequalities are established for rigid motion compatible Minkowski valuations of arbitrary degree. These extend classical log-concavity properties of intrinsic volumes and generalize seminal results of Lutwak and others. Two different approaches which refine previously employed techniques are…
Sacks-Uhlenbeck's result on metric spaces expanded.
problem Existence of non-trivial harmonic 2-spheres in metric spaces.
method Developed a metric approach to generalize Sacks-Uhlenbeck's result.
result Generalized Sacks-Uhlenbeck's result to a broader class of compact metric spaces.
Study shows infinite dimensional zero norm subspace in bounded cohomology of acylindrically hyperbolic groups.
problem Understanding the zero norm subspace in bounded cohomology of acylindrically hyperbolic groups.
method Introduced combinatorial volume forms and a new seminorm on exact bounded cohomology to construct non-trivial classes.
result Shows infinite dimensional zero norm subspace in degree 3 bounded cohomology of acylindrically hyperbolic groups.
We survey developments arising from Milnor's 1958 paper, "On the existence of a connection with curvature zero" and his 1977 paper, "On fundamental groups of complete affinely flat manifolds".
New normal subgroups found in mapping class groups.
problem Finding normal subgroups in mapping class groups.
method New windmill apparatus tailored to projection complexes.
result Normal subgroups isomorphic to non-free right-angled Artin groups.
The paper honors Lai's contributions to multi-armed bandits and establishes new regret bounds.
problem Improving regret bounds in multi-armed bandit problems.
method Establishes non-asymptotic regret bounds for upper confidence bound indices.
result New regret bounds match Lai-Robbins lower bound.
This paper proves a converse to Thurston's 1976 observation for taut foliations on hyperbolic 3-manifolds.
problem Proving a converse to Thurston's 1976 observation for taut foliations on hyperbolic 3-manifolds.
method Analyzing the Euler class and using hyperbolic geometry properties.
result For a taut foliation on a hyperbolic 3-manifold, if the Euler characteristic of a closed leaf equals the Euler class, then there exists another taut foliation with the same Euler class.
The paper extends submanifold reach to less regular classes.
problem Extending submanifold reach to less regular classes.
method Using -reach and regularity to quantify reach.
result Intermediate regularities induce quantitative results on reach.
Survey of recent progress in Sasakian geometry.
problem Topological properties and existence of Sasakian structures.
method Review of recent research findings.
result Recent progress in Sasakian geometry.
The paper "Minimal unit vector fields" by O. Gil-Medrano and E. Llinares-Fuster \cite{GilLli1}. is a seminal paper in the field that has been cited by many authors. It contains, however, a minor technical mistake in Theorem 14 that is important to fix. In this short note, we will provide a correction to that result.
We develop theory and applications of forward characteristic processes in discrete time following a seminal paper of Jan Kallsen and Paul Krühner. Particular emphasis is placed on the dynamics of volatility surfaces which can be easily formulated and implemented from the chosen discrete point of view. In mathematical t…
New metrics found with specific curvature properties on 4D manifolds.
problem Constructing metrics with specific curvature properties on closed manifolds.
method Using Aubin's deformation method to find metrics with pinched Bach tensor and scalar curvature.
result Existence of metrics with Bach tensor pinched by scalar curvature on 4D manifolds.
This paper solves a problem in 3D geometry by defining a canonical partition for certain manifolds.
problem Building an explicit canonical decomposition for orientable 3-manifolds defined by vector-colourings of 3-polytopes.
method Analysis of results from previous studies on similar problems.
result A complete answer to the problem of decomposing orientable 3-manifolds defined by vector-colourings of 3-polytopes.
Develops a model for causal discovery in path spaces.
problem Discover causal relationships in path spaces using asymmetric independence.
method Theory linking E-separation in DMGs to conditional independence in SDEs, proving global Markov property, characterizing equivalence classes of graphs.
result Each equivalence class of graphs has a greatest element as a parsimonious representation, which can be identified from data.
A new method shrinks a complex structure without much change.
problem Understanding the geometry of Bing's wild involution.
method Producing a counterintuitive construction to shrink the Bing decomposition without much change.
result A method to shrink a complex structure (Bing's decomposition) without much change.
In a remark in his seminal 1987 paper, Jones describes a way to define the Burau matrix of a positive braid using a metaphor of bowling a ball down a bowling alley with braided lanes. We extend this definition to allow multiple bowling balls to be bowled simultaneously. We obtain the Iwahori-Hecke algebra and a cabled …
Reduces energy for 4D submanifolds in R^n.
problem Energy reduction for 4D submanifolds in R^n.
method Connected sum energy reduction for fourth-order Willmore energy.
result Established a connected sum energy reduction for the fourth-order Willmore energy.
New method optimizes hierarchical multi-label classification results.
problem Optimizing classification results respecting class hierarchy and classifier scores.
method Introducing CATCH objective function and mLPR metric to rank multi-label classification results.
result HierRank algorithm optimizes CATCH, improving decision accuracy.
In this paper we settle Thurston's old question of whether the Weber-Seifert dodecahedral space is non-Haken, a problem that has been a benchmark for progress in computational 3-manifold topology over recent decades. We resolve this question by combining recent significant advances in normal surface enumeration, new he…
Study proves global existence and decay for complex wave equations.
problem Global existence and decay for quasilinear wave equations with weak-null condition.
method Novel decoupling of higher order energy estimates, focusing on tangential components.
result Established global existence and decay for solutions with small data.
Note on advancements in nonlinear elliptic equations' regularity theory.
problem Nonlinear elliptic equations and their regularity.
method De Giorgi-Nash-Moser theory, Krylov-Safonov theory, Evans-Safonov theory.
result Contributions to Hilbert's 19th problem and fully nonlinear equations.
Discusses the tight versus overtwisted dichotomy in 3D contact geometry.
problem Understanding the tight versus overtwisted dichotomy in 3D contact geometry.
method Reviews Eliashberg's seminal work and contributions to the theory.
result Explains the genesis and importance of the tight versus overtwisted dichotomy.
Experimental evidence for curve ratios on genus two surfaces.
problem Determining the ratio of topological curve types on surfaces.
method Experimental statistics applied to Mirzakhani's genus two surface results.
result Separating and non-separating curves occur in the ratio 1:48.
We extend Bayes' theorem for upper probabilities considering likelihood uncertainty.
problem Addressing uncertainty in likelihood for upper probability bounds.
method Generalization of Wasserman and Kadane's result, considering both prior and likelihood uncertainty.
result A sufficient condition for the upper bound to become an equality.
This paper removes the finite variance assumption for deep convolutional neural networks.
problem Removing the finite variance assumption for deep convolutional neural networks.
method Assuming iid parameters distributed according to a stable distribution, the paper shows that the infinite-channel limit of a deep feed-forward convolutional neural network is a multivariate stable stochastic process.
result The infinite-channel limit of a deep feed-forward convolutional neural network, under suitable scaling, is a multivariate stable stochastic process.
Explain Arnold's proof of the Morse index theorem using Maslov index.
problem Proving the Morse index theorem in Riemannian geometry.
method Using symplectic arguments and the Maslov index.
result Self-contained exposition of Arnold's proof.
Study improves ERM for heavy-tailed data with dependent inputs.
problem Empirical Risk Minimization with dependent and heavy-tailed data.
method Extending risk bounds for ERM with heavy-tailed, dependent data.
result Established risk bounds for ERM with dependent and heavy-tailed data.
The "bag-of-frames" approach (BOF), which encodes audio signals as the long-term statistical distribution of short-term spectral features, is commonly regarded as an effective and sufficient way to represent environmental sound recordings (soundscapes) since its introduction in an influential 2007 article. The present …
Improved kernel ridge regression for large datasets using weighted random binning.
problem Efficiently approximating kernel matrices for large-scale datasets.
method Introduced weighted random binning features for locality sensitive hashing.
result Weighted random binning features generate Gaussian processes of any desired smoothness.
The paper introduces elliptic quasi-modular forms via moduli spaces.
problem Developing a theory of elliptic quasi-modular forms.
method Using moduli spaces and the Gauss-Manin connection.
result Presented a succinct theory of elliptic quasi-modular forms.
We prove some ergodic-theoretic rigidity properties of the action of SL(2,R) on moduli space. In particular, we show that any ergodic measure invariant under the action of the upper triangular subgroup of SL(2,R) is supported on an invariant affine submanifold. The main theorems are inspired by the results of several a…
Corrected an error in Caponnetto and de Vito's 2007 minimax rates proof.
problem Error in bound on effective dimensionality for kernel ridge regression.
method Explained and corrected the proof of minimax rates.
result Main theorem remains true with corrected bound.
Unified binary and multiclass margin-based classification methods.
problem No consensus on multiclass loss functions analogous to binary margin loss.
method Showed multiclass loss functions can be expressed in relative margin form.
result Extended classification-calibration result to multiclass.
New algorithm reduces dynamic regret without prior function change knowledge.
problem Non-stationary stochastic optimization with bandit feedback.
method Fixed step sizes combined with multi-scale sampling framework.
result Achieves optimal dynamic regret without prior function change knowledge.
New method improves adversarial learning with unlabeled data.
problem Poor quality of pseudo labels on unlabeled data.
method Robust Co-training (RCT) using deep co-training.
result RCT significantly outperforms baselines in adversarial robustness.
Unified view on random walk and Weisfeiler-Leman kernels, improving accuracy.
problem Improving graph kernel methods for better classification accuracy.
method Define and analyze walk-based node refinement methods, relate to Weisfeiler-Leman test, and introduce new walk-based kernels.
result Walk-based kernels are as expressive as Weisfeiler-Leman subtree kernel but support non-strict neighborhood comparison.
Generalizes Thurston's asymmetric metric to flat metrics.
problem Defining an asymmetric metric on flat metrics.
method Defined an asymmetric metric on the space of unit-area flat metrics.
result Discussed two different topologies from the asymmetry.
Our purpose is to pursue the rigorous construction of Liouville Quantum Field Theory on Riemann surfaces initiated by F. David, A. Kupiainen and the last two authors in the context of the Riemann sphere and inspired by the 1981 seminal work by Polyakov. In this paper, we investigate the case of simply connected domains…
This paper extends the Gaussian process interpretation of deep networks to more varied weight distributions.
problem Understanding the impact of different weight initialization schemes on deep learning dynamics.
method Extending the Gaussian process interpretation to PSEUDO-IID weight distributions, including sparse and low-rank networks.
result PSEUDO-IID initialized networks are effectively equivalent up to variance, enabling tractable posterior distributions.
The paper proves asymptotic normality for multinomial logistic regression on null covariates.
problem Classical asymptotic normality results fail in high-dimensional multinomial logistic models.
method Developed asymptotic normality and chi-square results for multinomial logistic MLE on null covariates.
result Validated new methodology to test feature significance in high-dimensional classification problems.
New method uses variational inference for better reinforcement learning policies.
problem Improving policy gradient methods in reinforcement learning.
method Explicitly minimizing KL divergence to match target distribution for Bayesian parameterization.
result Enhanced performance in TRPO, PPO, and Bayesian Neural Network reinforcement learning.
Study on existence of weighted extremal Kähler metrics on projective bundles.
problem Existence of weighted extremal Kähler metrics on projective bundles.
method General existence result for weighted extremal metrics on admissible projective bundles.
result Existence of conformally Kähler, Einstein-Maxwell metrics of complex dimension .
New framework for ranking distributions using variable fractional parameters.
problem Ordering distributions with varying steepness and local non-concavities.
method Introducing a function to replace the fixed parameter in fractional SD.
result Enables ranking of a broader range of distributions and incorporates dynamic greediness.
Paper refines Chen-Cheng's estimates for Kähler metrics.
problem Uniform boundedness of scalar curvature assumption.
method Replacing uniform boundedness with -boundedness.
result Improved estimates for Kähler metrics under -boundedness.
Faster online calibration without randomization: interval forecasts and the power of two choicescs.LG
New method achieves faster calibration without randomization.
problem Calibrating probabilistic forecasts in adversarial settings.
method Using interval forecasts and the power of two choices.
result Achieves calibration error rate without randomization.