Paper develops theory of transverse generalized complex structures and proves a key lemma.
problem Proving equivalent conditions to the basic ddJ-lemma. method Describing transverse symplectic structure and relating the lemma to the Lefschetz map.
result Justified approach and proved equivalent conditions to the basic ddJ-lemma. We produce examples of generalized complex structures on manifolds by generalizing results from symplectic and complex geometry. We produce generalized complex structures on symplectic fibrations over a generalized complex base. We study in some detail different invariant generalized complex structures on compact Lie g…
A locally conformally Kahler (LCK) manifold is a complex manifold with a Kahler structure on its covering and the deck transform group acting on it by holomorphic homotheties. One could think of an LCK manifold as of a complex manifold with a Kahler form taking values in a local system L, called the conformal weight …
We consider the problem of extending a conformal metric of negative curvature, given outside a neighbourhood of 0 in the unit disk $\DD$, to a conformal metric of negative curvature in $\DD$. We give conditions under which such an extension is possible, and also give obstructions to such an extension. The methods we us…
In this lecture, we review some of the concepts of generalized geometry, as introduced by Hitchin and developed in the speaker's thesis. We also prove a Hodge decomposition for the twisted cohomology of a compact generalized Kähler manifold, as well as a generalization of the ddc-lemma of Kähler geometry.
The paper studies cohomologies of hypercomplex manifolds and their dimensions.
problem Understanding cohomologies and dimensions of invariant and anti-invariant subgroups.
method Proving a compact hypercomplex manifold is C∞-pure-and-full under certain conditions and studying dimensions of subgroups. result Characterization of hyperkähler with torsion metrics in terms of the dimension of the Jˉ-invariant subgroup. A Hermitian symplectic manifold is a complex manifold endowed with a symplectic form ω, for which the bilinear form ω(I⋅,⋅) is positive definite. In this work we prove ddc-lemma for 1- and (1,1)-forms for compact Hermitian symplectic manifolds of dimension 3. This shows that Albanese map for such manifol…
We study degenerate complex Monge-Ampère equations on a compact Kähler manifold (X,ω). We show that the complex Monge-Ampère operator (ω+ddc⋅)n is well-defined on the class E(X,ω) of ω-plurisubharmonic functions with finite weighted Monge-Ampère energy. The class E(X,ω) is the la…
The paper proves Hodge decompositions and partial bar partial lemmas for G2 and Calabi-Yau manifolds.
problem Proving Hodge decompositions and partial bar partial lemmas for G2 and Calabi-Yau manifolds.
method Defining cohomology spaces analogous to Bott-Chern cohomology and relating them to harmonic forms on the manifolds.
result Geometric interpretation of cohomology classes in terms of submanifolds and gerbes for G2 manifolds.
Study complex Monge-Ampère operator on weighted pluricomplex energy classes.
problem Characterize the range of the Complex Monge-Ampère Operator on weighted pluricomplex energy classes.
method Characterizations and a priori estimates on sub-level sets of solutions.
result A non-negative Borel measure is the Monge-Ampère of a unique function in \(\mathcal E_χ\) if and only if \(χ(\mathcal E_χ) \subset L^1(dμ)\).
Adapts Frolicher-type inequalities to foliations.
problem Determining foliations that can be made transversely Kahler.
method Adapting Frolicher-type inequalities to transversely holomorphic and symplectic foliations.
result Main results extend to orbifolds.
Let (X,ω) be a compact Kähler manifold of dimension n, and fix 1≤m≤n. We prove that the complex Hessian equation (ω+ddcφ)m∧ωn−m=fωn, with 0<f∈C∞(X) has a smooth admissible solution φ∈C∞(X). This was previously known to hold when $(X,…
Suppose M is a non-compact connected n-manifold without boundary, DD(M) is the group of C^\infty-diffeomorphisms of M endowed with the Whitney C^\infty-topology and DD_0(M) is the identity connected component of DD(M), which is an open subgroup in the group DD_c(M) \subset DD(M) of compactly supported diffeomorphisms o…
Researchers prove a stability result for a 3-sphere inequality, extending previous work.
problem Quantitative stability of nonlinear Yamabe-type inequalities on the 3-sphere.
method Proved a two-term refinement of the Schur lemma inequality in the conformal class of the 3-sphere.
result Deduced quantitative stability of an entire family of nonlinear Yamabe-type inequalities.
Paper accelerates nonlinear mapping in online systems with lower time complexity.
problem Speeding up nonlinear mapping in online systems.
method Integrates an acceleration module into Dendrite Net (DD) to reduce time complexity.
result DD with AC has lower time complexity while maintaining nonlinear mapping and system identification properties.
Study Drinfeld centralizers and Rouquier complexes in homotopy categories.
problem Interplay between Drinfeld centralizers and Rouquier complexes in homotopy categories.
method Study Drinfeld centralizers and Rouquier complexes in homotopy categories.
result Proved folklore facts about conjugation by Rouquier complexes in the Hecke category.
We study viscosity solutions to complex hessian equations. In the local case, we consider Ω a bounded domain in Cn, β the standard Kähler form in Cn and 1≤m≤n. Under some suitable conditions on F,g, we prove that the equation $(dd^c \varphi)^m\wedgeβ^{n-m}=F(x,\varphi)β^n,\ \f=…
We propose a generalization of the Hodge ddc-lemma to the case of hyperkähler manifolds. As an application of this result we derive the global construction of the fourth order transgression of the Chern character forms of hyperholomorphic bundles over compact hyperkähler manifolds. At the second part of the paper we…
Generalizes Thurston's jiggling lemma for piecewise smooth solutions.
problem Creating piecewise smooth solutions of differential relations without homotopical assumptions.
method Jiggling arbitrary sections of E to construct solutions of R. result Generalization of Thurston's lemma for piecewise smooth solutions of differential relations.
Corrects mistakes in convergence rate claims for SGD learning rate scheme.
problem Incorrect convergence rate claims for SGD learning rate scheme.
method Revised the convergence rate claims based on corrected test criterion for a series.
result Valid convergence rate of SGD is O(1/t), not O(1/t2) as previously stated. We derive weighted log-Sobolev inequalities from a class of super Poincaré inequalities. As an application, the Talagrand inequality with larger distances are obtained. In particular, on a complete connected Riemannian manifold, we prove that the $\log^\dd$-Sobolev inequality with $\dd\in (1,2)$ implies the $L^{2/(2-\d…
DD-SP uses ML to improve SP for Lorenz 96 systems, outperforming LR and DD-P.
problem Improving computational efficiency in weather/climate modeling.
method Data-driven super-parameterization using recurrent neural networks.
result DD-SP is more accurate and cheaper than SP, especially with scale separation.
Analyzes double descent in binary classification models with different losses.
problem Understanding the double descent phenomenon in binary classification models.
method Analytic study of gradient descent with logistic and square losses on binary linear classification models.
result The double descent phenomenon persists but with differences compared to logistic loss.
The paper generalizes a theorem about rectifiability of sets.
problem Understanding the rectifiability of sets in geometric analysis.
method Generalizing a classical theorem of Besicovitch to new contexts.
result Sets with certain properties are rectifiable.
DD algorithm tracks test error from train error without validation data.
problem Systematic generalization gap between train and test errors in modern model training.
method Decoupled descent (DD) algorithm that cancels data reuse biases via approximate message passing.
result DD algorithm rigorously demonstrates zero-cost validation and 100% data utilization.
Metric study on Kähler manifolds with prescribed singularities.
problem Defining a metric space for Kähler potentials with prescribed singularities.
method Introducing a distance d and dA on the relative finite energy class and showing convergence. result The space XA is complete and converges in Gromov-Hausdorff sense. The paper explores cohomologies on almost complex manifolds and their applications.
problem Distinguishing and understanding cohomologies on non-integrable almost complex structures.
method Defined and studied cohomologies HN∙(M) and HJ∙(M) using Nijenhuis-Lie derivations. result The J-cohomology encodes whether an almost complex structure satisfies the dLJ-lemma. Solves a complex Monge-Ampère equation on compact Hermitian manifolds.
problem Solving a specific Monge-Ampère equation on compact Hermitian manifolds.
method Uses complex Monge-Ampère equation and fixed potential approach.
result Shows the existence and uniqueness of a solution in a specific class.
Neural model detects DD risk in 5-year-olds, predicting 2 years ahead.
problem Early detection of developmental dyslexia for preventive teaching.
method Mixed neural model using auto-encoder and ordinal regression.
result System predicts DD risk 2 years before phonological processing is assessed.
DDS samples from noisy data by reversing diffusion, providing theoretical guarantees.
problem Sampling from unnormalized densities.
method Denoising diffusion process, score matching, optimal control, Schrödinger bridges.
result DDS provides theoretical guarantees for sampling.
Latent feature models are widely used to decompose data into a small number of components. Bayesian nonparametric variants of these models, which use the Indian buffet process (IBP) as a prior over latent features, allow the number of features to be determined from the data. We present a generalization of the IBP, the …
DD-VAE uses deterministic decoding for better latent code utilization in discrete data.
problem Inflexible decoders in VAEs lead to poor utilization of latent codes in discrete data.
method Proposed DD-VAE with deterministic decoding and new proposal distributions.
result DD-VAE improves latent code utilization and structure of learned manifold.
For any triple (W,L,ρ), where W is a closed connected and oriented 3-manifold, L is a link in W and ρ is a flat principal B-bundle over W (B is the Borel subgroup of $SL(2,\mc)$), one constructs a $\Dd$-scissors congruence class $\cG_{\Dd}(W,L,ρ)$ which belongs to a (pre)-Bloch group $\Pp (\Dd)$. The class $\cG_{\D…
End-to-end deep learning boosts IM/DD fiber communication over dispersive channels.
problem Improving data transmission over dispersive IM/DD channels with memory.
method Bidirectional recurrent neural network (BRNN) for end-to-end deep learning of the communication system.
result End-to-end SBRNN achieves significant bit-error-rate reduction compared to FFNNs.
Double descent found in DRL, improving generalization with model capacity.
problem Generalization in over-parameterized DRL models.
method Actor-Critic framework, Policy Entropy metric.
result Policy Entropy significantly reduces as model capacity increases, indicating improved generalization.
DDS uses a scorer to adaptively weigh data during training, improving model performance.
problem Efficiently optimizing data usage during machine learning training.
method Differentiable Data Selection (DDS) using a learnable scorer network and a reward signal.
result DDS delivers strong and consistent improvements over baselines on machine translation and image classification tasks.
Gaussian sketching preserves kernel inner products in low dimensions.
problem Preserving kernel inner products in low-dimensional spaces.
method Gaussian sketching of kernel Gram matrices and random projections in RKHS.
result Sketching yields a random projection operator that preserves weighted RKHS inner products.
SDD improves DD for estimating treatment effects by adjusting for confounding.
problem Estimating treatment effects in observational studies with confounding.
method Synthesized Difference in Differences (SDD) using RCT data to infer correct slopes.
result SDD achieves state-of-the-art performance across synthetic and real datasets.
Dead-Direction Signatures (DDS) provide a cheap, closed-form spectral reading of a network's singular complexity.
problem Estimating the complexity of deep networks through their loss singularities.
method DDS replaces the SGLD posterior chain with spectral linear algebra.
result DDS observables rank-track the network's singular complexity at the framework-predicted sign.
Improved UDA framework using f-divergence measures.
problem Addressing distribution shifts in machine learning.
method Refined f-divergence-based discrepancy and f-domain discrepancy. result Novel target error and sample complexity bounds.
Paper presents a universal baseline for binary prediction models.
problem Need a robust baseline to evaluate model performance.
method Dutch Draw (DD) baseline method for binary classification models.
result Reduces to almost always predicting zero or one in most situations.
Enhances machine learning generalization with DD-R-DRO method.
problem Improving machine learning generalization and reducing testing error.
method Doubly robust data-driven distributionally robust optimization (DD-R-DRO) method.
result Reduces testing error relative to state-of-the-art classifiers.
Study complex manifolds with negative curvature, finding either a current or Kähler property.
problem Characterizing compact complex manifolds with negative curvature operators.
method Proves properties using metrics and the pluriclosed flow.
result Classification of complex surfaces with negative curvature operators.
Survey examines distillation methods for large language models.
problem Efficiently compress large language models while preserving their capabilities.
method Knowledge Distillation and Dataset Distillation techniques.
result Integrating KD and DD can produce more effective and scalable compression strategies.
Study of Monge-Ampère volumes on hermitian manifolds, focusing on plurisigned metrics.
problem Behavior of Monge-Ampère volumes under hermitian variations.
method Analysis of hermitian forms and cohomology, studying boundedness and positivity.
result Existence and non-existence of plurisigned hermitian metrics on specific manifolds.
Paper tackles adapting multiple domains to a target domain using distillation and dictionary learning.
problem Adapting multiple heterogeneous labeled source domains to an unlabeled target domain.
method Combines Multi-Source Domain Adaptation and Dataset Distillation with Dataset Dictionary Learning.
result Achieves state-of-the-art adaptation performance even with minimal labeled data.
Efficiently calibrates Libor Market Model with SV-DD using Edgeworth expansions.
problem Calibrating the Libor Market Model with Stochastic Volatility and Displaced Diffusion.
method Combining Edgeworth and Gram-Charlier expansions with moments up to fourth order.
result 98% reduction in computational time for DD-SV-LMM calibration.
Starting from a sequence of independent Wright-Fisher diffusion processes on [0,1], we construct a class of reversible infinite dimensional diffusion processes on $\DD_\infty:= \{{\bf x}\in Let $MbeacompleteRiemnnianmanifoldandμthedistributionofthediffusionprocessgeneratedby\ff 1 2\DD+ZwhereZ$…