3d gauge theories encode 4d simplicial geometries, with quantum blocks approximating gravity.
problem Encoding 4d quantum gravity in 3d gauge theories.
method Applying Dimofte-Gaiotto-Gukov construction to graph complements in 3-manifolds.
result Holomorphic blocks approximate quantum 4d simplicial geometries.
Tangle blocks simplify knot theory by breaking down complex knots into manageable pieces.
problem Complex knot theory calculations are simplified by breaking down knots into tangle blocks.
method The approach involves expressing link polynomials as multilinear combinations of tangle blocks.
result Tangle blocks provide a powerful tool for understanding and calculating knot invariants.
Study of 3d-3d correspondence involving q q q -Weyl algebra and 3d-index.
problem Understanding the action of a q q q -Weyl algebra on the 3d-index of knots. method Investigation of the q q q -Weyl algebra's module action on the 3d-index, conjecturing structural properties. result Bilinear factorization, pair of linear q q q -difference equations, and rational function matrix for the 3d-index determination. We introduce a method to construct G 2 G_2 G 2 -instantons over compact G 2 G_2 G 2 -manifolds arising as the twisted connected sum of a matching pair of building blocks [Kov03,KL11,CHNP12]. Our construction is based on gluing G 2 G_2 G 2 -instantons obtained from holomorphic bundles over the building blocks via the first named author's wo…
Holomorphic immersions blocked in 9D real hypersurface with specific signature.
problem Existence of holomorphic immersions of bi-disks into a specific 9D real hypersurface.
method Application of Cartan's method to analyze the existence of immersions.
result Holomorphic immersions are obstructed by specific conditions on the hypersurface.
Study periodic points on genus two surfaces, solving dynamics and geometry problems.
problem Classifying and understanding periodic points on genus two surfaces.
method Analyzing GL(2, R)-equivariant point markings and using properties of hyperelliptic involution, Weierstrass points, and golden points.
result All GL(2, R)-equivariant point markings over orbit closures arise from specific point exchanges.
Motivated by gauge theory under special holonomy, we present techniques to produce holomorphic bundles over certain noncompact 3 − 3- 3 − folds, called building blocks, satisfying a stability condition `at infinity'. Such bundles are known to parametrise solutions of the Yang-Mills equation over the G 2 − \rm G_2- G 2 − manifolds obtain…
New proof of Grauert's theorem using differential geometry.
problem Proper holomorphic morphisms and coherence of higher direct images.
method Differential-geometric proof, antiholomorphic superconnection, Hironaka's desingularization.
result New proof of Grauert's theorem in both smooth and singular cases.
Quantum gravity model includes cosmological constant using Chern-Simons theory.
problem Incorporating a positive cosmological constant into quantum gravity models.
method Quantization of a complex SL(2,C) Chern-Simons theory.
result Inclusion of a cosmological constant in quantum gravity models.
Method constructs G2-instantons over twisted connected sums.
problem Constructing G2-instantons over complex manifolds.
method Gluing G2-instantons from holomorphic bundles over building blocks.
result Explicit examples from semi-Fano 3-folds.
Classifies GL(2,R) invariant markings over Abelian differential components.
problem Classifying GL(2,R) invariant point markings over Abelian differential components.
method Analyzing hyperelliptic components and their invariant markings.
result Invariant markings arise from specific points or involution exchanges, and can determine holomorphic sections.
New minimal 2-spheres found in hyperkähler 4-manifolds, unstable and not holomorphic.
problem Characterizing stable minimal surfaces in hyperkähler 4-manifolds.
method Gluing construction using Scherk and Taub-NUT surfaces, harmonic map parametrization.
result Existence of unstable minimal 2-spheres with degree-1 Gauss lift, not holomorphic.
New K-theory framework reveals exotic brane charges and conformal blocks.
problem Identifying exotic brane charges in string/M-theory.
method Twisted equivariant differential K-theory and flat complex line bundles.
result Observation of conformal blocks and their relation to brane charges.
In this paper we introduce a new algebraic device, which enables us to treat the quaternions as though they were a commutative field. This is of interest both for its own sake, and because it can be applied to develop an "algebraic geometry" of noncompact hypercomplex manifolds. The basic building blocks of the theory …
Study on Klt varieties with trivial canonical class, focusing on holonomy and stability.
problem Holonomy group of singular Kähler-Einstein metrics on klt varieties with numerically trivial canonical divisor.
method Investigation of holonomy group properties, finiteness of connected components, Bochner principle for holomorphic tensors, and connections between irreducibility of holonomy representations and stability of the tangent sheaf.
result Refinement of known decompositions for tangent sheaves of varieties with trivial canonical divisor, showing that up to finite quasi-étale covers, varieties with strongly stable tangent sheaves are either Calabi-Yau or irreducible holomorphic symplectic.
The classical Beauville-Bogomolov Decomposition Theorem asserts that any compact Kähler manifold with numerically trivial canonical bundle admits an étale cover that decomposes into a product of a torus, and irreducible, simply-connected Calabi-Yau-- and holomorphic-symplectic manifolds. The decomposition of the simply…
We use the 3d-3d correspondence together with the DGG construction of theories T n [ M ] T_n[M] T n [ M ] labelled by 3-manifolds M to define a non-perturbative state-integral model for SL(n,C) Chern-Simons theory at any level k, based on ideal triangulations. The resulting partition functions generalize a widely studied k=1 state-integ…
Classifies Kähler structures with special foliations using symplectic techniques.
problem Classifying Kähler structures with specific foliations.
method Weinstein's method of constructing symplectic bundles applied to Kähler data.
result New Kähler structures and metrics are obtained.
We construct from first principles the operator 'A-hat' that annihilates the partition functions (or wavefunctions) of three-dimensional Chern-Simons theory with gauge groups SU(2), SL(2,R), or SL(2,C) on a knot complement M. The operator 'A-hat' is a quantization of the knot complement's classical A-polynomial A(l,m).…
Proposes Teichmüller TQFT as a Chern-Simons theory with a new integration cycle.
problem Defining Teichmüller TQFT as a Chern-Simons theory with a novel integration cycle.
method Analytically continues Chern-Simons path-integral with an unusual integration cycle.
result Teichmüller TQFT is dual to complex SL(2,C) Chern-Simons theory at integer level k=1.
New algorithm detects and estimates block sparse signal supports and amplitudes.
problem Reconstructing block sparse signals with unknown block structures.
method Bayesian hypothesis testing for support detection and MMSE estimation for amplitude estimation.
result Demonstrated effectiveness through numerical experiments.
This paper presents a novel Block Iterative Bayesian Algorithm (Block-IBA) for reconstructing block-sparse signals with unknown block structures. Unlike the existing algorithms for block sparse signal recovery which assume the cluster structure of the nonzero elements of the unknown signal to be independent and identic…
Proposes a Nested Block Model to unify various network block models.
problem Lack of nested structure and differing parameter complexity among block models.
method Formulates a hierarchy of block models (NBM) that includes SBM, DCBM, and PABM as special cases.
result Allows clustering and estimation without preliminary testing, simplifying model selection.
Two-block ADMM outperformed multi-block ADMM in multi-task learning experiments.
problem Comparing two-block vs. multi-block ADMM in optimization performance.
method Compared two-block and multi-block ADMM on multi-task learning problems.
result Multi-block ADMM consistently outperformed two-block ADMM in optimization and prediction performance.
We examine the recovery of block sparse signals and extend the framework in two important directions; one by exploiting signals' intra-block correlation and the other by generalizing signals' block structure. We propose two families of algorithms based on the framework of block sparse Bayesian learning (BSBL). One fami…
New model allows some connections to be zero, improving network analysis.
problem Networks with block structure and sparsity.
method Sparse Popularity Adjusted Stochastic Block Model (PABM).
result Allows some probabilities of connections to be zero.
Bayesian framework for choosing number of blocks in stochastic block models.
problem Lack of principled statistical model selection criteria for stochastic block models.
method Bayesian framework for choosing the number of blocks and comparing to degree-corrected block models.
result Universal model selection framework capable of comparing multiple modeling combinations.
We theoretically investigate the convergence rate and support consistency (i.e., correctly identifying the subset of non-zero coefficients in the large sample limit) of multiple kernel learning (MKL). We focus on MKL with block-l1 regularization (inducing sparse kernel combination), block-l2 regularization (inducing un…
SympFormer accelerates attention blocks using inertial dynamics on density spaces.
problem Improving the efficiency of self-attention blocks in Transformers.
method Introduced accelerated attention blocks derived from inertial Nesterov dynamics on density spaces.
result Accelerated attention blocks converge faster than classical blocks while preserving oracle calls.
We introduce block-tree graphs as a framework for deriving efficient algorithms on graphical models. We define block-tree graphs as a tree-structured graph where each node is a cluster of nodes such that the clusters in the graph are disjoint. This differs from junction-trees, where two clusters connected by an edge al…
Paper studies non-tight reconstruction threshold in a 4-state model with different in/out block mutations.
problem Non-tight reconstruction threshold in a 4-state symmetric model with different in-block and out-block mutations.
method Inspired by the q 1 + q 2 q_1+q_2 q 1 + q 2 stochastic block model, rigorously analyzes conditions for non-tightness of the reconstruction threshold. result Rigorously gives conditions for the non-tightness of the reconstruction threshold in a 4-state symmetric model.
A new channel locality block improves CNN performance.
problem Improving the performance of convolutional neural networks.
method Proposed a variant of Squeeze-and-Excitation block using convolutional layers to learn nearby channel correlation.
result Our C-Local block achieved higher accuracy than the standard SE block on the cifar-10 dataset.
We simplify matrix computations for block matrices, especially useful for covariance and correlation matrices.
problem Complex computations for block matrices, especially for covariance and correlation matrices.
method Obtained a canonical representation for block matrices, facilitating computation of various matrix operations.
result Simplified computation of matrix operations for block matrices, particularly useful for covariance and correlation matrices.
Proposes BMME for optimizing nonsmooth nonconvex problems with block structure.
problem Optimizing nonsmooth nonconvex problems with block structure.
method Block Alternating Bregman Majorization Minimization with Extrapolation (BMME).
result Subsequential convergence to a first-order stationary point under mild assumptions, global convergence under stronger conditions.
Develops a multichannel deep network for faster, artifact-free image CS.
problem Block-wise sampling artifacts in image CS with multiple sampling rates.
method Multichannel deep network for block-based image CS, removing blocking artifacts.
result Significantly outperforms state-of-the-art CS methods in objective and subjective metrics.
Paper proposes ABDR for convex subspace clustering with adaptive block diagonal representation.
problem Subspace clustering with block diagonal structure for noisy data.
method ABDR explicitly pursues block diagonality without sacrificing convexity, using a specially designed convex regularizer.
result Experimental results show ABDR outperforms state-of-the-arts.
Holomorphic Jacobi structures enrich the theory of Poisson manifolds.
problem Holomorphic Poisson structures are limited; holomorphic Jacobi structures offer more.
method Developed holomorphic Jacobi structures and their relationship with other structures.
result Holomorphic Jacobi structures provide a broader framework than holomorphic Poisson structures.
New algorithms reduce bias in federated learning for non-i.i.d. data.
problem Client and block biases in federated learning with non-i.i.d. data.
method Multi-model parallel SGD (MM-PSGD) and multi-chain parallel SGD (MC-PSGD) algorithms.
result Achieve a linear speedup with convergence rate O ( 1 / N T ) O(1/\sqrt{NT}) O ( 1/ N T ) . Paper finds effective building blocks for CNNs, improving model performance and size.
problem Finding optimal deep model architectures and parameters.
method Search framework for discovering effective building blocks for CNNs.
result Discovered models are smaller and perform comparably to state-of-the-art models.
Geometric Block Model improves community detection in sparse graphs.
problem Improving community detection in sparse graphs.
method Proposes a new geometric block model and a triangle-counting algorithm.
result Triangle-counting algorithm performs near-optimal in sparse graphs.
New formula calculates loss from arbitrage in blockchain liquidity pools.
problem Calculating loss from arbitrage in Automated Market Makers (AMMs) under varying block times.
method Derived a closed-form approximation for expected loss using random walk theory.
result The formula approximates the loss from arbitrage with high accuracy and shows that constant block intervals minimize this loss.
This paper solves matrix blind joint block diagonalization with noise.
problem Identifying the diagonalizer and block diagonal structure of matrices under noise.
method Bi-block diagonalization method.
result The method can identify the exact solution under certain conditions.
Develops theory of d-holomorphic connections on Klein surfaces.
problem No specific problem stated; focuses on theory development.
method Constructs Atiyah exact sequence for d-holomorphic bundles and provides existence criterion.
result Established theory of d-holomorphic connections and existence criterion.
Study holomorphic last multipliers on complex manifolds.
problem Equivalence between holomorphic and real ODE systems.
method Analyzing last multipliers in complex manifold context.
result Relate holomorphic last multipliers to real last multipliers.
Researchers developed a new Riemannian manifold for SPD matrix-valued optimal transport problems.
problem Optimal transport between SPD matrix-valued measures.
method Formulated as a generalized optimal transport problem with block SPD matrices, endowed with a novel Riemannian manifold structure.
result The novel Riemannian manifold allows solving SPD matrix-valued optimal transport problems using Riemannian optimization.
Corrects a mistake in h v hv h v -block cross-validation consistency claim.
problem Incorrectly assumed h v hv h v -block cross-validation is a BIBD. method Demonstrates h v hv h v -block is not a BIBD and reopens consistency question. result Theoretical consistency of h v hv h v -block remains unresolved. Study provides selective inference method for latent block models.
problem Challenges in constructing a test on a block structure selected by clustering algorithms.
method Developed a selective inference method for latent block models using squared residue minimization and simulated annealing.
result Proposed tests effectively handle selective bias in block structures compared to naive tests.
Paper introduces a new edge exchangeable block model for complex networks.
problem Limitations of the stochastic block model in analyzing complex networks.
method Develops a Bayesian nonparametric edge exchangeable block model.
result The new model outperforms state-of-the-art SBMs for link prediction.