Proves symplectomorphism of noncompact manifolds with embedded rays.
problem Symplectic manifolds with embedded rays.
method Constructs explicit symplectomorphisms using excision trick.
result Symplectomorphic to complement of the ray in standard Euclidean space.
Excises interesting subsets from symplectic manifolds.
problem Excision of interesting closed subsets from symplectic manifolds.
method Time-independent incomplete Hamiltonian flows.
result Generalizes a result about excision of a ray.
An excision theorem connects Heegaard Floer homology of 3-manifolds.
problem Establishing a connection between Heegaard Floer homology of related 3-manifolds.
method Using Heegaard Floer homology and excision construction.
result Isomorphic twisted Heegaard Floer homology groups of related 3-manifolds.
In this article, we introduce the notion of a functor on coarse spaces being coarsely excisive- a coarse analogue of the notion of a functor on topological spaces being excisive. Further, taking cones, a coarsely excisive functor yields a topologically excisive functor, and for coarse topological spaces there is an ass…
Develops a categorified excision principle for elliptic symbol families.
problem Expressing the index class in K-theory using differential-topological data.
method Categorical index calculus, excision principle.
result Allows comparison of categorified index problems on different manifolds.
New instanton Floer homology for sutured manifolds with tangles defined.
problem Defining instanton Floer homology for complex geometric structures.
method Extending excision theorems to handle singularities and tangles.
result Detection of unlink and trivial braid closures in annular Khovanov homology.
Study calculates ring structure in instanton homology for a surface with points.
problem Calculating the ring structure in instanton homology for a surface with points.
method Used singular instanton Floer homology and excision formula.
result Proved an excision formula for instanton Floer homology when n=1.
Study contact structures and sutured monopole Floer homology, computing and deriving formulas.
problem Understanding the relationship between contact structures and sutured monopole Floer homology.
method Exploring contact elements under Floer excisions, computing sutured monopole Floer homology, and deriving formulas.
result Obtained exact triangles and connected sum formulas for sutured monopole Floer homology.
Relates two types of skein algebras using explicit correspondences.
problem Defining and relating stated and internal skein algebras.
method Explicit correspondence between stated and internal skein algebras, distinguishing between left and right boundary edges, proving excision properties.
result Agrees with excision properties of stated skein algebras under specific conditions.
Satellite formula connects knot concordance invariants to surgery.
problem Understanding knot concordance invariants.
method Excision theorem for real Floer homotopy types.
result Concordance invariants depend only on zero-framed surgery.
We observe that the main theorem in \cite{KMsuture} immediately implies its analogue for closed 3--manifolds.
We develop monopole and instanton Floer homology groups for balanced sutured manifolds. Applications include a new proof of Property P for knots.
Smooth groupoid algebras are H-unital, with implications for algebraic and homological properties.
problem Understanding the structure of convolution algebras on Lie groupoids.
method Analyzing smooth functions and invariant subsets to prove H-unitality.
result H-unitality of groupoid algebras and their quotients, leading to excision properties.
We prove that the instanton knot homology KHI(K) as defined by Kronheimer and Mrowka (Knots, sutures and excision, preprint), recovers the Alexander polynomial for knots K in the 3-sphere.
Formulae prove equivalence of two invariants on specific 4-manifolds.
problem Equivalence of two invariants on homology S1imesS3. method Surgery and excision formulas for Furuta-Ohta invariant λFO. result Computes λFO of certain families of manifolds. New mapping class group actions on Hochschild complexes for modular categories.
problem Understanding actions of mapping class groups on Hochschild complexes of modular categories.
method Construction of a symmetric monoidal functor with excision property.
result Homotopy coherent projective action of mapping class groups on Hochschild complexes.
Paper compares higher torsions and removes fiberwise Morse function assumption.
problem Comparing higher torsions from analytic and topological perspectives.
method Introduced fiberwise generalized Morse functions (GMFs) and excised neighborhoods around birth-death points.
result Established a generalized version of the higher Cheeger-Müller/Bismut-Zhang theorem.
Paper defines embolic volume and relates it to Betti number using the covering trick.
problem Relating embolic volume to topological invariants.
method Covering trick from systolic geometry applied to Berger's inequality.
result Relates embolic volume to the first Betti number.
Study of skateboard flips as continuous curves in SO(3) group.
problem Characterize skateboard flip tricks as continuous motions.
method Model flips as curves in SO(3), analyze lifts to S3, derive formulas. result There are only four distinct flip tricks up to continuous deformation.
A new method centers outliers in robust PCA without manual intervention.
problem Outliers in robust PCA require manual centering, complicating the analysis.
method Introduces a 'bias trick' to automatically center non-outliers.
result First optimal RPCA algorithm with automatic centering.
Nash's theorem proved with Günther's trick
problem Proving Nash's smooth embedding theorem
method Using Günther's trick
result Nash's theorem proved
Explains Conway's tangle trick and its mathematical origins.
problem Understanding the relationship between braids and elliptic curves.
method Discusses the tangle trick, its mathematical underpinnings, and historical context.
result Establishes the connection between braids and elliptic curves.
Unified framework for gradient estimation in combinatorial spaces.
problem Scaling relaxed gradient estimators to large combinatorial distributions.
method Introducing stochastic softmax tricks within the perturbation model framework.
result Stochastic softmax tricks improve model performance and discover more latent structure.
We prove all knots can be transformed into a trefoil using special diagrams.
problem Transforming any knot into a trefoil using magic tricks.
method Introducing knotholder diagrams to encode transformations.
result All knots can be transformed into a trefoil.
Geometric trick simplifies link homotopy and concordance.
problem Homotopy and concordance of links in homology spheres.
method Relative Whitney trick to remove double points.
result Links in homology spheres can be simplified to topologically slice links.
The Gumbel-max trick and its extensions simplify sampling from categorical distributions in machine learning.
problem Sampling from categorical distributions with unnormalized probabilities.
method Extensions of the Gumbel-max trick for various applications.
result Simplified and efficient methods for sampling and gradient estimation.
Tricks improve retail product image classification accuracy.
problem Retail Product Image Classification
method Various tricks including a new LCA layer, Instagram-pretrained Convnet, and Maximum Entropy loss.
result Increased accuracy of fine-tuned convnets by a large margin.
A new gradient estimator for categorical distributions reduces bias and variance.
problem Intractability of gradients for categorical distributions in discrete latent variable models.
method CatLog-Derivative trick and IndeCateR gradient estimator.
result IndeCateR reduces bias and variance of gradients for categorical distributions.
Establishes necessary and sufficient conditions for smooth triviality of Lie subalgebras and Lie ideals, and proves Moser's trick for foliations.
problem Smooth triviality of Lie subalgebras and Lie ideals
method Establishing necessary and sufficient conditions and proving Moser's trick for foliations
result Direct proof of Moser's trick for foliations
Triple-point Whitney trick classifies ornaments of 3-manifolds.
problem Classifying ornaments of 3-manifolds in high dimensions.
method Triple-point Whitney trick applied to orientable manifolds.
result Classification of ornaments by the μ-invariant.
Alexander trick applied to homology spheres for manifold homeomorphisms.
problem Group of homeomorphisms of contractible manifolds.
method Strong uniqueness statement for one-sided h-cobordisms.
result Group of homeomorphisms is contractible for d≥6. Expands Bredon's trick for applications in geometry and topology.
problem Local-to-global extension principles in geometric and topological contexts.
method Novel applications and frameworks for stratified pseudomanifolds, Ricci flow, and persistent homology.
result Establishes Bredon's trick as a unifying framework.
Paper generalizes reparameterization trick for broader applicability.
problem Limited applicability of reparameterization trick to specific distributions.
method Introduces a generalized transformation-based gradient method.
result Proposed model combines advantages of control variates and generalized reparameterization.
Bredon's trick helps extend local properties to global topological spaces.
problem Extending local properties to global topological spaces.
method Bredon's trick for local properties to global spaces.
result Bredon's trick allows for natural alternative demonstrations of classic results.
We develop an epsilon-controlled algebraic L-theory, extending our earlier work on epsilon-controlled algebraic K-theory. The controlled L-theory is very close to being a generalized homology theory; we study analogues of the homology exact sequence of a pair, excision properties, and the Mayer--Vietoris exact sequence…
We observe that gradients computed via the reparameterization trick are in direct correspondence with solutions of the transport equation in the formalism of optimal transport. We use this perspective to compute (approximate) pathwise gradients for probability distributions not directly amenable to the reparameterizati…
The Gumbel trick is a method to sample from a discrete probability distribution, or to estimate its normalizing partition function. The method relies on repeatedly applying a random perturbation to the distribution in a particular way, each time solving for the most likely configuration. We derive an entire family of r…
An online reinforcement learning algorithm is anytime if it does not need to know in advance the horizon T of the experiment. A well-known technique to obtain an anytime algorithm from any non-anytime algorithm is the "Doubling Trick". In the context of adversarial or stochastic multi-armed bandits, the performance of …
We prove a generalization of Bennequin's inequality for Legendrian knots in a 3-dimensional contact manifold (Y,xi), under the assumption that Y is the boundary of a 4-dimensional manifold M and the version of Seiberg-Witten invariants introduced by Kronheimer and Mrowka [Invent. Math. 130 (1997) 209-255] is nonvanishi…
In this paper, we study a certain cohomology attached to a smooth function, which arose naturally in Poisson geometry. We explain how this cohomology depends on the function, and we prove that it satisfies both the excision and the Mayer-Vietoris axioms. For a regular function we show that the cohomology is related to …
Inference in popular nonparametric Bayesian models typically relies on sampling or other approximations. This paper presents a general methodology for constructing novel tractable nonparametric Bayesian methods by applying the kernel trick to inference in a parametric Bayesian model. For example, Gaussian process regre…
New methods for estimating gradients of expectations using pairwise interactions.
problem Estimating gradients of expectations for complex models.
method Introducing new pairwise stochastic gradient estimators based on the log-derivative trick and reparameterisation.
result New estimators are unbiased and offer variance reduction compared to the log-derivative estimator.
New trick builds hyperbolic manifolds from compact ones, proving some don't virtually fiber.
problem Proving some hyperbolic manifolds don't virtually fiber.
method Hyperbolic reflection group trick, embedding theory, manifold topology.
result Constructed Gromov hyperbolic 7-manifolds that don't virtually fiber over a circle.
The study offers conditions for Darboux charts on specific types of manifolds.
problem Existence of Darboux charts on weakly symplectic manifolds.
method Using Moser's trick to find sufficient conditions.
result Sufficient conditions for Darboux charts on weakly symplectic manifolds.
We define a cobordism category of topological manifolds and prove that if d=4 its classifying space is weakly equivalent to Ω∞−1MTTop(d), where MTTop(d) is the Thom spectrum of the inverse of the canonical bundle over BTop(d). We also give versions with tangential structures and boundary. The pro…
Boltzmann machines (BMs) are appealing candidates for powerful priors in variational autoencoders (VAEs), as they are capable of capturing nontrivial and multi-modal distributions over discrete variables. However, non-differentiability of the discrete units prohibits using the reparameterization trick, essential for lo…
Link framings can only change when a 3-manifold has a non-separating sphere.
problem Understanding how framings of links can change in 3-manifolds.
method Using McCullough's work on mapping class groups and the Dirac trick.
result Link framings can only change in specific 3-manifolds with a non-separating sphere.
New method for simplifying knots with specific properties.
problem Understanding knots with a specific unknotting number.
method Derive and apply the Montesinos trick for proper rational tangle replacement.
result Prove that knots with proper rational unknotting number one are prime and classify certain types.