Derives localization formulas in Batalin-Vilkovisky formalism.
problem Localization in Batalin-Vilkovisky formalism.
method Equivariant localization formulas in Batalin-Vilkovisky formalism.
result Derives localization formulas in Batalin-Vilkovisky formalism.
Local biquandles link link coloring to tribracket theory.
problem Link coloring and tribracket theory.
method Introduced local biquandles and defined their cohomology.
result Local biquandle cohomology is isomorphic to Niebrzydowski's tribracket cohomology.
We give axioms which characterize the local Reidemeister trace for orientable differentiable manifolds. The local Reidemeister trace in fixed point theory is already known, and we provide both uniqueness and existence results for the local Reidemeister trace in coincidence theory.
Unified Higgs bundle vacua from M-theory on Spin(7) spaces.
problem Unifying Higgs bundle vacua from different string compactifications.
method Developed formalism for M-theory on local Spin(7) spaces and constructed explicit solutions.
result Unified 3D effective field theory from 4D M- and F-theory vacua.
The paper characterizes boundaries in Turaev-Viro TQFTs and Dijkgraaf-Witten theories.
problem Characterizing boundaries in Turaev-Viro TQFTs and Dijkgraaf-Witten theories.
method Identifying explicit boundary locality conditions and proving consistency with state sum models.
result Turaev-Viro and Dijkgraaf-Witten theories with boundary defects admit a state sum description.
It is argued that the enlargement of the gauge group found in non-commutative gauge theory is more fundamentally thought of as a consequence of the non-locality of the construction and that it was already encountered in an earlier discussion of a non-local gauge theory.
Generalizing local Gromov-Witten theory, in this paper we define a local version of symplectic field theory. When the symplectic manifold with cylindrical ends is four-dimensional and the underlying simple curve is regular by automatic transversality, we establish a transversality result for all its multiple covers and…
Improved lattice field theory simulations with local-Autoregressive Conditional Normalizing Flow.
problem Efficiently sampling lattice field theories with computational challenges.
method Integrates locality into autoregressive conditional normalizing flows.
result Autocorrelation times improved by orders of magnitude for φ4 theory on a 2D lattice. Study S-duality and supersymmetry on curved manifolds using localization.
problem Understanding S-duality and supersymmetry on curved manifolds.
method Localization and Fourier transform interpretation of S-duality.
result Evidence for interpreting S-duality as Fourier transform.
Paper derives localization formula for S1-action and equivariant η-invariants.
problem Localization formula for S1-action and equivariant η-invariants. method Constructing a pre-λ-ring structure in differential K-theory.
result Established a localization formula for equivariant η-invariants.
The paper studies cohomological Donaldson-Thomas theory for local systems on a 3-torus.
problem Cohomological Donaldson-Thomas theory for local systems on the 3-torus.
method Using exponential maps and the tripled Jordan quiver, the paper proves cohomological integrality for GL_n and SL_n local systems.
result The paper proves Langlands duality statements for SL_n and PGL_n cohomological Donaldson-Thomas invariants for prime n.
New approach to Lagrangian field theories using pro-finite structures and L-infinity algebras.
problem Formulating Lagrangian field theories with locality constraints.
method Using pro-finite structures and L-infinity algebras to define local observables and a pre-multisymplectic form.
result Definition of L-infinity algebra of local observables based on Lagrangian cohomology.
Local gluing connects flow lines in finite time intervals.
problem Connecting flow lines in finite time intervals.
method Functional analytic approach to define local gluing map.
result Explicit construction of local gluing map in Euclidean case; intricate construction in non-Euclidean case.
Unified treatment of gauge theories and Yang-Mills theory duality.
problem Unified treatment of gauge theories and Yang-Mills theory duality.
method Cohomological localization techniques and Atiyah-Singer index theorem.
result Unified framework and simplified derivations of localization formulas.
Developed a theory of local convexity for second order differential equations on Lie algebroids.
problem Analyzing convexity in differential equations on Lie algebroids.
method Theory development for local convexity of SODEs on Lie algebroids.
result Extensive discussion of homogeneous quadratic SODEs on Lie algebroids.
Study geodesics on strong Kropina spaces, global and local aspects.
problem Geodesics behavior on strong Kropina spaces.
method Global and local analysis of geodesics.
result Illustrated geodesics with examples.
Explores local structure of morphisms and formal submanifolds in formal manifolds theory.
problem Understanding the local structure of morphisms and formal submanifolds in formal manifolds.
method Study of formal manifolds, including local structure of constant rank morphisms and formal submanifolds.
result Developed the local structure of constant rank morphisms and formal submanifolds.
We introduce a notion of non-local almost minimal boundaries similar to that introduced by Almgren in geometric measure theory. Extending methods developed recently for non-local minimal surfaces we prove that flat non-local almost minimal boundaries are smooth. This can be viewed as a non-local version of the Almgren-…
Proves local existence of quaternionic contact structures using Cartan-Kähler theory.
problem Proving the existence of quaternionic contact structures locally.
method Utilized Cartan-Kähler theory to prove local existence.
result Depends on 2n+2 real analytic functions of 2n+3 variables modulo diffeomorphisms.
The paper connects machine learning interpretability with learning theory.
problem Performance and explanation generalization in local machine learning models.
method Theoretical analysis and empirical validation of local approximation explanations.
result Theoretical bounds on test-time accuracy and explanation generalization.
Researchers tackle the globalization problem of locally cosymplectic Hamiltonian dynamics.
problem Globalization problem of locally cosymplectic Hamiltonian dynamics.
method Investigate the geometry of locally conformally cosymplectic manifolds and provide a geometric Hamilton-Jacobi theory.
result Provide a geometric Hamilton-Jacobi theory on locally conformally cosymplectic manifolds.
Study local control of a specific mechanism with growth vector (4,7).
problem Local control of a mechanism with a specific growth vector.
method Controllability and extremal trajectories on the nilpotent approximation.
result Solutions of the system and examples of local extremal trajectories.
Shadow biquandles and local biquandles have similar homology and invariants.
problem Comparing homology and invariants of shadow biquandles and local biquandles.
method Defined local biquandle structure on a shadow biquandle and showed isomorphic (co)homology groups and invariant equivalence.
result Homology and invariants of shadow biquandles and local biquandles are equivalent.
Develops a new theory of localization in algebraic geometry.
problem Localization of cohomological theories on closed subsets.
method Categorical and algebro-geometric approach, focusing on torsors and translation groupoids.
result Found that localization often results in a torsor of supported refinements rather than a localized class.
We view strict ring spectra as generalized rings. The study of their algebraic K-theory is motivated by its applications to the automorphism groups of compact manifolds. Partial calculations of algebraic K-theory for the sphere spectrum are available at regular primes, but we seek more conceptual answers in terms of lo…
Theory explains creativity in diffusion models generating novel images.
problem Diffusion models generate highly original images far from training data.
method Identified locality and equivariance as inductive biases to prevent optimal score-matching.
result Analytic models predict diffusion model outputs with high accuracy.
Develops local theory for singular spacetimes becoming asymptotically self-similar.
problem Construction of singular spacetimes in all dimensions.
method Local theory and construction of exact self-similar solutions.
result Construction of exact self-similar solutions corresponding to formal asymptotic expansions.
Following the analogies between 3-dimensional topology and number theory, we study an idèlic form of class field theory for 3-manifolds. For a certain set K of knots in a 3-manifold M, we first present a local theory for each knot in K, which is analogous to local class field theory, and then,…
Study Morse complexity of manifolds and homology classes, proving bounds and implications.
problem Understanding Morse complexity of manifolds and homology classes.
method Used surgery theory and index theory to prove upper and lower bounds.
result Locally symmetric spaces of Lie groups with discrete series representations do not admit open book decompositions.
Formulates a new class of gravity theories coupled to scalar and gauge fields.
problem Global formulation of theories of gravity coupled to scalar and gauge fields.
method Global mathematical formulation of a class of generalized four-dimensional theories of gravity.
result Global solutions can be interpreted as classical U-folds.
Deep neural networks perform well on local tasks but struggle with global tasks.
problem Understanding the limitations of overparameterized deep neural networks in learning global functions.
method Introduced k-local and k-global functions to study the interplay between depth and function locality. result Depth is beneficial for learning local functions but detrimental to learning global functions.
Survey on singularity theory and its relation to minimal model program.
problem Understanding singularities and their role in minimal model program.
method Construction and analysis of dual complexes, proof of ACC conjecture, local stability theory.
result Recent progress on local stability theory of Kawamata log terminal singularities.
Local index formula for Lorentzian Dirac operators on spacetimes.
problem Index theory for Lorentzian Dirac operators with nontrivial dynamics.
method Local index formula based on microlocal analysis.
result Established a local index formula for Lorentzian Dirac-type operators.
GridPyM handles grid diagrams for knot theory.
problem Handling grid diagrams for knot theory.
method Generates and simplifies grids, models local transformations.
result Models local transformations between grid diagrams.
The paper generalizes Hodge theory to semisimple local systems and proves a geometric Decomposition theorem.
problem Generalizing Hodge theory to semisimple local systems.
method Establishing a canonical isomorphism and proving a global invariant cycle theorem.
result A new geometric proof of the Decomposition theorem for semisimple local systems.
Study virtual fundamental classes of derived manifolds, proving invariant vanishes.
problem Computing virtual fundamental classes for derived manifolds.
method Combining derived differential geometry and cosection localization.
result Stable pair invariants of hyperkähler fourfolds are zero.
For a closed locally symmetric space M=Γ\G/K and a representation of G we consider the push-forward of the fundamental class in the homology of the linear group and a related invariant in algebraic K-theory. We discuss the nontriviality of this invariant and we generalize the construction to cusped locally symmetric sp…
New neural network theory mimics physics laws, making computations more plausible.
problem Neural networks lack biological plausibility in computations.
method Variational framework of the least action principle, local in space and time.
result SpatioTemporal Local Propagation (STLP) scheme is biologically plausible.
The study finds minimal hypersurfaces in wedge-shaped manifolds with boundary.
problem Finding minimal hypersurfaces in wedge-shaped manifolds with boundary.
method Developed a min-max theory for locally wedge-shaped manifolds with boundary.
result Proved existence of smooth free boundary minimal hypersurfaces in wedge-shaped manifolds.
Improved local SGD analysis for identical and heterogeneous data.
problem Analyzing local SGD performance in identical and heterogeneous data regimes.
method Developed a new variance measure specific to local SGD and improved existing theory.
result Tightened stepsize and number of local iterations for optimal performance.
Improved Local SGD convergence for general convex objectives with bounded second-order heterogeneity.
problem Understanding when and why Local SGD outperforms alternatives in distributed optimization.
method Established improved convergence guarantees for Local SGD on general convex objectives under bounded second-order heterogeneity.
result Upper bounds for Local SGD are nearly tight, providing a sharper convergence theory.
Develops a new theory of localization in algebraic geometry.
problem Understanding localizations in cohomological theories with open-closed structures.
method Categorical and algebro-geometric approach, focusing on torsors and refinements.
result Establishes compatibility with various algebraic operations and recovers classical results.
Study finds strictly convex surfaces with specific curvature and boundary in space forms.
problem Finding strictly locally convex hypersurfaces with prescribed curvature and boundary in space forms.
method Using C2 a priori estimates and degree theory arguments, the study establishes existence results. result Existence of strictly locally convex hypersurfaces with prescribed curvature and boundary in space forms.
New bounds for non-convex estimators without Bernstein condition.
problem Sharp excess risk bounds for non-convex and improper estimators.
method Exponential-tail local Rademacher complexity risk bounds with offset condition.
result Sharp bounds for non-convex and improper estimators without Bernstein condition.
We introduce characteristics into chromatic homotopy theory. This parallels the prime characteristics in number theory as well as in our earlier work on structured ring spectra and unoriented bordism theory. Here, the K(n)-local Hopkins-Miller classes ζn take the places of the prime numbers, and this allows us to di…
Paper resolves decades-old problem about L-spectra.
problem Identifying L-spectra local information with geometric data. method Proved equivalence of L-orientations and characteristic classes. result Levitt-Ranicki's theory equivalent to Brumfiel-Morgan's classes.
Introduces a framework for rational homotopy theory in diffeological spaces.
problem Challenges in rational homotopy theory for smooth spaces with arbitrary fundamental groups.
method Utilizes local systems over simplicial sets and a model structure for diffeological spaces.
result Establishes an equivalence between fibrewise rational diffeological spaces and algebraic local systems.
We consider the notion of stable isomorphism of bundle gerbes. It has the consequence that the stable isomorphism classes of bundle gerbes over a manifold M are in bijective correspondence with H^3(M, Z). Stable isomorphism sheds light on the local theory of bundle gerbes and enables us to develop a classifying theory …