Study pin manifolds using Clifford linear Dirac operator and KO-theory.
problem Index theory on Pin manifolds.
method Clifford linear Dirac operator and differential KO-theory.
result Systematic treatment of index theory on Pin manifolds.
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48 results for “manifold theory”
New class of singular complex manifolds studied with degenerate theory.
problem Understanding singular complex manifolds.
method Developed degenerate Kodaira-Hodge theory for new class.
result New degenerate theory for singular complex manifolds.
Extends index theory results to manifolds with boundaries.
problem Applying index theory to manifolds with boundaries.
method Extends results of Llarull and Goette-Semmelmann.
result Results extended to manifolds with boundaries.
Main mathematical applications of Frobenius manifolds are in the theory of Gromov - Witten invariants, in singularity theory, in differential geometry of the orbit spaces of reflection groups and of their extensions, in the hamiltonian theory of integrable hierarchies. The theory of Frobenius manifolds establishes rema…
Introduction to gauge theorymath.DG
Lecture notes on gauge theory for manifold invariants.
problem Constructing invariants of manifolds using gauge theory.
method Gauge-theoretic approach to manifold invariants.
result Introduction to Seiberg-Witten gauge theory.
New insights into manifold properties using Seiberg-Witten and harmonic theories.
problem Characterizing properties of 4-manifolds with specific geometric conditions.
method Combining Seiberg-Witten theory on compact manifolds and harmonic theory on non-compact manifolds, with a new argument for asymptotic properties.
result Found a pair of homeomorphic 4-manifolds with distinct geometric properties under Riemannian metrics.
Developed a real sutured Heegaard Floer theory.
problem Real sutured manifolds and their Floer homology.
method Real nice diagrams, combinatorial computation.
result New structural properties distinguishing from sutured Floer homology.
Study of gauge theories on manifolds, including instantons and Chern-Simons.
problem Investigate gauge theories on manifolds.
method Analyze -Yang-Mills and higher-order -Chern-Simons theories on manifolds.
result Emergence of manifolds from anti-self-dual instantons and calculation of partition functions.
Uniform first order interpretation of the second order theory of countable groups of homeomorphismsmath.GR
Uniform interpretation of group theory in manifold homeomorphisms.
problem Understanding group properties in manifold homeomorphisms.
method First order theory interpretation of second order group theory.
result Many group theory problems encoded in homeomorphism groups.
Semisimple 4D field theories can't distinguish smooth 4-manifolds.
problem Detecting exotic smooth structures in 4-manifolds.
method Proving field theories lead to stable invariants, distinguishing only homeomorphic and homotopy equivalent manifolds.
result Semisimple 4D field theories can't distinguish homotopy equivalent 4-manifolds.
Gauge theory for familiesmath.GT
Gauge theory for families helps compare 4-manifold groups.
problem Comparing diffeomorphism and homeomorphism groups of 4-manifolds.
method Gauge theory applied to families of 4-manifolds.
result Improved understanding of group comparisons in 4-manifolds.
We develop a theory of Nobeling manifolds similar to the theory of Hilbert space manifolds. We show that it reflects the theory of Menger manifolds developed by M. Bestvina and is its counterpart in the realm of complete spaces. In particular, the Nobeling manifold characterization conjecture is proven.
Twistor theory applied to special holonomy manifolds.
problem Understanding special holonomy manifolds.
method Application of twistor theory to and Spin(7) manifolds.
result Natural Riemannian twistorial structures on and Spin(7) manifolds.
Survey of spectral theory and dynamics for infinite volume hyperbolic manifolds.
problem Understanding infinite volume asymptotically hyperbolic manifolds.
method Survey of geometry, spectral theory, dynamics, and quantum/classical mechanics.
result Recent results, ideas, and conjectures discussed.
Develops Hodge theory on ALG manifolds, proving existence and vanishing results.
problem Existence and vanishing of certain cohomology groups on ALG manifolds.
method Fredholm Theory for Hodge Laplacian in weighted spaces on ALG manifolds.
result Non-existence of ALG manifolds with non-negative Ricci curvature at infinity.
Survey of gauge theory for families of 4-manifolds.
problem Understanding gauge theory for families of 4-manifolds.
method Developed gauge theory for diffeomorphism groups of 4-manifolds.
result New applications of gauge theory to diffeomorphism groups.
Researchers construct an index map for contact manifolds using K-theory.
problem Constructing an index for maximally hypoelliptic operators on contact manifolds.
method Using Higson's construction for symbol class in K-theory, they derive a series of maps whose induced map in K-theory is the Heisenberg Atiyah-Singer index map.
result Explicit construction of a series of maps leading to the Heisenberg Atiyah-Singer index map.
The paper develops a new theory for knots and 3-manifolds with involutions.
problem Developing a new theory for knots and 3-manifolds with involutions.
method Establishing a version of Seiberg-Witten Floer K-theory for knots and 3-manifolds with involutions.
result 10/8-type inequalities for knots and involutions, yielding lower bounds on stabilizing numbers and relative genera.
This paper extends de Rham theory of smooth manifolds to exploded manifolds. Included are versions of Stokes' theorem, De Rham cohomology, Poincare duality, and integration along the fiber. The resulting cohomology theory is used to define Gromov Witten invariants of exploded manifolds in a separate paper.
Almost complex Hodge theorymath.DG
Study on Hodge theory for almost complex manifolds.
problem Determining Hodge numbers for almost complex manifolds.
method Review and analysis of recent developments in Hodge theory for almost complex manifolds.
result Hodge numbers are almost complex, almost Kähler, or birational invariants in dimension four.
Proves rigidity for maps between manifolds using degree theory and current developments.
problem Lipschitz-volume rigidity for maps between metric manifolds and Riemannian manifolds.
method Degree theory and recent developments of Lipschitz-volume rigidity for integral currents.
result Proves a Lipschitz-volume rigidity result for 1-Lipschitz maps.
New -holonomy manifolds from 5d N=1 theories domain walls.
problem Geometrizing domain walls in 5d N=1 theories.
method Constructing 7-manifolds by fibering a Calabi-Yau over a real line.
result 7-manifolds with -holonomy from domain walls in 5d theories.
We study orientability issues of moduli spaces from gauge theories on Calabi-Yau manifolds. Our results generalize and strengthen those for Donaldson-Thomas theory on Calabi-Yau manifolds of dimensions 3 and 4. We also prove a corresponding result in the relative situation which is relevant to the gluing problem in DT …
Study on deformation theory of nearly G2 manifolds with obstructions.
problem Deformation theory of nearly G2 manifolds with obstructions.
method Study of real Killing spinors and cohomology of nearly G2 manifolds.
result Infinitesimal deformations of nearly G2 structures are obstructed in general.
Monoidally graded manifoldsmath.DG
Extends manifold theory to -graded manifolds.
problem Generalizing manifold theory to non-integer grading.
method Introduces -graded manifolds and proves Batchelor's theorem.
result Proves Batchelor's theorem for -graded manifolds.
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 of knots in a 3-manifold , we first present a local theory for each knot in , which is analogous to local class field theory, and then,…
Unified framework for 3D and 4D manifold and knot theory.
problem Unified understanding of manifold and knot theory.
method Unified correspondence between different subfields of low-dimensional topology.
result Unified algebraic manifestations of 3D and 4D manifold and knot theory.
We use tools from generalized complex geometry to develop the theory of SKT (a.k.a. pluriclosed Hermitian) manifolds and more generally manifolds with special holonomy with respect to a metric connection with closed skew-symmetric torsion. We develop Hodge theory on such manifolds showing how the reduction of the holon…
-covering of a supermanifoldmath.DG
Theory of -graded manifolds and coverings of supermanifolds.
problem Developing a theory for -graded manifolds and coverings of supermanifolds.
method Using tools from representation theory, we introduce and investigate -graded coverings of supermanifolds.
result Theory of -graded coverings of supermanifolds introduced and investigated.
We present a Donaldson-Witten type field theory in eight dimensions on manifolds with holonomy. We prove that the stress tensor is BRST exact for metric variations preserving the holonomy and we give the invariants for this class of variations. In six and seven dimensions we propose similar theories on Calabi…
The paper explores the index theory of sub-Laplacians on higher nilpotent Carnot manifolds.
problem Characterizing the index theory of sub-Laplacians on higher nilpotent Carnot manifolds.
method Analyzes the structure of hypoelliptic sub-Laplacian type operators and provides examples where the index theory is trivial.
result Provides examples where the index theory of sub-Laplacians is trivial in higher degrees of nilpotency.
The paper studies deformations of cohesive modules on complex manifolds.
problem Deformation theory of cohesive modules on compact complex manifolds.
method Development of Kuranishi maps and obstructions for deformations of cohesive modules.
result Generalization of deformation theory for holomorphic vector bundles and coherent sheaves.
Gem theory helps estimate trisection genus of 4-manifolds.
problem Estimating the trisection genus of 4-manifolds.
method Using gem theory, a type of edge-colored graphs dual to colored triangulations.
result Regular genus is an upper bound for trisection genus of closed 4-manifolds.
In anomaly-free quantum field theories the integrand in the bosonic functional integral--the exponential of the effective action after integrating out fermions--is often defined only up to a phase without an additional choice. We term this choice ``setting the quantum integrand''. In the low-energy approximation to M-t…
The abstract discusses constructing 3d N=2 gauge theories using surgeries and M5-branes.
problem Constructing geometrically 3d N=2 gauge theories.
method Using surgeries and M5-branes wrapping on plumbing three-manifolds.
result Various dualities of 3d theories can be interpreted as Kirby moves and equivalent surgeries.
E-string theory reveals modular properties of 4-manifold invariants.
problem Understanding the topology of 4-dimensional manifolds.
method Computing partition function on and verifying its modular properties.
result The partition function of the E-string theory is modular and can be lifted to a topological modular form.
Study on homeomorphism groups of manifolds using set theory.
problem Relationship between set theory and homeomorphism groups of manifolds.
method First-order rigidity, type versus conjugacy, axiom of constructibility, projective determinacy.
result Under V=L, homeomorphism groups of manifolds are first-order rigid and conjugacy class is determined by type.
M-theory compactified on -holonomy manifolds results in 4d supersymmetric gauge theories coupled to gravity. In this paper we focus on the gauge sector of such compactifications by studying the Higgs bundle obtained from a partially twisted 7d super Yang-Mills theory on a supersymmetric three-cycle…
Corners in M-theoryhep-th
M-theory can be defined on closed manifolds as well as on manifolds with boundary. As an extension, we show that manifolds with corners appear naturally in M-theory. We illustrate this with four situations: The lift to bounding twelve dimensions of M-theory on Anti de Sitter spaces, ten-dimensional heterotic string the…
A surgery classification theory is introduced for manifolds of bounded geometry up to quasi-isometry. The Borel conjecture for this theory is proven for flat Euclidean space.
Extends 4D cornered skein theory to surfaces, proving gluing formulas.
problem Formulating gluing formulas for 4-manifolds with corners and boundaries.
method Develops a categorical framework and introduces bicategories for closed surfaces.
result Proves gluing formulas for categories associated with 3-manifolds with boundary.
Proof of orientable 3-manifolds parallelizability using knot theory.
problem Proving all orientable 3-manifolds are parallelizable.
method Using knot theory and relationships between tangent and normal bundles.
result Completion and modification of a proof of Stiefel's theorem.
Theory proves existence of hypersurfaces with prescribed curvature.
problem Existence of hypersurfaces with prescribed mean curvature in noncompact manifolds.
method Developed min-max theory for noncompact manifolds.
result Proved existence of closed and finite area hypersurfaces.
Construct M-Theory lifts of type IIA orientifolds.
problem Lift type IIA orientifolds to M-Theory.
method Construct M-Theory on twisted connected sum G2 manifolds.
result Two building blocks correspond to open and closed string sectors.
Motivated by the definition of homotopy spaces, we develop a new theory of Kuranishi manifolds, closely related to Joyce's recent theory. We prove that Kuranishi manifolds form a -category with invertible -morphisms, and that certain fiber product property holds in this -category. In a subsequent pa…
On topological M-theoryhep-th
We construct a gauge fixed action for topological membranes on -manifold such that its bosonic part is the standard membrane theory in a particular gauge. We prove that quantum mechanically the path-integral in this gauge localizes on associative submanifolds. Moreover on the theory naturally reduces…
We propose a dictionary between geometry of triangulated 3-manifolds and physics of three-dimensional N=2 gauge theories. Under this duality, standard operations on triangulated 3-manifolds and various invariants thereof (classical as well as quantum) find a natural interpretation in field theory. For example, independ…
Link homology theories connect to 4-manifold invariants and TQFTs.
problem Connecting link homology theories to 4-manifold invariants and TQFTs.
method Functorial link homologies, skein modules, handle decompositions.
result Link homology theories become diagram-independent and furnish invariants of 4-manifolds.