T-duality is extended to include exchange of momentum and winding.
problem Exchange of momentum and winding in T-duality.
method Extended Hori isomorphism to exotic differential forms, compatible with Courant algebroids.
result Exchange of momentum and winding numbers in T-duality is achieved.
We define exotic twisted S1-equivariant cohomology for the loop space LZ of a smooth manifold Z via the invariant differential forms on LZ with coefficients in the (typically non-flat) holonomy line bundle of a gerbe, with differential an equivariantly flat superconnection. We introduce the twisted Bismut-Cher…
This paper extends T-duality to exotic chiral de Rham complexes.
problem Extending T-duality to new mathematical structures.
method Introducing exotic chiral de Rham complexes and isomorphisms.
result Established an isomorphism between chiral de Rham complexes.
This work is dedicated to some new exotic homological constructions associated with the different Morse-type inequalities for differential forms and vector fields. It contains also survey of ideas developed by the present author in 1986 for this goal.
The theory of differential forms began with a discovery of Poincare who found conservation laws of a new type for Hamiltonian systems - The Integral Invariants. Even in the absence of non-trivial integrals of motion, there exist invariant differential forms: a symplectic two-form, or a contact one-form for geodesic flo…
New spanning tree model connects knot homology, s-invariant, and exotic discs.
problem Understanding exotic discs in the 4-ball for knots.
method Explicitly defined differential in spanning tree complex, described Rasmussen's s-invariant.
result Identified new infinite family of knots bounding exotic discs.
Different types of nonstandard homology groups based on the various subcomplexes of differential forms are considered as a continuation of the recent authors works. Some of them reflect interesting properties of dynamical systems on the compact manifolds. In order to study them a Special Perturbation Theory in the form…
It is known that the long line supports 2ℵ1 many non-diffeomorphic differential structures. We show that the long plane supports a similar number of exotic differential structures, ie structures which are not merely diffeomorphic to the product of two structures on the factor spaces.
New exotic structures found on 4-manifolds with specific groups.
problem Finding exotic smooth structures on 4-manifolds with certain fundamental groups.
method Constructing new smooth structures using definite intersection forms and fundamental groups.
result Infinitely many exotic smooth structures on 4-manifolds with fundamental group Z/2Z.
Study exotic smoothness structures on R^4 using set-theoretic forcing.
problem Characterize small exotic smooth R^4 structures.
method Assign set-theoretic forcing to Casson handles and use models in Grothendieck toposes.
result Obtain deformation of usual complex functions to noncommutative operators.
The paper finds small exotic 4-manifolds with free abelian groups.
problem Finding small exotic 4-manifolds with specific fundamental groups.
method Pairwise non-diffeomorphic closed irreducible 4-manifolds with free abelian fundamental group of rank less than three.
result Explicit mechanism to compute equivariant intersection form and stabilization result for exotic structures.
An explicit example of an exotic symplectic R6 is given. Together with an earlier known example on R4, this yields an explicit exotic symplectic form on R2n for all n≥2.
Study finite group actions on exotic aspherical space forms.
problem Classify finite group actions on M#Σ where M is a closed aspherical space form and Σ is an exotic n-sphere. method Combines geometric and topological rigidity results with smoothing theory and spectral sequence computations.
result Classification of free actions of finite groups on M#Σ when M is 7-dimensional. The paper calculates a cosmological constant from a small exotic R^4.
problem Determining the cosmological constant from a small exotic R^4.
method Low dimensional differential topology and canonical constructions.
result The cosmological constant is a topological invariant.
Study on stochastic flows on exotic spheres, exploring their properties.
problem Investigating stochastic processes on exotic (m+n+1)-dimensional spheres. method Constructing exotic manifolds from disjoint unions and identifying points using maps.
result Explicit homeomorphisms and stochastic processes on exotic spheres.
An explicit (-1)^n-quadratic form over Z[Z^{2n}] representing the surgery problem E_8 x T^{2n} is obtained, for use in the Bryant-Ferry-Mio-Weinberger construction of 2n-dimensional exotic homology manifolds.
The paper finds infinite families of exotic spheres with free actions.
problem Detecting smooth free S1 and S3 actions on exotic spheres. method Topological modular forms.
result Infinite families of very exotic spheres with free actions.
Study exotic tori and their SL_d(Z) actions, proving many do not admit nontrivial actions.
problem Characterize exotic tori admitting SL_d(Z) actions.
method Compute mapping class groups, analyze homology actions, and prove non-existence of nontrivial actions.
result Many exotic tori do not admit nontrivial SL_d(Z) actions.
The paper provides a differential form interpretation of a theorem about the dimensions of rational homotopy groups of Diff(D^4).
problem Lower bounds of dimensions of rational homotopy groups of Diff(D^4) in terms of graph homology.
method Differential form interpretation and extension to arbitrary even dimensions.
result The proof can be extended to arbitrary even dimensions and made accessible to more readers.
We give a short proof of the (known) result that there are no Kaehler structures on exotic tori. This yields a negative solution to a problem posed by Benson and Gordon. W discuss the symplectic version of the problem and analyze results which yield an evidence for the conjecture that there are no symplectic structures…
A physicist explores exotic 7-spheres using fibre bundles and instantons.
problem Investigating the geometry and topology of exotic 7-spheres.
method Analytical and computational tools from theoretical physics, including fibre bundles and instantons.
result Derives an analytic expression for a family of Riemannian metrics on the Gromoll-Meyer sphere.
We give an algorithm which produces infinitely many pairwise exotic Stein fillings of the same contact 3-manifolds, applying positive allowable Lefschetz fibrations over the disk. As a corollary, for a large class of Stein fillings, we realize the topological invariants (i.e. fundamental group, homology group, homology…
Paper explores using 0-surgery to find exotic 4-manifolds.
problem Distinguishing smooth structures on 4-manifolds.
method Uses 0-surgery homeomorphisms to generate exotic pairs.
result Found 5 topologically slice knots potentially leading to exotic spheres.
Study on 4-manifolds with exotic smooth structures and Z_2 fundamental group.
problem Exploring smooth structures on 4-manifolds with specific fundamental groups.
method Construction of irreducible, smooth, oriented, closed, definite 4-manifolds with Z_2 fundamental group and specific Betti numbers.
result Proves existence of infinitely many smooth structures on definite 4-manifolds with positive second Betti number and Z_2 fundamental group.
Kirby diagrams for exotic R^4's constructed from specific knot complements.
problem Identifying and visualizing exotic R4's using Kirby diagrams. method Provided Kirby diagrams for a family of exotic R4's constructed from specific knot complements. result Generalized Kirby diagrams for a broader family of exotic R4's. This paper explores twisted Lagrangian tori in C^2 and their Hamiltonian stationarity.
problem Understanding the Hamiltonian stationarity of twisted Lagrangian tori in C^2.
method Investigation of differential geometry of twisted tori, including product and Chekanov's exotic tori.
result Only product tori are minimal under Hamiltonian deformations, indicating Chekanov's exotic tori are not area minimal.
New proof of Kondo-Tanaka theorem using geometric measure theory.
problem Existence of special systems of Whitney flat 1-forms on homology manifolds.
method Geometric measure theory and tools from non-smooth analysis.
result Simple new proof of Kondo-Tanaka theorem and its converse.
Study on stochastic flows on 7-dimensional spheres.
problem Stochastic processes on 7-dimensional spheres.
method Isometric stochastic flows of Stratonovich SDE on spheres.
result Properties of stochastic processes on Gromoll-Meyer exotic sphere.
The real form Spin(6,H) in End(R^{32}) of Spin(12,C) in End(C^{32}) is absolutely irreducible and thus satisfies the algebraic identities (40) and (41). Therefore, it also occurs as an exotic holonomy and the associated supermanifold M_g admits a SUSY-invariant polynomial. This real form has been erroneously omitted in…
We present an infinite sequence of smooth embeddings of a connected sum of 6 projective planes in the 4-sphere, which are all ambient homeomorphic, but pairwise ambient non-diffeomorphic. The double covers of the 4-sphere ramified along these surfaces form a family of the exotic $\Bbb CP^2#5\bar{\Bbb CP^2}$ constructed…
Extends exotic embeddings of RP^2 to a larger family and produces homotopy spheres.
problem Constructing exotic embeddings of RP^2 and homotopy spheres.
method Using Montesinos knots and roll-spun knots to prove the existence of homotopy spheres.
result An infinite family of homotopy spheres and homotopy CP^2s are produced.
The paper bounds the genus of surfaces in four-manifolds with indefinite forms.
problem Bounding the genus of surfaces in four-manifolds with indefinite intersection forms.
method Using adjunction inequalities and the 10/8 + 4 theorem, the paper provides methods to bound the genus of surfaces.
result The set of H-slice knots can detect exotic smooth structures on closed 4-manifolds.
There are many theorems in the differential geometry literature of the following sort. Let M be a complete Riemannian manifold with some conditions on various curvatures, diameters, volumes, etc. Then M is homotopy equivalent to a finite CW complex, or M is the interior of a compact, topological manifold with boundary.…
Classifies 4-manifolds with specific properties and fundamental group.
problem Classifying 4-manifolds with fundamental group Z and boundary. method Topological and smooth classification methods, including Hermitian forms and 2-handlebody constructions. result Every Hermitian form over Z[t±1] arises as the equivariant intersection form of exotic smooth 4-manifolds. Synthetic Differential Geometry modifies local space structures with infinitesimal curvature.
problem Infinitesimal curvature in categorical spaces.
method Developed differential geometry on infinitesimal formal manifolds, constructed models and studied their properties.
result Riemann curvature tensor is infinitesimal in infinitesimal level.
New 4-manifolds with exotic diffeomorphisms found.
problem Existence of exotic diffeomorphisms in 4-manifolds.
method Proves existence of infinitely many contractible 4-manifolds with exotic diffeomorphisms.
result Infinitely many contractible 4-manifolds with absolutely exotic diffeomorphisms.
We prove that there exist diffeomorphisms of tori, supported in a disc, which are not isotopic to symplectomorphisms with respect to any symplectic structure. This yields a partial negative answer to a question of Benson and Gordon about the existence of symplectic structures on tori with exotic differential structure.
Convex-cocompact groups in infinite hyperbolic space are deformable.
problem Understanding deformability of convex-cocompact groups in infinite hyperbolic spaces.
method Proving convex-cocompact representations form an open set and using bending to deform them.
result Deformable convex-cocompact representations of surface groups not conjugate to exotic PSL(2,R) representations.
Study shows infinitely many nonnegatively curved metric spaces on exotic 7-manifolds.
problem Investigating nonnegatively curved metrics on exotic 7-manifolds.
method Using Kreck-Stolz invariant and Atiyah-Patodi-Singer index theorem for orbifolds with boundary.
result Moduli space of nonnegatively curved metrics has infinitely many connected components.
The paper proves no exotic actions of diffeomorphism groups on 1-manifolds.
problem The study tackles the exotic actions of diffeomorphism groups on 1-dimensional manifolds.
method The approach involves showing any nontrivial homomorphism has a standard form, with countably many embeddings and conjugate actions.
result The groups Diff^r_c(M) have no countable index subgroups, solving a conjecture of Matsumoto.
Exotic submanifolds in 4-manifolds remain exotic after stabilizations.
problem Constructing exotic codimension-1 submanifolds in 4-manifolds.
method Constructing pairs of exotic codimension-1 submanifolds with diffeomorphic complements and showing they remain exotic after stabilizations.
result Exotic submanifolds remain exotic after any number of stabilizations.
In this paper, we are concerned with interactions between isoparametric theory and differential topology. Two foliations are called equivalent if there exists a diffeomorphism between the foliated manifolds mapping leaves to leaves. Using differential topology, we obtain several results towards the classification probl…
Develops path-dependent optimal transport for exotic derivatives calibration.
problem Calibrating volatility models to exotic derivatives prices.
method Semimartingale optimal transport in path-dependent setting, duality results, dimension reduction via semifiltrations.
result Exact calibration of volatility models to path-dependent derivative prices.
As our main theorem, we prove that a Lipschitz map from a compact Riemannian manifold M into a Riemannian manifold N admits a smooth approximation via immersions if the map has no singular points on M in the sense of F.H. Clarke, where dimM≤dimN. As its corollary, we have that if a bi-Lipschitz homeomo…
Second-order optimization speeds up deep hedging for complex options.
problem Hedging exotic options with market frictions in realistic markets.
method Second-order optimization scheme leveraging pathwise differentiability and Kronecker-factoring.
result Our method optimizes the policy in 1/4 the steps of standard optimization.
Completes classification of high-dimensional manifolds up to exotic sums.
problem Classifying high-dimensional manifolds up to exotic sums.
method Algebraic refinement of intersection forms and Witten genus.
result All but finitely many dimensions are resolved, completing the classification.
The paper proves manifold diffeomorphism under certain curvature conditions.
problem Proving diffeomorphism between manifolds with specific curvature properties.
method Using radial curvatures and L1-norm comparison to establish diffeomorphism. result Closed Riemannian manifolds with single cut points are diffeomorphic under certain curvature conditions.
Study exotic Dehn twists in 4-manifolds, producing first known exotic diffeomorphisms.
problem Detecting exotic diffeomorphisms in 4-manifolds.
method 2-parameter families Seiberg-Witten theory over RP2. result Constructed the smallest closed 4-manifold with exotic diffeomorphisms.