Paper defines end Khovanov homology to detect exotic planes.
problem Detecting exotic Lagrangian and symplectic planes in 4-space.
method Define and apply end Khovanov homology to surfaces in 4-space.
result First known examples of exotic Lagrangian and symplectic planes found.
Study of symplectic manifolds degenerating into singular spaces.
problem Understanding degenerations of symplectic manifolds into singular spaces.
method Topological framework focusing on Lagrangian manifolds and holomorphic membranes.
result Degenerations into singular toric varieties yield exotic Lagrangian tori.
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.
In dimensions congruent to 1 modulo 4, we prove that the cotangent bundle of an exotic sphere which does not bound a parallelisable manifold is not symplectomorphic to the cotangent bundle of the standard sphere. More precisely, we prove that such an exotic sphere cannot embed as a Lagrangian in the cotangent bundle of…
New surgery operation preserves monotonicity of Lagrangians.
problem Preserving monotonicity of Lagrangians in surgery operations.
method BSP surgery, wall-crossing formula for disk-potentials.
result BSP surgery can preserve monotonicity of Lagrangians.
The paper proves symplectic neighbourhood theorems for stratified subspaces.
problem Finding symplectic neighbourhoods of stratified subspaces.
method Analogy with Weinstein's neighbourhood theorem, strong version of Moser's trick, and tubular neighbourhood theorem.
result Generalization of existing constructions for exotic Lagrangians.
We introduce and discuss notions of regularity and flexibility for Lagrangian manifolds with Legendrian boundary in Weinstein domains. There is a surprising abundance of flexible Lagrangians. In turn, this leads to new constructions of Legendrians submanifolds and Weinstein manifolds. For instance, many closed n-mani…
We compute the classical and quantum cohomology rings of the twistor spaces of 6-dimensional hyperbolic manifolds and the eigenvalues of quantum multiplication by the first Chern class. Given a half-dimensional totally geodesic submanifold we associate, after Reznikov, a monotone Lagrangian submanifold of the twistor s…
We describe an iterative construction of Lagrangian tori in the complex Grassmannian Gr(k,n), based on the cluster algebra structure of the coordinate ring of a mirror Landau-Ginzburg model proposed by Marsh-Rietsch. Each torus comes with a Laurent polynomial, and local systems controlled by the k-va…
Develops a framework for consistent pricing of interest rate derivatives.
problem Consistent pricing of bivariate interest rate exotics across interconnected markets.
method Schrödinger optimal transport problem with constraints.
result Demonstrates practical applicability and no-arbitrage bounds computation.
We show that, for certain families φs of diffeomorphisms of high-dimensional spheres, the commutator of the Dehn twist along the zero-section of T∗Sn with the family of pullbacks φs∗ gives a noncontractible family of compactly-supported symplectomorphisms. In particular, we find example…
We show that, for any two orientable smooth open 4-manifolds X0,X1 which are homeomorphic, their cotangent bundles T∗X0,T∗X1 are symplectomorphic with their canonical symplectic structure. In particular, for any smooth manifold R homeomorphic to R4, the standard Stein structure on T∗R is Stei…
It is known since 40 years old paper by M. Keane that minimality is a generic (i.e. holding with probability one) property of an irreducible interval exchange transformation. If one puts some integral linear restrictions on the parameters of the interval exchange transformation, then minimality may become an "exotic" p…
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.
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.
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.
New methods distinguish exotic 4-manifolds using Heegaard Floer homology.
problem Distinguishing exotic 4-manifolds from standard ones.
method Using Heegaard Floer homology to define new invariants.
result New invariants remain distinct in covers and can distinguish exotic manifolds.
Constructs exotic proper 2-knots from open 2-handles.
problem Proving topological equivalence of exotic proper knotted surfaces.
method Flexible construction of exotic open 2-handles, distinguishing through genus functions and end Floer homology.
result Constructs infinite families of irreducible exotic proper knotted surfaces.
The paper presents a new method to create exotic 4-manifolds and surfaces that remain exotic after stabilization.
problem Stabilization of exotic 4-dimensional phenomena and knotted surfaces.
method Elementary approach to constructing exotic 4-manifolds and surfaces, including examples in closed, simply connected 4-manifolds.
result The construction yields exotic surfaces in the 4-ball that remain exotic after stabilization, detected by Khovanov homology.
New actions found on exotic spheres using group theory.
problem Understanding smooth transformations on exotic spheres.
method Recent progress in stable homotopy groups of spheres and group theory.
result Smooth circle and cyclic group actions on exotic spheres produced.
The paper constructs exotic knotted surfaces and curves in 4-manifolds.
problem Understanding exotic knotted surfaces and curves in 4-manifolds.
method Local knotting and construction of surfaces and curves.
result First examples of exotically knotted complex curves and symplectic 2-spheres.
Paper finds exotic diffeomorphisms on specific 4-manifolds.
problem Understanding exotic diffeomorphisms on 4-manifolds with b2+=2. method Comparing winding numbers of parameter families.
result 2\mathbb{C}\mathbb{P}^2 \# 10 (-\mathbb{C}\mathbb{P}^2) admits exotic diffeomorphisms.
Detects exotic embeddings in 4-manifolds with boundary.
problem Detecting exotic diffeomorphisms in 4-manifolds with boundary.
method Defined family versions of Kronheimer-Mrowka invariants and used them to find exotic embeddings.
result First example of exotic 3-spheres in a smooth closed 4-manifold with diffeomorphic complements.
Study exotic phenomena in small 4-manifolds using diffeomorphisms.
problem Understanding exotic smooth structures and embeddings in 4-manifolds.
method Using constraints from Seiberg-Witten theory on diffeomorphisms.
result New methods to detect and bound exotic phenomena.
We show how to construct absolutely exotic smooth structures on compact 4-manifolds with boundary, including contractible manifolds. In particular, we prove that any compact smooth 4-manifold W with boundary that admits a relatively exotic structure contains a pair of codimension-zero submanifolds homotopy equivalent t…
Since the first work on exotic smoothness in physics, it was folklore to assume a direct influence of exotic smoothness to quantum gravity. In the second paper, we calculate the "smoothness structure" part of the path integral in quantum gravity for the exotic R^4 as non-compact manifold. We discuss the influence of th…
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.
4-manifolds can be exotic after connected sum with S^2 x S^2.
problem Existence of exotic contractible 4-manifolds.
method Constructed a cork that remains exotic after connected sum with S^2 x S^2.
result Existence of exotic pair of contractible 4-manifolds.
New exotic 4-manifolds found with small trisection genus.
problem Finding exotic 4-manifolds with small trisection genus.
method Constructing genus-3 relative trisections for contractible 4-manifolds.
result Exotic 4-manifolds with boundary have trisection genus of 4.
This paper constructs exotic diffeomorphisms on reducible 4-manifolds with odd b+.
problem Detecting exotic diffeomorphisms on reducible 4-manifolds with odd b+.
method Using a families Bauer-Furuta invariant and pin-cobordism.
result Constructs exotic diffeomorphisms on reducible 4-manifolds with odd b+.
In this paper we introduce exotic twisted T-equivariant K-theory of loop space LZ depending on the (typically non-flat) holonomy line bundle LB on LZ induced from a gerbe with connection B on Z. We also define exotic twisted T-equivariant Chern character that maps the exotic tw…
Classifies geodesic planes outside convex core of geometrically finite 3-manifolds.
problem Classifying geodesic planes in geometrically finite 3-manifolds.
method Constructive proof involving exotic rays and roofs.
result Existence of exotic roofs depends on the existence of exotic rays and bending lamination properties.
Exotic diffeomorphisms found on complex surfaces and 4-manifolds.
problem Finding exotic diffeomorphisms on 4-manifolds.
method Minimal complex surfaces and spin 4-manifolds with S3 boundary. result First known instances of exotic diffeomorphisms of irreducible 4-manifolds.
New exotic 4D spaces found using knot slicing techniques.
problem Finding non-diffeomorphic exotic 4D spaces.
method Using RBG links to create slice knots with non-diffeomorphic complements.
result Distinguished new exotic 4D spaces using end Floer homology.
Study of torus surgeries on knot traces, finding exotic surfaces and traces.
problem Understanding exotic surfaces and traces through torus surgeries.
method Realizing annulus twisting as torus surgery, using key technical insight.
result Exotic elliptic surfaces and traces discovered, improving known geography.
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.
In this article we construct a minimal symplectic 4-manifold R that has small Euler characteristic (e(R)=8) and two essential Lagrangian tori with nice properties. These properties make R particularly suitable for constructing interesting examples of symplectic manifolds with small Euler characteristic. In particular, …
New results on localization of exotic diffeomorphisms in 4-manifolds.
problem Understanding when groups of exotic diffeomorphisms can be localized to smaller embedded submanifolds.
method Analyzing families Seiberg-Witten theory, Dehn twists, and properties of Seifert fibered homology spheres.
result Exotic diffeomorphisms cannot be localized to topologically embedded rational homology balls or homology spheres.
New exotic 4D spaces with nontrivial mappings.
problem Creating exotic copies of R4 with nontrivial mapping class groups. method Modification of diffeomorphism corks to make exteriors simply-connected.
result Construct exotic copies of R4 with mapping class groups of arbitrarily large rank. Construct non-split 2-component links in S4 to produce exotic pairs of 4-manifolds
problem Construct non-split 2-component links in S4 method Construct non-split 2-component links in S4 result Obtain exotic pairs of simply connected, definite 4-manifolds with boundary
New method uses Khovanov homology to distinguish exotic 4-manifolds.
problem Distinguishing exotic compact, orientable 4-manifolds.
method Introduces a new, simple way of using Khovanov homology.
result First analysis-free proof of existence of exotic Mazur manifolds.
The paper creates exotic 4-manifold structures with a specific group.
problem Producing exotic structures on 4-manifolds with infinite dihedral fundamental group.
method Using specific conditions on b2+ and b2−, the paper constructs these structures. result The existence of infinite exotic structures on 4-manifolds with infinite dihedral fundamental group.
We consider the action of symplectic monodromy on chain-level enhancements of quantum cohomology. First, we construct a family version of A∞-structure on quantum cohomology (this should morally correspond to Hochschild cohomology of a "family of Fukaya categories over the circle"). Following Kaledin, we look at…
Exotic diffeomorphism survives stabilizations on a contractible 4-manifold.
problem Finding exotic diffeomorphisms on contractible 4-manifolds.
method Developed a Pin(2) × Z_2-equivariant refinement for computing Seiberg-Witten Floer homotopy types.
result Constructed an exotic diffeomorphism that survives two stabilizations.
Exotic hypercomplex structures on a torus are proven to not exist.
problem Existence of exotic hypercomplex structures on a torus.
method Classification of complete flat affine structures on real tori using the Obata connection.
result Exotic hypercomplex structures on a torus do not exist.
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.
New module constructs exotic surfaces in 4-manifolds.
problem Constructing exotic surfaces in 4-manifolds.
method Using Bar-Natan Khovanov homology and skein lasagna gluing maps.
result Exotic surfaces require at least one internal stabilization.
Paper extends Miyazawa's construction to more 4-manifolds, finding exotic involutions and embeddings.
problem Finding exotic involutions and embeddings in 4-manifolds.
method Real Seiberg-Witten theory.
result Many infinite families of exotic involutions and embeddings.