Invariants of knot complements remain unchanged under a specific transformation.
problem Invariance of knot complements under a special transformation.
method Bordered Heegaard Floer invariant and elliptic involution on knot complements.
result The invariant remains unchanged under the elliptic involution.
Researchers compute cohomology of Lie groups using Lie algebras.
problem Computing cohomology of left-invariant elliptic and hypocomplex structures on compact Lie groups.
method Used the Leray spectral sequence connecting Lie algebras to Dolbeault cohomology of homogeneous manifolds.
result Cohomology can be computed purely algebraically.
Every element of PU(2,1) can be decomposed into at most 4 special elliptic isometries.
problem Understanding the length of elements in PU(2,1) relative to special elliptic isometries.
method Generalizing the involution length of the complex hyperbolic plane, calculating the α-length of PU(2,1) and describing decompositions of isometries. result Every element of PU(2,1) can be decomposed into at most 4 special elliptic isometries.
We generalize McShane's identity for the length series of simple closed geodesics on a cusped hyperbolic surface to hyperbolic cone-surfaces (with all cone angles ≤π), possibly with cusps and/or geodesic boundary. In particular, by applying the generalized identity to the orbifolds obtained from taking the quotien…
By carrying out a rational transformation on the base curve CP1 of the Seiberg-Witten curve for N=2 supersymmetric pure SU(2)-gauge theory, we obtain a family of Jacobian elliptic K3 surfaces of Picard rank 17. The isogeny relating the Seiberg-Witten curve for pure SU(2)-ga…
Let M be a Spin-manifold with S1-action and let σ∈S1 be of finite order. We show that the indices of certain twisted Dirac operators vanish if the action of σ has sufficiently large fixed point codimension. These indices occur in the Fourier expansion of the elliptic genus of M in one of its cusps. As …
The elliptic Hall algebra governs torus link homology.
problem Proving the elliptic Hall algebra's role in torus link homology.
method Developed a rational Shareshian-Wachs involution to prove the symmetry of generating functions.
result Resolved a conjecture by establishing the elliptic Hall algebra's role in torus link homology.
The paper studies quadratic Poisson structures on Lie algebras, finding a 10-parametric family.
problem Compatibility of quadratic Poisson structures with linear structures on Lie algebras.
method Developed general theory and studied families of functions in involution.
result Found a 10-parametric family of quadratic Poisson structures on $\gl(3)^*$.
New method simplifies cohomology computation for specific Lie group structures.
problem Computing cohomology for hypocomplex structures on compact Lie groups.
method Dual of Fréchet-Schwartz spaces theory applied to hypocomplex structures.
result Top-degree cohomology can be computed using only left-invariant forms.
New examples show non-commensurable 3-manifolds with similar geodesic lengths.
problem Whether spectrally similar hyperbolic 3-manifolds are commensurable.
method Constructing non-commensurable 3-manifolds with shared length spectra up to a large portion.
result Found examples of incommensurable 3-manifolds with length spectra agreeing up to length n.
The paper classifies discrete complex hyperbolic triangle groups.
problem Classifying discrete complex hyperbolic triangle groups.
method Analyzing complex hyperbolic spaces and isometries.
result Classifies discrete complex hyperbolic (n,∞,∞)-triangle groups for n=3,4,5. We construct a class of stable SU(5) bundles on an elliptically fibered Calabi-Yau threefold with two sections, a variant of the ordinary Weierstrass fibration, which admits a free involution. The bundles are invariant under the involution, solve the topological constraint imposed by the heterotic anomaly equation and …
We prove that the sequence of projective representations of the mapping class group obtained from the projective flat connection in the SU(n)-Verlinde bundles over Teichmuller space is asymptotically faithful, that is the intersection over all levels of the kernels of these representations is trivial, whenever the genu…
New exotic 4-manifolds with even b2+ and Z/2Z fundamental group.
problem Creating new exotic smooth structures on 4-manifolds with specific fundamental groups.
method Using double node surgery and rational blowdown constructions on elliptic fibrations with a free involution.
result Construction of infinitely many irreducible exotic smooth structures.
Classifies involutions on spherical 3-manifolds.
problem Classifying involutions on spherical 3-manifolds.
method Geometric approach to conjugacy classification.
result Insights into topological properties of involutions.
Study classifies Calabi-Yau threefolds with non-Gorenstein involutions.
problem Understanding non-Gorenstein involutions on Calabi-Yau threefolds.
method Classification of Calabi-Yau threefolds with specific properties.
result Classification of Calabi-Yau threefolds with Picard rank one and non-Gorenstein involutions.
The paper studies circular evolutes and involutes of framed curves in Euclidean space.
problem Investigating properties of framed curves and their evolutes and involutes.
method Definition and analysis of circular evolutes and involutes of framed curves, properties of normal surfaces, and their relations.
result Circular evolutes and involutes of framed curves are opposite operations under suitable assumptions, similar to fronts in the Euclidean plane.
Complex hyperbolic isometries are decomposed into involutions, revealing specific lengths.
problem Understanding the structure of complex hyperbolic isometries.
method Analyzing decompositions of complex hyperbolic isometries into products of involutions.
result PU(2,1) has involution length 4 and commutator length 1, and PU(n,1) has involution length at most 8 for all n ≥ 3.
New formula for dual knots using involutions.
problem Understanding dual knots and their transformations.
method Involutive analog of knot surgery formula.
result Computed local equivalence class for involutive dual knots.
The study classifies involutions on del Pezzo surfaces.
problem Classifying involutions on del Pezzo surfaces.
method Mapping class group theory and hyperbolic reflection groups.
result A complete classification of involutions on del Pezzo surfaces.
Characterizes the Legendre involution on generic frontals.
problem Identifying the Legendre involution on a specific class of frontals.
method Analyzes generic frontals under mild assumptions and uses complexification.
result Any involution with the same fixed points as the Legendre involution is the Legendre involution.
The paper defines conditions for good involutions in generalized Alexander quandles.
problem Determining conditions for good involutions in generalized Alexander quandles.
method Analyzing the structure of generalized Alexander quandles and their involutions.
result Classification of all good involutions in connected generalized Alexander quandles.
Study proves naturality and functoriality in a type of Heegaard Floer homology.
problem Proving naturality and functoriality in a specific type of Heegaard Floer homology.
method Used the doubling model for the involution and variations to prove results.
result First-order naturality of involutive Heegaard Floer homology proved.
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.
The study proves symplectic quandles cannot have good involutions.
problem Existence of good involutions in symplectic quandles.
method Investigation of necessary and sufficient conditions for good involutions.
result Nonexistence of good involutions in symplectic quandles.
Involution algebroids extend Lie algebroids to tangent categories.
problem Extending Lie algebroid theory to tangent categories.
method Defining involution algebroids that replace the Jacobi identity with a Yang-Baxter-like equation.
result Every Lie algebroid is an involution algebroid and every involution algebroid admits a Lie bracket.
Classifies dissecting involutions on symmetric spaces.
problem Identifying dissecting involutions on symmetric spaces.
method Analyzing properties of involutions and fixed point sets.
result Characterizes dissecting involutions on specific symmetric spaces.
Involutive bordered Floer homology computes 3-manifold invariants.
problem Computing involutive Heegaard Floer homology for 3-manifolds.
method Bordering technique to extend involutive HF-hat, algorithmic approach, mapping class group action computation.
result Involutive HF-hat satisfies a surgery exact triangle, computed for many knots.
Involutions generate mapping class groups of infinite surfaces.
problem Generating involutions for mapping class groups of infinite surfaces.
method Analyzing infinite surfaces with n ends, showing involutions generate groups for n ≥ 6 and n ≥ 3.
result Involutions generate mapping class groups for n ≥ 6 and n ≥ 3.
Minimal involutions generate a subgroup of nonorientable surfaces.
problem Generating a minimal set of involutions for a specific subgroup.
method Obtained a minimal generating set of involutions.
result Minimal involutions for the level 2 subgroup of a nonorientable surface.
Knot concordance linked to involutive knot Floer homology equivalence.
problem Understanding knot concordance through algebraic topology.
method Involutive knot Floer homology and stable equivalence.
result Concordant knots have equivalent involutive knot Floer complexes.
Study exact surgery formula in involutive Heegaard Floer homology.
problem Understanding integer homology spheres through knot surgery.
method Using doubling model of involution and mapping cone formula.
result Examples of non-homology cobordant integer homology spheres.
Three involutions generate mapping class groups of large surfaces.
problem Generating mapping class groups with minimal involutions.
method Proved using group theory for surfaces of genus ≥8.
result Mapping class groups are generated by three involutions for large surfaces.
Real slices of parabolic opers on Riemann surfaces are studied.
problem Understanding the fixed-point locus of involutions on parabolic opers.
method Investigated the space of parabolic SL(r,C)-opers and their involutions.
result Fixed-point loci of involutions on different descriptions of parabolic opers coincide.
Anti-symplectic involutions connect a sphere in a symplectic surface.
problem Understanding involutions on Lagrangian spheres in symplectic quadrics.
method Using Hamiltonian isotopy to show connections between involutions.
result Anti-symplectic involutions are Hamiltonian isotopic.
Classifies periodic diffeomorphisms on surfaces commuting with specific involutions.
problem Classifying periodic automorphisms on surfaces that commute with certain involutions.
method Analyzes irreducible periodic automorphisms on surfaces Σg that commute with hyperelliptic involutions. result A classification up to conjugacy for irreducible periodic automorphisms of a surface Σg commuting with involutions ι such that Σg/⟨ιangle is homeomorphic to T2. New knot invariants from Floer theory help bound knot three-genus.
problem Bounding the three-genus of knots.
method Involutive Heegaard Floer knot theory.
result Two new concordance invariants defined.
Paper studies involutions generating the twist subgroup of nonorientable surfaces.
problem Generating the twist subgroup by involutions on nonorientable surfaces.
method Analyzes involutions to find the smallest generating sets.
result Provides generating sets of involutions with minimal elements.
Let Σg,b denote a closed orientable surface of genus g with b punctures and let Mod(Σg,b) denote its mapping class group. In [Luo] Luo proved that if the genus is at least 3, Mod(Σg,b) is generated by involutions. He also asked if there exists a universal upper bound, indepe…
Computed involutive knot invariants for specific pretzel knots.
problem Computing involutive knot invariants for a specific class of knots.
method Computed involutive knot invariants for pretzel knots of the form P(-2,m,n) with m and n odd and ≥ 3.
result Computed involutive invariants for a specific class of knots.
Presentations for involutions on non-orientable surfaces up to genus 5.
problem Representing involutions on non-orientable surfaces.
method Dehn twist--crosscap slide presentations.
result Presentations for involutions on non-orientable surfaces of genera up to 5.
Three involutions generate the mapping class group for surfaces of genus 6 or more.
problem Generating the mapping class group with minimal involutions.
method Proving the group is generated by three involutions for surfaces of genus 6 or more.
result The mapping class group is generated by three involutions for surfaces of genus 6 or more.
The paper defines evolutes and involutes for framed curves and their properties.
problem Defining evolutes and involutes for framed curves with singular points.
method Using the theory of framed curves and Bertrand type curves.
result Conditions for evolutes and involutes being inverse operations of framed curves.
In this article, we classify all involutions on S^6 with 3-dimensional fixed point set. In particular, we discuss the relation between the classification of involutions with fixed point set a knotted 3-sphere and the classification of free involutions on homotopy CP^3's.
This thesis proves how to generate a specific group using involutions.
problem Generating the mapping class group of non-orientable surfaces by involutions.
method Using a set of involutions, the thesis provides the number of elements needed for generation based on the surface's genus and punctures.
result The mapping class group of non-orientable surfaces can be generated by a fixed number of involutions, independent of the surface's genus and punctures.
New supergroups created from odd involutions in supergeometry.
problem Creating new supergroups from odd involutions.
method Constructing ν−Grassmannians by gluing ν−domains with an odd involution. result Introduced a supergroup associated with odd involutions.
Study of K3 surfaces with special involutions and singularities.
problem Characterizing K3 surfaces with specific symmetries.
method Analysis of K3 surfaces with commuting non-symplectic involutions.
result Identification of 320 examples of K3 surfaces with various singularities.
Finite groups act on 3-manifolds with specific involution properties.
problem Understanding finite groups acting on hyperelliptic 3-manifolds.
method Analyzing finite groups containing hyperelliptic involutions with specific fixed-point sets.
result Simple groups containing hyperelliptic involutions are isomorphic to PSL(2,q) or four other small groups.