arXiv research

Study shows non-equivariant and equivariant non-orientable 4-genus of periodic knots can differ.

Study on equivariant Heegaard genus of reducible 3-manifolds with group actions.

The paper defines new knot genera and finds bounds for stabilization distances.

New invariants refine link homology, showing large genus differences.

Study on knots, genera, and algebraic concordance groups.

Study equivariant 4-genus of knots in symmetric 4-manifolds.

Study knot Floer homology to create concordance invariants and slice genus bounds.

The Witten class is derived from equivariant cohomology of a conformal loop space.

Added examples of S^1-manifolds with finite 2nd homotopy group and non-zero A-genus.

New findings on knot genera using advanced techniques.

In this paper we study equivariant constrained Willmore tori in the 3-sphere. These tori admit a 1-parameter group of Möbius symmetries and are critical points of the Willmore energy under conformal variations. We show that the associated spectral curve of an equivariant torus is given by a double covering of $\mathbb …

Defines a new homomorphism for strongly invertible knots, proving equivariant algebraic concordance.

Study on invariant Seifert surfaces for strongly invertible knots, showing large gaps in genus.

Generalizes Molino's theory for Riemannian foliations.

Constructs a unique surface in a ball with specific properties.

We show that real and imaginary parts of equivariant spherical harmonics on $S^3$ have almost surely a single nodal component. Moreover, if the degree of the spherical harmonic is $N$ and the equivariance degree is $m$, then the expected genus is proportional to $m \left(\frac{N^2 - m^2}{2} + N\right)$. Hence if $\fra…

The paper calculates the equivariant genus for a specific type of knot.

Nonorientable spanning surfaces of periodic knots can have arbitrarily high first Betti number.

Equivariant trisections for group actions on 4-manifolds are introduced and studied.

New minimal surfaces found in 3D sphere with low genus.

By using the equivariant localization formula of toric varieties. We prove the vanishing of the Witten genus of some string complete intersections in smooth toric varieties.

Characterizes area-minimizing maps for surfaces of genus ≥ 2.

New knot concordance invariants from Seiberg-Witten theory bound slice genus.

Develops equivariant grid homology for strongly invertible knots.

We show that all GL(2, R)-equivariant point markings over orbit closures of primitive genus two translation surfaces arise from marking pairs of points exchanged by the hyperelliptic involution, Weierstrass points, or the golden points in the golden eigenform locus. As corollaries, we classify the holomorphically varyi…

Equivariant localization techniques give a rigorous interpretation of the Witten genus as an integral over the double loop space. This provides a geometric explanation for its modularity properties. It also reveals an interplay between the geometry of double loop spaces and complex analytic elliptic cohomology. In part…

New minimal hypersphere found in 4-sphere solving Bernstein problem.

We develop an equivariant min-max theory as proposed by Pitts-Rubinstein in 1988 and then show that it can produce many of the known minimal surfaces in $\mathbb{S}^3$ up to genus and symmetry group. We also produce several new infinite families of minimal surfaces in $\mathbb{S}^3$ proposed by Pitts-Rubinstein. These …

A knot $K$ is definite if $|σ(K)| = 2g(K)$. We prove that the quotient of a definite periodic knot is definite by considering equivariant minimal genus Seifert surfaces.

We construct geometric generators of the effective $S^1$-equivariant Spin- (and oriented) bordism groups with two inverted. We apply this construction to the question of which $S^1$-manifolds admit invariant metrics of positive scalar curvature. It turns out that, up to taking connected sums with several copies of the …

Study cyclic group actions on spin 4-manifolds with boundary using Seiberg-Witten theory.

The paper proves an adjunction inequality for Real embedded surfaces in 4-manifolds.

We define and study the signature, A-hat genus and higher signatures of the quotient space of an $S^1$-action on a closed oriented manifold. We give applications to questions of positive scalar curvature and to an Equivariant Novikov Conjecture.

Monoidal categorifies genus zero skein algebra using K-theory.

Witt algebra acts on Khovanov-Rozansky homology of links.

The paper finds a special hypersurface in a manifold with positive Ricci curvature.

The paper studies almost complex torus manifolds using graphs and Hirzebruch genera, proving properties of their fixed points and cohomology.

A projection maps geodesic currents to Teichmüller space.

In this paper, we construct spines, i.e., $\Mod_g$-equivariant deformation retracts, of the Teichmüller space $\T_g$ of compact Riemann surfaces of genus $g$. Specifically, we define a $\Mod_g$-stable subspace $S$ of positive codimension and construct an intrinsic $\Mod_g$-equivariant deformation retraction from $\T_g$…

Maps on surfaces can be embedded into spheres with minimal dimensions.

In this article we introduce a definition for the moduli space of equivariant minimal immersions of the Poincaré disc into a non-compact symmetric space, where the equivariance is with respect to representations of the fundamental group of a compact Riemann surface of genus at least two. We then study this moduli space…

Study circle actions on unitary manifolds with discrete fixed points.

Consider a compact prequantizable symplectic manifold M on which a compact Lie group G acts in a Hamiltonian fashion. The ``quantization commutes with reduction'' theorem asserts that the G-invariant part of the equivariant index of M is equal to the Riemann-Roch number of the symplectic quotient of M, provided the quo…

Given two Fuchsian representations $ρ_l$ and $ρ_r$ of the fundamental group of a closed oriented surface $S$ of genus $\geq 2$, we study the relation between Lagrangian submanifolds of $M_ρ=(\mathbb{H}^2/ρ_l(π_1(S)))\times (\mathbb{H}^2/ρ_r(π_1(S)))$ and $ρ$-equivariant embeddings $σ$ of $\widetilde S$ into Anti-de Sit…

This paper shows how to approximate CAT(-1) representations by Fuchsian ones.

The data of a "2D field theory with a closed string compactification" is an equivariant chain level action of a cell decomposition of the union of all moduli spaces of punctured Riemann surfaces with each component compactified as a pseudomanifold with boundary. The axioms on the data are contained in the following ass…

Let $(M,Q)$ be a compact, three dimensional manifold of strictly negative sectional curvature. Let $(Σ,P)$ be a compact, orientable surface of hyperbolic type (i.e. of genus at least two). Let $θ:π_1(Σ,P)\toπ_1(M,Q)$ be a homomorphism. Generalising a recent result of Gallo, Kapovich and Marden concerning necessary and …

For each integer $g\geq 1$ we use variational methods to construct in the unit $3$-ball $B$ a free boundary minimal surface $Σ_g$ of symmetry group $\mathbb{D}_{g+1}$. For $g$ large, $Σ_g$ has three boundary components and genus $g$. As $g\rightarrow\infty$ the surfaces $Σ_g$ converge as varifolds to the union of the d…