arXiv research

Study nearly parallel G2-structures with torus symmetry using multi-moment maps.

Generalizes symmetries of curved manifolds.

Ancient solutions to Ricci flow on torus bundles have additional symmetries.

Classifies Legendrian torus and cable links, revealing symmetries and invariants.

New categorified homology expressions for torus knots and links.

The study connects matroids with torus representations and positive curvature.

Curved 10-manifolds with torus symmetry are spheres or complex projective spaces.

We propose a spectral curve describing torus knots and links in the B-model. In particular, the application of the topological recursion to this curve generates all their colored HOMFLY invariants. The curve is obtained by exploiting the full Sl(2, Z) symmetry of the spectral curve of the resolved conifold, and should …

We prove that if a closed, smooth, simply-connected 4-manifold with a circle action admits an almost non-negatively curved sequence of invariant Riemannian metrics, then it also admits a non-negatively curved Riemannian metric invariant with respect to the same action. The same is shown for torus actions of higher rank…

Metrics of exceptional holonomy are vacuum solutions to the Einstein equation. In this paper we describe manifolds with holonomy contained in Spin(7) preserved by a three-torus symmetry in terms of tri-symplectic geometry of four-manifolds. These complement examples that have appeared in the context of domain wall prob…

The paper explores hidden torus symmetries in integrable systems and their stability.

In 1965, S.-S. Chern posed a question concerning the extent to which fundamental groups of manifolds admitting positive sectional curvature look like spherical space form groups. The original question was answered in the negative by Shankar in 1998, but there are a number of positive results in the presence of symmetry…

Let $\mathcal{M}_{0}^n$ be the class of closed, simply-connected, non-negatively curved Riemannian manifolds admitting an isometric, effective, isotropy-maximal torus action. We prove that if $M\in \mathcal{M}_{0}^n$, then $M$ is equivariantly diffeomorphic to the free linear quotient by a torus of a product of spheres…

In this paper we summarize our recent work in the construction of Lagrangian torus fibrations for Calabi-Yau hypersurfaces in toric varieties and the symplectic Strominger-Yau-Zaslow conjecture, together with some new development. It is submittded to the Proceedings of the Conference in Symplectic Geometry and Mirror S…

Geodesic rays constructed in Kähler metrics with torus symmetry.

We study fixed points of smooth torus actions on closed manifolds using fixed point formulas and equivariant elliptic genera. We also give applications to positively curved Riemannian manifolds with symmetry.

Proves uniqueness and existence of toric gravitational instantons.

We consider nearly Kähler 6-manifolds with effective 2-torus symmetry. The multi-moment map for the $T^2$-action becomes an eigenfunction of the Laplace operator. At regular values, we prove the $T^2$-action is necessarily free on the level sets and determines the geometry of three-dimensional quotients. An inverse con…

Study of MHD equilibria with orientation-reversing symmetry, showing all orbits are periodic.

Explains examples of Lagrangian flow with circle symmetry.

Defines discrete symmetry of manifolds and proves bounds on its value.

We show that a closed simply connected 8-manifold (9-manifold) of positive sectional curvature on which a 3-torus (4-torus) acts isometrically is homeomorphic to a sphere, a complex projective space or a quaternionic projective plane (sphere). We show that a closed simply connected 2m-manifold (m>4) of positive section…

A new method analyzes topological B-model on a torus using doubled geometry.

Classifies knots that bound equivariant surfaces with free symmetries.

New R-equivalence classes found for torus knot diagrams.

Homological mirror symmetry for toric Fano surfaces using Morse homotopy.

Symmetry of geometrical figures is reflected in regularities of their algebraic invariants. Algebraic regularities are often preserved when the geometrical figure is topologically deformed. The most natural, intuitively simple but mathematically complicated, topological objects are Knots. We present in this papers seve…

This is a companion paper to earlier work of the authors, which interprets the Heegaard Floer homology for a manifold with torus boundary in terms of immersed curves in a punctured torus. We prove a variety of properties of this invariant, paying particular attention to its relation to knot Floer homology, the Thurston…

Study minimizes CR surfaces in Heisenberg group with rotational symmetry.

This is the first part of a series of papers where we compute Euler characteristics, signatures, elliptic genera, and a number of other invariants of smooth manifolds that admit Riemannian metrics with positive sectional curvature and large torus symmetry. In the first part, the focus is on even-dimensional manifolds i…

Proves Willmore conjecture for surfaces with specific symmetries.

In this paper we discuss two major conjectures in Mirror Symmetry: Strominger-Yau-Zaslow conjecture about torus fibrations, and the homological mirror conjecture (about an equivalence of the Fukaya category of a Calabi-Yau manifold and the derived category of coherent sheaves on the dual Calabi-Yau manifold). Our point…

The study characterizes embedded minimal hypersurfaces in $S^{n+1}$ with symmetries.

In this paper we prove a mirror symmetry conjecture based on the work of Brini-Eynard-Mariño \cite{BEM} and Diaconescu-Shende-Vafa \cite{DSV}. This conjecture relates open Gromov-Witten invariants of the conifold transition of a torus knot to the topological recursion on the B-model spectral curve.

We prove that certain Riemannian manifolds can be isometrically embedded inside Calabi-Yau manifolds. For example we prove that given any real-analytic one parameter family of Riemannian metrics $g_t$ on a 3-dimensional manifold $Y$ with volume form independent of $t$ and with a real-analytic family of nowhere vanishin…

A quasitoric manifold $M$ is a $2n$-dimensional manifold which admits an action of an $n$-dimensional torus which has some nice properties. We determine the isomorphism type of a maximal compact connected Lie-subgroup $G$ of $\text{Homeo}(M)$ which contains the torus. Moreover, we show that this group is unique up to c…

Inspired by the work of Gross on topological Mirror Symmetry we construct candidate Lagrangian torus fibration models for the 105 families of smooth Fano threefolds. We prove, in the case the second Betti number is one, that the total space of each fibration is homeomorphic to the expected Fano threefold, and show that…

We prove that the three-dimensional Iwasawa manifold $X$, viewed as a locally holomorphically trivial fibration by elliptic curves over its two-dimensional Albanese torus, is self-dual in the sense that the base torus identifies canonically with its dual torus under a sesquilinear duality, the Jacobian torus of $X$, wh…

We construct infinitely many complete, immersed self-shrinkers with rotational symmetry for each of the following topological types: the sphere, the plane, the cylinder, and the torus.

In this paper we give a construction of Lagrangian torus fibration for Calabi-Yau hypersurface in toric variety via the method of gradient flow. Using our construction of Lagrangian torus fibration, we are able to prove the symplectic topological version of SYZ mirror conjecture for generic Calabi-Yau hypersurface in t…

In this paper we develop a relative version of T-duality in generalized complex geometry which we propose as a manifestation of mirror symmetry. Let M be an n-dimensional smooth real manifold, V a rank n real vector bundle on M, and nabla a flat connection on V. We define the notion of a nabla-semi-flat generalized com…

We classify closed, simply connected $n$-manifolds of non-negative sectional curvature admitting an isometric torus action of maximal symmetry rank in dimensions $2\leq n\leq 6$. In dimensions $3k$, $k=1,2$ there is only one such manifold and it is diffeomorphic to the product of $k$ copies of the 3-sphere.

Researchers prove a Willmore conjecture for surfaces with specific symmetries.

Study electric field and potential of torus knots, focusing on z-axis.

We describe the basic Dolbealut cohomology algebra of the canonical foliation on a class of complex manifolds with a torus symmetry group. This class includes complex moment-angle manifolds, LVM- and LVMB-manifolds and, in most generality, complex manifolds with a maximal holomorphic torus action. We also provide a dga…

We associate a root system to a finite set in a free abelian group and prove that its irreducible subsystem is of type A, B or D. We apply this general result to a torus manifold, where a torus manifold is a $2n$-dimensional connected closed smooth manifold with a smooth effective action of an $n$-dimensional compact t…

We show that any effective isometric torus action of maximal rank on a compact Riemannian manifold with positive (sectional) curvature and maximal symmetry rank, that is, on a positively curved sphere, lens space, complex or real projective space, is equivariantaly diffeomorphic to a linear action. We show that a compa…

Proves a 1930s Hopf conjecture about positive curvature manifolds.