arXiv research

The paper verifies a conjecture about the index of symmetric spaces.

Study links' arc index and Turaev genus, proving conjectures.

New geometric proof shows index of umbilic points on analytic surfaces is at most one.

New index theory proves Gromov's dihedral conjectures.

Counterexample disproves key index computation in Gromov's conjecture paper.

New findings confirm parallels to De Giorgi's conjecture for phase transitions in higher dimensions.

Paper solves Carathéodory's conjecture for $C^2$-regular convex surfaces.

Let $γ$ be a non-degenerate Ustilovsky geodesic in $Ham (M, ω)$ generated by $H$. We give a simple proof of a generalization of the conjecture stated in \cite{virtmorse}, relating the Morse index of $γ$, as a critical point of the Hofer length functional, with the Conley Zehnder index of the extremizers of $H$, consid…

The paper refines the stability index for a specific minimal hypersurface and verifies Yau's conjecture.

Classifies totally geodesic submanifolds in exceptional symmetric spaces.

Using equivariant Toeplitz operator calculus, we give a new proof of the Atiyah-Weinstein conjecture on the index of Fourier integral operators and the relative index of CR structures.

The paper explores knots with equal bridge and braid index, conjecturing they have a unique equilibrium state.

The Allen-Cahn equation is a semilinear PDE which is deeply linked to the theory of minimal hypersurfaces via a singular limit. We prove curvature estimates and strong sheet separation estimates for stable solutions (building on recent work of Wang-Wei) of the Allen-Cahn equation on a 3-manifold. Using these, we are ab…

Generalizes Novikov conjecture results to infinite-dimensional bundles.

Handles verify a manifold conjecture.

The paper proves a conjecture about the minimum number of closed geodesics on a Finsler 3-sphere.

The index of a Riemannian symmetric space is the minimal codimension of a proper totally geodesic submanifold (Onishchik, 1980). There is a conjecture by the first two authors for how to calculate the index. In this paper we give an affirmative answer to this conjecture for the exceptional Riemannian symmetric spaces a…

We first introduce an invariant index for G-equivariant elliptic differential operators on a locally compact manifold M admitting a proper cocompact action of a locally compact group G. It generalizes the Kawasaki index for orbifolds to the case of proper cocompact actions. Our invariant index is used to show that an a…

The 3D-index connects to Turaev-Viro invariant and knot periods.

The paper proves stability of minimal embeddings in spheres and relates it to Yau's conjecture.

New graph invariant measures embeddability in 3D.

We prove that positive knots and knots with braid index three in the 3-sphere satisfy the Property P conjecture.

An index formula is proposed for contact transformations between contact manifolds equipped with CR structures or with fillings by symplectic manifolds. The formula generalizes the Atiyah-Singer formula and gives a conjectured formula for the index of Fourier integral operators, as well as Epstein's relative index for …

Proves a quantitative index theorem for positive scalar curvature metrics.

It has been conjectured that the algebraic crossing number of a link is uniquely determined in minimal braid representation. This conjecture is true for many classes of knots and links. The Morton-Franks-Williams inequality gives a lower bound for braid index. And sharpness of the inequality on a knot type implies the …

We prove a generalization of the Jones-Kawamuro conjecture that relates the self-linking number and the braid index of closed braids, for planar open books with certain additional conditions and modifications. We show that our result is optimal in some sense by giving several counter examples for naive generalizations …

The equivariant coarse index is well-understood and widely used for actions by discrete groups. We extend the definition of this index to general locally compact groups. We use a suitable notion of admissible modules over $C^*$-algebras of continuous functions to obtain a meaningful index. Inspired by work by Roe, we t…

Study of 3d-3d correspondence involving $q$-Weyl algebra and 3d-index.

Paper proves a noncompact version of Gromov's band-width estimate.

The paper explores index theory for Dirac operators to understand scalar curvature properties.

We prove that orientable index one minimal surfaces in spherical space forms with large fundamental group have genus at most two. This confirms a conjecture of R. Schoen for an infinite class of 3-manifolds.

Counterexample disproves Borde-Sorkin conjecture on causal continuity of Morse spacetimes.

Paper extends Fenchel's conjecture to non-Euclidean crystallographic groups.

Study proves bridge and braid indices match for twist positive knots.

We show that in a closed 3-manifold with a generic metric of positive Ricci curvature, there are minimal surfaces of arbitrary large Morse index, which partially confirms a conjecture by F. Marques and A. Neves. We prove this by analyzing the lamination structure of the limit of minimal surfaces with bounded Morse inde…

Local constancy of index for certain gradient mappings proved.

This is an expository paper about Seiberg-Witten Floer stable homotopy types. We outline their construction, which is based on the Conley index and finite dimensional approximation. We then describe several applications, including the disproof of the high-dimensional triangulation conjecture.

For an immersed minimal surface in $\mathbb{R}^3$, we show that there exists a lower bound on its Morse index that depends on the genus and number of ends, counting multiplicity. This improves, in several ways, an estimate we previously obtained bounding the genus and number of ends by the index. Our new estimate resol…

Proves the index of a Möbius band in 4D ball equals 5.

Counter-examples to the famous conjecture of Caratheodory, as well as the bound on umbilic index proposed by Hamburger, are constructed with respect to Riemannian metrics that are arbitrarily close to the flat metric on Euclidean 3-space. In particular, Riemannian metrics with a smooth strictly convex 2-sphere containi…

In 1987, Kalai proved that stacked spheres of dimension $d\geq 3$ are characterised by the fact that they attain equality in Barnette's celebrated Lower Bound Theorem. This result does not extend to dimension $d=2$. In this article, we give a characterisation of stacked $2$-spheres using what we call the {\em separatio…

We show that for an immersed two-sided minimal surface in $R^3$, there is a lower bound on the index depending on the genus and number of ends. Using this, we show the nonexistence of an embedded minimal surface in $R^3$ of index $2$, as conjectured by Choe. Moreover, we show that the index of a immersed two-sided mini…

This article surveys the relations among local and nonlocal invariants in Atiyah-Singer index theory. We discuss the local invariants that arise from the heat equation approach to the index theorem for geometric operators, as well as the nonlocal invariants (the eta invariant, the determinant of the Laplacian/analytic …

New minimal surface theory disproves a conjecture in symmetric spaces.

We present a condition for towers of fiber bundles which implies that the fundamental group of the total space has a nilpotent subgroup of finite index whose torsion is contained in its center. Moreover, the index of the subgroup can be bounded in terms of the fibers of the tower. Our result is motivated by the conject…

Study on divergence and thickness for Coxeter groups, generalizing previous work.

Extends width estimates to family case using index theory.

The Rosenberg index vanishes if a manifold admits a wide Riemannian band or cube-like domain.