arXiv research

Characterizes values of slice-torus invariants related to knot genus.

Proves Arnold-Thom conjecture for surfaces' arrival times.

New findings contradict the Thom conjecture for high degree hypersurfaces in $CP^3$.

Paper confirms Thom's conjecture for nonlinear evolutions on manifolds.

In [5] I solved the Thom's conjecture that a proper Thom map is triangulable. In this paper I drop the properness condition in the semialgebraic case and, moreover, in the definable case in an o-minimal structure.

This paper has been withdrawn by the author due a crucial sign error in Theorem B. We present a geometric proof of Thom conjecture, which uses Khovanov homology. Our approach doesn't use any analytic methods and is quite different from proof given by Kronheimer and Mrowka in 1994.

We prove conjectures of Rene Thom and Vladimir Arnold for C^2 solutions to the degenerate elliptic equation that is the level set equation for motion by mean curvature. We believe these results are the first instances of a general principle: Solutions of many degenerate equations behave as if they are analytic, even wh…

Proves Thom's conjecture for parabolic flows on Hilbert spaces.

In this paper, we demonstrate a relation among Seiberg-Witten invariants which arises from embedded surfaces in four-manifolds whose self-intersection number is negative. These relations, together with Taubes' basic theorems on the Seiberg-Witten invariants of symplectic manifolds, are then used to prove the symplectic…

The paper proves a conjecture linking two metrics on manifold cohomology.

In this paper, we develop new techniques for understanding surfaces in $\mathbb{CP}^2$ via bridge trisections. Trisections are a novel approach to smooth 4-manifold topology, introduced by Gay and Kirby, that provide an avenue to apply 3-dimensional tools to 4-dimensional problems. Meier and Zupan subsequently develope…

This note is an exposition of the proof of Thom's conjecture by Kronheimer and Mrowka, using the new Seiberg-Witten invariants.

We demonstrate the usage of explicit form of the Thom class found by Mathai and Quillen for the definition of generating functional of a simple supersymmetric quantum mechanical model.

Develops a Thom-Mather theory for corank 1 frontals.

Koszul duality for manifold modules proven.

By results of Loeffler and Comezana, the Pontrjagin-Thom map from geometric G-equivariant bordism to homotopy theoretic equivariant bordism is injective for compact abelian G. If G = S^1 x ... x S^1, we prove that the associated fixed point square is a pull back square, thus confirming a recent conjecture of D. Sinha. …

Study on slow convergence in geometric variational problems.

Paper proves a symplectic inequality using trisections and contact geometry.

In this article we define Lagrangian concordance of Legendrian knots, the analogue of smooth concordance of knots in the Legendrian category. In particular we study the relation of Lagrangian concordance under Legendrian isotopy. The focus is primarily on the algebraic aspects of the problem. We study the behavior of t…

Let G be a compact Lie group. Let M be a smooth G-manifold and V --> M be an oriented G-equivariant vector bundle. One defines the spaces of equivariant forms with generalized coefficients on V and M. An equivariant Thom form $θ$ on V is a compactly supported closed equivariant form such that its integral along the fib…

The paper establishes criteria for symplectic surfaces in 4-manifolds.

We will present proofs for two conjectures stated in arXiv:1808.08073. The first one is that for an arbitrary manifold $W$, the homotopy classes of proper maps $W\times\mathbb{R}^n\to\mathbb{R}^{k+n}$ stabilise as $n\to\infty$, and the second one is that in a stable range there is a Pontryagin--Thom type bijection for …

Given a smooth compact manifold with boundary, we show that the subcomplex of the deformed de Rham complex consisting of eigenspaces of small eigenvalues of the Witten Laplacian is canonically isomorphic to the Thom-Smale complex constructed by Laudenbach. Our proof is based on Bismut-Lebeau's analytic localization tec…

In the note, we give a proof, based on the Generalized Thom Conjecture, of Bennequin's Theorem on upper bound for the Euler number of a link which is considered as a closed braid. A lower bound for the Euler number of a link is also given.

We prove a new inequality relating volume to length of closed geodesics on area minimizers for generic metrics on the complex projective plane. We exploit recent regularity results for area minimizers by Moore and White, and the Kronheimer--Mrowka proof of the Thom conjecture.

This article is a survey of recent work of the author, together with Markus Banagl, Eric Leichtnam, Rafe Mazzeo, and Paolo Piazza, on the Hodge theory of stratified spaces. We discuss how to resolve a Thom-Mather stratified space to a manifold with corners with an iterated fibration structure and the generalization of …

Study local moduli of Sasaki-Einstein metrics on specific polynomial links.

We study the curvature of metric spaces and branched covers of Riemannian manifolds, with applications in topology and algebraic geometry. Here curvature bounds are expressed in terms of the CAT(k) inequality. We prove a general CAT(k) extension theorem, giving sufficient conditions on and near the boundary of a locall…

Let S(D) be the surface produced by applying Seifert's algorithm to the oriented link diagram D. I prove that if D has no negative crossings then S(D) is a quasipositive Seifert surface, that is, S(D) embeds incompressibly on a fiber surface plumbed from positive Hopf annuli. This result, combined with the truth of the…

Enhances Pontryagin-Thom theorem for manifold maps.

An oriented link L in a 3-sphere S in complex 2-space is a C-boundary if it bounds a piece of algebraic curve in the 4-ball bounded by S. Using Kronheimer and Mrowka's proof of the Thom Conjecture, we construct many oriented knots which are not concordant to a C-boundary. We use the two-variable HOMFLY polynomial to gi…

For an oriented link $L \subset S^3 = \Bd\!D^4$, let $χ_s(L)$ be the greatest Euler characteristic $χ(F)$ of an oriented 2-manifold $F$ (without closed components) smoothly embedded in $D^4$ with boundary $L$. A knot $K$ is {\it slice} if $χ_s(K)=1$. Realize $D^4$ in $\C^2$ as $\{(z,w):|z|^2+|w|^2\le1\}$. It has been c…

The paper computes a tau-invariant for holomorphic curves in Stein domains and links.

The paper connects cut locus, Thom space, and Morse-Bott functions in Riemannian geometry.

Establishes jet transversality for regular maps from flexible manifolds.

Relative Thom polynomials for maps around boundaries established.

The paper explores density of stable mappings and their properties.

New approach simplifies topological T-duality for torus bundles.

Proves a Thom Isotopy Theorem for nonproper semialgebraic maps.

Researchers describe a new Thom form for mapping cones.

We present a cocycle model for elliptic cohomology with complex coefficients in which methods from 2-dimensional quantum field theory can be used to rigorously construct cocycles. For example, quantizing a theory of vector bundle-valued fermions yields a cocycle representative of the elliptic Thom class. This construct…

Paper constructs Thom-Smale complex using instantons from Morse functions.

The paper explores $ρ$-regularity for real analytic maps and its relation to Milnor fibrations.

New Sasaki-Einstein 7-manifolds found, including rational homology 7-spheres and connected sums.

Analytic realization of Thom-Smale complex for G-manifolds.

In this work we analyze the behavior of Massey products of closed manifolds under the blow-up construction. The results obtained in the article are applied to the problem of constructing closed symplectic non-formal manifolds. The proofs use Thom spaces as an important technical tool. This application of Thom spaces is…

We develop a theory of parametrized geometric cobordism by introducing smooth Thom stacks. This requires identifying and constructing a smooth representative of the Thom functor acting on vector bundles equipped with extra geometric data, leading to a geometric refinement of the the Pontrjagin-Thom construction in stac…

We prove the $K(π,1)$ conjecture for affine Artin groups: the complexified complement of an affine reflection arrangement is a classifying space. This is a long-standing problem, due to Arnol'd, Pham, and Thom. Our proof is based on recent advancements in the theory of dual Coxeter and Artin groups, as well as on sever…