arXiv research

Smooth but not symplectic embeddings of rational balls in complex projective plane found.

We generalise theorems of Cochran-Lickorish and Owens-Strle to the case of links with more than one component. This enables the use of linking forms on double branched covers, Heegaard Floer correction terms, and Donaldson's diagonalisation theorem to complete the table of unlinking numbers for nonsplit prime links wit…

The paper constrains families of smooth 4-manifolds using Seiberg-Witten invariants.

Diagonalizes tautological classes of definite 4-manifolds.

Algorithm finds ribbon disks for alternating knots, resolving sliceness for most prime knots.

Extends method for solving certain hydrodynamic systems.

The slicing number of a knot, $u_s(K)$, is the minimum number of crossing changes required to convert $K$ to a slice knot. This invariant is bounded above by the unknotting number and below by the slice genus $g_s(K)$. We show that for many knots, previous bounds on unknotting number obtained by Ozsvath and Szabo and b…

We provide a partial classification of the 3-strand pretzel knots $K = P(p,q,r)$ with unknotting number one. Following the classification by Kobayashi and Scharlemann-Thompson for all parameters odd, we treat the remaining families with $r$ even. We discover that there are only four possible subfamilies which may satis…

We give a short elementary proof that a Khovanov-type link homology constructed from a diagonalisable Frobenius algebra is degenerate.

Geodesic flows with diagonalisable integrals are orthogonal.

New proofs of Donaldson-Uhlenbeck-Yau theorem using geodesic rays.

The paper studies symmetries and conservation laws of non-diagonalisable hydrodynamic systems.

Proof of Donaldson's theorem using Seiberg-Witten equations for multiple spinors.

In this paper, we provide an alternative proof of Donaldson's almost-holomorphic section theorem and symplectic Lefschetz pencil theorem, through constructions of certain special kind of Donaldson-type sections of the line bundle based on properties of exponential sums.

Penrose limit results for specific 3-surfaces in space-time.

A proof via the Seiberg-Witten moduli space of Donaldson's theorem on smooth 4-manifolds with definite intersection forms.

We use Donaldson invariants of regular surfaces with p_g >0 to make quantitative statements about modulispaces of stable rank 2 sheaves. We give two examples: a quantitative existence theorem for stable bundles, and a computation of the rank of the canonical holomorphic two forms on the moduli space. The results are in…

New proof of Donaldson-Uhlenbeck-Yau theorem using variational approach.

New approach to proving Chen-Donaldson-Sun theorem with examples.

The paper proves a Donaldson-Uhlenbeck-Yau theorem for normal varieties and semistable bundles.

This note contains two remarks about the application of the d-invariant in Heegaard Floer homology and Donaldson's diagonalization theorem to knot theory. The first is the equivalence of two obstructions they give to a 2-bridge knot being smoothly slice. The second carries out a suggestion by Stefan Friedl to replace t…

Paper uses flow to prove theorem on Higgs bundles.

Survey on embedding 3-manifolds in definite 4-manifolds, focusing on Donaldson's theorem.

This paper constructs a symplectic manifold from a Riemannian one.

Constructs a new field theory for 4-manifolds.

We wish to attack the problems that H.~Anciaux and K.~Panagiotidou posed in [1], for non-degenerate real hypersurfaces in indefinite complex projective space. We will slightly change these authors' point of view, obtaining cleaner equations for the almost contact metric structure. To make the theory meaningful, we cons…

We consider a perturbed Hermitian-Einstein equation, which we call the Donaldson-Thomas equation, on compact Kähler threefolds. In arXiv:0805.2195, we analysed some analytic properties of solutions to the equation, in particular, we proved that a sequence of solutions to the Donaldson-Thomas equation has a subsequence …

Extends Donaldson's techniques to symplectic orbifolds, proving existence of sections and computing cohomology.

New findings on knot genera using advanced techniques.

We generalize Witten's conjectured formula relating Donaldson and Seiberg-Witten invariants to manifolds of non-simple type, via equivariant localization techniques. This approach does not use the theory of non-abelian monopoles, but works directly on the Donaldson-Witten and Seiberg-Witten moduli spaces. We give a for…

Study smooth embeddings of line configurations in complex projective plane.

Proves existence of Kähler-Einstein metrics in big cohomology classes.

We prove that for an algebraic curvature tensor on a pseudo-Euclidean space, the Jordan-Osserman condition implies the Rakić duality principle, and that the Osserman condition and the duality principle are equivalent in the diagonalisable case.

In arXiv:0805.2192, we set up a gauge-theoretic equation on symplectic 6-manifolds, which is a version of the Hermitian-Einstein equation perturbed by Higgs fields, and call Donaldson-Thomas equation, to analytically approach the Donaldson-Thomas invariants. In this article, we consider the equation on compact Kähler t…

We extend the notion of basic classes (for the Donaldson invariants) to 4-manifolds with $b^+>1$ which are (potentially) not of simple type or satisfy $b_1 >0$. We also give a structure theorem for the Donaldson invariants of 4-manifolds with $b^+>1$, $b_1>0$ and of strong simple type.

Continuity of delta invariant leads to uniform Kähler-Einstein metrics.

The paper proves that rational concordance of double twist knots is reciprocal.

We discuss a conjecture of Donaldson on a version of Yau's Theorem for symplectic forms with compatible almost complex structures and survey some recent progress on this problem. We also speculate on some future possible directions, and use a monotonicity formula for harmonic maps to obtain a new local estimate in the …

Study shows surgeries on certain knots bound rational homology 4-balls.

Let L be a nonunimodular definite lattice. Using a theorem of Elkies we show that whether L embeds in the standard definite lattice of the same rank is completely determined by a collection of lattice correction terms, one for each metabolizing subgroup of the discriminant group. As a topological application this gives…

Uniform proof of Kähler-Einstein metrics with arbitrary polarizations.

Paper proves uniqueness of special Lagrangian pair in Calabi-Yau 3-fold.

Study of instantons and knot representations in 3-manifolds.

Study proves obstructions to equivariantly slice strongly negative amphichiral knots.

In this paper, we prove a generalized Donaldson-Uhlenbeck-Yau theorem on Higgs bundles over a class of non-compact Gauduchon manifolds.

We prove an analogue of the Kotschick-Morgan conjecture in the context of SO(3) monopoles, obtaining a formula relating the Donaldson and Seiberg-Witten invariants of smooth four-manifolds using the SO(3)-monopole cobordism. The main technical difficulty in the SO(3)-monopole program relating the Seiberg-Witten and Don…

The paper proves a new version of dimensional reduction in cohomological Donaldson-Thomas theory.

We extend Donaldson's diagonalization theorem to intersection forms with certain local coefficients, under some constraints. This provides new examples of non-smoothable topological 4-manifolds.