arXiv research

We introduce $Z$-critical connections for holomorphic vector bundles and prove their existence under stability conditions.

Introduces stability conditions and their connection to Artin groups.

Surveying progress on deformed Hermitian-Yang-Mills equation and its connections to stability.

Inspired by mirror symmetry, we investigate some differential geometric aspects of the space of Bridgeland stability conditions on a Calabi-Yau triangulated category. The aim is to develop theory of Weil-Petersson geometry on the stringy Kähler moduli space. A few basic examples are studied. In particular, we identify …

Non-asphericity of strata of genus-one differentials

We consider Bridgeland stability conditions for three-folds conjectured by Bayer-Macrì-Toda in the case of Picard rank one. We study the differential geometry of numerical walls, characterizing when they are bounded, discussing possible intersections, and showing that they are essentially regular. Next, we prove that w…

This is a survey on two closely related subjects. First, we review the study of topological structure of `finite type' components of spaces of Bridgeland's stability conditions on triangulated categories. The key is to understand Happel-Reiten-Smalo tilting as tiling of cells. Second, we review topological realizations…

We provide a novel proof that the set of directions that admit a saddle connection on a meromorphic quadratic differential with at least one pole of order at least two is closed, which generalizes a result of Bridgeland and Smith, and Gaiotto, Moore, and Neitzke. Secondly, we show that this set has finite Cantor-Bendix…

Introduces stability conditions for polarized varieties, linking to K-stability.

A (meromorphic) quadratic differential is a (meromorphic) section of the tensor square of the canonical bundle of a Riemann surface. They arose in the study of quasiconformal mappings in the works of Oswald Teichmüller, and have played a mayor role in the study of the Riemann moduli, where they can be identified with c…

We determine the image of the monodromy map for meromorphic projective structures with poles of orders greater than two. This proves the analogue of a theorem of Gallo-Kapovich-Marden, and answers a question of Allegretti and Bridgeland. Our proof uses coordinates on the moduli space of framed representations arising f…

Paper calculates Donaldson-Thomas invariants for a specific category.

Defines special Joyce structures for ASK manifolds encoding real HK structures.

Study special Lagrangian sections in Calabi-Yau threefolds, showing stability conditions imply isomorphism to special Lagrangians.

We show that the Borel sums of the Voros symbols considered in the theory of exact WKB analysis arise naturally as Fock-Goncharov coordinates of framed $PGL_2(\mathbb{C})$-local systems on a marked bordered surface. Using this result, we show that these Borel sums can be meromorphically continued to any point of $\math…

We study the deformed Hermitian-Yang-Mills (dHYM) equation, which is mirror to the special Lagrangian equation, from the variational point of view via an infinite dimensional GIT problem mirror to Thomas' GIT picture for special Lagrangians. This gives rise to infinite dimensional manifold $\mathcal{H}$ mirror to Solom…

Introduces relative stability conditions on triangulated categories.

New map connects stability conditions to Teichmüller space.

Characterizes periodic elements in Artin-Tits groups via stability conditions.

We introduce the notions of categorical systoles and categorical volumes of Bridgeland stability conditions on triangulated categories. We prove that for any projective K3 surface, there exists a constant C depending only on the rank and discriminant of its Picard group, such that $$\mathrm{sys}(σ)^2\leq C\cdot\mathrm{…

We study a class of flat bundles, of finite rank $N$, which arise naturally from the Donaldson-Thomas theory of a Calabi-Yau threefold $X$ via the notion of a variation of BPS structure. We prove that in a large $N$ limit their flat sections converge to the solutions to certain infinite dimensional Riemann-Hilbert prob…

The paper compactifies stability conditions on curves, akin to Teichmüller theory.

Null Kähler metrics are characterized by Painlevé I or II ODEs.

Proves stability condition for Lagrangian sections in toric weak Fano manifolds.

For a punctured surface $S$, we characterize the representations of its fundamental group into $\mathrm{PSL}_2 (\mathbb{C})$ that arise as the monodromy of a meromorphic projective structure on $S$ with poles of order at most two and no apparent singularities. This proves the analogue of a theorem of Gallo-Kapovich-Mar…

Study higher rank deformed Hermitian-Yang-Mills equations for stable vector bundles.

The study of special Lagrangian classes and semistable Mukai vectors on K3 surfaces.

Unified approach to stability conditions on surfaces with quadratic differentials.

For $G$ a finite subgroup of ${\rm SL}(3,{\mathbb C})$ acting freely on ${\mathbb C}^3{\setminus} \{0\}$ a crepant resolution of the Calabi-Yau orbifold ${\mathbb C}^3\!/G$ always exists and has the geometry of an ALE non-compact manifold. We show that the tautological bundles on these crepant resolutions admit rigid H…

This paper gives a description of the full space of Bridgeland stability conditions on the bounded derived category of a contraction algebra associated to a 3-fold flop. The main result is that the stability manifold is the universal cover of a naturally associated hyperplane arrangement, which is known to be simplicia…

Study of decorated surfaces with vortices and their group structures.

Constructs stable Hilbert bundles on curves using Diophantine approximation.

We introduce the cluster exchange groupoid associated to a non-degenerate quiver with potential, as an enhancement of the cluster exchange graph. In the case that arises from an (unpunctured) marked surface, where the exchange graph is modelled on the graph of triangulations of the marked surface, we show that the univ…

Characterizes solutions to Z-critical equations on surfaces using effective conditions.

The resolved conifold geometry is linked to a special Kähler manifold and an instanton-corrected hyperkähler manifold.

Proves existence of Lagrangian mean curvature flow solutions.

We further develop the asymptotic analytic approach to the study of scattering diagrams. We do so by analyzing the asymptotic behavior of Maurer-Cartan elements of a differential graded Lie algebra constructed from a (not-necessarily tropical) monoid-graded Lie algebra. In this framework, we give alternative differenti…

Let $M$ be a Calabi-Yau $m$-fold, and consider compact, graded Lagrangians $L$ in $M$. Thomas and Yau math.DG/0104196, math.DG/0104197 conjectured that there should be a notion of "stability" for such $L$, and that if $L$ is stable then Lagrangian mean curvature flow $\{L^t:t\in[0,\infty)\}$ with $L^0=L$ should exist f…

The thesis explores stability conditions and metrics in differential geometry.

The article extends Thurston's Grafting Theorem to signed spaces and defines a framed monodromy map.

The paper characterizes when numerical criteria for PDE solvability fail and provides effective criteria for existence.

Compactifies stability space for $A_2$ category, introducing $q$-deformed rational numbers.

Let X be a Calabi-Yau 3-fold, T=D^b(coh(X)) the derived category of coherent sheaves on X, and Stab(T) the complex manifold of Bridgeland stability conditions Z on T. It is conjectured that one can define rational numbers J^a(Z) for Z in Stab(T) and a in the numerical Grothendieck group K(T) generalizing Donaldson-Thom…

The deformed Hermitian Yang-Mills (dHYM) equation is a special Lagrangian type condition in complex geometry. It requires the complex analogue of the Lagrangian phase, defined for Chern connections on holomorphic line bundles using a background Kähler metric, to be constant. In this paper we introduce and study dHYM eq…

Mathematical structures link Gromov-Witten to Donaldson-Thomas invariants.

Compactifies stability conditions on triangulated categories, inspired by Teichmüller theory.

This paper constructs hyper-Kähler metrics and hyper-Lagrangian foliations for a class of complex manifolds.

Joyce's criterion for sLag smoothings extended to non-compact, non-transverse intersections.