arXiv research

Compact Ricci-flat manifolds with conical singularities are found.

Constructs Lagrangians in Calabi-Yau threefolds using tropical curves.

Algorithm computes eigenvalues and eigenforms on Calabi-Yau threefolds.

We study the variety of Poisson structures and compute Poisson cohomology for two families of Fano threefolds - smooth cubic threefolds and the del Pezzo quintic threefold. Along the way we reobtain by a different method earlier results of Loray, Pereira and Touzet in the special case we are considering.

This paper focuses on a topological version on the Strominger-Yau-Zaslow mirror symmetry conjecture. Roughly put, the SYZ conjecture suggests that mirror pairs of Calabi-Yau manifolds are related by the existence of dual special Lagrangian torus fibrations. We explore this conjecture without reference to the special La…

After Bershadsky-Cecotti-Ooguri-Vafa, we introduce an invariant of Calabi-Yau threefolds, which we call the BCOV invariant and which we obtain using analytic torsion. We give an explicit formula for the BCOV invariant as a function on the compactified moduli space, when it is isomorphic to a projective line. As a corol…

We study log canonical thresholds on quartic threefolds, quintic fourfolds, and double spaces. As an application, we show that they have a Kaehler-Einstein metric if they are general.

Neural nets approximate Ricci flat metrics for Calabi-Yau manifolds.

Study new invariants in complex geometry using Bott-Chern hypercohomology.

The paper proves K-stability of special Gushel-Mukai manifolds.

New metrics solve complex equations on special 3D shapes.

Lecture notes on conifold transitions between Calabi-Yau manifolds.

Gauge theory and G2-geometry on Calabi-Yau links are explored.

We prove the factoriality of the following nodal threefolds: a complete intersection of hypersurfaces $F$ and $G\subset\mathbb{P}^{5}$ of degree $n$ and $k$ respectively, where $G$ is smooth, $|\mathrm{Sing}(F\cap G)|\leqslant(n+k-2)(n-1)/5$, $n\geqslant k$; a double cover of a smooth hypersurface $F\subset\mathbb{P}^{…

Study of K-moduli of prime Fano threefolds of genus twelve, proving boundary purely divisorial.

For a generic anti-canonical hypersurface in each smooth toric Fano 4-fold with rank 2 Picard group, we prove there exist three isolated rational curves in it. Moreover, for all these 4-folds except one, the contractions of generic anti-canonical hypersurfaces along the three rational curves can be deformed to smooth t…

This thesis explores algebraic cycles and moduli spaces over real numbers.

Determined the balanced cone of a specific geometric space.

In this paper we give a construction of Lagrangian torus fibration for Fermat type quintic \cy hypersurfaces via the method of gradient flow. We also compute the monodromy of the expected special Lagrangian torus fibration and discuss structures of singular fibers.

Neural networks approximate Calabi-Yau metrics and curvature.

Quantum K-theory of quintic 3-fold conjectured with non-polynomial coefficients.

In this article we construct Lagrangian torus fibrations for general quintic \cy hypersurfaces near the large complex limit and their mirror manifolds using gradient flow method. Then we prove the Strominger-Yau-Zaslow mirror conjecture for this class of \cy manifolds in symplectic category.

Model captures SPX and VIX volatility surfaces and skew-stickiness ratio.

Mathematicians prove a conjecture about mirror symmetry at genus one.

New complete Calabi-Yau metric on C^3 with maximal volume growth.

We compute the Chen-Ruan orbifold cohomology ring of the Batyrev mirror orbifold of a smooth quintic hypersurface in 4-dimensional projective space. We identify the obstruction bundle for this example by using the Riemann bilinear relations for periods. We outline a general method of computing the Chen-Ruan ring for Ca…

Proves resurgent nature of a series solution to deformed Painlevé I equation.

The study examines nodal geometry on smooth manifolds, providing bounds and insights.

The paper classifies diffeomorphism classes of Calabi-Yau threefolds constructed by gluing methods.

Study finds nodal solutions of Yamabe equation constant along isoparametric levels.

Optimizes bounds for threefold singularity volumes.

Eigenfunction maxima inside high-d nodal domains.

The study examines nodal geometry, heat diffusion, and Brownian motion on Riemannian manifolds.

New bounds found for nodal sets on special manifolds.

Study classifies Calabi-Yau threefolds with non-Gorenstein involutions.

A polynomial curve of degree 5, $α$, is a helix, if and only if both $||α^'||$ and $||α^'\wedge α^{''}||$ are polynomial functions.

The study finds infinite nodal solutions for equations on positive Ricci curvature manifolds.

The study classifies holomorphic projective connections on complex threefolds.

K-stability proven for a specific type of Fano threefold.

New nodal domain theorems for symmetric matrices via signed graphs.

Study proves most Fano threefolds are Kähler-Einstein.

Study how nodal domains change on surfaces under perturbations.

In this paper we investigate the properties of series of vacua in the string theory landscape. In particular, we study minima to the flux potential in type IIB compactifications on the mirror quintic. Using geometric transitions, we embed its one dimensional complex structure moduli space in that of another Calabi-Yau …

The paper studies eigenfunctions and nodal sets of the Witten-Laplacian.

Abstract reviews applications of parabolic structures to nodal curve sheaves.

We prove by Hilbert-Mumford criterion that a slope stable polarized weighted pointed nodal curve is Chow asymptotic stable. This generalizes the result of Caporaso on stability of polarized nodal curves, and of Hasset on weighted pointed stable curves polarized by the weighted dualizing sheaves. It also solved a questi…

Study noncommutative deformations of Calabi-Yau threefolds.

The study examines non-Kähler threefolds with specific metrics and finds they are quasi-bundles over surfaces.