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We study asymptotic estimates of the dimension of cohomology on possibly non-compact complex manifolds for line bundles endowed with Hermitian metrics with algebraic singularities. We give a unified approach to establishing singular…

In this article, we construct an orbifold quantum cohomology twisted by a flat gerbe. Then we compute these invariants in the case of a smooth manifold and a discrete torsion on a global quotient orbifold.

We address the basic question in discrete Morse theory of combining discrete gradient fields that are partially defined on subsets of the given complex. This is a well-posed question when the discrete gradient field $V$ is generated using a…

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…

Homology is a tool in topological data analysis which measures the shape of the data. In many cases, these measurements translate into new insights which are not available by other means. To compute homology, we rely on mathematical…

This paper gives a detailed construction of Seiberg-Witten-Floer homology for a closed oriented 3-manifold with a non-torsion $\spinc$ structure. Gluing formulae for certain 4-dimensional manifolds splitting along an embedded 3-manifold are…

We outline a strategy for computing intersection numbers on smooth varieties with torus actions using a residue formula of Bott. As an example, Gromov-Witten numbers of twisted cubic and elliptic quartic curves on some general complete…

Fold maps are smooth maps at each singular point of which it is represented as the product map of a Morse function and the identity map. Round fold maps are, in short, such maps the sets of all singular points of which are embedded…

As a higher dimensional version of the theory of Morse functions, there have been various studies of smooth manifolds using generic smooth maps. As fundamental results, in these studies, they have found that inverse images of such maps…

For Morse-Smale pairs on a smooth, closed manifold the Morse-Smale-Witten chain complex can be defined. The associated Morse homology is isomorphic to the singular homology of the manifold and yields the classical Morse relations for Morse…

We present an algorithm for computing line bundle valued cohomology classes over toric varieties. This is the basic starting point for computing massless modes in both heterotic and Type IIB/F-theory compactifications, where the manifolds…

We introduce a model for Hermitian holormorphic Deligne cohomology on a projective algebraic manifold which allows to incorporate singular hermitian structures along a normal crossing divisor. In the case of a projective curve, the…

We study computable topological spaces and semicomputable and computable sets in these spaces. In particular, we investigate conditions under which semicomputable sets are computable. We prove that a semicomputable compact manifold $M$ is…

We compute numerically the homology of several graph complexes in low loop orders, extending previous results.

We develop a new approach to Lagrangian-Floer gluing. The construction of the gluing map is based on the intersection theory in some Hilbert manifold of paths $\mathcal{P} $. We consider some moduli spaces of perturbed holomorphic curves…

We introduce the Mixed-Integer Quadratically Constrained Quadratic Programming framework for the quantum compilation problem and apply it in the context of topological quantum computing. In this setting, quantum gates are realized by…

A fundamental way to study 3-manifolds is through the geometric lens, one of the most prominent geometries being the hyperbolic one. We focus on the computation of a complete hyperbolic structure on a connected orientable hyperbolic…

In this note, we construct invariant and coinvariant Morse chain complexes with integer coefficients for any compact effective orbifold. We show that the homologies of these two chain complexes are invariants of the orbifold. We conjecture…

Cross-manifold clustering is a hard topic and many traditional clustering methods fail because of the cross-manifold structures. In this paper, we propose a Multiple Flat Projections Clustering (MFPC) to deal with cross-manifold clustering…

The goal of this paper is the construction of a compact manifold with G$_2$ holonomy and nodal singularities along circles using twisted connected sum method. This paper finds matching building blocks by solving the Calabi conjecture on…