Andrew Schaug - Academia.edu (original) (raw)

Papers by Andrew Schaug

arXiv: Algebraic Geometry, 2015

In the early 1990s, Borcea-Voisin orbifolds were some of the ear- liest examples of Calabi-Yau th... more In the early 1990s, Borcea-Voisin orbifolds were some of the ear- liest examples of Calabi-Yau threefolds shown to exhibit mirror symmetry. However, their quantum theory has been poorly investigated. We study this in the context of the gauged linear sigma model, which in their case encom- passes Gromov-Witten theory and its three companions (FJRW theory and two mixed theories). For certain Borcea-Voisin orbifolds of Fermat type, we calculate all four genus zero theories explicitly. Furthermore, we relate the I-functions of these theories by analytic continuation and symplectic transfor- mation. In particular, the relation between the Gromov-Witten and FJRW theories can be viewed as an example of the Landau-Ginzburg/Calabi-Yau correspondence for complete intersections of toric varieties.

arXiv: Mathematical Finance, 2019

Stochastic bridges are commonly used to impute missing data with a lower sampling rate to generat... more Stochastic bridges are commonly used to impute missing data with a lower sampling rate to generate data with a higher sampling rate, while preserving key properties of the dynamics involved in an unbiased way. While the generation of Brownian bridges and Ornstein-Uhlenbeck bridges is well understood, unbiased generation of such stochastic bridges subject to a given extremum has been less explored in the literature. After a review of known results, we compare two algorithms for generating Brownian bridges constrained to a given extremum, one of which generalises to other diffusions. We further apply this to generate unbiased Ornstein-Uhlenbeck bridges and unconstrained processes, both constrained to a given extremum, along with more tractable numerical approximations of these algorithms. Finally, we consider the case of drift, and applications to geometric Brownian motions.

arXiv: Algebraic Geometry, 2015

Borcea-Voisin threefolds provided some of the first examples of mirror pairs in the Hodge-theoret... more Borcea-Voisin threefolds provided some of the first examples of mirror pairs in the Hodge-theoretic sense, but their mirror symmetry at the quantum level have not previously been shown. We prove a Givental-style quantum mirror theorem for certain Borcea-Voisin threefolds: by means of certain birational models, we show that their Gromov-Witten J-functions are related by a mirror map to solutions of the multi-parameter Picard-Fuchs equations coming from the variation of Hodge structures of their mirror partners.

In the early 1990s, Borcea-Voisin orbifolds were some of the earliest examples of Calabi-Yau thre... more In the early 1990s, Borcea-Voisin orbifolds were some of the earliest examples of Calabi-Yau threefolds shown to exhibit mirror symmetry, but at the quantum level this has been poorly understood. Here the enumerative geometry of this family is placed in the context of a gauged linear sigma model which encompasses the threefolds’ Gromov-Witten theory and three companion theories (FJRW theory and two mixed theories). For certain Borcea-Voisin orbifolds of Fermat type, all four genus zero theories are calculated explicitly. Furthermore, the I-functions of these theories are related by analytic continuation and symplectic transformation. In particular, it is shown that the relation between the Gromov-Witten and FJRW theories can be viewed as an example of the Landau-Ginzburg/Calabi-Yau correspondence for complete intersections of toric varieties. For certain mirror families, the corresponding Picard-Fuchs systems are then derived and the I-functions are shown to solve them, thus demonst...

Illinois Journal of Mathematics

FJRW theory is a formulation of physical Landau-Ginzburg models with a rich algebraic structure, ... more FJRW theory is a formulation of physical Landau-Ginzburg models with a rich algebraic structure, rooted in enumerative geometry. As a consequence of a major physical conjecture, called the Landau-Ginzburg/Calabi-Yau correspondence, several birational morphisms of Calabi-Yau orbifolds should correspond to isomorphisms in FJRW theory. In this paper, we exhibit some of these isomorphisms that are related to Borcea-Voisin mirror symmetry. In particular, we develop a modified version of BHK mirror symmetry for certain LG models. Using these isomorphisms, we prove several interesting consequences in the corresponding geometries.

arXiv: Algebraic Geometry, 2015

In the early 1990s, Borcea-Voisin orbifolds were some of the ear- liest examples of Calabi-Yau th... more In the early 1990s, Borcea-Voisin orbifolds were some of the ear- liest examples of Calabi-Yau threefolds shown to exhibit mirror symmetry. However, their quantum theory has been poorly investigated. We study this in the context of the gauged linear sigma model, which in their case encom- passes Gromov-Witten theory and its three companions (FJRW theory and two mixed theories). For certain Borcea-Voisin orbifolds of Fermat type, we calculate all four genus zero theories explicitly. Furthermore, we relate the I-functions of these theories by analytic continuation and symplectic transfor- mation. In particular, the relation between the Gromov-Witten and FJRW theories can be viewed as an example of the Landau-Ginzburg/Calabi-Yau correspondence for complete intersections of toric varieties.

arXiv: Mathematical Finance, 2019

Stochastic bridges are commonly used to impute missing data with a lower sampling rate to generat... more Stochastic bridges are commonly used to impute missing data with a lower sampling rate to generate data with a higher sampling rate, while preserving key properties of the dynamics involved in an unbiased way. While the generation of Brownian bridges and Ornstein-Uhlenbeck bridges is well understood, unbiased generation of such stochastic bridges subject to a given extremum has been less explored in the literature. After a review of known results, we compare two algorithms for generating Brownian bridges constrained to a given extremum, one of which generalises to other diffusions. We further apply this to generate unbiased Ornstein-Uhlenbeck bridges and unconstrained processes, both constrained to a given extremum, along with more tractable numerical approximations of these algorithms. Finally, we consider the case of drift, and applications to geometric Brownian motions.

arXiv: Algebraic Geometry, 2015

Borcea-Voisin threefolds provided some of the first examples of mirror pairs in the Hodge-theoret... more Borcea-Voisin threefolds provided some of the first examples of mirror pairs in the Hodge-theoretic sense, but their mirror symmetry at the quantum level have not previously been shown. We prove a Givental-style quantum mirror theorem for certain Borcea-Voisin threefolds: by means of certain birational models, we show that their Gromov-Witten J-functions are related by a mirror map to solutions of the multi-parameter Picard-Fuchs equations coming from the variation of Hodge structures of their mirror partners.

In the early 1990s, Borcea-Voisin orbifolds were some of the earliest examples of Calabi-Yau thre... more In the early 1990s, Borcea-Voisin orbifolds were some of the earliest examples of Calabi-Yau threefolds shown to exhibit mirror symmetry, but at the quantum level this has been poorly understood. Here the enumerative geometry of this family is placed in the context of a gauged linear sigma model which encompasses the threefolds’ Gromov-Witten theory and three companion theories (FJRW theory and two mixed theories). For certain Borcea-Voisin orbifolds of Fermat type, all four genus zero theories are calculated explicitly. Furthermore, the I-functions of these theories are related by analytic continuation and symplectic transformation. In particular, it is shown that the relation between the Gromov-Witten and FJRW theories can be viewed as an example of the Landau-Ginzburg/Calabi-Yau correspondence for complete intersections of toric varieties. For certain mirror families, the corresponding Picard-Fuchs systems are then derived and the I-functions are shown to solve them, thus demonst...

Illinois Journal of Mathematics

FJRW theory is a formulation of physical Landau-Ginzburg models with a rich algebraic structure, ... more FJRW theory is a formulation of physical Landau-Ginzburg models with a rich algebraic structure, rooted in enumerative geometry. As a consequence of a major physical conjecture, called the Landau-Ginzburg/Calabi-Yau correspondence, several birational morphisms of Calabi-Yau orbifolds should correspond to isomorphisms in FJRW theory. In this paper, we exhibit some of these isomorphisms that are related to Borcea-Voisin mirror symmetry. In particular, we develop a modified version of BHK mirror symmetry for certain LG models. Using these isomorphisms, we prove several interesting consequences in the corresponding geometries.