arXiv subjects

Amanda Francis

Publications and source records attributed to Amanda Francis.

10 recordsLinked to original sources

arXiv2023

Semisimple FJRW theory of polynomials with two variables

We study the Dubrovin-Frobenius manifold in the Fan-Jarvis-Ruan-Witten theory of Landau-Ginzburg pairs $(W, \ )$, where $W$ is an invertible nondegenerate quasihomogeneous polynomial with two variables and $\ $ is the minimal admissible group of $W$. We conjecture that the Dubrovin-Frobenius manifolds from these FJRW theory are semisimple. We show the conjecture holds true for simple singularities and almost all Brieskorn-Pham polynomials. For Brieskorn-Pham polynomials, the result follows from the calculation of a quantum Euler class in the FJRW theory. As a consequence, our result shows that for the FJRW theory of these Landau-Ginzburg pairs, both a Dubrovin type conjecture and a Virasoro conjecture hold true.

math.AG↗

arXiv2019

Borcea-Voisin Mirror Symmetry for Landau-Ginzburg models

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 it is shown that not only does this claim prove to be the case, but is a special case of a wider FJRW isomorphism theorem, which in turn allows for a proof of mirror symmetry for a new class of cases in the Landau-Ginzburg setting. We also obtain several interesting geometric applications regarding the Chen-Ruan cohomology of certain Calabi-Yau orbifolds.

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arXiv2018

General Equitable Decompositions for Graphs with Symmetries

Using the theory of equitable decompositions it is possible to decompose a matrix $M$ appropriately associated with a given graph. The result is a collection of smaller matrices whose collective eigenvalues are the same as the eigenvalues of the original matrix $M$. This is done by decomposing the matrix over a graph symmetry. Previously it was shown that a matrix can be equitably decomposed over any uniform, basic, or separable automorphism. Here we extend this theory to show that it is possible to equitably decompose a matrix over any automorphism of a graph, without restriction. Moreover, we give a step-by-step procedure which can be used to generate such a decomposition. We also prove under mild conditions that if a matrix $M$ is equitably decomposed the resulting divisor matrix, which is the divisor matrix of the associated equitable partition, will have the same spectral radius as the original matrix $M$.

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arXiv2018

Curriculum Guidelines for Undergraduate Programs in Data Science

The Park City Math Institute (PCMI) 2016 Summer Undergraduate Faculty Program met for the purpose of composing guidelines for undergraduate programs in Data Science. The group consisted of 25 undergraduate faculty from a variety of institutions in the U.S., primarily from the disciplines of mathematics, statistics and computer science. These guidelines are meant to provide some structure for institutions planning for or revising a major in Data Science.

stat.OT↗

arXiv2017

Extensions and Applications of Equitable Decompositions for Graphs with Symmetries

We extend the theory of equitable decompositions, in which, if a graph has a particular type of symmetry, i.e. a uniform or basic automorphism $ϕ$, it is possible to use $ϕ$ to decompose a matrix $M$ appropriately associated with the graph. The result is a number of strictly smaller matrices whose collective eigenvalues are the same as the eigenvalues of the original matrix $M$. We show here that a large class of automorphisms, which we refer to as \emph{separable}, can be realized as a sequence of basic automorphisms, allowing us to equitably decompose $M$ over any such automorphism. We also show that not only can a matrix $M$ be decomposed but that the eigenvectors of $M$ can also be equitably decomposed. Additionally, we prove under mild conditions that if a matrix $M$ is equitably decomposed the resulting divisor matrix, which is the divisor matrix of the associated equitable partition, will have the same spectral radius as the original matrix $M$. Last, we describe how an equitable decomposition effects the Gershgorin region $Γ(M)$ of a matrix $M$, which can be used to localize the eigenvalues of $M$. We show that the Gershgorin region of an equitable decomposition of $M$ is contained in the Gershgorin region $Γ(M)$ of the original matrix. We demonstrate on a real-world network that by a sequence of equitable decompositions it is possible to significantly reduce the size of a matrix' Gershgorin region.

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arXiv2016

Equitable Decompositions of Graphs

We investigate connections between the symmetries (automorphisms) of a graph and its spectral properties. Whenever a graph has a symmetry, i.e. a nontrivial automorphism $ϕ$, it is possible to use $ϕ$ to decompose any matrix $M\in\mathbb{C}^{n \times n}$ appropriately associated with the graph. The result of this decomposition is a number of strictly smaller matrices whose collective eigenvalues are the same as the eigenvalues of the original matrix $M$. Some of the matrices that can be decomposed are the graph's adjaceny matrix, Laplacian matrix, etc. Because this decomposition has connections to the theory of equitable partitions it is referred to as an equitable decomposition. Since the graph structure of many real-world networks is quite large and has a high degree of symmetry, we discuss how equitable decompositions can be used to effectively bound both the network's spectral radius and spectral gap, which are associated with dynamic processes on the network. Moreover, we show that the techniques used to equitably decompose a graph can be used to bound the number of simple eigenvalues of undirected graphs, where we obtain sharp results of Petersdorf-Sachs type.

math.CO↗

arXiv2016

A Brief Survey of FJRW Theory

In this paper we describe some of the constructions of FJRW theory. We also briefly describe its relation to Saito-Givental theory via Landau-Ginzburg mirror symmetry and its relation to Gromov-Witten theory via the Landau-Ginzburg/Calabi-Yau correspondence. We conclude with a discussion of some of the recent results in the field.

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arXiv2014

Computational Techniques in FJRW Theory with Applications to Landau-Ginzburg Mirror Symmetry

The Landau-Ginzburg A-model, given by FJRW theory, defines a cohomological field theory, but in most examples is very difficult to compute, especially when the symmetry group is not maximal. We give some methods for finding the A-model structure. In many cases our methods completely determine the previously unknown A-model Frobenius manifold structure. In the case where these Frobenius manifolds are semisimple, this can be shown to determine the structure of the higher genus potential as well. We compute the Frobenius manifold structure for 27 of the previously unknown unimodal and bimodal singularities and corresponding groups, including 13 cases using a non-maximal symmetry group.

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arXiv2014

Witten's D_4 Integrable Hierarchies Conjecture

We prove that the total descendant potential functions of the theory of Fan-Jarvis-Ruan-Witten for D_4 with symmetry group and D_4^T with symmetry group G_{max}, respectively, are both tau-functions of the D_4 Kac-Wakimoto/Drinfeld-Sokolov hierarchy. This completes the proof, begun in [FJR], of the Witten Integrable Hierarchies Conjecture for all simple (ADE) singularities.

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arXiv2011

Landau-Ginzburg Mirror Symmetry for Orbifolded Frobenius Algebras

We prove the Landau-Ginzburg Mirror Symmetry Conjecture at the level of (orbifolded) Frobenius algebras for a large class of invertible singularities, including arbitrary sums of loops and Fermats with arbitrary symmetry groups. Specifically, we show that for a quasi-homogeneous polynomial W and an admissible group G within the class, the Frobenius algebra arising in the FJRW theory of [W/G] is isomorphic (as a Frobenius algebra) to the orbifolded Milnor ring of [W^T/G^T], associated to the dual polynomial W^T and dual group G^T.

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