arXiv subjects

Jack Smith

Publications and source records attributed to Jack Smith.

17 recordsLinked to original sources

arXiv2024

Quantum cohomology and Fukaya summands from monotone Lagrangian tori

Let $L$ be a monotone Lagrangian torus inside a compact symplectic manifold $X$, with superpotential $W_L$. We show that a geometrically-defined closed-open map induces a decomposition of the quantum cohomology $\operatorname{QH}^*(X)$ into a product, where one factor is the localisation of the Jacobian ring $\operatorname{Jac} W_L$ at the set of isolated critical points of $W_L$. The proof involves describing the summands of the Fukaya category corresponding to this factor -- verifying the expectations of mirror symmetry -- and establishing an automatic generation criterion in the style of Ganatra and Sanda, which may be of independent interest. We apply our results to understanding the structure of quantum cohomology and to constraining the possible superpotentials of monotone tori

math.SG↗

arXiv2024

Improved stability for the size and structure of sumsets

Let $A \subset \mathbb{Z}^d$ be a finite set. It is known that the sumset $NA$ has predictable size ($\vert NA\vert = P_A(N)$ for some $P_A(X) \in \mathbb{Q}[X]$) and structure (all of the lattice points in some finite cone other than all of the lattice points in a finite collection of exceptional subcones), once $N$ is larger than some threshold. In previous work, joint with Shakan, the first and third named authors established the first effective bounds for both of these thresholds for an arbitrary set $A$. In this article we substantially improve each of these bounds, coming much closer to the corresponding lower bounds known.

math.CO↗

arXiv2023

Hamiltonian isotopies of relatively exact Lagrangians are orientation-preserving

Given a closed, orientable Lagrangian submanifold $L$ in a symplectic manifold $(X, \omega)$, we show that if $L$ is relatively exact then any Hamiltonian diffeomorphism preserving $L$ setwise must preserve its orientation. In contrast to previous results in this direction, there are no spin hypotheses on $L$. Curiously, the proof uses only mod-2 coefficients in its singular and Floer cohomology rings.

math.SG↗

arXiv2023

Hochschild Cohomology of the Fukaya Category via Floer Cohomology with Coefficients

Given a monotone Lagrangian $L$ in a compact symplectic manifold $X$, we construct a commutative diagram relating the closed-open string map $\mathcal{CO}_\lambda \colon \operatorname{QH}^*(X) \to \operatorname{HH}^*(\mathcal{F} (X)_\lambda)$ to a variant of the length-zero closed-open map on $L$ incorporating $\mathbf{k}[\operatorname{H}_1(L; \mathbb{Z})]$ coefficients, denoted $\mathcal{CO}^0_\mathbf{L}$. The former is categorically important but very difficult to compute, whilst the latter is geometrically natural and amenable to calculation. We further show that, after a suitable completion, injectivity of $\mathcal{CO}^0_\mathbf{L}$ implies injectivity of $\mathcal{CO}_\lambda$. Via Sheridan's version of Abouzaid's generation criterion, this gives a powerful tool for proving split-generation of the Fukaya category. We illustrate this by showing that the real part of a monotone toric manifold (of minimal Chern number at least 2) split-generates the Fukaya category in characteristic 2. We also give a short new proof (modulo foundational assumptions in the non-monotone case) that the Fukaya category of an arbitrary compact toric manifold is split-generated by toric fibres.

math.SG↗

arXiv2022

Homological Lagrangian monodromy for some monotone tori

Given a Lagrangian submanifold $L$ in a symplectic manifold $X$, the homological Lagrangian monodromy group $\mathcal{H}_L$ describes how Hamiltonian diffeomorphisms of $X$ preserving $L$ setwise act on $H_*(L)$. We begin a systematic study of this group when $L$ is a monotone Lagrangian $n$-torus. Among other things, we describe $\mathcal{H}_L$ completely when $L$ is a monotone toric fibre, make significant progress towards classifying the groups than can occur for $n=2$, and make a conjecture for general $n$. Our classification results rely crucially on arithmetic properties of Floer cohomology rings.

math.SG↗

arXiv2021

Viability of a Dyson Swarm as a Form of Dyson Sphere

First conceptualised in Olaf Stapledon's 1937 novel 'Star Maker', before being popularised by Freeman Dyson in the 1960s, Dyson Spheres are structures which surround a civilisation's sun to collect all the energy it radiates. Through discussion of the features of such a feat of engineering, the viability, scale and likely design of a Dyson structure is evaluated, before details about each stage of its construction and operation are investigated. It is found that a Dyson Swarm, a large array of individual satellites orbiting another planetary body, is the ideal design for such a structure over the solid sun-surrounding structure which is typically associated with the Dyson Sphere. In our solar system, such a structure based around Mars would be able to account for the Earth's 2019 global power consumption of 18.35 TW within fifty years once its construction has begun, which itself could start by 2040. The swarm of over 5.5 billion satellites would be constructed on the surface of Mars before being launched by electromagnetic accelerators into a Martian orbit. Efficiency of the Dyson Swarm ranges from around 0.74 - 2.77% of the Sun's $3.85\times10^{26}$ W output, with large potential for growth as both current technologies improve and future concepts are brought to reality in the time before and during the swarm's construction. Not only would a Dyson Swarm provide a near infinite, renewable power source for Earth, it would allow for significant expansions in human space exploration and for our civilisation as a whole.

physics.pop-ph↗

arXiv2021

Strategies for Democratization of Supercomputing: Availability, Accessibility and Usability of High Performance Computing for Education and Practice of Big Data Analytics

There has been an increasing interest in and growing need for high performance computing (HPC), popularly known as supercomputing, in domains such as textual analytics, business domains analytics, forecasting and natural language processing (NLP), in addition to the relatively mature supercomputing domains of quantum physics and biology. HPC has been widely used in computer science (CS) and other traditionally computation intensive disciplines, but has remained largely siloed away from the vast array of social, behavioral, business and economics disciplines. However, with ubiquitous big data, there is a compelling need to make HPC technologically and economically accessible, easy to use, and operationally democratized. Therefore, this research focuses on making two key contributions, the first is the articulation of strategies based on availability, accessibility and usability for the demystification and democratization of HPC, based on an analytical review of Caliburn, a notable supercomputer at its inception. The second contribution is a set of principles for HPC adoption based on an experiential narrative of HPC usage for textual analytics and NLP of social media data from a first time user perspective. Both, the HPC usage process and the output of the early stage analytics are summarized. This research study synthesizes expert input on HPC democratization strategies, and chronicles the challenges and opportunities from a multidisciplinary perspective, of a case of rapid adoption of supercomputing for textual analytics and NLP. Deductive logic is used to identify strategies which can lead to efficacious engagement, adoption, production and sustained usage for research, teaching, application and innovation by researchers, faculty, professionals and students across a broad range of disciplines.

cs.CY↗

arXiv2019

Superfiltered $A_\infty$-deformations of the exterior algebra, and local mirror symmetry

The exterior algebra $E$ on a finite-rank free module $V$ carries a $\mathbb{Z}/2$-grading and an increasing filtration, and the $\mathbb{Z}/2$-graded filtered deformations of $E$ as an associative algebra are the familiar Clifford algebras, classified by quadratic forms on $V$. We extend this result to $A_\infty$-algebra deformations $\mathcal{A}$, showing that they are classified by formal functions on $V$. The proof translates the problem into the language of matrix factorisations, using the localised mirror functor construction of Cho-Hong-Lau, and works over an arbitrary ground ring. We also compute the Hochschild cohomology algebras of such $\mathcal{A}$. By applying these ideas to a related construction of Cho-Hong-Lau we prove a local form of homological mirror symmetry: the Floer $A_\infty$-algebra of a monotone Lagrangian torus is quasi-isomorphic to the endomorphism algebra of the expected matrix factorisation of its superpotential.

math.SG↗

arXiv2019

A monotone Lagrangian casebook

We present an array of new calculations in Lagrangian Floer theory which demonstrate observations relating to symplectic reduction, grading periodicity, and the closed-open map. We also illustrate Perutz's symplectic Gysin sequence and the quilt theory of Wehrheim and Woodward.

math.SG↗

arXiv2019

Homological Berglund-H\"ubsch mirror symmetry for curve singularities

Given a two-variable invertible polynomial, we show that its category of maximally-graded matrix factorisations is quasi-equivalent to the Fukaya-Seidel category of its Berglund-H\"ubsch transpose. This was previously shown for Brieskorn-Pham and $D$-type singularities by Futaki-Ueda. The proof involves explicit construction of a tilting object on the B-side, and comparison with a specific basis of Lefschetz thimbles on the A-side.

math.SG↗

arXiv2018

Monotone Lagrangians in $\mathbb{CP}^n$ of minimal Maslov number $n+1$

We show that a monotone Lagrangian $L$ in $\mathbb{CP}^n$ of minimal Maslov number $n + 1$ is homeomorphic to a double quotient of a sphere, and thus homotopy equivalent to $\mathbb{RP}^n$. To prove this we use Zapolsky's canonical pearl complex for $L$ with coefficients in $\mathbb{Z}$, and various twisted versions thereof, where the twisting is determined by connected covers of $L$. The main tool is the action of the quantum cohomology of $\mathbb{CP}^n$ on the resulting Floer homologies.

math.SG↗

arXiv2018

Generating the Fukaya categories of compact toric varieties

Let $X$ be a compact toric variety. The quantum cohomology of $X$ decomposes as a direct sum, and associated to each summand $Q$ is a toric fibre $L_Q$ with rank $1$ local system. By building an explicit twisted-complex-like object, we show that on $Q$ the Kodaira-Spencer isomorphism of Fukaya-Oh-Ohta-Ono factors through the closed-open string map to the Hochschild cohomology of $L_Q$. We deduce that the latter is injective and hence, assuming an appropriate version of Abouzaid's criterion, that $L_Q$ split generates the corresponding summand of the Fukaya category.

math.SG↗

arXiv2018

Quantum cohomology and closed-string mirror symmetry for toric varieties

We give a short new computation of the quantum cohomology of an arbitrary smooth toric variety $X$, by showing directly that the Kodaira-Spencer map of Fukaya-Oh-Ohta-Ono defines an isomorphism onto a suitable Jacobian ring. The proof is based on the purely algebraic fact that a class of generalised Jacobian rings associated to $X$ are free as modules over the Novikov ring. In contrast to previous results of this kind, $X$ need not be compact. When $X$ is monotone the presentation we obtain is completely explicit, using only well-known computations with the standard complex structure.

math.SG↗

arXiv2017

Discrete and continuous symmetries in monotone Floer theory

This paper studies the self-Floer theory of a monotone Lagrangian submanifold $L$ of a symplectic manifold $X$ in the presence of various kinds of symmetry. First we suppose $L$ is $K$-homogeneous and compute the image of low codimension $K$-invariant subvarieties of $X$ under the length-zero closed-open string map. Next we consider the group $\mathrm{Symp}(X, L)$ of symplectomorphisms of $X$ preserving $L$ setwise, and extend its action on the Oh spectral sequence to coefficients of arbitrary characteristic, incorporating its action on the classes of holomorphic discs. This imposes constraints on the differentials which force them to vanish in certain situations. These techniques are combined to study a family of homogeneous Lagrangians in products of projective spaces, which exhibit some unusual properties.

math.SG↗

arXiv2015

Floer cohomology of Platonic Lagrangians

We analyse holomorphic discs on Lagrangian SU(2)-orbits in a family of quasihomogeneous threefolds of SL(2, C), previously studied by Evans-Lekili, introducing several techniques that should be applicable to wider classes of homogeneous Lagrangians. By studying the closed-open map we place strong restrictions on the self-Floer cohomology of these Lagrangians, which we then compute using the Biran-Cornea pearl complex.

math.SG↗

arXiv1994

Exclusive W^+ + photon production in proton-antiproton collisions II: results

We present results for total cross sections, single and double differential distributions and correlations between pairs of outgoing particles in the reactions p + antip --> W^+ + photon and p + antip --> W^+ + photon + jet at sqrt(S)=1.8 TeV. Order alpha-strong QCD corrections and leading logarithm photon bremsstrahlung contributions are included in the MS-bar mass factorization scheme for three experimental scenarios: 1) 2-body inclusive production of W^+ and photon, 2) exclusive production of W^+, photon and 1 jet and 3) exclusive production of W^+ and photon with 0 jet. The latest CTEQ parton distribution functions, which fit the newly released HERA data, are used in our analysis. The dependence of our results on the mass factorization scale is used to place error bars on our predictions for the single differential distributions and correlations.

hep-ph↗

arXiv1994

Exclusive W + photon production in proton-antiproton collisions I: general formalism

We present a detailed computation of the fully exclusive cross section of p + antip --> W + photon + X with X = 0 and 1 jet in the framework of the factorization theorem and dimensional regularization. Order alpha-strong and photon bremsstrahlung contributions are discussed in the MS-bar mass factorization scheme. The resulting expressions are ready to be implemented numerically using Monte Carlo techniques to compute single and double differential cross sections and correlations between outgoing pairs of particles.

hep-ph↗