ORCID Profile
0000-0003-2340-5257
Current Organisation
University of Nottingham
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Publisher: Informa UK Limited
Date: 08-12-2016
Publisher: Cambridge University Press (CUP)
Date: 10-06-2016
DOI: 10.1017/S0013091515000115
Abstract: In previous work by Coates, Galkin and the authors, the notion of mutation between lattice polytopes was introduced. Such mutations give rise to a deformation between the corresponding toric varieties. In this paper we study one-step mutations that correspond to deformations between weighted projective planes, giving a complete characterization of such mutations in terms of T -singularities. We also show that the weights involved satisfy Diophantine equations, generalizing results of Hacking and Prokhorov.
Publisher: American Physiological Society
Date: 05-2011
Publisher: Cambridge University Press (CUP)
Date: 26-09-2012
DOI: 10.1017/S0004972711002577
Abstract: For a d -dimensional convex lattice polytope P , a formula for the boundary volume vol( ∂P ) is derived in terms of the number of boundary lattice points on the first ⌊ d /2⌋ dilations of P . As an application we give a necessary and sufficient condition for a polytope to be reflexive, and derive formulas for the f -vector of a smooth polytope in dimensions three, four, and five. We also give applications to reflexive order polytopes, and to the Birkhoff polytope.
Publisher: arXiv
Date: 2022
Publisher: Springer Science and Business Media LLC
Date: 08-09-2023
Publisher: Springer International Publishing
Date: 2022
Publisher: Informa UK Limited
Date: 29-08-2020
Publisher: Springer Science and Business Media LLC
Date: 12-04-2022
DOI: 10.1038/S41597-022-01232-6
Abstract: Fano manifolds are basic building blocks in geometry – they are, in a precise sense, atomic pieces of shapes. The classification of Fano manifolds is therefore an important problem in geometry, which has been open since the 1930s. One can think of this as building a Periodic Table for shapes. A recent breakthrough in Fano classification involves a technique from theoretical physics called Mirror Symmetry. From this perspective, a Fano manifold is encoded by a sequence of integers: the coefficients of a power series called the regularized quantum period. Progress to date has been hindered by the fact that quantum periods require specialist expertise to compute, and descriptions of known Fano manifolds and their regularized quantum periods are incomplete and scattered in the literature. We describe databases of regularized quantum periods for Fano manifolds in dimensions up to four. The databases in dimensions one, two, and three are complete the database in dimension four will be updated as new four-dimensional Fano manifolds are discovered and new regularized quantum periods computed.
Publisher: The Royal Society
Date: 10-2021
Abstract: We introduce a class of Laurent polynomials, called maximally mutable Laurent polynomials (MMLPs), which we believe correspond under mirror symmetry to Fano varieties. A subclass of these, called rigid, are expected to correspond to Fano varieties with terminal locally toric singularities. We prove that there are exactly 10 mutation classes of rigid MMLPs in two variables under mirror symmetry these correspond one-to-one with the 10 deformation classes of smooth del Pezzo surfaces. Furthermore, we give a computer-assisted classification of rigid MMLPs in three variables with reflexive Newton polytope under mirror symmetry these correspond one-to-one with the 98 deformation classes of three-dimensional Fano manifolds with very le anti-canonical bundle. We compare our proposal to previous approaches to constructing mirrors to Fano varieties, and explain why mirror symmetry in higher dimensions necessarily involves varieties with terminal singularities. Every known mirror to a Fano manifold, of any dimension, is a rigid MMLP.
Publisher: No publisher found
Date: 2013
Publisher: Springer Science and Business Media LLC
Date: 28-11-2017
Publisher: The Electronic Journal of Combinatorics
Date: 08-03-2019
DOI: 10.37236/7780
Abstract: Recent work has focused on the roots $z\in\mathbb{C}$ of the Ehrhart polynomial of a lattice polytope $P$. The case when $\Re{z}=-1/2$ is of particular interest: these polytopes satisfy Golyshev's "canonical line hypothesis". We characterise such polytopes when $\mathrm{dim}(P)\leq 7$. We also consider the "half-strip condition", where all roots $z$ satisfy $-\mathrm{dim}(P)/2\leq\Re{z}\leq \mathrm{dim}(P)/2-1$, and show that this holds for any reflexive polytope with $\mathrm{dim}(P)\leq 5$. We give an ex le of a $10$-dimensional reflexive polytope which violates the half-strip condition, thus improving on an ex le by Ohsugi–Shibata in dimension $34$.
Publisher: Royal College of General Practitioners
Date: 12-2011
Publisher: Mathematical Sciences Publishers
Date: 29-02-2016
Publisher: American Mathematical Society (AMS)
Date: 24-09-2015
DOI: 10.1090/PROC/12876
Publisher: Mathematical Institute, Tohoku University
Date: 03-2006
Publisher: Canadian Mathematical Society
Date: 14-12-2010
Abstract: An inductive approach to classifying all toric Fano varieties is given. As an application of this technique, we present a classification of the toric Fano threefolds with at worst canonical singularities. Up to isomorphism, there are 674,688 such varieties.
Publisher: Informa UK Limited
Date: 23-04-2022
Publisher: International Press of Boston
Date: 2019
Publisher: Cambridge University Press (CUP)
Date: 2022
DOI: 10.1017/FMS.2022.93
Abstract: We give an upper bound on the volume $\\operatorname {vol}(P^*)$ of a polytope $P^*$ dual to a d -dimensional lattice polytope P with exactly one interior lattice point in each dimension d . This bound, expressed in terms of the Sylvester sequence, is sharp and achieved by the dual to a particular reflexive simplex. Our result implies a sharp upper bound on the volume of a d -dimensional reflexive polytope. Translated into toric geometry, this gives a sharp upper bound on the anti-canonical degree $(-K_X)^d$ of a d -dimensional Fano toric variety X with at worst canonical singularities.
Publisher: arXiv
Date: 2022
Publisher: Elsevier BV
Date: 04-2022
Publisher: Wiley
Date: 09-02-2008
DOI: 10.1112/S1461157008000387
Abstract: Toric log del Pezzo surfaces correspond to convex lattice polygons containing the origin in their interior and having only primitive vertices. Upper bounds on the volume and on the number of boundary lattice points of these polygons are derived in terms of the index ℓ . Techniques for classifying these polygons are also described: a direct classification for index two is given, and a classification for all ℓ ≤16 is obtained.
Publisher: European Mathematical Society Publishing House
Date: 30-11-2013
DOI: 10.4171/120
Publisher: WORLD SCIENTIFIC
Date: 12-2012
Publisher: The Electronic Journal of Combinatorics
Date: 19-07-2012
DOI: 10.37236/2366
Abstract: We introduce reflexive polytopes of index $l$ as a natural generalisation of the notion of a reflexive polytope of index $1$. These $l$-reflexive polytopes also appear as dual pairs. In dimension two we show that they arise from reflexive polygons via a change of the underlying lattice. This allows us to efficiently classify all isomorphism classes of $l$-reflexive polygons up to index $200$. As another application, we show that any reflexive polygon of arbitrary index satisfies the famous "number $12$" property. This is a new, infinite class of lattice polygons possessing this property, and extends the previously known sixteen instances. The number $12$ property also holds more generally for $l$-reflexive non-convex or self-intersecting polygonal loops. We conclude by discussing higher-dimensional ex les and open questions.
Publisher: Tokyo Institute of Technology, Department of Mathematics
Date: 06-2009
Publisher: The Electronic Journal of Combinatorics
Date: 16-10-2014
DOI: 10.37236/4288
Abstract: We characterise mutations between fake weighted projective spaces, and give explicit formulas for how the weights and multiplicity change under mutation. In particular, we prove that multiplicity-preserving mutations between fake weighted projective spaces are mutations over edges of the corresponding simplices. As an application, we analyse the canonical and terminal fake weighted projective spaces of maximal degree.
Publisher: Wiley
Date: 2012
DOI: 10.1112/S1461157013000041
Abstract: We exhibit seven linear codes exceeding the current best known minimum distance $d$ for their dimension $k$ and block length $n$ . Each code is defined over ${ \\mathbb{F} }_{8} $ , and their invariants $[n, k, d] $ are given by $[49, 13, 27] $ , $[49, 14, 26] $ , $[49, 16, 24] $ , $[49, 17, 23] $ , $[49, 19, 21] $ , $[49, 25, 16] $ and $[49, 26, 15] $ . Our method includes an exhaustive search of all monomial evaluation codes generated by points in the $[0, 5] \\times [0, 5] $ lattice square.
Publisher: American Physiological Society
Date: 12-2005
Publisher: Universidad Catolica del Norte - Chile
Date: 04-2022
DOI: 10.22199/ISSN.0717-6279-5279
Abstract: We survey the approach to mirror symmetry via Laurent polynomials, outlining some of the main conjectures, problems, and questions related to the subject. We discuss: how to construct Landau–Ginzburg models for Fano varieties how to apply them to classification problems and how to compute invariants of Fano varieties via Landau–Ginzburg models.
Publisher: Cambridge University Press (CUP)
Date: 2017
DOI: 10.1017/FMS.2017.10
Abstract: We introduce a concept of minimality for Fano polygons. We show that, up to mutation, there are only finitely many Fano polygons with given singularity content, and give an algorithm to determine representatives for all mutation-equivalence classes of such polygons. This is a key step in a program to classify orbifold del Pezzo surfaces using mirror symmetry. As an application, we classify all Fano polygons such that the corresponding toric surface is qG-deformation-equivalent to either (i) a smooth surface or (ii) a surface with only singularities of type $1/3(1,1)$ .
Publisher: The Royal Society
Date: 03-2015
Abstract: We find at least 527 new four-dimensional Fano manifolds, each of which is a complete intersection in a smooth toric Fano manifold.
Publisher: SIGMA (Symmetry, Integrability and Geometry: Methods and Application)
Date: 08-12-2012
Publisher: Springer Science and Business Media LLC
Date: 04-08-2010
Location: United Kingdom of Great Britain and Northern Ireland
Location: United Kingdom of Great Britain and Northern Ireland
Location: United Kingdom of Great Britain and Northern Ireland
Location: United Kingdom of Great Britain and Northern Ireland
Location: United Kingdom of Great Britain and Northern Ireland
Start Date: 2016
End Date: 2021
Funder: Engineering and Physical Sciences Research Council
View Funded ActivityStart Date: 2018
End Date: 2021
Funder: University of Sydney
View Funded Activity