All Outputs (39)

On K-moduli of quartic threefolds (2024)
Journal Article

The family of smooth Fano 3-folds with Picard rank 1 and anticanonical volume 4 consists of quartic 3-folds and of double covers of the 3-dimensional quadric branched along an octic surface. They can all be parametrised as complete intersections of a... Read More about On K-moduli of quartic threefolds.

Machine learning detects terminal singularities (2023)
Conference Proceeding

Algebraic varieties are the geometric shapes defined by systems of polynomial equations; they are ubiquitous across mathematics and science. Amongst these algebraic varieties are Q-Fano varieties: positively curved shapes which have Q-factorial termi... Read More about Machine learning detects terminal singularities.

Polytopes and machine learning (2023)
Journal Article

We introduce machine learning methodology to the study of lattice polytopes. With supervised learning techniques, we predict standard properties such as volume, dual volume, reflexivity, etc, with accuracies up to 100%. We focus on 2d polygons and 3d... Read More about Polytopes and machine learning.

The Rapid Rise of Generative AI: Assessing risks to safety and security (2023)
Report

This CETaS Research Report presents the findings from a major project exploring the implications of generative AI for national security. It is based on extensive engagement with more than 50 experts across government, academia, industry, and civil so... Read More about The Rapid Rise of Generative AI: Assessing risks to safety and security.

Machine learning the dimension of a Fano variety (2023)
Journal Article

Fano varieties are basic building blocks in geometry – they are ‘atomic pieces’ of mathematical shapes. Recent progress in the classification of Fano varieties involves analysing an invariant called the quantum period. This is a sequence of integers... Read More about Machine learning the dimension of a Fano variety.

Machine Learning: The Dimension of a Polytope (2023)
Book Chapter

We use machine learning to predict the dimension of a lattice polytope directly from its Ehrhart series. This is highly effective, achieving almost 100% accuracy. We also use machine learning to recover the volume of a lattice polytope from its Ehrha... Read More about Machine Learning: The Dimension of a Polytope.

Toric Sarkisov Links of Toric Fano Varieties (2023)
Conference Proceeding

We explain a web of Sarkisov links that overlies the classification of Fano weighted projective spaces in dimensions 3 and 4, extending results of Prokhorov.

On the maximum dual volume of a canonical Fano polytope (2022)
Journal Article

We give an upper bound on the volume 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 is achieved... Read More about On the maximum dual volume of a canonical Fano polytope.

Recent Developments in Algebraic Geometry: To Miles Reid for his 70th Birthday (2022)
Book

On the Fine Interior of Three-Dimensional Canonical Fano Polytopes (2022)
Conference Proceeding

The Fine interior ∆FI of a d-dimensional lattice polytope ∆ is a rational subpolytope of ∆ which is important for constructing minimal birational models of non-degenerate hypersurfaces defined by Laurent polynomials with Newton polytope ∆. This paper... Read More about On the Fine Interior of Three-Dimensional Canonical Fano Polytopes.

Interactions with Lattice Polytopes (2022)
Conference Proceeding

This book collects together original research and survey articles highlighting the fertile interdisciplinary applications of convex lattice polytopes in modern mathematics. Covering a diverse range of topics, including algebraic geometry, mirror symm... Read More about Interactions with Lattice Polytopes.

Databases of quantum periods for Fano manifolds (2022)
Journal Article

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 th... Read More about Databases of quantum periods for Fano manifolds.

Laurent polynomials in Mirror Symmetry: why and how? (2022)
Journal Article

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 cl... Read More about Laurent polynomials in Mirror Symmetry: why and how?.

Hilbert series, machine learning, and applications to physics (2022)
Journal Article

We describe how simple machine learning methods successfully predict geometric properties from Hilbert series (HS). Regressors predict embedding weights in projective space to ∼1 mean absolute error, whilst classifiers predict dimension and Gorenstei... Read More about Hilbert series, machine learning, and applications to physics.

Maximally mutable Laurent polynomials (2021)
Journal Article

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 wi... Read More about Maximally mutable Laurent polynomials.

Laurent inversion (2019)
Journal Article

We describe a practical and effective method for reconstructing the deformation class of a Fano manifold X from a Laurent polynomial f that corresponds to X under Mirror Symmetry. We explore connections to nef partitions, the smoothing of singular to... Read More about Laurent inversion.

Gorenstein Formats, Canonical and Calabi–Yau Threefolds (2019)
Journal Article

Gorenstein formats present the equations of regular canonical, Calabi–Yau and Fano varieties embedded by subcanonical divisors. We present a new algorithm for the enumeration of these formats based on orbifold Riemann-Roch and knapsack packing-type a... Read More about Gorenstein Formats, Canonical and Calabi–Yau Threefolds.

Quantum Periods For Certain Four-Dimensional Fano Manifolds (2018)
Journal Article

We collect a list of known four-dimensional Fano manifolds and compute their quantum periods. This list includes all four-dimensional Fano manifolds of index greater than one, all four-dimensional toric Fano manifolds, all four-dimensional products o... Read More about Quantum Periods For Certain Four-Dimensional Fano Manifolds.

Fano 3-folds in P2xP2 format, Tom and Jerry (2017)
Journal Article

We study Q-factorial terminal Fano 3-folds whose equations are modelled on those of the Segre embedding of P^2xP^2. These lie in codimension 4 in their total anticanonical embedding and have Picard rank 2. They fit into the current state of classific... Read More about Fano 3-folds in P2xP2 format, Tom and Jerry.

Minimality and mutation-equivalence of polygons (2017)
Journal Article

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 o... Read More about Minimality and mutation-equivalence of polygons.