DANGER: Data, Numbers and Geometry

Thursday 8 August

Arrival

Computer-aided conjecture generation in maths and physics

Proposing good conjectures is at least as valuable as proving theorems. Good conjectures capture our attention and orient our efforts, acting like guide posts on the 'mazy paths to hidden truths' (Hilbert). This talk will touch on three aspects of computer-aided conjecture generation: matching numerical values, symbolic regression and generative models. I will present a Zipf-type law for Physics equations, discuss how genetic algorithms can be used to identify new identities of Rogers-Ramanujan type and present a number of conjectural generating functions for holomorphic line bundle cohomology on certain complex projective varieties.

Lunch

Asymptotic formulae for the regularized quantum period

In this talk, we explore the asymptotic behaviour a certain sequence, which contains the coefficients of a series called the regularized quantum period. This is conjectured to be a complete invariant of Fano varieties and its can be considered as a numerical fingerprint of each Fano variety. We discuss how these results originate from the study of large datasets using data analysis and machine learning techniques and show how they can be used to visualise the landscape of these objects. This is joint work with Tom Coates and Alexander Kasprzyk.

Modelling machine learning systems

How can we make sure that a machine learning system gives results as it should, is not biased or is safe to use–as we would like for self-driving cars, for example? We cannot. Not fully. But we can try! This talk will be an incursion in theoretical computer science and formal verification. We will explain ways one can model machine learning systems to try to obtain partial guarantees on them, and how to enforce some safety property.

Formalising topological data analysis in Cubical Agda

Topological data analysis (TDA) offers a plethora of methods and tools to study the “shape of data”. This talk is about work towards implementing a TDA tool inside a theorem prover, leading to software that is formally proved to implement the intricate mathematical theory underlying TDA. The theorem prover of choice for this project is Cubical Agda, which implements ideas from Homotopy Type Theory and offers a logic to directly reason about (homotopy types of) topological spaces. We give a taste of what it’s like to reason in this theorem prover, presenting a formalisation of discrete Morse theory for graphs, and discuss challenges for turning this formalisation into a fully fledged tool for TDA.

Break

Machine Learning of the Prime Distribution

We provide a theoretical argument explaining the experimental observations of Yang-Hui He about the learnability of primes, and posit that the Erdős-Kac law would very unlikely be discovered by current machine learning techniques. Numerical experiments that we perform corroborate our theoretical findings. This is joint work with A. Alistair Rocke.

Precision string phenomenology

Calabi-Yau compactifications of string theory lead to quantum field theories in four dimensions with chiral matter. Calculating parameters of the low-energy effective theory in general compactifications requires the Ricci-flat metric on the Calabi-Yau manifold. Such metrics are not known analytically. In this talk, we discuss how to approximate the Ricci-flat metric using neural networks. The accuracy of the numerical metrics is assessed for K3 and the quintic threefold. In the standard embedding, we calculate Yukawa couplings for compactifications on various Calabi-Yau geometries. This is an initial step toward a first principles calculation of particle masses from string theory.

Friday 9 August

Math+AI=AGI

In this talk, we explore the transformative potential of off-the-shelf reinforcement learning (RL) algorithms in accelerating solutions to complex, research-level mathematical challenges. We begin by illustrating how these algorithms have achieved a 10× improvement in areas where previous advances of the same magnitude required many decades. A comparative analysis of different network architectures is presented to highlight their performance in this context. We then delve into the application of RL algorithms to exceptionally demanding tasks, such as those posed by the Millennium Prize problems and the smooth Poincaré conjecture in four dimensions. Drawing on our experiences, we discuss the prerequisites for developing new RL algorithms and architectures that are tailored to these high-level challenges.

Reproducing kernel Hilbert C*-module for data analysis

Reproducing kernel Hilbert C*-module (RKHM) is a generalisation of Reproducing kernel Hilbert space (RKHS) and is characterised by a C*-algebra-valued positive definite kernel and the inner product induced by this kernel. The advantages of applying RKHMs to data analysis instead of RKHSs are that we can enlarge representation spaces, construct positive definite kernels using the product structure in the C*-algebra, and use the operator norm for theoretical analyses. We show fundamental properties in RKHMs for data analysis, such as a minimization property of the orthogonal projection and representer theorems. Then, we propose a deep RKHM, which is constructed as the composition of multiple RKHMs. This framework is valid, for example, for analysing image data.

Lunch

Machine models for weighted spaces

We explore the idea of using machine learning to determine rational points on weighted projective varieties. Our work leads to many interesting questions on the geometry of weighted projective spaces and how machine learning models can be used to study weighted spaces. While mathematically it seems that such models would be superior to more traditional models, it is still an open question to provide computational evidence to support such claims.

Break

Machine learning in the moduli space of genus two curves

We use machine learning to study the moduli space of genus two curves. During the talk, we will focus on the locus Lₙ of genus two curves with (n,n)-split Jacobian. More precisely, we design a transformer model which, given moduli points (i.e., values for the Igusa invariants), determines if the corresponding genus two curve is in the locus Lₙ, for n = 2, 3, 5, and 7. During this study we discover some interesting arithmetic properties which seem difficult to guess otherwise. For example we show that there are no rational points p∈ Lₙ with weighted moduli height ≤ 2 in any of L₂, L₃, and L₅.

Break

Applying a computational algebra and data science pipeline to Clifford algebra and cluster algebra data

Rather than relying on existing data sets to data mine, the combination of computational algebra and data science techniques allows one to simultaneously create algebraic data sets and to then apply standard data science techniques in order to try to find patterns in this generated data through exploratory data analysis. Ultimately, this could help complement mathematical intuition in guiding hypothesis formulation and theorem proving. We will show examples from recent work on cluster algebras and Clifford algebras, such as machine learning interesting new Clifford invariants of linear functions, using the example of Coxeter elements as transformations of interest with links to other, well-known, geometric invariants.