Pillar III · The mathematical lens

Explore mathematics and data science for scientific discovery

Mathematical Data Science and Frontier Exploration

We investigate mathematical and data-science questions and develop methods that reveal structure in complex observations. Our work spans mathematical modelling, statistical inference, scientific computing, and the analysis of complex data.

Scientific problems help us develop and test these methods. New abstractions, findings, and limitations can in turn reshape our models, evidence strategies, and research questions. We examine what can be inferred, computed, and transferred under explicit assumptions, and how these insights can support scientific understanding and design.

Core question
What mathematical and data-science methods can reveal important structure and open new scientific possibilities?
Research focus
Investigate mathematical and data-science questions and develop methods that reveal structure in complex observations.
Intended contribution
Abstractions, statistical and computational tools, and testable stress-test frameworks that reveal structure, limits, and new scientific possibilities.

Explore what the observations identify

Explore the limits of inference

When does a good fit become a unique answer?

Two unknown quantities can explain the same observation. Change the available constraints and inspect which answers remain compatible.

Illustrative linear model on 0 ≤ a, b ≤ 4. Exact values and error bounds are stipulated, not experimental data or confidence intervals.

Compare all four constraint sets

One sum

A single exact constraint

a + b = 4

A line of possible answers

Every point on the blue line fits the sum. The marked pairs are examples; infinitely many pairs remain compatible within the stated domain.

The same relation again

A redundant exact constraint

a + b = 4

2a + 2b = 8

Still a line of possible answers

Doubling the same equation adds no independent relation. Repeated noisy measurements can improve precision, but this exact redundant constraint does not resolve structural ambiguity.

An independent relation

Add the exact difference

a + b = 4

a − b = 2

One answer within this model

The independent sum and difference lines meet once. The answer is unique under these stipulated equations; this does not prove that the equations describe a unique physical mechanism.

a = 3, b = 1

Allow measurement error

Bounded error in both relations

|a + b − 4| ≤ 0.4

|a − b − 2| ≤ 0.4

A region of compatible answers

The shaded bands overlap in a region. Values throughout this region satisfy both error bounds. Its shape shows compatibility, not probability or confidence.

What does uniqueness depend on?

Identifiability is conditional on a model and its observation rules. More repetitions of the same relation need not add an independent constraint. Uncertainty, nearly dependent measurements, model mismatch and experimental feasibility must be assessed separately. This two-variable example does not describe a completed inference system.

View the original linear example

A good fit need not identify a unique explanation.

Identifiability asks whether different underlying explanations can produce the same observations. This simple example shows why the kind of measurement matters, even when it is exact.

The measurement a plus b equals four leaves a line of possible pairs, including (1,3), (2,2) and (3,1). Adding the independent measurement a minus b equals two leaves one intersection at (3,1).
Illustrative linear model with exact measurements, not experimental data. The blue solid line represents the sum measurement; the green dashed line represents the difference measurement. Uniqueness here depends on the assumed equations and does not establish a physical mechanism.
Read the calculation and its limits

In this example, a and b are two unknown quantities between zero and four. Observing only their sum, a + b = 4, cannot distinguish (1,3), (2,2), (3,1), or the other points on the line.

Measuring the same sum more precisely does not remove this structural ambiguity. A second relation, a − b = 2, supplies independent information. Adding the two equations gives 2a = 6, so a = 3 and b = 1.

With measurement uncertainty, the lines become regions of compatible values. Nearly dependent measurements can also make estimates unstable. In scientific settings, model assumptions and whether such independent measurements are feasible must be examined separately.

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An open blue surface and a closed navy ring are crossed by a translucent teal plane, with three amber points highlighting selected relationships.

A conceptual view of mathematical structures and constraints as tools for exploring possible explanations. These abstract forms are illustrative; the worked example on this page gives an exact identifiability argument.

Mathematical-lens conceptual framework

From a scientific bottleneck to a testable method

A scientific bottleneck leads through assumptions and method development to stress testing and a new constraint or possibility. A feedback path returns stress tests to the assumptions.

View full-size framework: Wide layout Vertical layout

Read the framework step by step
  1. 01

    Scientific bottleneck

    Identify a question that existing abstractions, data structures, or computations cannot resolve adequately.

  2. 02

    Abstraction and assumptions

    Define the mathematical objects, invariances, constraints, and conditions under which an answer is possible.

  3. 03

    Method or algorithm

    Develop a statistical, computational, geometric, dynamical, or information-theoretic tool.

  4. 04

    Stress test and uncertainty

    Challenge identifiability, sensitivity, computation, failure modes, and alternative explanations.

  5. 05

    New constraint or possibility

    Return a verified limit, method, question, or capability target to the wider research system.

This diagram describes the exploratory logic of Pillar III. It is a conceptual framework, not evidence of completed cross-domain transfer or a validated frontier theory.

Scope

Mathematical structure becomes a scientific instrument.

This Pillar begins where a new abstraction, method, computational bottleneck, or consequential transfer question can change what a scientific system is able to resolve.

01

Mathematical foundations

Develop useful abstractions through probability, statistics, optimisation, geometry, topology, graphs, dynamical systems, and information theory.

02

Structure discovery

Study clustering, dimensionality reduction, anomaly discovery, representation diagnostics, and visual analytics for complex scientific observations.

03

Complex data systems

Organise multimodal, multiscale, spatiotemporal, and network data while preserving context, uncertainty, and meaningful relations.

04

Scientific computing and translation

Address simulation, surrogate modelling, numerical methods, computational bottlenecks, and carefully evidenced transfer across domains.

Research contributions

Methods that reveal structure in complex observations.

We examine what can be inferred, computed, and transferred under explicit assumptions, and how these insights can support scientific understanding and design.

Method families under study

  • Applied and foundational mathematical modelling
  • Probability, statistics, optimisation, and information theory
  • Geometry, topology, graph methods, and dynamical systems
  • Data mining, clustering, dimensionality reduction, anomaly discovery, and visual analytics
  • Multimodal, multiscale, spatiotemporal, and network analysis
  • Simulation, surrogate modelling, numerical methods, uncertainty, and causal analysis

What we aim to develop

Mathematical specification
A precise statement of objects, assumptions, invariances, identifiability conditions, and limits.
Method or computational tool
A documented algorithm, analysis, or implementation connected to a defined scientific bottleneck.
Stress-test benchmark
An evaluation framework with alternative explanations, controls, uncertainty checks and challenging cases.
Transfer study
Evidence showing where a method transfers, where it fails, and which new constraint or question returns to the research system.

Representative testbeds

Frontier settings expose mathematical bottlenecks.

Consequential domains are used to stress assumptions, computation, uncertainty, and transfer. They are testbeds for method development, not an applications bucket.

  • 01Molecular and biomedical systems
  • 02Materials science
  • 03Aging, longevity, and neurotechnology
  • 04Quantum and complex physical systems

Research horizon

Explore broadly while keeping present capability explicit.

These are planning windows. Expansion will follow research progress and validation.

  1. Now

    Mechanistic questions in molecular and materials science

    We develop and test AI methods for mechanism understanding, discovery, and design through scientific questions and feedback from experiments and simulation.

  2. Over approximately five years

    Building reusable scientific capabilities

    We aim to develop reusable representation and model methods, scientific system architectures, research infrastructure, evidence-acquisition strategies, and evaluation systems that support mechanism understanding and reliable design.

  3. Over approximately five to ten years

    Testing principles across systems

    We aim to extract mathematical, computational, and system-design principles and test how these methods and capabilities transfer and combine across broader scientific fields.

Scientific principles

A frontier method is credible when its assumptions and limits are visible.

State assumptions before interpretation

Define the mathematical objects, boundary conditions, data regime, and computational approximations before assigning scientific meaning.

Test identifiability and alternatives

Mechanistic, invariant, causal, or interpretable language requires evidence that challenges plausible competing explanations.

Expose uncertainty and failure

Report sensitivity, approximation error, computational limits, negative transfer, and out-of-distribution behaviour as part of the method.

Demonstrate transfer

A method is not cross-domain merely because it can be executed elsewhere; its assumptions, baselines, and scientific value must be re-tested.

Continue through the system

Connect mathematical questions to learning, evidence and published work.

ResearchReturn to the system mapSee how all three Pillars work together. OutputsExplore publicationsDiscover the work behind our research. CollaborateBring a testable questionStart a scientific or experimental conversation.