Departments / Extreme-Scale, Nano & Fundamental Engineering / D09

Mathematics & Computational Foundations

The mathematical foundations the whole department stands on: numerical methods, convergence and stability, optimisation, probability, and the modelling support every other division draws on.

Mathematical Theoretical research 10^-100 m to 10^100 m 11 open

What this work is, and what it is not

A numerical result without a convergence statement is an opinion with decimal places. This division reports the order of a scheme, the conditions under which it is stable, and the regime where its error estimate no longer holds. Where a method is used outside the range it was analysed for, that is stated rather than hoped over.

Theoretical research. Mathematical and theoretical work. Derivation and internal consistency are the standard here; experimental confirmation is not currently available at this scale.

Part of this division's range sits where no experiment reaches: Ultra-fundamental and theoretical, Planck, fundamental and high-energy theory, Cosmological, theoretical and computational. Work in those bands is theory and computation, and every posting says which side of that line it falls on.

Scale bands are a research classification, not a claim of experimental reach. Most of this span cannot be probed by any apparatus that exists: nothing below about 10^-19 m has been measured directly, and anything at 10^26 m or beyond is inferred from observation rather than engineered. Every opportunity states the kind of work it actually is.

Where this division sits

Scale range 10^-100 m to 10^100 m

10-100 m 100 m 10100 m
Experiments reach here 3 bands no experiment reaches This role
B01

Ultra-fundamental and theoretical (10^-100 to 10^-35)
Below the Planck length. No experiment reaches here and none is proposed; work in this band is mathematics and theory about what a physical theory would have to look like at all.

B02

Planck, fundamental and high-energy theory (10^-35 to 10^-18)
From the Planck length up to the smallest distances probed by collider experiments. Almost all of this band is inferred from theory rather than measured.

B03

Subatomic, particle and nuclear (10^-18 to 10^-12)
Quarks, nucleons and nuclei. Studied through accelerators, detectors, and the statistical analysis of very large datasets.

B04

Atomic and quantum (10^-12 to 10^-9)
Atoms, ions, electronic structure and quantum states. Directly measured with spectroscopy, traps and quantum devices.

B05

Nano (10^-9 to 10^-6)
Molecules, nanostructures, two-dimensional materials and quantum dots. Fabricated, imaged and characterised routinely.

B06

Micro (10^-6 to 10^-3)
Microelectronics, MEMS, microfluidics and cellular biology. Mature fabrication and metrology.

B07

Meso and human-scale devices (10^-3 to 10^0)
Components, instruments and devices a person can hold. Where most engineering prototypes are built and tested.

B08

Macro, infrastructure and planetary surface (10^0 to 10^6)
Machines, structures, vehicles, buildings and infrastructure networks, up to regional scale.

B09

Space, planetary and astronomical (10^6 to 10^13)
Planetary bodies, orbits and the Solar System. Reached by spacecraft and observed directly.

B10

Stellar and galactic modelling (10^13 to 10^26)
Stars, star systems, galaxies and large-scale structure. Observed remotely and modelled computationally. Nothing at this scale is engineered.

B11

Cosmological, theoretical and computational (10^26 to 10^100)
At and beyond the edge of the observable universe. A statement about this band is a statement about a cosmological model, not about anything that can be observed.

Charter

This division develops the mathematics other divisions use. It builds numerical methods and proves what they converge to, designs optimisation algorithms, and supports modelling across every scale band the department declares. It is a service division and a research division at once, and it is mandatory to the department because a method nobody has analysed is a method nobody should trust.

Domains

Applied mathematicsNumerical analysisMathematical modellingOptimisationProbability and statisticsDynamical systemsInformation theoryPartial differential equationsDifferential geometryTopology

Works with

Divisions this one collaborates with routinely.

D01D02D08D10D11D15

11 open opportunities

Grouped by career rung. Every posting states its own classification, its scale range and what it expects you to have already done.

Level 1 3

Applied Mathematics Intern

A modelling question from another division formulated mathematically and checked for well-posedness.

Mathematical Theoretical research L1 · Research and Engineering Intern Internship On-site / Hybrid — Kolkata, West Bengal, India

Scale 10^-100 m to 10^100 m

Can state and use the definition of convergence for a numerical scheme, not just cite the order. · Linear algebra to the level of conditioning, decompositions and why an ill-conditioned problem is not a bug in the solver. · Real analysis: limits, continuity, and what makes a problem well-posed. · Has implemented a numerical method from a paper and verified its convergence order empirically. + 5 more

Mathematical Modelling Intern

Building a model of a real system, stating its assumptions, and testing where it stops describing the system.

Mathematical Theoretical research L1 · Research and Engineering Intern Internship On-site / Hybrid — Kolkata, West Bengal, India

Scale 10^-100 m to 10^100 m

Can state and use the definition of convergence for a numerical scheme, not just cite the order. · Linear algebra to the level of conditioning, decompositions and why an ill-conditioned problem is not a bug in the solver. · Real analysis: limits, continuity, and what makes a problem well-posed. · Has implemented a numerical method from a paper and verified its convergence order empirically. + 5 more

Mathematics Research Intern

One mathematical result reconstructed in full, including the steps its source omitted.

Mathematical Theoretical research L1 · Research and Engineering Intern Internship On-site / Hybrid — Kolkata, West Bengal, India

Scale 10^-100 m to 10^100 m

Can state and use the definition of convergence for a numerical scheme, not just cite the order. · Linear algebra to the level of conditioning, decompositions and why an ill-conditioned problem is not a bug in the solver. · Real analysis: limits, continuity, and what makes a problem well-posed. · Has implemented a numerical method from a paper and verified its convergence order empirically. + 5 more

Level 3 3

Applied Mathematician

Whole modelling problems from other divisions: formulation, analysis, and an honest statement of validity.

Mathematical Theoretical research L3 · Engineer and Scientist Full-Time On-site — Kolkata, West Bengal, India

Scale 10^-100 m to 10^100 m

Can state and use the definition of convergence for a numerical scheme, not just cite the order. · Linear algebra to the level of conditioning, decompositions and why an ill-conditioned problem is not a bug in the solver. · Real analysis: limits, continuity, and what makes a problem well-posed. · Has implemented a numerical method from a paper and verified its convergence order empirically. + 5 more

Computational Mathematician

Where the mathematics and the machine meet: conditioning, floating point, and what the computed answer is worth.

Computational Computational research L3 · Engineer and Scientist Full-Time On-site — Kolkata, West Bengal, India

Scale 10^-100 m to 10^100 m

Can state and use the definition of convergence for a numerical scheme, not just cite the order. · Linear algebra to the level of conditioning, decompositions and why an ill-conditioned problem is not a bug in the solver. · Real analysis: limits, continuity, and what makes a problem well-posed. · Has implemented a numerical method from a paper and verified its convergence order empirically. + 5 more

Numerical Mathematics Engineer

Numerical methods implemented, verified against their theoretical order, and handed on as reference code.

Computational Computational research L3 · Engineer and Scientist Full-Time On-site — Kolkata, West Bengal, India

Scale 10^-100 m to 10^100 m

Can state and use the definition of convergence for a numerical scheme, not just cite the order. · Linear algebra to the level of conditioning, decompositions and why an ill-conditioned problem is not a bug in the solver. · Real analysis: limits, continuity, and what makes a problem well-posed. · Has implemented a numerical method from a paper and verified its convergence order empirically. + 5 more

Other divisions in this department