Abeliuk Institute

Privately funded · Research and education

The Institute funds theoretical research on information, learning and complex systems.

We study systems that are fully specified and still not understood — trained networks, evolving populations, folding chains — using the methods of statistical mechanics, information theory and dynamical systems. The Institute funds undergraduate fellows to work on these questions, carries out its own theoretical programme, and works with university and industrial researchers whose questions their own funding will not cover. The Institute's mission is also to educate: to publish content that makes these ideas accessible beyond specialists, and to support initiatives that spread them and open them to discussion.

Funding
Private; topics chosen for interest, not fundability
Open call
Undergraduate fellowships, read on a rolling basis
Output
Proofs, models and reproducible numerical studies
Reach
Physics, chemistry, biology and mathematics
computed at load
Fig. I — One line of arithmetic, x ↦ r x(1 − x), iterated for 1,024 values of r. Nothing in the rule names the doubling, and no amount of reading it tells you where the order ends. The gap between a complete specification and what the system does is the subject the Institute funds.

Programmes

Three ways in: a fellowship for students, the Institute's own research, and joint work with groups elsewhere.

  1. I

    Undergraduate fellowships

    Open call · rolling

    For an undergraduate with a question of their own: a stipend that lets them work on it rather than take other work, compute time, travel to the workshop where the people who care about the problem will be, and the costs of a preprint.

    Any institution and any department. No advisor, publication record, transcript or reference letters required. Applications are read by the people who would work with the fellow, and everyone who applies gets an answer.

    Read the call and apply →

  2. II

    Institute research

    Ongoing

    The Institute's own theoretical work. Large questions are cut into small ones stated precisely, and each is carried to a theorem or to a simulation whose code is short enough to read. Results appear as preprints, with the code that produced them.

  3. III

    Collaborations

    By correspondence

    Joint work with university groups, and with researchers inside industrial labs who have a question their roadmap will not let them chase. Proposals, objections and counterexamples all arrive the same way.

    Write to the Institute →

Areas of inquiry

The core is information and learning: what a system can extract from data, what that costs, and what the result can be known to do. Modern machine learning produced objects that behave like physical systems, with phases, thresholds and collective behaviour no line of their code mentions, and we think their useful theory is closer to statistical mechanics than to software engineering. The same questions reach into physics, chemistry, biology and mathematics, wherever the rules are known and their consequences are not.

AREA 1

Learning in networks and populations

Why do enormous networks generalise, and what stays fixed while a population adapts?

Networks large enough to memorise their data still work on data they have never seen. Their error falls as a clean power of data and compute, and full-batch gradient descent drives their curvature to the value at which training should break down, then keeps training there. None of this follows from the rules as written, and no known measure of a solution yet says reliably whether it will generalise. Evolution poses the same question from the other side: what a theory of selection could predict rather than recount, and how rugged real fitness landscapes are.

Fig. II — Two descents begun 0.15 apart on the same surface end in different valleys at different depths.

AREA 2

Phases, thresholds and coarse-graining

Where does a system change character, and which parameter is the threshold in?

A network that has only memorised its data can suddenly generalise, and each layer of a deep network may be doing what a physicist does when coarse-graining. Matter has thresholds of its own that are measured far better than they are explained: crystals melt when their atoms' vibrations reach roughly the same fraction of the spacing between them, very different liquids gain nearly the same entropy when they boil, and a system started hotter sometimes reaches equilibrium sooner than one started cooler.

Fig. III — A lattice occupied at p ≈ pc, clusters labelled by exhaustive search. Just below, dust; just above, a single sheet.

AREA 3

Information, energy and time

What does it cost to predict, and to learn, a system whose every rule is known?

Determinism is not predictability: the gap has a rate, a horizon and a cost in bits. Around it sit questions about the physical price of information: how much energy training must use, as opposed to how much it happens to use; whether computing with spikes is a different kind of computing or only a cheaper one; why the energy levels of chaotic quantum systems fluctuate like the eigenvalues of random matrices; and why cause and effect appear at human scale when the underlying physics runs equally well backwards.

Fig. IV — Two integrations of the same three equations, started 10⁻⁵ apart. Every constant is shared; the histories are not.

AREA 4

From rules to structure

The rules settle the answer. Why can it not be read off them?

A protein's sequence settles its fold, the equations of quantum mechanics settle an atom's shells and a molecule's properties, and a crystal's composition settles its phases. Chemistry still runs on rules that are used every day and not derived: the order in which atoms fill their shells, why the highest-spin state usually lies lowest, the 4n + 2 rule for aromatic rings, Pauling's rules for ionic crystals. In biology we ask what protein language models and structure predictors have learned rather than remembered, how far back a sequence can remember its history, and whether the genetic code was optimised or frozen by accident. The work is models, bounds and computation, not the bench.

Fig. V — A fourteen-bead toy protein. Every fold is enumerated and exactly one reaches the lowest energy: the sequence alone selects the fold, and here the only way to find it was to check them all.

AREA 5

Problems computation can settle

Which open problems will yield, at least in part, to careful computation at scale?

Many celebrated open problems in mathematics and mathematical physics can be attacked in part by exhaustive search, exact arithmetic and machine-checked proof: the 3n + 1 problem, the six-state busy beaver, the Ramsey number R(5, 5), Hadamard matrices of every order, four mutually unbiased bases in dimension six, the number of central configurations of n bodies. We are interested both in what a computation can establish and in making its result trustworthy enough to count.

Fig. VI — Steps to reach 1 under n ↦ n/2 or 3n + 1, for every starting number up to 10,000. Every one of them gets there; nobody has proved that every number does.

How the Institute works

Correspondence

Enquiries, collaboration proposals, notes, counterexamples and objections reach the Institute here. We read everything that arrives and answer what we can. Fellowship applications go through the fellowship page.