by Kurusch Ebrahimi-Fard, Hans Munthe-Kaas and Cordian Riener
For much of the twentieth century, mathematics evolved along two largely separate trajectories.
On one side stood abstraction: algebra, analysis, geometry, topology, and the pursuit of internal
structural understanding. On the other stood computation: algorithms, numerical methods,
and the practical demands of scientific modelling.
Today, this division is rapidly dissolving, along with the notion that computation can be cleanly separated from structure.
Contemporary science relies on computation at a fundamental level, yet increasingly it is
clear that computation itself is not merely a technical tool.
Algorithms behave differently depending on the geometry of the spaces they act on,
the symmetries they respect or violate, and the algebraic laws that govern composition and interaction.
At the same time, new computational challenges reshape the questions mathematicians ask,
revealing structures that might otherwise remain hidden.
Computation is not neutral: it amplifies some structures, suppresses others, and exposes the geometry hidden in problems.
The Lie–Størmer Center (LSC)n bears in its name the legacy of two Norwegian mathematicians whose work epitomized the unity between computation and abstraction: Sophus Lie (1842–1899) and Carl Størmer (1874–1957). Størmer’s computation of charged particle trajectories in the Earth’s magnetic field—reinterpreted today through modern Lie group integration—lies at the heart of the LSC logo, symbolically linking these two giants of Norwegian mathematics. LCS is founded on the conviction that structure and computation must be understood in tandem, not only as a research challenge, but as a guiding principle for how mathematics is developed, taught, and communicated.
Rather than treating computation as a necessary consequence of theory, the Center and its research program view it as a structured phenomenon in its own right, shaped by algebraic relations, geometric constraints, and analytical frameworks, and further organised through categorical principles.
From this perspective, understanding computation means uncovering the laws that make
complex processes coherent rather than fragile, stable rather than ad hoc,
and meaningful rather than opaque.
A central scientific component of the Center’s mission is to develop a unified mathematical foundation
for computation based on three intertwined ideas: symmetry, order, and structure.
These are not auxiliary concepts, but organizing principles:
they determine what can be computed, how efficiently, and with what degree of robustness.
They appear across mathematics and its applications,
from the composition laws governing dynamical systems,
to the geometric invariants that capture persistent features in data,
to the structural parameters that decide whether a computational problem
admits scalable solutions or becomes intractable.
By studying these ideas across both continuous and discrete settings,
the Center seeks to bring symbolic reasoning, numerical approximation,
and data-driven methods into a coherent conceptual framework.
This vision aligns with major developments across modern computational science and builds on research areas where members of the Lie–Størmer Center have made foundational contributions. Advances in Hopf-algebraic, Lie-theoretic, and operadic frameworks reveal deep algebraic principles underlying compositionality and numerical algorithms, while emerging directions in algebraic and combinatorial geometry, such as positive geometry and tropical methods, provide powerful tools for modelling and data analysis. At the algorithmic level, structure-aware computation and symmetry-based complexity theory are reshaping our understanding of scalable computation. These structural perspectives are increasingly transforming applications, from topological data analysis in neuroscience and geometric optimisation in quantum chemistry to differential-geometric approaches in deep learning. Together, these developments illustrate a unified structural–computational paradigm that underpins the Center’s mission and positions Norway to play a leading role in shaping the mathematical foundations of computation.
Rather than organizing research along traditional discipline-driven lines,
the Center brings together complementary perspectives.
Algebraic and combinatorial structures explain how complex operations compose.
Geometric and topological ideas provide invariants that remain meaningful
under noise, approximation, or model reduction. Analytical methods clarify stability, convergence, and quantitative behaviour across scales.
Categorical methods ensure that structure is preserved when models change representation.
Structure-aware algorithms exploit these insights to produce stable,
interpretable, and efficient computational methods.
Together, these perspectives shift the focus from isolated algorithms
to coherent computational systems.
A defining feature of the Center is its emphasis on interaction between mathematical disciplines,
between theory and computation, and between mathematics and the broader scientific landscape
in which it operates.
Progress in one direction naturally feeds into others:
new algebraic insights suggest novel algorithms;
computational constraints reveal previously unseen mathematical structure.
This circular workflow reflects a broader shift in the mathematical sciences,
where theory and computation evolve in continuous dialogue.
The LSC also serves as an institutional framework for sustained collaboration.
By providing continuity beyond individual projects, it creates space for ideas to mature,
for long-term partnerships to form, and for mathematical perspectives on computation
to influence neighboring disciplines.
In this way, the Center functions not only as a site of research,
but as an intellectual infrastructure for the mathematics of computation.
This is further strengthen through strategic partnerships between the Lie–Størmer Center and leading international institutions, including the Max Planck Institute for Mathematics in the Sciences in Leipzig, the Max Planck Institute of Molecular Cell Biology and Genetics in Dresden, and the Heidelberg Institute for Theoretical Studies. By connecting LSC researchers with world-class expertise the Center fosters a vibrant exchange of ideas.
Alongside its research activities supported by seminars, workshops, and conferences, the Center places strong emphasis on training and mentoring as a core part of its mission.
Its goal is to educate a new generation of mathematicians who are fluent
both in abstract reasoning and in computational practice,
and who can move comfortably between theory, algorithms, and applications.
Through advanced schools across all research themes, international collaboration, open dissemination of results,
and a strong culture of exchange, the Center contributes to the long-term
development of the mathematical foundations of computation.
In this sense, the Lie–Størmer Center is not defined by a single method or application.
It is defined by a viewpoint: that computation becomes more powerful when guided by structure,
that mathematical structure comes fully alive when tested against the demands of computation,
and that both gain lasting significance when developed within a shared scientific
and educational culture.
This is the conviction that defines the Lie–Størmer Center.