Important Mathematics Institutions | Institut des Hautes Études Scientifiques (IHES)

The simplest useful definition of IHES

IHES is an institution built around a radical allocation of a scarce resource: uninterrupted intellectual attention.

Universities have many jobs. They educate students, award degrees, maintain curricula, operate laboratories, provide public service, supervise examinations and conduct research. A specialised institute can narrow the operating problem. IHES states that its researchers are given the possibility of devoting themselves entirely to research without teaching or administrative obligations.

This does not mean researchers work alone. Quite the opposite. IHES combines a small number of permanent professors with a large international visitor programme. Official IHES material states that more than 200 scientists come to the Institute each year for research visits. The result is a structure with two properties that are difficult to obtain simultaneously:

• continuity, because permanent researchers and institutional memory remain; and
• renewal, because visitors continually bring new questions, methods and collaborations into the same environment.

Permanent depth + rotating frontier = a research institution that can remember and change at the same time.

Why IHES belongs immediately after IAS in this series

The first article in this series examined the Institute for Advanced Study. IHES is a natural second node because its founding story is directly connected to that model.

Official IHES history says Léon Motchane wanted to build a European counterpart to the Institute for Advanced Study. Robert Oppenheimer, then Director of IAS, supported the new institute and joined its Scientific Committee. Yet the relationship should not be read as simple imitation. An institution becomes historically significant only if it develops enough scientific identity to attract researchers for its own reasons.

IHES did that remarkably quickly. Jean Dieudonné and Alexander Grothendieck joined at its birth. René Thom arrived in the early 1960s. Later permanent professors included Pierre Deligne, Dennis Sullivan, Alain Connes, Jean Bourgain, Mikhail Gromov, Laurent Lafforgue and others whose work reshaped major areas of modern Mathematics.

The important comparison is therefore not “IAS versus IHES.” The more useful question is: what happens when several institutions around the world independently protect high-autonomy fundamental research and then exchange people between them?

The answer is a network. Researchers move between Paris-Saclay, Princeton, Bonn, Oxford, Chicago, MIT, Paris, Zürich, Kyoto, Bengaluru and many other centres. A theorem may be conceived in one place, clarified in another, presented at a third and completed through a collaboration whose participants belong to several institutions.

1958: the institution begins with Mathematics at the centre

IHES was created in 1958 by Léon Motchane, a businessman with a deep personal interest in Mathematics who had himself completed a doctoral thesis. The Institute’s official mission describes his goal as creating a research centre modelled partly on Princeton’s IAS and devoted to theoretical sciences, particularly Mathematics and theoretical physics.

The early appointments were extraordinary. Jean Dieudonné became the first permanent professor, and official IHES history records that he accepted on the condition that Alexander Grothendieck also be recruited.

That decision mattered far beyond staffing.

Dieudonné was a major analyst, algebraist and expositor and had been part of the Bourbaki collective. Grothendieck was building a new architecture for algebraic geometry. Their collaboration helped create Éléments de Géométrie Algébrique, while Grothendieck’s seminars at IHES became the celebrated Séminaires de Géométrie Algébrique.

This is a useful example of institutional leverage. IHES did not “invent algebraic geometry.” Algebraic geometry existed long before 1958. What it provided was an environment in which a new foundational program could be pursued at enormous depth and then distributed through papers, seminars, students and collaborators.

Official historical sources: The Foundation, Science history, and Alexander Grothendieck at IHES.

Grothendieck shows how an institution can help Mathematics change language

Many advances in school Mathematics can be explained as solving harder problems with familiar objects. Research Mathematics sometimes advances differently: mathematicians discover that the existing language itself is preventing the underlying structure from becoming visible.

Grothendieck’s work is one of the strongest historical examples. Instead of treating algebraic geometry as a collection of isolated geometric problems, he helped rebuild the subject around schemes, functorial methods, cohomology and a new way of moving between arithmetic, geometry and topology. The point was not abstraction for its own sake. The new language allowed relationships that had been hidden inside older formulations to become explicit and reusable.

A frontier sometimes advances because somebody proves a new theorem. Sometimes it advances because somebody invents a better world in which many theorems can be stated.

This is why institutional history matters to students of Mathematics. If we record only the final theorem and not the research environment, we miss the fact that large mathematical frameworks require sustained communities of readers, critics, collaborators and successors.

For an entry into the underlying modern subject, see the Bukit Timah Tutor routes on Affine Varieties and Algebraic Sets, Ideals and Coordinate Rings, Projective Varieties and Morphisms, Singularities and Dimension.

IHES sits deliberately between Mathematics and theoretical physics

IHES has never been only a Mathematics institute. Its scientific identity is built around Mathematics, theoretical physics and their interactions. This matters because several of the most powerful twentieth- and twenty-first-century mathematical developments have moved through the mathematics–physics boundary.

Quantum field theory has generated structures that later became mathematical objects in their own right. Statistical mechanics has driven probability, dynamical systems and geometric questions. Symmetry in physics feeds representation theory. Gauge theory has transformed topology and geometry. String theory has influenced algebraic geometry, symplectic geometry and representation theory. Conformal field theory creates bridges between analysis, probability, algebra and quantum theory.

The important mechanism is not that physicists supply applications while mathematicians supply proofs. That picture is too simple. At the frontier, the influence runs both ways. Physics can suggest structures before rigorous definitions exist. Mathematics can reveal constraints that change physical theory. A conjectural calculation can become a theorem years later. A theorem can become a tool in a physical model.

IHES has been unusually comfortable with this traffic.

The current permanent Mathematics faculty shows several kinds of frontier

As of 7 September 2026, IHES lists a compact permanent faculty including mathematicians working in harmonic analysis, algebraic K-theory and homotopy theory, probability and statistical physics, mathematical analysis, algebraic and arithmetic geometry, and mathematical structures connected to quantum theory.

The value of a small faculty is not breadth by headcount. It is the ability of each appointment to define a substantial scientific direction while the visitor programme supplies much of the changing breadth.

Hong Wang: harmonic analysis and the three-dimensional Kakeya problem

Hong Wang joined IHES as a permanent professor in 2025. The Institute describes her as a specialist in harmonic analysis whose work includes major advances around the restriction problem and related questions. In 2025 she and Joshua Zahl announced a proof of the three-dimensional Kakeya conjecture, a longstanding problem at the intersection of analysis and geometry.

In July 2026, Hong Wang received a Fields Medal. IHES’s awards page now includes her among the Institute’s permanent Mathematics professors who have received the Medal.

The Kakeya problem is a useful illustration of frontier Mathematics because its statement can be motivated geometrically while its solution requires sophisticated harmonic analysis, combinatorics and geometric structure. A line segment turning through every direction sounds elementary. The sets capable of containing those directions can be extraordinarily difficult to measure and understand.

Official profile: Hong Wang.

Dustin Clausen: algebraic K-theory, homotopy and condensed Mathematics

Dustin Clausen has been a permanent professor at IHES since 2023. IHES describes his work as specialising in algebraic K-theory and connections between number theory and homotopy theory. Together with Peter Scholze, he developed condensed Mathematics, a framework designed to handle topological and analytic objects in a more algebraically robust way.

This is another case where frontier work changes the infrastructure of thought. Condensed Mathematics is not one answer to one problem. It is an attempt to rebuild certain foundations so that difficult constructions become better behaved.

Official profile: Dustin Clausen.

Hugo Duminil-Copin: probability and phase transitions

Hugo Duminil-Copin, a permanent professor since 2016 and a Fields Medalist in 2022, works in probability and mathematical statistical physics. His IHES profile highlights models such as Ising, Potts, self-avoiding walks and percolation and the use of probabilistic methods to understand critical behaviour and phase transitions.

These models show how Mathematics can study large-scale behaviour emerging from simple local rules. A lattice may contain only binary or finite local states, yet the collective system can exhibit sharp transitions, long-range correlation and universal phenomena.

Official profile: Hugo Duminil-Copin.

Laure Saint-Raymond: partial differential equations and asymptotic limits

Laure Saint-Raymond has been a permanent professor since 2021. IHES describes her work as focusing on asymptotic analysis for systems of partial differential equations, especially equations describing gases, plasmas and fluids.

Asymptotic analysis asks a powerful question: when a system contains many scales or parameters, what simpler mathematical law emerges in an extreme regime? This is one of the central ways Mathematics connects microscopic rules to macroscopic behaviour.

Maxim Kontsevich: geometry, topology and quantum structures

Maxim Kontsevich has been a permanent professor at IHES since 1995. His work is famously wide-ranging across geometry, topology, deformation theory and mathematical physics. IHES describes him as part of a generation of mathematicians able to integrate aspects of quantum theory into Mathematics and open new perspectives.

Kontsevich’s career demonstrates why rigid subject labels can become misleading at the frontier. A single research program may pass through symplectic geometry, algebraic geometry, topology, category theory and quantum field theory because the underlying structures demand it.

Emmanuel Ullmo: arithmetic geometry and institutional continuity

Emmanuel Ullmo has served as Director of IHES since 2013. His own research lies in algebraic and arithmetic geometry. The combination of mathematician and institutional director matters because leadership in a research institute involves scientific judgement: recruiting permanent professors, selecting visitors, maintaining the research environment and deciding how the Institute remains open to emerging fields without losing its identity.

A current visitor list reveals what the permanent roster cannot

A permanent faculty page shows continuity. A visitor page shows motion.

On 7 September 2026, IHES’s invited-researcher list included mathematicians working in partial differential equations, geometry, representation theory, algebraic geometry, harmonic analysis, number theory, arithmetic geometry, modular forms, L-functions and other areas. Yu Deng, a 2026 Fields Medalist and University of Chicago professor, was listed as a visitor from 5 to 12 September 2026. Richard Schwartz was visiting in geometry. Alexandre Samokhin was visiting in representation theory and algebraic geometry. Other listed researchers were arriving for longer periods in analysis, number theory and related fields.

This snapshot will age quickly, which is exactly why it is useful. The permanent professor list tells us what IHES owns over years. The visitor list tells us what the Institute is touching this week.

Permanent faculty defines memory. Visitors define permeability.

Current roster: IHES Invited Researchers.

The Scientific Council is part of the mathematical machine

Visitors do not simply arrive because they are famous or because a topic is fashionable. IHES’s Scientific Council is responsible for recruiting permanent professors, selecting invited researchers and organising the scientific programme. Official governance information says the Council meets twice a year to select visitors and discuss scientific policy.

This means selection itself is a mathematical function of the institution.

A frontier institute has finite offices, finite funding and finite researcher time. It cannot host every promising person. The selection system therefore shapes the future network: which fields meet, which problems receive concentration, which young researchers enter contact with established ones, and which ideas receive time to mature.

Official governance: IHES Governance and Scientific Council.

Why a small permanent faculty can be an advantage

A large university department can cover enormous disciplinary breadth. A small institute must use a different architecture.

IHES keeps the permanent core compact and imports breadth through visitors, joint researchers, chairs, junior positions and collaborations. This creates a high ratio of movement to permanence. A few appointments can define long-term directions while hundreds of visitors prevent the research environment from becoming closed.

The model creates risks as well. A small faculty makes each appointment disproportionately important. A narrow sequence of appointments could unintentionally shrink the intellectual field. The visitor programme and Scientific Council therefore become essential balancing mechanisms.

This is a general lesson for frontier institutions: small does not automatically mean focused; small works only when the network around the core remains large and porous.

Publications Mathématiques de l’IHÉS: the institution also distributes Mathematics

Research institutions do not matter only because of what happens on site. They also matter because of how mathematical results leave the building.

Publications Mathématiques de l’IHÉS was founded in 1959 under Léon Motchane and Jean Dieudonné. It became one of the world’s highly regarded research journals, known for publishing major work across Mathematics.

The publication history is especially relevant in 2026. IHES says the journal has been diamond open access without article-processing charges since 2021 and, from 1 January 2026, is distributed through Centre Mersenne, a diamond open-access platform.

This matters because frontier Mathematics has two separate problems:

• creation: produce correct, important new Mathematics; and
• distribution: make that Mathematics accessible enough that the global community can inspect, challenge, cite and extend it.

A theorem hidden behind an inaccessible channel can still be true, but the rate at which it becomes shared mathematical infrastructure may be lower. Open distribution therefore affects how quickly knowledge can travel, especially across institutions with unequal library budgets.

Official source: The Publications mathématiques de l’IHES.

The journal is not merely a communications department

It is tempting to imagine a journal as an accessory to an institute. In Mathematics, that underestimates the role of publication.

Mathematical knowledge has unusual durability. A paper can remain structurally important decades after publication. The publication channel therefore becomes part of institutional memory. Editorial standards, referee processes, archives and stable citations help convert private mathematical discovery into public mathematical infrastructure.

IHES’s journal has historically published work by researchers far beyond the Institute itself. That gives IHES a second kind of reach. The visitor programme brings the world inward. The journal sends Mathematics outward.

Visitors import active problems; publications export validated results.

Nine Fields Medals are evidence of density, not a complete measure of value

IHES currently highlights nine Fields Medal recipients among its permanent Mathematics professors across its history: René Thom, Alexander Grothendieck, Pierre Deligne, Alain Connes, Jean Bourgain, Maxim Kontsevich, Laurent Lafforgue, Hugo Duminil-Copin and Hong Wang.

The number is extraordinary for such a small institution, but it must be interpreted with care.

A Fields Medal is awarded to an individual, not to an institution. Some prize-winning work begins before an appointment; some develops across several institutions; some collaborations depend on researchers elsewhere. The correct conclusion is therefore not that IHES “produces” medals as a factory produces objects.

The stronger conclusion is that IHES has repeatedly been able to recruit, retain or host mathematicians working at extremely high levels and to remain part of the international network through which frontier Mathematics develops.

Official current list: IHES Awards.

The 2026 Fields Medalists make the network visible

July 2026 gives a particularly clear snapshot. Hong Wang, who joined IHES as a permanent professor in 2025, received a Fields Medal. Yu Deng, another 2026 Fields Medalist, is a regular IHES visitor and was listed at the Institute again in September 2026.

These two examples show two different institutional relationships:

• permanent ownership: a mathematician becomes part of the long-term faculty; and
• network participation: a mathematician repeatedly visits without the Institute becoming their home university.

Both matter. Frontier Mathematics would become brittle if institutions valued only permanent capture. A healthy ecosystem needs circulation.

Harmonic analysis shows how geometry, analysis and combinatorics meet

Hong Wang’s presence gives us an opportunity to see how modern subfields overlap.

Harmonic analysis began historically from ideas about decomposing functions into frequencies. Modern harmonic analysis reaches much further: oscillatory integrals, Fourier restriction, geometric measure theory, partial differential equations, incidence geometry and additive combinatorics all interact.

The Kakeya problem illustrates this network. Its geometric statement concerns sets containing a unit line segment in every direction. But proving strong dimensional results requires analytical estimates and combinatorial organisation of how tubes can overlap.

This is important for the “institution map” because a label like analysis can hide the fact that the actual research touches several mathematical communities. A frontier institution is valuable when those communities can coexist without needing the problem to choose one department first.

Condensed Mathematics shows the frontier can be foundational rather than problem-specific

Dustin Clausen’s work with Peter Scholze on condensed Mathematics provides a different model of frontier research.

Some research asks, “Can we prove this conjecture?” Foundational research may ask, “Are we using the best category of objects in which to formulate the subject?” This shift can feel more abstract, but it can be transformative.

In ordinary Mathematics, topological objects and algebraic objects often interact awkwardly. Completion, infinite products, duality and functional-analytic constructions can misbehave in categories that were designed for simpler settings. Condensed Mathematics changes the ambient framework in an attempt to make these operations more systematic.

The lesson for students is not to learn condensed Mathematics early. The lesson is to understand that representation choices remain important all the way to the frontier. The same principle that helps a Secondary student switch from words to a graph can, at a vastly more advanced level, motivate mathematicians to redesign the objects through which an entire theory is expressed.

Probability at IHES connects pure Mathematics to collective behaviour

Hugo Duminil-Copin’s work represents another frontier: extracting rigorous macroscopic behaviour from random microscopic systems.

Percolation models ask whether randomly open connections create large connected structures. Ising-type models describe local spins whose interactions can produce global magnetisation. Self-avoiding walks model constrained random paths. Each system has simple local rules but can generate complicated collective phenomena.

At criticality, small changes in parameters can correspond to dramatic changes in behaviour. Mathematicians study whether a phase transition occurs, where it occurs, how correlations decay and which features are universal across different microscopic models.

This style of research demonstrates why Mathematics and theoretical physics are natural neighbours at IHES. The physical intuition supplies classes of phenomena. Probability and analysis turn them into precise mathematical statements. Rigorous results then sharpen what physicists can safely claim.

Partial differential equations connect scales, motion and continuum models

Laure Saint-Raymond’s work brings another mathematical language into the institution: partial differential equations and asymptotic analysis.

Many physical systems can be described at several scales. A gas can be treated as an enormous collection of particles or as a continuum governed by equations for density, momentum and energy. The mathematical difficulty is not simply to write both models. It is to show when one model genuinely emerges from the other and how errors behave during the transition.

This is frontier Mathematics because limiting procedures can hide singularities, instabilities or loss of information. A formal approximation may be physically persuasive and still mathematically false outside a particular regime.

The deeper habit is familiar throughout Mathematics: state assumptions, preserve scale, control error and know where the model stops being trustworthy.

Topology and geometry are part of IHES’s long institutional memory

IHES’s historical roster includes René Thom, Dennis Sullivan, Mikhail Gromov and other mathematicians whose work transformed topology and geometry. These subjects provide another example of cumulative institutional memory.

Topology studies features of spaces that persist under continuous deformation. Differential geometry studies smooth structure, metrics, curvature and related local-to-global behaviour. Modern geometry branches further into symplectic geometry, algebraic geometry, geometric group theory, geometric analysis and more.

The boundaries between these areas move. A topological invariant can solve a geometric problem. A geometric flow can reveal topology. A physical theory can produce new invariants. An algebraic construction can classify geometric spaces.

Bukit Timah Tutor routes into this territory include Topological Spaces and Continuity, Fundamental Groups and Covering Spaces, Riemannian Geometry and Geodesics and Connections and Curvature.

The mathematics–physics interface is not an application section at the end

In school, applications often appear after the theory: first learn the formula, then use it in a physical problem. At the frontier, the order can reverse.

A physical theory may generate a mathematical structure before mathematicians know how to formalise it. Quantum field theory, statistical mechanics and string theory have repeatedly produced conjectures, invariants and correspondences that later became major mathematical research programs.

IHES’s permanent theoretical physicists and mathematical physicists therefore matter to the Mathematics story. Slava Rychkov works on strongly coupled quantum and conformal field theories. Julio Parra-Martinez works on quantum field theory, scattering amplitudes, gravitation and related topics. Dalimil Mazáč works on quantum field theory and its connections with harmonic analysis and number theory.

The subject boundary is productive precisely because it is not completely settled.

Seminars convert private thought into shared mathematical pressure

IHES runs seminars, lecture series, summer schools and conferences throughout the year, with many events recorded and placed online. The official event calendar in September 2026 lists seminars in theoretical physics and geometry and discrete groups, following summer Mathematics seminars on topics ranging from perfectoid Shimura varieties to higher categories and interacting particle systems.

A seminar is not merely a presentation format. It performs several mathematical functions at once:

• forces a researcher to compress a problem into communicable form;
• exposes definitions and assumptions to expert questioning;
• allows neighbouring specialists to recognise unexpected connections;
• transmits new techniques before a textbook exists;
• creates opportunities for collaboration and correction; and
• preserves part of the intellectual event through notes or recordings.

Current calendar: IHES Events.

The video archive changes who can enter the room

IHES states that since 2013 it has uploaded its conferences, lectures and public events to its video channel. This changes the distribution geometry of advanced Mathematics.

Historically, a seminar was highly local. If a mathematician was not in Paris, Princeton, Bonn or another centre on the right day, the intellectual event could be lost except through notes and later papers. Recorded seminars create another layer of access. A researcher in Singapore can watch a lecture from Bures-sur-Yvette without being physically present.

Video does not replace residence. Informal discussion before and after a lecture remains valuable. But it widens the perimeter of the institution and allows frontier Mathematics to travel beyond the people who could afford to enter the room.

IHES is part of Paris-Saclay, but its identity remains unusually independent

IHES operates within the wider Paris-Saclay scientific environment and is associated with Université Paris-Saclay, while retaining its own legal and institutional identity. Since 1981 it has held the status of a foundation recognised as serving the public interest.

This combination matters. Independence can protect scientific autonomy. Proximity to a dense university and research ecosystem provides students, seminars, collaborators, laboratories and intellectual traffic.

An institute can therefore be independent without being isolated. The strongest version of independence is not separation from the world; it is enough autonomy to choose difficult long-term work while remaining connected to the world’s researchers.

Public and private support meet inside the research model

The founding history of IHES is also interesting because support came from both public and private sectors, in France and internationally. Its governance and funding environment has continued to involve governments, research organisations, foundations, corporations and individual supporters.

Fundamental Mathematics creates an awkward financing problem. Its eventual consequences can be enormous, but its short-term output is often difficult to price. Nobody funding algebraic geometry in 1960 could have produced a reliable commercial spreadsheet for what its full intellectual value would be sixty years later.

Institutions like IHES therefore depend on patrons and public systems willing to support knowledge before application is guaranteed. This is one of the major differences between a frontier Mathematics institute and a commercial research laboratory.

Why companies still belong in the same frontier map

Although IHES is not a company, the larger series will eventually connect frontier Mathematics institutions to companies because modern Mathematics moves through both kinds of organisation.

A technology company may employ mathematicians to work on cryptography, optimisation, machine learning, verification, computer graphics or quantum algorithms. A financial company may employ stochastic analysts, statisticians and optimisation researchers. A semiconductor or aerospace company may depend on numerical analysis, control theory and geometry. AI laboratories increasingly use probability, information theory, optimisation and high-dimensional geometry.

The difference is not that companies do “applied” work and institutes do “pure” work. The more useful distinction is the return path. A company usually has to return mathematical capability to a product, system, customer or market. IHES can allow a question to remain fundamental for much longer without requiring that commercial return.

Why individuals also need their own nodes

Grothendieck, Thom, Deligne, Connes, Bourgain, Gromov, Kontsevich, Lafforgue, Duminil-Copin and Wang cannot be reduced to “IHES mathematicians.” Each has a mathematical identity that exceeds one employer.

This is why the larger frontier Mathematics map should use at least three distinct object types:

• Institutions — preserve environments, archives, selection systems and research continuity.
• Companies — convert mathematical capability into operational systems, technologies and products.
• Individuals — originate, combine and transmit mathematical ideas across organisational boundaries.

The same individual may pass through several institutions and companies over a career. The map should preserve those relationships rather than forcing one canonical owner for a person.

A research institute is not a university ranking

One danger in an “Important Institutions” series is that it can accidentally become a prestige list. That would be less useful than a systems map.

IHES does not need to be ranked against Cambridge, MIT, Princeton, ETH Zürich, the Max Planck Institute for Mathematics, SLMath, RIMS Kyoto or the Fields Institute for the page to be meaningful. Different institutions perform different jobs.

A university department trains large numbers of students. A research institute may host visitors. A national centre can build long-term programs. A mathematical sciences institute can organise thematic semesters. A society can coordinate journals and conferences. A funding foundation can create collaborations spanning many institutions.

The question is not “Who is number one?” The question is “What mathematical work becomes more likely because this institution exists?”

The IHES operating model in one diagram

Permanent professors → scientific continuity
Visitors → new problems and methods
Scientific Council → selection and direction
Seminars → rapid exchange and criticism
Publications → durable mathematical record
Open video → wider access
External universities and institutes → circulation back into the world

No single line explains the institution. The power comes from the closed loop.

A frontier institute must know what not to optimise

Many organisations are built around measurable throughput: students graduated, products shipped, papers counted, patents filed, revenue generated, projects completed.

Fundamental Mathematics does not always behave well under short-term throughput optimisation.

A researcher may spend months on a path that fails but teaches the community which obstruction is real. A conjecture may resist proof for decades. A new framework may initially look overcomplicated before later becoming standard. A seminar question may generate the useful idea only years afterward.

An institution like IHES is valuable partly because it can preserve a zone where not every hour must be converted into an immediate measurable deliverable.

The absence of immediate output is not automatically the absence of mathematical progress.

But freedom still needs standards

Research freedom is not the same as absence of standards. Mathematical research is eventually constrained by unusually hard requirements: definitions must be coherent, proofs must survive expert checking, examples must agree with claims, counterexamples must be addressed and published results must become inspectable by others.

IHES therefore combines freedom with selection and criticism. The Scientific Council selects. Seminars challenge. Journals referee. Collaborators verify. The wider community decides whether a new idea becomes useful.

This is an important institutional pattern:

High autonomy works when the verification system is also strong.

What a Secondary or JC student can learn from IHES

The Mathematics at IHES is far beyond a school syllabus, but the institution reveals several truths about mathematical learning.

1. Mathematics continues after the answer book ends

School Mathematics presents problems with known mathematical ecosystems. Research Mathematics begins with uncertainty about whether a useful theorem, method or definition even exists.

2. Representation remains central

Grothendieck’s geometry, condensed Mathematics and modern harmonic analysis all show that choosing the right representation can be as important as calculation.

3. Proof is a social object as well as an individual act

A mathematician may write the proof, but its reliability emerges through seminars, collaborators, referees and later readers who try to reuse it.

4. Difficult work can require long attention

Mathematical strength is not measured only by finishing quickly. Research often rewards the capacity to remain with a difficult structure for a long time without losing precision.

5. Fields connect

Algebra, geometry, topology, probability, analysis, theoretical physics and computation repeatedly meet. School chapters are useful learning containers, not permanent walls around Mathematics.

From school algebra to algebraic geometry

The distance from Secondary algebra to Grothendieck’s algebraic geometry is enormous, but there is a recognisable conceptual thread.

A school student begins by solving polynomial equations and studying their graphs. Later Mathematics asks what entire systems of polynomial equations define, how the solution spaces fit together, how algebraic properties of equations correspond to geometric properties of spaces and how these structures behave over fields other than the real numbers.

The abstraction increases because the questions increase. The underlying habit remains: represent relationships in a form that preserves enough structure to reason about them.

From school probability to statistical physics

School probability begins with sample spaces, events, conditional probability and expected behaviour. Research probability can study huge random systems whose global behaviour is not obvious from their local rules.

Percolation is one example. Imagine a large network in which connections are randomly open or closed. At low connection probability, large-scale connectivity may fail. Above a critical threshold, a giant connected structure can emerge. The frontier problem is not simply to simulate the system but to prove what happens, locate thresholds and understand universal behaviour.

The difference in level is vast, but the school foundation still matters: events, independence, conditional structure, expectation and limiting behaviour remain part of the dependency chain.

From calculus to PDE and asymptotic analysis

School calculus studies rates of change and accumulation, usually for functions of one variable before moving toward multivariable settings. Partial differential equations ask how functions of several variables evolve under local differential laws.

Fluid equations, wave equations, diffusion equations and kinetic equations are all examples of mathematical descriptions built from rates of change. At the research frontier, the central questions include whether solutions exist, whether they are unique, whether singularities form, how numerical approximation behaves and what simpler laws emerge under scaling limits.

The calculus is no longer an isolated chapter. It has become infrastructure.

What the IHES history teaches about mathematical succession

One of the most important institutional functions is succession.

Grothendieck left IHES in 1970. The Mathematics did not end. Pierre Deligne, who had worked within the emerging framework, became a permanent professor and pushed arithmetic geometry further. Later generations developed new cohomology theories, categorical structures, arithmetic techniques and connections far beyond what the founders could have predicted.

This demonstrates a general principle:

A strong institution should not preserve one person’s Mathematics unchanged. It should preserve enough memory that the next generation can transform it intelligently.

Why institutional memory should not become intellectual conservatism

Prestigious institutions face a specific risk. Success in one era can make an institution overprotect the fields and styles that created that success.

IHES’s current appointments suggest an effort to avoid this. Harmonic analysis, condensed Mathematics, statistical physics, PDE and modern theoretical physics sit alongside long institutional traditions in algebraic geometry, topology and mathematical physics.

The healthy version of memory says: “Know where we came from.” The unhealthy version says: “Only continue what worked before.” Frontier institutions need the first without becoming trapped by the second.

The visitor programme is a hedge against institutional blindness

No Scientific Council can perfectly predict which subject will matter most in ten years. A visitor programme reduces the need to predict perfectly.

By hosting researchers from many institutions and fields for periods ranging from days to years, IHES can expose its permanent community to emerging work without immediately converting every new direction into a permanent chair.

This is a flexible strategy:

• permanent appointments represent high-conviction long-term commitments;
• visitors create lower-commitment access to changing frontiers;
• seminars test whether connections are productive; and
• future appointments can then be informed by several years of observed mathematical movement.

IHES and the global Mathematics network

The current and historical IHES community connects to universities and institutes around the world. Dustin Clausen came through Harvard, MIT, Copenhagen and Bonn. Hong Wang studied at Peking University, École Polytechnique and MIT and later worked at IAS, UCLA and the Courant Institute. Visitors arrive from Chicago, Brown, Bielefeld, Tokyo, Bonn, King’s College London and many other institutions.

This mobility is not incidental. Modern frontier Mathematics is distributed. No institute contains every technique, every student and every problem. The ecosystem works because people carry mathematical knowledge between nodes.

That is why this series will link institutions rather than present them as isolated monuments.

IHES institutional map

Entity	Institut des Hautes Études Scientifiques (IHES)
Type	Independent advanced research institute / public-interest foundation
Founded	1958
Founder	Léon Motchane
Location	Bures-sur-Yvette, France, in the Paris-Saclay research environment
Core domains	Mathematics, theoretical physics and related sciences
Operating model	Small permanent faculty + international visitors + seminars + publications
Annual visitor scale	IHES states more than 200 scientists are invited each year
Historical Mathematics anchors	Grothendieck, Dieudonné, Thom, Deligne, Sullivan, Connes, Bourgain, Gromov, Kontsevich, Lafforgue and others
Current Mathematics anchors checked	Hong Wang, Dustin Clausen, Laure Saint-Raymond, Hugo Duminil-Copin, Maxim Kontsevich, Emmanuel Ullmo
Publication channel	Publications Mathématiques de l’IHÉS
Current open-access status	Diamond open access; distributed by Centre Mersenne from 1 January 2026
Verification date	7 September 2026

How IHES connects to the existing Bukit Timah Tutor Mathematics estate

This page owns the institutional node. It should not duplicate the detailed Mathematics already owned by topic pages. Use these routes when a concept mentioned in the institutional history becomes the actual learning object:

• Affine Varieties and Algebraic Sets — first route into modern algebraic geometry.
• Ideals and Coordinate Rings — the algebra–geometry dictionary.
• Projective Varieties and Homogeneous Coordinates — geometry at infinity and compactification.
• Morphisms, Singularities and Dimension — local algebraic geometry.
• Topological Spaces and Continuity — foundational topology.
• Homology and Cohomology — algebraic invariants of topology.
• Riemannian Geometry and Geodesics — metric geometry and curvature.
• Connections and Curvature — covariant change and geometric structure.
• Geometric Representation Theory — a modern geometry–symmetry bridge.
• Quantum Complexity Theory — one route into Mathematics at the computation–physics boundary.

Return to the Singapore Mathematics Hub for the wider Mathematics library, or return to the previous institutional node, the Institute for Advanced Study.

Official IHES sources

• IHES Mission
• The Foundation
• Science History
• IHES Professors
• IHES Governance
• Scientific Council
• Invited Researchers
• IHES Events
• Publications Mathématiques de l’IHÉS
• IHES Awards
• Alexander Grothendieck
• Hong Wang
• Dustin Clausen
• Hugo Duminil-Copin

The larger lesson

IHES demonstrates that a Mathematics institution can be small in permanent population and enormous in mathematical consequence.

Its historical power came partly from people such as Grothendieck, Dieudonné, Thom, Deligne, Sullivan, Connes, Bourgain, Gromov and Kontsevich. Its current frontier includes researchers such as Wang, Clausen, Duminil-Copin and Saint-Raymond. But the institution is more than the sum of those names.

The durable machinery is the combination of autonomy, selection, visitors, seminars, publishing, memory and circulation.

That machinery lets a new language for algebraic geometry grow over years. It lets a harmonic analyst carry a geometric problem into a new regime. It lets a probabilist study phase transitions with methods that cross field boundaries. It lets mathematical physicists and geometers recognise that they are handling related structures before either field possesses a final theory.

Frontier Mathematics is not produced by institutions instead of people. Institutions change the conditions under which people can discover, test, transmit and extend Mathematics that does not exist yet.

That is the mathematical job IHES has performed since 1958, and why it belongs as a central node in the map of Important Mathematics Institutions.