Important Mathematics Institutions | Institute for Computational and Experimental Research in Mathematics (ICERM)

The simple answer: what mathematical job does ICERM perform?

ICERM makes mathematical experimentation institutionally legitimate.

A traditional caricature says that mathematicians prove and computers calculate.

Modern research is more complicated.

A computer can enumerate millions of examples, search for counterexamples, visualise a geometric object, test an identity to high precision, compute invariants, solve a huge linear system, manipulate symbolic expressions, explore parameter spaces and assist in rigorous verification.

None of this automatically becomes a theorem.

But it can change what theorem the mathematician knows to ask.

Computation can discover the pattern. Mathematics must still explain why the pattern survives every admissible case.

2010: Brown University and the NSF build an institute around computation

ICERM was created through a major National Science Foundation grant awarded to Brown University in 2010.

The founding group combined pure Mathematics and applied Mathematics from the beginning.

Jill Pipher worked in harmonic analysis and cryptography. Jeffrey Brock worked in geometry and topology. Jeffrey Hoffstein worked in number theory and cryptography. Jan Hesthaven and Bjorn Sandstede brought computational and applied mathematical depth.

This mix encoded the institutional thesis: computation should not be isolated inside numerical analysis while pure mathematicians work elsewhere. The frontier increasingly requires both.

Official history: About ICERM.

The NSF treats ICERM as national research infrastructure

ICERM is one of the NSF-supported Mathematical Sciences Institutes in the United States.

Its core support has been renewed repeatedly. The most recent core NSF award, announced in August 2025, provides US$16.5 million through 2030.

This funding model matters because fundamental computational Mathematics does not always have a short commercial return path.

A new algorithm for arithmetic geometry may later influence cryptography or databases, but its immediate value may be simply that it lets mathematicians test conjectures that were previously inaccessible.

Public research infrastructure can support that uncertainty.

Official support history: ICERM Core NSF Grant.

Brendan Hassett leads ICERM in 2026

The current ICERM leadership page lists Brendan Hassett as Director, Caroline Klivans as Deputy Director and Jeffrey Hoffstein as Consulting Associate Director.

Hassett is an algebraic geometer whose research includes moduli spaces, rationality questions and higher-dimensional algebraic geometry.

Klivans works in combinatorics and related discrete Mathematics.

Hoffstein works in number theory and cryptography.

The leadership therefore spans geometry, combinatorics, arithmetic and computation—the same interfaces ICERM is designed to strengthen.

Official roster: ICERM Directors and Staff.

Semester programmes create the long research timescale

ICERM’s major mechanism is the semester programme.

A strong proposal must explain the central scientific challenges and, importantly, the experimental and computational aspects of the programme.

This requirement distinguishes ICERM from institutes where computation may be present but is not part of the institutional identity.

A semester programme is expected to create a critical mass of senior scientists, long-term participants, postdoctoral researchers, graduate students and workshops.

The proposal guidance explicitly requires concrete plans for involving and mentoring early-career mathematicians.

Official proposal architecture: Propose a Program.

25 September 2026: K3 surfaces are the live mathematical frontier

The current semester programme is Computations on K3 Surfaces and Related Varieties, running from 9 September through 11 December 2026.

On 25 September, the active programme schedule includes LMFDB Fridays, linking the semester directly to one of the most important databases in computational number theory and arithmetic geometry.

The programme combines geometry, arithmetic and explicit computation.

Associated workshops include arithmetic and geometry of K3 surfaces, mirror symmetry and Calabi–Yau threefolds, K3 surfaces over finite fields, cubic and Gushel–Mukai fourfolds, hyperkähler manifolds and moduli of K3 surfaces.

Official current programme: Computations on K3 Surfaces and Related Varieties.

What is a K3 surface?

A K3 surface is a special kind of complex algebraic surface with remarkably rich geometry.

K3 surfaces sit at a crossroads.

• Algebraic geometers study their divisors, moduli and rational points.
• Number theorists study K3 surfaces over finite fields and number fields.
• Differential geometers encounter hyperkähler structures.
• Mathematical physicists encounter K3 surfaces in string compactifications and mirror symmetry.
• Computational mathematicians attempt to calculate explicit models, zeta functions and arithmetic invariants.

The object is therefore ideal for ICERM because computation does not sit outside the theory. It is one of the ways the theory becomes visible.

Arithmetic geometry needs databases because examples are too rich to hold in one head

Arithmetic geometry contains families of curves, surfaces, number fields, modular forms and Galois representations whose invariants can be computed and compared.

Large databases such as the L-functions and Modular Forms Database make these objects searchable.

This changes research.

A mathematician can search thousands of examples for a pattern, compare local factors, test whether a conjectural relationship appears systematically and find exceptional cases worth understanding.

The database does not prove the conjecture.

It changes the evidential landscape from “I computed three examples by hand” to “the pattern survives a large, structured family.”

Experiment increases confidence in the question. Proof establishes confidence in the answer.

The Brauer–Manin obstruction shows how computation and abstraction meet

The current semester includes study of the Brauer–Manin obstruction.

In arithmetic geometry, a variety may have solutions over every completion of a number field—real numbers and all relevant p-adic fields—yet still have no global rational solution.

The Brauer–Manin obstruction provides a refined mechanism that can sometimes explain this failure of the local-to-global principle.

This is a striking frontier because the question begins with solving equations and ends in cohomological algebra and arithmetic geometry.

Computation then returns: explicit examples require calculations of local points, Brauer groups and arithmetic invariants.

The Hasse principle is a lesson in local versus global reasoning

The Hasse principle asks whether a global solution exists whenever compatible local solutions exist everywhere.

For some classes of equations, the principle works beautifully. For others, it fails.

This creates a reusable mathematical habit:

analyse the object locally → compare all local views → identify the obstruction preventing them from assembling globally.

The local–global tension appears throughout number theory, geometry and topology.

28 September 2026: mirror symmetry and Calabi–Yau threefolds arrive next

The next associated semester workshop begins on 28 September 2026: Mirror Symmetry, Calabi–Yau Threefolds, and Connections to Physics.

Mirror symmetry arose from string theory and predicts deep relationships between seemingly different geometric spaces.

A difficult enumerative problem on one Calabi–Yau space can sometimes be translated into a period or variation-of-Hodge-structure problem on a mirror partner.

This is a powerful form of mathematical duality:

hard problem on Space A ↔ transformed problem on Space B.

The duality does not merely simplify calculation. It can reveal that two mathematical languages are describing one deeper structure.

Experimental Mathematics does not mean lowering proof standards

The word experimental can mislead readers trained to think of Mathematics as purely deductive.

Experimental Mathematics uses computation to explore examples, formulate conjectures, discover invariants, test algorithms and locate counterexamples.

The standard of final mathematical truth does not change.

If a theorem is claimed, the proof must still be valid.

What changes is the route to the theorem.

Deduction decides truth; experiment can decide where deduction should look.

Computer-assisted proof creates a third category

Some computations can become part of a rigorous proof.

This happens when every computational step is controlled tightly enough that the result carries a mathematical guarantee.

Examples include:

• interval arithmetic with certified error bounds;
• formal proof assistants;
• exhaustive finite-case verification;
• exact integer or rational arithmetic;
• certified optimisation bounds.

The relevant Bukit Timah Tutor routes include Construction, Verification, Witnesses and Impossibility and Iteration, Convergence, Error Control and Stopping Criteria.

Visualization can reveal structure invisible in symbolic notation

ICERM’s mission explicitly includes simulation and visualisation.

This matters because some mathematical objects are too complicated to understand only through formulas.

A visualisation can reveal:

• symmetry;
• singularity;
• bifurcation;
• topological change;
• distribution of points;
• unexpected clustering;
• failure of a conjectured monotonic pattern.

Again, the image is not proof. It is an instrument for seeing the conjecture.

AI-assisted coding for mathematical exploration arrives in December 2026

ICERM’s 2026 programme calendar includes a December Hot Topics Workshop titled AI-assisted Coding for Mathematical Exploration.

This is a frontier whose institutional timing is significant.

AI systems can already assist with code generation, symbolic manipulation, literature search and conjecture exploration. But mathematical use introduces unusually demanding reliability requirements.

A plausible-looking function that silently mishandles one boundary case can destroy a computation. A symbolic simplification can be invalid outside an assumed domain. A generated proof sketch can cite a false lemma.

The productive frontier is therefore not “let AI do Mathematics.” It is:

use AI to increase exploratory bandwidth while designing verification systems strong enough that increased speed does not increase mathematical error.

Hot Topics workshops give ICERM a faster response time

Semester programmes require long planning. Mathematics and computation can move faster than that.

ICERM therefore also accepts Hot Topics proposals with shorter lead times.

This institutional flexibility matters when a new computational method, theorem or technology changes the field rapidly.

The 2026 calendar includes a late-September Hot Topics workshop on extreme and rare events and the December AI-assisted-coding meeting.

An institute needs both planning speeds: long enough to build deep semester communities and fast enough to respond to emerging frontiers.

Graduate Training Workshops make computational fluency teachable

ICERM also runs Graduate Training Workshops.

These programmes explicitly combine lectures with interactive activities such as working groups, software tutorials and coding sprints.

This matters because computational Mathematics is not learned only through definitions.

A student has to learn how to:

• represent a mathematical object in software;
• choose stable algorithms;
• distinguish floating-point evidence from exact calculation;
• reproduce experiments;
• document assumptions;
• benchmark against known cases.

These are research skills, not merely programming skills.

Collaborate@ICERM supports small-group production

The Collaborate@ICERM format supports small teams of researchers working together intensively for a week.

Its proposal criteria ask a particularly useful question: why will meeting at ICERM enable research that would not happen otherwise?

That question should govern every research institute.

Travel and residence are expensive. The justification cannot simply be that researchers enjoy meeting in person.

The meeting should change the probability of mathematical progress.

Brown University gives ICERM a dense local ecosystem

ICERM benefits from Brown’s departments in Mathematics and Applied Mathematics and from wider university activity in computation, data science, engineering and scientific modelling.

The institute is physically located in Providence close to the university and the broader New England research network.

This allows visiting researchers to enter an institution focused on computation without being isolated from a major university environment.

The institute changes what counts as mathematical evidence during discovery

At the theorem stage, evidence is binary: the argument is valid or it is not.

At the discovery stage, evidence has degrees.

• Three examples may suggest a pattern weakly.
• Ten thousand exact computations may suggest it strongly.
• A random search may find a counterexample quickly.
• A numerical experiment may reveal instability.
• A database may reveal an exceptional family.

ICERM institutionalises this exploratory layer without confusing it with final proof.

What a Secondary or JC student can learn from ICERM

1. Computation is not cheating

A calculator can hide thinking if used blindly. A computer can deepen Mathematics if used to expose structure and test conjectures.

2. Evidence and proof are different

A billion matching examples do not prove an infinite statement. They may still tell you what theorem is worth proving.

3. Representation matters

A difficult geometric object may become computationally accessible after choosing coordinates, invariants or a database representation.

4. Error analysis is Mathematics

Numerical output has meaning only when rounding, conditioning, convergence and approximation error are understood.

5. AI increases the need for verification

Faster code and faster conjectures are valuable only if the checking system grows stronger too.

From school Mathematics toward ICERM frontiers

• Algebraic equations → algebraic varieties → arithmetic geometry → K3 surfaces and rational points.
• Graphs and coordinates → visualisation → computational geometry and topology.
• Number theory → modular forms and L-functions → searchable arithmetic databases.
• Iteration → numerical algorithms → convergence and certified computation.
• Linear algebra → large matrix computation → scientific computing and optimisation.
• Programming → mathematical software → experimentation, formalisation and AI-assisted exploration.

ICERM institutional map

Entity	Institute for Computational and Experimental Research in Mathematics (ICERM)
Type	NSF Mathematical Sciences Institute at Brown University
Founded	2010
Operational launch	2011
Location	Providence, Rhode Island, United States
Founding team	Jill Pipher, Jeffrey Brock, Jan Hesthaven, Jeffrey Hoffstein, Bjorn Sandstede
Current Director checked	Brendan Hassett
Current Deputy Director checked	Caroline Klivans
Core mission	Broaden the relationship between Mathematics and computation/experiment
Current NSF core award	US$16.5M, announced 2025, supporting operations through 2030
Current semester programme	Computations on K3 Surfaces and Related Varieties, 9 September–11 December 2026
Next workshop	Mirror Symmetry, Calabi–Yau Threefolds, and Connections to Physics, 28 September–2 October 2026
Late-2026 frontier	AI-assisted Coding for Mathematical Exploration
Verification date	25 September 2026

Connections into the Bukit Timah Tutor Mathematics estate

• Affine Varieties and Algebraic Sets
• Projective Varieties and Homogeneous Coordinates
• Morphisms, Singularities and Dimension
• Homology and Cohomology
• Galois Representations
• Automorphic Representations, L-Functions and the Langlands Program
• Iteration, Convergence, Error Control and Stopping Criteria
• Conditioning, Ill-Posedness, Sensitivity and Stable Answers

Return to the Singapore Mathematics Hub.

Official ICERM sources

• ICERM Homepage
• About ICERM
• Directors, Staff and Emeritus Leadership
• Core NSF Grant
• Propose a Program
• Computations on K3 Surfaces and Related Varieties
• Current and Upcoming Programs

The larger lesson

ICERM demonstrates that computation can expand mathematical imagination without weakening mathematical proof.

Its institutional architecture protects the difference between exploration and verification.

Computation searches. Visualisation reveals. Databases organise. Simulation tests. AI accelerates. Computer-assisted proof certifies selected steps. Human mathematical reasoning still decides what statement matters and why the result is true.

The 2026 K3 programme shows the model at full power. Algebraic surfaces become explicit computational objects. Arithmetic invariants can be searched. Moduli spaces can be compared. Mirror symmetry connects geometry to physics. Databases let thousands of examples speak collectively.

ICERM matters because it gives Mathematics permission to experiment aggressively while demanding that whatever survives must still meet the standards of proof.

That makes ICERM one of the essential institutions in the modern frontier where Mathematics meets computation.