Do not compete with AI in producing more pages. Become the mathematician who knows which page is necessary. 

A letter to Master's and PhD students.

Dear Students,

You are beginning research at a strange and promising moment. For most of the history of mathematics, obtaining a good explanation was expensive. One had to find the right book, wait for a seminar, travel to a workshop, or write to an expert and hope for a reply. Today, an AI system can explain a standard theorem at several levels, translate notation, suggest examples, draft code, reorganize a proof, and produce exercises within minutes.

This is a genuine gift.  A researcher can enter an adjacent subject faster. Language is less of a barrier. Routine calculations and first drafts need not consume the same amount of time.


But what is Research in Mathematics?  It is deciding which words are true, which questions are coherent, which hypotheses are natural, which examples actually exist, which obstruction has been overlooked, and which direction deserves several years of one's life.   AI has sharply reduced the cost of a first answer. It has not reduced the cost of a satisfying answer by the same amount.

A few months ago, I wrote an article titled  "Two Characters on One Punctured Riemann Surface". In that work,  motivated by the work of Weber and Wolf,  I developed an abstract framework for the coupled period-realization of two meromorphic one-forms. Under three substantial conditions—Teichmüller regularity, degeneration detection, and pushability—the framework produces a punctured Riemann surface carrying differentials that realize a prescribed restricted pair of characters.
            The theorem was conditional.    When I asked and researched with the help of  AI,   even  AI could not help me to remove those conditions, every time AI translates my condition to some other condition-- but eventually that was the game of changing the conditions in a Fancy languages.    I am just sharing the current limitations of AI. 

We should not deny the speed of recent progress. AlphaGeometry combined a learned model with symbolic deduction and solved 25 of a benchmark set of 30 Olympiad geometry problems.[1] AlphaGeometry2 subsequently reported much broader coverage and stronger performance on geometry problems from the 2000–2024 International Mathematical Olympiads.[2] In 2025, an advanced general-purpose reasoning system scored 35 out of 42 on that year's IMO problems—the official gold cutoff was also 35.[3][4] Formal systems are growing too: Lean's mathlib now contains a very large body of machine-checked mathematics across algebra, analysis, topology, probability, and other areas.[5] 

Which kinds of mathematical work will retain value?

The old categories will not disappear, but their meaning will change.

Exposition

Generic exposition will be cheap. Excellent exposition will remain rare.

Easy generalization

Mechanical transport from one category to a nearby category will be increasingly automatable. It will matter when the transfer uncovers a new obstruction, produces a useful example, connects distant theories, or opens a genuine application. 

“Replace X by Y throughout” will rarely sustain a research identity.

New methods and results

A technique that changes what can be proved will remain central. The premium may rise because AI will quickly propagate the consequences of a method once it exists. Creating the method—and knowing why it is the right one—is the hard part.

Examples and obstructions

Constructing an object that satisfies several global constraints, or proving that no such object can exist, is exactly where plausible language often fails. Expect carefully verified examples, counterexamples, moduli boundary cases, and non-emptiness theorems to become more—not less—important.

Formalization and research infrastructure

Libraries of definitions, machine-checked lemmas, reliable datasets of examples, and reusable software will be mathematical contributions when they organize a field and enable results that were previously impractical.

What should students, postdocs, and faculty do now?

For an M.Sc. student: build a mind that can detect a false answer

• Work through proofs in algebra, topology, real and complex analysis, differential geometry, and measure/PDE according to your direction. Use AI after your own attempt, not before every attempt.

• Read a proof, close the source, and reproduce its architecture. Ask which hypotheses are used at each step.

• For every definition, keep one standard example, one non-example, one boundary case, and one calculation.

• Become comfortable with Python or Julia, one computer algebra system, LaTeX, and Git.

• Reproduce a theorem, compute a family of examples, and write a clear note stating exactly what is yours and what comes from the literature.

For a PhD student: create a thesis with a mathematical moat

• AI makes breadth easier for everyone. Your advantage must come from understanding a small territory better than a generic assistant does.

• Even if the first result is a generalization, identify the new obstruction, example, invariant, or method that makes the setting genuinely different.

• Talk to people. AI cannot replace the moment when an expert says, “We tried this ten years ago; here is where it breaks.” Attend seminars, ask precise questions, and write to authors after doing the necessary homework.

• Own every line. An AI-generated lemma can enter your thesis only after you can prove it, explain it at a board, and locate it within the literature.

For a postdoctoral researcher

• Several disconnected results are less durable than a clear program whose successive papers address different aspects of a single geometric phenomenon.

• If you are primarily algebraic, learn enough PDE, dynamics, or computation to test geometry. If you are primarily analytic, learn the moduli or representation-theoretic structure governing your equations.

• A trustworthy codebase, a catalog of examples, a formalized lemma library, or a carefully curated set of conjectures can make your program legible and collaborative.

• The goal is not to outrun AI in volume but to publish work that experts could not obtain by asking for a routine extension.

The Future of Mathematics Research and Jobs (My opinion- just story) 

By 2028:  AI becomes a Normal research assistant. 

By 2028, most active researchers will probably use AI regularly.

AI will help researchers to:  search the literature;  summarize papers;  suggest examples and counterexamples;  perform symbolic calculations; write computer programs;  translate informal arguments into precise proofs; Also: check parts of proofs; improve exposition and LaTeX; and prepare preliminary drafts of papers.  

A researcher may describe an idea, and AI may produce several possible proof strategies. However, AI-generated arguments will still require careful human checking. Hallucinated lemmas, incorrect citations, missing hypotheses, and circular arguments will remain serious problems.    Researchers who completely avoid AI may become slower than researchers who use it intelligently. At the same time, researchers who trust AI without verification will produce unreliable work.

Academic hiring will not collapse by 2028, but expectations will begin to change. Universities may value candidates who combine serious mathematical knowledge with computation, formal proof, data analysis, or scientific applications.

By 2030: Research becomes Human-AI collaboration: 

An AI mathematics assistant may be able to work on a problem for hours or days, test many approaches, and return a structured report.

It may produce:   a survey of known results;   possible conjectures;  computational evidence; candidate proofs; formal verification of selected arguments; and a first draft of a paper.

This will increase the amount of mathematical writing. Journals and arXiv may receive many more papers. Therefore, producing a paper will become easier, but attracting attention will become harder.   The important question will no longer be only, “Can you prove something?” It will also be:

Is the result genuinely new?   Is the problem important? Does the proof reveal a new idea?  Can the result be placed in a larger theory?  Can independent experts or formal systems verify it?   A routine paper containing a small generalization may have less career value because AI will be able to generate many such extensions. Researchers will need stronger mathematical taste and a clearer research identity.

Traditional permanent academic jobs may remain highly competitive. However, mathematical employment outside pure academic research may grow in areas such as statistics, data science, operations research, cryptography, verification, scientific computing, finance, biotechnology and AI safety. 

By 2035: Routine Mathematical Research is heavily automated

AI may be able to complete a large proportion of technically routine research.

It may be especially strong at:  finding consequences of known theories;  generalizing a theorem to nearby cases; checking long calculations;  searching large spaces of examples;  formalizing existing proofs;  combining standard techniques;  and producing polished research manuscripts.

Some papers that currently require several months of work may be produced in days. This does not necessarily mean that every generated paper will be valuable. Mathematics may face an abundance problem: too many correct results and too little human attention.

The researcher’s role may shift from being primarily a proof producer to being a research director. A mathematician may guide several AI agents, compare their approaches, reject meaningless directions, and combine useful fragments into a coherent theory.

The strongest researchers will probably have one or more of the following qualities: A deep understanding of a difficult area, the ability to formulate original questions, the skill in connecting different fields, the ability to build useful definitions and frameworks, knowledge of formal verification, the ability to work with scientists and engineers, and excellent mathematical exposition.

Teaching will also change. Basic explanation and exercise generation may be largely automated. Human teachers will remain important for mentoring, diagnosing conceptual misunderstandings, designing curricula, motivating students, and developing mathematical maturity.

By 2040: Mathematics may become a mixed Human-Machine Science. Predictions for 2040 are highly uncertain.

In a moderate scenario, AI will be an extremely powerful mathematical collaborator but will still require human direction. Researchers will choose goals, evaluate significance, and connect mathematical results with human scientific questions.

In a more disruptive scenario, AI systems may independently formulate conjectures, develop theories, and produce verified proofs that few humans can understand completely. Human mathematicians may then act as interpreters, evaluators, and curators of machine-generated mathematics.

Even in this scenario, human mathematics is unlikely to disappear. Mathematics is not only the production of correct proofs. It is also the search for understanding.

A proof may be formally correct but conceptually unhelpful. Mathematicians will still ask:

Why is the theorem true?  What is the central mechanism?  Which definition is the natural one?  How does the result connect with geometry, physics, computation or another field?  What should we investigate next?  Mathematical understanding, not merely mathematical output, will remain valuable.

The Future of Mathematics Jobs

The number of purely traditional research positions may not grow as quickly as the number of qualified researchers. Competition for permanent university positions may therefore remain severe.

Some routine tasks may require fewer people. These include elementary computation, standard literature summaries, basic coding, routine proof checking and production of minor variations of known results.

At the same time, new roles are likely to develop:

formal mathematics researcher;   AI-assisted mathematics researcher;  theorem-proving engineer;  mathematical verification specialist;  research evaluator;  scientific AI researcher;   mathematical data curator;   AI safety researcher.

How a mathematics researcher should prepare 

First, develop strong foundations. 

Learn definitions, examples, counterexamples, and major proof techniques deeply. A researcher who only knows how to ask AI for an answer will not be able to recognize when the answer is wrong.

Second, develop mathematical taste.

Learn to distinguish an important problem from a routine problem. Read influential papers and ask why they changed the subject. Choosing the right question will become more valuable as proving routine statements becomes easier.

Third, learn some programming.

Python, Julia, SageMath, Mathematica, or another computational system can help with experiments and applications. One does not need to become a full software engineer, but basic computational independence will be increasingly important.

Fourth, learn formal proof.

Become familiar with a proof assistant such as Lean, Coq, or Isabelle. Formal mathematics is likely to become more important because machine-generated proofs need reliable verification. DeepMind’s AlphaProof already works through the Lean formal language.

Fifth, learn to use AI critically.

Use AI to explore ideas, locate references, test calculations, and improve writing. Never accept a proof or citation merely because it looks convincing. Check every important statement against definitions, original sources and independent arguments.

Sixth, build a second area of competence.

Combine pure mathematics with at least one neighboring area, such as computer science, statistics, physics, biology, economics, cryptography, optimization, or machine learning. This provides both research flexibility and wider employment options.

Ninth, maintain several career possibilities.

A person preparing only for a permanent pure-mathematics position takes a significant risk. It is reasonable to pursue pure mathematics seriously while also developing skills useful in teaching, industry, computing or interdisciplinary research.

Refrences: 

1. Trieu H. Trinh et al., “Solving olympiad geometry without human demonstrations,” Nature 625 (2024). Nature article.

2. Yuri Chervonyi et al., “Gold-medalist performance in solving olympiad geometry with AlphaGeometry2” (2025). arXiv abstract.

3. Google DeepMind, “Advanced version of Gemini with Deep Think officially achieves gold-medal standard at the International Mathematical Olympiad” (21 July 2025). Official report.

4. International Mathematical Olympiad, 2025 results and medal thresholds. Official IMO results.

5. Lean FRO, overview of Lean and mathlib. Lean official site.