Tag Archives for Mathematics

From Proof Scarcity to Proof Abundance

tl;dr anyone who previously used or currently uses mathematical tools to solve problems in their field (whether in sciences, pure or applied mathematics, engineering, economics, and so on) needs to pay attention to discussions around the use of AI to solve mathematically precise problems. The revolution has begun!

Here’s what I believe: once AI-assisted proofs become abundant, their scarcity value will fall and attention will shift toward the harder tasks of formulating compelling conjectures, building theories, and creating definitions that organize rapidly expanding bodies of results. The analogy is economic: just as an influx of currency can produce inflation rather than lasting wealth, an influx of proofs can reduce the value attached to any individual proof without necessarily producing a comparable increase in understanding.

Firsthand Experience

Honestly, for me, the question of how useful AI could be in solving math problems did not begin in 2026. In November of 2022, ChatGPT was released. I had just begun my postdoc. With support from the Simons Foundation, I started thinking seriously in 2022 about what mathematics (and intellectual work more broadly) might look like in a world with advanced AI. At the time, much of this remained speculative: What would happen if mathematical reasoning became inexpensive? Which parts of research would retain their value? Would the bottleneck shift from proving statements to identifying the right questions, definitions, and conceptual frameworks?

At ICM (International Congress of Mathematicians) 2026, Terence Tao gave a lecture on Mathematics in the Age of AI. I’d encourage everyone to, at least, read the slides! Tao disusses the implications of the abundance of proofs (not necessarily correct). Recent events have made this transition feel concrete rather than hypothetical. While I was watching the 2026 World Cup final, Levent Alpöge announced a counterexample (credited to the AI system Fable) to the Jacobian conjecture. More precisely, the construction disproves the conjecture in dimension three, and therefore in every dimension n\geq 3, while the two-dimensional case remains open. The counterexample has a nonzero constant Jacobian determinant but maps three distinct points to the same output, directly contradicting the conjectured invertibility.

In recent days have also tested advanced AI systems on several problems from my own active research agenda. These were not merely exercises with known answers: in multiple cases, the systems made progress that appeared genuinely novel, uncovering arguments or directions that I had not previously considered. The important question is no longer whether AI can occasionally imitate mathematical reasoning. It is now clear that AI can perform at least some research-level mathematics. The remaining questions concern the breadth and reliability of that ability, the amount of human guidance and computation it requires, and, most importantly, how the mathematical community should evaluate, explain, verify, and build upon the resulting work.

Terence Tao on Mathematics in the Age of AI

In the rest of this blog post, I will highlight a few parts of his talk which I found illuminating!

Tao identifies the central challenge posed by mathematical AI: not merely generating correct proofs, but transforming those proofs into shared human understanding. That is, the most important question is no longer whether artificial intelligence can prove difficult theorems. I believe that Tao deliberately set that question aside. Instead, he asked what may be a more consequential question:

If AI systems become capable of performing a substantial fraction of research-level mathematical work, what exactly should the mathematical community be trying to accomplish?

We all need to think of that shift (from technological capability to institutional purpose). Most discussions of AI and mathematics focus on benchmark performance: Can a model solve Olympiad problems? Can it formalize a proof in Lean? Can it make progress on an open conjecture? Tao’s lecture asks what happens after the answer to some of those questions becomes yes. Tao asks the audience to provisionally assume that reasonably capable mathematical AI is coming. He then investigates what follows for proof, exposition, reviewing, education, authorship, and the organization of mathematical knowledge.

What I appreciate from the talk is that the resulting picture is neither utopian nor apocalyptic. (Although a large fraction of folks see it as a warning.) Clearly, we are headed towards an era in which proofs are scarce to one in which proofs are abundant; our present institutions are designed almost entirely for the former world, where proofs are not abundant.

A crisis of values rather than a crisis of logic

In the late nineteenth and early twentieth centuries, mathematicians were forced to examine assumptions that had previously remained largely implicit. Russell’s paradox, the emergence of competing foundational programs, and Gödel’s incompleteness theorems created a period of uncertainty about sets, numbers, infinity, axioms, and formal proof. This led to the development of more explicit and standardized foundations that gave mathematicians a trusted environment in which to work.

Tao suggests that AI is producing a comparable period of turbulence, although the foundations now at issue are not primarily logical. They are the foundations of mathematical values and practices: what counts as progress, who deserves credit, what authors are responsible for, why proofs matter, and what the profession is ultimately trying to produce. A disruptive development can expose assumptions that were previously invisible because they rarely came into conflict. Before AI, several goals of mathematics tended to advance together. A mathematician who solved a difficult problem often developed a new technique, trained students, wrote an influential paper, helped organize a research community, and contributed to a larger theory. Because those outputs were correlated, institutions could reward one visible quantity, such as the publication of a paper with one or more important theorems, and assume that the others would follow.

AI threatens to break those correlations.

The two questions that are often confused

Tao separates the AI debate into two questions.

The first is a capability question: Will AI tools, at an acceptable cost and with some level of human supervision, be able to perform meaningful research-level mathematical tasks with a nontrivial rate of success?

Tao formulates this as a family of conjectures because almost every term requires qualification. Which tools? Which fields? What counts as research-level? How much supervision? At what cost? What success rate? What standard of correctness? What standard of exposition? A system that autonomously proves 80 percent of new theorems in algebraic geometry is very different from one that occasionally completes a technical lemma after several days of expert prompting. Both might satisfy some version of an AI capability claim. Tao also stresses that public evidence is affected by reporting bias, commercial incentives, undisclosed compute budgets, selective demonstrations, and ambiguous levels of human assistance. The truth of a capability claim must also be separated from whether that capability is desirable.

The second question is a question about goals and values: What are the mathematical community’s actual objectives, including the implicit objectives encoded in publication, hiring, funding, prizes, and professional prestige? Tao’s key methodological move is to condition on a working hypothesis: suppose that a reasonably strong version of the capability claim becomes true reasonably soon. He does not ask the audience to endorse or welcome the hypothesis. He asks what the community should do if it holds. Very reasonable! This is a useful form of reasoning under uncertainty. To continue to do mathematics, we cannot wait until every capability forecast is resolved before examining whether our institutions are robust to rapid automation. Even a moderate probability of large changes can justify preparing publishing policies, authorship standards, review infrastructure, and educational norms.

First Proof

Tao briefly points to the First Proof project as one of the better-controlled pieces of evidence about current AI capabilities. For its second batch, First Proof assembled ten unpublished research-level problems originating in the work of research mathematicians. Four systems were tested in a controlled environment. The outputs were evaluated using a double-blind journal-review model involving approximately thirty experts, with each submission examined by multiple referees. Across the four systems, seven of the ten problems received at least one solution judged essentially correct or in need of only minor revisions. One system found a novel solution to a stochastic partial differential equations problem that differed from the human proof and impressed the referees. Other problems saw little or no meaningful progress.

The results are substantial, but their limitations are equally informative. The systems often handled routine portions of an argument in meticulous detail while moving too quickly through the genuinely difficult step. Referees found unsupported appeals to supposedly standard arguments, incorrect or missing citations, unnecessary notation, and lengthy explanations of facts that an expert would normally omit. Some submissions reproduced distinctive language and notation from earlier work without appropriate attribution (i.e., behavior that would raise serious plagiarism concerns in a human submission).

The computational costs also varied dramatically. In the reported runs, per-problem costs ranged from single-digit dollar amounts to nearly one thousand dollars, depending on the system and problem. This reinforces Tao’s point that “AI solved the problem” is not a complete scientific statement without information about models, harnesses, prompts, search procedures, human intervention, tokens, time, and cost.

Most importantly, First Proof evaluated only one part of mathematical research: generating proofs of questions selected and formulated by humans. Its organizers explicitly distinguish solving questions within existing frameworks from the equally important work of asking new questions and developing the frameworks in which those questions become meaningful.

Mathematics has more than one objective

What is mathematical research for?

Tao identifies several answers:

  • solving pure and applied problems;
  • building theories and techniques;
  • understanding the natural and social world;
  • training future mathematicians (part of my job);
  • sustaining a mathematical community;
  • contributing to a shared network of knowledge;
  • creating works of enduring intellectual or aesthetic value.

Historically, progress along these dimensions was positively correlated. Solving an important problem often required theory building. Theory building produced reusable techniques. Writing and teaching those techniques helped train students and build a community. This made it tempting to treat theorem production as a proxy for mathematical progress. Tao invokes Goodhart’s law: once a proxy becomes an explicit target, optimization can destroy the relationship that made the proxy useful. AI makes this danger especially severe because it can lower the cost of producing objects that look like the outputs rewarded by current metrics.

Suppose an institution rewards:

  • the number of problems solved;
  • the number of papers published;
  • being first to announce a proof;
  • benchmark scores;
  • citation counts;
  • the number of formalized theorems.

An AI-intensive workflow may increase each of these without proportionately increasing understanding, theory, pedagogy, reliability, or long-term usefulness.

A thousand isolated proofs are not necessarily more valuable than one conceptual framework explaining why all thousand theorems are true. A technically correct argument is not necessarily a useful argument. A formally verified statement is not necessarily the statement that mathematicians intended to prove. A highly cited result is not necessarily well understood.

The deeper point is that mathematical knowledge is not a warehouse of theorem statements. It is an organized system of concepts, explanations, analogies, techniques, examples, counterexamples, and judgments about what matters.

Tao’s proposed “oral defense” standard

Tao offers a rule of thumb: a result should not be published unless its human authors can convincingly present a clear, correct, properly attributed expert-level explanation of it. The proposal addresses several problems simultaneously.

An author capable of defending the result is more likely to have:

  • checked that the theorem says what it is supposed to say;
  • located the crucial step;
  • understood the relationship to prior work;
  • detected misleading AI-generated exposition;
  • assumed meaningful responsibility for correctness.

The idea resembles a thesis defense. Producing a document is not enough; the author must demonstrate command of its content. As a literal universal publication requirement, however, the rule would need careful implementation. It is bound to be controversial: Oral presentation ability varies, and a rigid talk requirement could disadvantage researchers with disabilities, non-native speakers, or authors whose strengths lie in writing rather than performance. Large collaborations may also have genuinely distributed understanding.

The underlying principle is stronger than any particular format: publication should require evidence of human intellectual ownership and accountability. That evidence might take the form of a seminar, an annotated proof map, a technical interview, an expository companion, or an independent human audit. (I’m hoping to implement the oral defense standard within my research group.)

Selected references and further reading

Terence Tao, Mathematics in the Age of AI, ICM public-lecture slides, July 24, 2026. (Teorth)

Mohammed Abouzaid et al., First Proof: Second Batch Report, 2026. (First Proof Project)

William P. Thurston, On Proof and Progress in Mathematics, Bulletin of the American Mathematical Society, 1994. (arXiv)

Tanya Klowden and Terence Tao, Mathematical Methods and Human Thought in the Age of AI, 2026. (arXiv)

Johan Commelin, Mateja Jamnik, Rodrigo Ochigame, Lenny Taelman, and Akshay Venkatesh, Shaping the Future of Mathematics in the Age of AI, 2026. (arXiv)

Jeremy Avigad, Mathematicians in the Age of AI, 2026. (arXiv)

Leiden Declaration on Artificial Intelligence and Mathematics, June 2026. (Leiden AI & Math Declaration)

The Existence of Error-Correcting Codes Implies Privacy Lower Bounds

Excited to share this recent IEEE BITS article in the Special Issue on Error-Correcting Codes [1].

Abstract

We discuss how the lens of error-correcting codes yields lower bounds on the privacy-utility tradeoff in differential privacy (DP). Reconstruction attacks, packing and covering arguments, and fingerprinting codes can all be interpreted as coding-theoretic tools: they allow for analysis of families of datasets whose query-answer patterns form (possibly randomized) codewords with large pairwise distance. Any DP mechanism answering these queries too accurately reveals enough structure to enable “decoding”; that is, identifying a user or reconstructing large parts of the dataset. By presenting each lower-bound technique through an explicit coding viewpoint, this survey unifies classical results on counting queries with recent advances in statistical estimation and high-dimensional learning, including Gaussian covariance estimation. We conclude with open problems at the intersection of coding theory and differential privacy.

[1] https://ieeexplore.ieee.org/document/11481645

(Scalable) Multiterminal Key Agreement

In cryptography, secret key agreement is usually framed as a two-party story: Alice and Bob want to agree on a shared secret while Eve listens. But many real systems are not two-party systems. They involve teams of devices, servers, users, or sensors that need to establish the same secret key, often in the presence of public discussion and potentially powerful adversaries. A recent paper of mine, with Benjamin Kim (lead student author) and Lav Varshney, asks a simple question:

Can ideas from coding theory and secret sharing give us a clean, scalable way to do multiterminal secret key agreement?

Our answer is yes! The paper develops a simple multiterminal key agreement scheme built from maximum distance separable (MDS) codes (specifically Reed-Solomon codes) and then analyzes its secrecy and rate through the lens of secret key capacity and multivariate mutual information (MMI). The key point is that the same threshold structure that makes Reed-Solomon codes powerful for secret sharing also makes them surprisingly natural for multiterminal key agreement.

The Big Picture

The paper sits at the intersection of three ideas.

First, there is secret sharing. In a threshold secret sharing scheme, a secret is split into n shares so that any k shares can reconstruct the secret, but any fewer than k reveal nothing about it. Shamir Secret Sharing schemes (via use of Reed-Solomon codes) are a well-studied class of techniques to realize this threshold property.

Second, there is secret key agreement (SKA). In the multiterminal SKA setting, multiple parties start with correlated private observations, publicly discuss information over an authenticated public channel, and aim to end with the same shared key while leaking essentially nothing useful to an eavesdropper. The information-theoretic performance limit is measured by the secret key capacity.

Third, there is coding theory. Error-correcting codes already encode redundancy in a structured way. MDS codes are optimal in a precise sense: they achieve the maximum possible distance for their block length and dimension. That is, they give the strongest threshold-reconstruction behavior for a given amount of redundancy.

What our paper does is connect these three threads in a particularly direct way: use an MDS codeword as the object being partially distributed across terminals, then exploit the threshold property to guarantee recoverability for the legitimate users and secrecy against the wiretapper.

Why this Connection is Interesting

At a high level, secret sharing and multiterminal key agreement share many goals/targets/techniques.

In secret sharing, you start with a secret and want to distribute it safely across many users.

In multiterminal key agreement, you start with distributed private information and want users to end up with a shared secret key.

These are not identical tasks, but they share the same core geometry: some subsets of information should be enough to reconstruct, and smaller subsets should reveal nothing. The paper’s main conceptual contribution is to push this analogy far enough to get an explicit multiterminal SKA protocol out of it. (The abstract states this clearly: the work explores the connection between secret sharing and secret key agreement, yielding “a simple and scalable multiterminal key agreement protocol” based on Reed-Solomon codes with threshold reconstruction.)

The threshold semantics of MDS codes are exactly the right structure for a scalable multiterminal key agreement design.

The Model

The paper works in the standard multiterminal communication setup. There is a set of terminals V, a subset A\subseteq V active users who must recover the final key, and possibly helpers in V\setminus A. Each user has access to a private correlated source Z_i​, and everyone can participate in public discussion, which an eavesdropper also sees. The key must satisfy two conditions:

  • Secrecy: the public discussion should reveal essentially no information about the key.
  • Recoverability: every active user should be able to reconstruct the key from their private source and the public discussion.

The paper also recalls the standard definition of secret key capacity, the maximum achievable key rate over all allowed protocols in this model.

The Coding Idea

Now for the protocol idea.

Suppose we take a secret and pad it with additional uniform randomness, forming a vector K=(s, u_1, \ldots, u_{k-1}),

where s is the actual secret and the u_i‘s are random pads. Then we encode K using an (n, k) Reed-Solomon code with generator matrix G, obtaining a codeword Z=KG. Each terminal receives one symbol z_i​ of this codeword, and then the protocol publicly reveals exactly k-1 additional symbols. Since any k symbols of an (n,k) MDS code suffice to reconstruct the message, each legitimate terminal can combine its private symbol with the k-1 public symbols and recover the full encoded message K.

This is the reconstruction side.

On the secrecy side, the protocol masks the privately delivered shares using independent random values r_i​. In the “secret-sharing-inspired” version of the scheme, terminal Z_1​ has a distinct random shared value r_i​ with each terminal Z_i​, and broadcasts e_i = z_i + r_i.. Each intended recipient can subtract its own mask r_i​ and recover the corresponding code symbol, but the eavesdropper only sees the masked version.

So the core protocol can be summarized as follows:

  1. Encode the secret-plus-padding with an MDS code,
  2. Privately give each user one share (masked),
  3. Publicly reveal exactly k additional shares,
  4. Let each legitimate user reconstruct,
  5. Rely on the threshold property to ensure that the eavesdropper still stays below the reconstruction threshold.

The Main Security Statements

In the paper, we show that the eavesdropper learns nothing about the secret from the entire visible transcript.

The first main theorem captures the information-theoretic secrecy of this setup. There is also a computational variant. If one uses an IND-CPA-secure public-key encryption scheme to distribute the shares to terminals, then the protocol becomes computationally secure against PPT adversaries rather than information-theoretically secure against unbounded ones. That computational extension is practically important because it shows the scheme is not tied to idealized pre-shared masks. If you are willing to step down from unconditional secrecy to computational secrecy, you can distribute shares using standard encrypted channels.

If I had to summarize the paper in one sentence, I would say this:

It shows that MDS codes are not just a convenient implementation tool for multiterminal key agreement; they are a structural bridge between secret sharing, error correction, and SKA capacity.

For further details, take a look at the paper. Feedback/comments are welcome!