10 Unsolved Problems in Math and Science: How AI is Breaking the Boundaries

Unsolved problems in the history of mathematics and science are not mere objects of idle curiosity. They are the keys to expanding the boundaries of human knowledge and unlocking entirely new paradigms. Recently, the explosive growth of artificial intelligence has begun to provide fresh breakthroughs for these profound questions that have eluded humanity for centuries.

In this post, we introduce 10 of the most famous unsolved problems in math and science and explore how modern AI agents and machine learning are solving, bypassing, or helping us understand them to push the frontiers of human intelligence.

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📐 Mathematical Dilemmas and AI’s Progress

1. Riemann Hypothesis

• The Problem: The holy grail of number theory. It asserts that all non-trivial zeros of the Riemann zeta function ($\zeta(s)$) lie on the critical line with a real part of $1/2$. Proving this would unlock the exact distribution of prime numbers.
• The AI Connection: Moving past the era of manual calculation, AI is now utilized to search for microscopic patterns in the distribution of zeta function zeros using deep learning. In addition, LLM-based theorem-proving agents act as intellectual partners for mathematicians, proposing candidate lemmas and auxiliary theories to aid in formal proofs.

2. Birch and Swinnerton-Dyer Conjecture

• The Problem: An algebraic geometry puzzle concerning elliptic curves. It suggests a deep, elegant link between the group of rational points on an elliptic curve and the behavior of its associated L-function at $s=1$.
• The AI Connection: Google DeepMind collaborated with pure mathematicians to train neural networks on geometric and knot theory datasets. By detecting unseen structural patterns across mathematical dimensions, the AI has generated new conjectures and formulated mathematical links that pure intuition had missed, guiding the path toward a formal proof.

3. Hodge Conjecture

• The Problem: A major question in algebraic geometry and topology. It asks whether certain topological classes (Hodge cycles) on smooth complex algebraic varieties are always algebraic—meaning they can be built from simpler algebraic subvarieties.
• The AI Connection: Geometric Neural Networks (GNNs), designed to handle complex non-Euclidean structures, are being used to classify topological data on complex manifolds. AI acts as a multidimensional map, visualizing the intricate topology of Kähler manifolds and reducing compute complexity for pure algebraic verification.

4. Navier–Stokes Existence and Smoothness

• The Problem: The equations describing fluid flow. The challenge is to mathematically prove whether smooth, physically reasonable solutions always exist in three dimensions without developing singularities (blow-ups).
• The AI Connection: Physics-Informed Neural Networks (PINNs) are revolutionizing this domain by approximating solutions to fluid flow in real-time. Where traditional numerical solvers break down due to computational instability near potential singularities, PINNs maintain stability, helping researchers map out turbulence boundaries and identify structural limits.

5. Yang-Mills Existence and Mass Gap

• The Problem: A fundamental quantum field theory problem. It requires proving that quantum Yang-Mills theory exists mathematically and that the lightest excitation (particle) predicted by the theory has a strictly positive mass (the mass gap).
• The AI Connection: In Lattice Gauge Theory, machine learning is mitigating the heavy computational bottlenecks of Quantum Chromodynamics (QCD) simulations. Generative AI architectures, specifically Normalizing Flows, accelerate the configuration sampling of lattice states by over 100x, allowing physicists to calculate the numerical bounds of the mass gap with unprecedented precision.

6. P vs NP Problem

• The Problem: The cornerstone of computer science. It asks whether every problem whose solution can be quickly verified (NP) can also be solved quickly (P).
• The AI Connection: AI-driven automated theorem provers (like Lean and Coq integrated with neural models) accelerate the search for formal proofs. While solving P vs NP itself remains incredibly difficult, AI provides heuristic shortcuts for NP-hard problems, optimizing SAT solvers and solving complex combinatorial optimization tasks for practical industry use.

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🧪 Scientific Mysteries and AI’s Solutions

7. Dark Matter

• The Problem: Approximately 85% of the matter in the universe is dark matter—unseen, non-interactive with light, but exerting massive gravitational influence that dictates galactic rotation. Its identity remains a complete mystery.
• The AI Connection: Cosmologists use Convolutional Neural Networks (CNNs) to analyze petabytes of weak gravitational lensing data from space telescopes. AI filters out cosmic noise and cosmic dust to build precise 3D maps of dark matter distribution, narrowing down the potential properties of the elusive dark matter particles.

8. Dark Energy

• The Problem: The universe is not just expanding; it is accelerating. Dark energy is the placeholder name for the mysterious repulsive force driving this acceleration, constituting about 68% of the universe.
• The AI Connection: By processing massive supernova catalogs and accelerating cosmological simulations, AI is help mapping the rate of cosmic expansion. Through inverse modeling, AI algorithms test whether this acceleration behaves as a static cosmological constant (Einstein’s $\Lambda$) or points to a dynamic field representing new physics.

9. Quantum Gravity

• The Problem: The grand unification. It seeks to combine quantum mechanics (which explains the subatomic realm) with general relativity (which explains gravity and the cosmos) into a single, cohesive theory.
• The AI Connection: In String Theory, researchers face the landscape of $10^{500}$ possible Calabi-Yau manifolds. Machine learning classifiers are deployed to screen these configurations, identifying which manifolds yield physical constants matching our universe and reverse-engineering the geometry required for unified quantum gravity.

10. Protein Folding Problem

• The Problem: Predicting how a linear sequence of amino acids folds into its three-dimensional, functional structure. This stood as a 50-year-old bottleneck in biology.
• The AI Connection: The most successful conquest of a scientific mystery by AI to date. Google DeepMind’s AlphaFold bypassed decades of crystallography labs by predicting 3D structures from sequence data with over 90% accuracy. This has permanently accelerated drug discovery, enzyme engineering, and synthetic biology.

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💡 Conclusion: The New Intellectual Partnership

Historically, resolving these grand mysteries depended entirely on the flash of insight from a once-in-a-generation genius. Today, AI has emerged as an indispensable intellectual compass that scans mathematical spaces and a computational engine that dissolves simulation bottlenecks.

Just as AlphaFold dismantled the protein folding problem, the future will witness human-AI collaborations cracking open the secrets of the Riemann Hypothesis or the equations of quantum gravity. With every mystery solved, humanity takes a permanent leap forward into a new scientific civilization.