Mathematics and applied mathematics

Mathematics provides the rigorous logical framework underlying all quantitative sciences, developing abstract concepts and structures that describe patterns across diverse domains.

The field ranges from pure mathematical investigations of number theory, geometry, and algebra to applied mathematics that models real-world phenomena in engineering, physics, and biology.

Central challenges include resolving famous conjectures that have resisted proof for decades or centuries, and developing new mathematical tools to handle increasingly complex systems. Mathematical breakthroughs often have unexpected applications that transform other fields and enable new technologies.

The 10 mathematics problems*

* These are just preliminary ideas and do not represent final problems of the Berkeley 100 Challenge. The final problems will be determined by our Scientific Committees.

  • P versus NP Problem

  • Riemann Hypothesis

  • Navier-Stokes Existence and Smoothness

  • Birch and Swinnerton-Dyer Conjecture

  • Hodge Conjecture

  • Efficient Algorithms for Factoring Large Integers

  • Langlands Program Unification

  • Four-Color Theorem Simplified Proof

  • Goldbach Conjecture

  • Optimal Transport Theory Unification

Mathematics problem sample

* These are just preliminary ideas and do not represent final problems of the Berkeley 100 Challenge. The final problems will be determined by our Scientific Committees.

Riemann Hypothesis

Problem Statement:

Prove or disprove the Riemann Hypothesis, which states that all non-trivial zeros of the Riemann zeta function have real part equal to 1/2.

Evaluation Criteria:

  • Complete mathematical proof published in peer-reviewed journal(s)

    Verification by multiple independent mathematical review committees

    Proof accepted by consensus among experts in number theory and analysis

    Clarification of connections to prime number theory

    If proven true, classification of all exceptions if any exist

    Broad implications for related conjectures and number theory clearly established

Feasibility Assessment: 

Extremely challenging, potentially requiring 10-30 years. Despite over 160 years of attempts, a proof remains elusive. Likely requires novel connections between different branches of mathematics. Recent advancements in analytical techniques and computational verification provide some optimism.

Impact on the Field: 

Would resolve one of the most significant open problems in mathematics with profound implications for prime number theory, cryptography, and quantum chaos. Would enable new classes of mathematical proofs that rely on the hypothesis and provide deeper understanding of the distribution of prime numbers.

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