| Hilbert 2025 Problems: A Vision for the Future of Mathematics |
[May. 21st, 2025|11:43 pm] |
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Решил спросить у deepseek список задач типа 23 задач Гильберта, но для современной математики. Что думаете?
deepseek
Hilbert 2025 Problems: A Vision for the Future of Mathematics
1. The P vs NP Problem (Extended)
Formulate a constructive roadmap toward resolving P vs NP, including implications for cryptography, optimization, and artificial intelligence. If P = NP, provide an explicit polynomial-time algorithm for an NP-complete problem.
2. Quantum Complexity Theory
Develop a rigorous mathematical framework for quantum computation that classifies problems solvable by quantum computers (e.g., BQP vs PH) and establishes limits of quantum advantage.
3. The Riemann Hypothesis (Quantized)
Prove the Riemann Hypothesis or its quantum analogue (e.g., connections to random matrices, quantum chaos, or non-commutative geometry).
4. The Langlands Program (Global Unification)
Complete the geometric Langlands correspondence and establish its connections to quantum field theory, string theory, and number theory.
5. The Navier-Stokes Millennium Problem (Resolved Constructively)
*Construct a global smooth solution or prove blow-up for the 3D Navier-Stokes equations, with physical and computational implications.*
6. The Artificial General Intelligence (AGI) Formalization Problem
Develop a rigorous mathematical theory of intelligence, including complexity-theoretic, dynamical, and information-theoretic foundations for AGI.
7. The Hodge Conjecture (Generalized)
Extend the Hodge conjecture to non-algebraic varieties, derived categories, or infinity-categorical settings.
8. The Dark Matter/Dark Energy Mathematical Structure Problem
Discover new mathematical structures (e.g., exotic geometries, higher symmetries) that explain dark matter and dark energy within theoretical physics.
9. The Homotopy Type Theory & Foundation of Mathematics
Construct a fully formalized, computable foundation for mathematics unifying homotopy theory, category theory, and proof assistants (e.g., via univalent foundations).
10. The Optimal Climate Modeling Problem
Develop mathematically rigorous, scalable, and predictive models for climate systems, incorporating stochastic PDEs, network theory, and AI.
11. The Black Hole Information Paradox (Mathematically Settled)
Resolve the black hole information paradox within a mathematically consistent quantum gravity framework (e.g., AdS/CFT, causal sets).
12. The Erdős Discrepancy Problem (Generalized)
Extend discrepancy theory to high-dimensional and quantum settings, with applications to pseudorandomness and algorithm design.
13. The Neuroscience-Mathematics Interface
Formulate a mathematical theory of consciousness, learning, and neural computation, bridging algebraic topology, dynamical systems, and deep learning.
14. The Optimal Inequality Problem
Discover sharp inequalities in analysis, geometry, and combinatorics that unify disparate areas (e.g., entropy bounds, curvature constraints).
15. The Synthetic Biology & Algebraic Systems Biology Problem
Develop algebraic and topological methods to model, predict, and engineer biological systems (e.g., gene regulatory networks, protein folding).
16. The Cryptography Post-Quantum Problem
Construct cryptographic protocols with provable security against quantum and classical attacks, based on new hardness assumptions.
17. The Mathematical Theory of Emergence
Define and quantify emergence in complex systems (e.g., phase transitions, consciousness, swarm intelligence) using category theory, renormalization group methods, or sheaf theory.
18. The Singularity Avoidance Problem
Prove or disprove the existence of finite-time singularities in general relativity under modified (e.g., non-local, higher-curvature) gravitational theories.
19. The Universal Topological Quantum Computer
Classify topological phases of matter that permit universal fault-tolerant quantum computation.
20. The Mathematics of Explainable AI (XAI)
Develop a rigorous theory of interpretability in machine learning, linking geometric deep learning, logic, and causal inference.
21. The Global Dynamics of Pandemics
Formulate a universal mathematical framework for pandemic spread, incorporating human behavior, network adaptation, and multi-scale modeling.
22. The Geometric Langlands for p-adic Fields
Extend the Langlands program to p-adic and function-field settings with explicit computational tools.
23. The Mathematical Ethics Problem
Define a formal mathematical framework for ethical reasoning in AI, economics, and governance, incorporating game theory, logic, and social choice.
24. The Infinity-Categorical Derived Geometry Problem
Develop a computable theory of derived geometry and infinity-categories for applications in physics and algebraic geometry.
25. The Unification of Discrete and Continuous Mathematics
Discover deep connections between combinatorics, number theory, and analysis (e.g., additive combinatorics meets harmonic analysis).
chatGPT
Foundations, Logic, and Infinity
1Foundational Unification of Set Theory, Type Theory, and Category Theory
Develop a foundational framework unifying these three pillars into a single coherent basis for all of mathematics.
2Ultimate Status of the Continuum Hypothesis
Determine whether new axioms (e.g., from large cardinal hypotheses or forcing axioms) can naturally resolve the continuum hypothesis and settle its place in the universe of sets.
3Understanding the Landscape of Incompleteness
Classify and characterize mathematical theories according to their degree of incompleteness, extending Gödel’s insights into more practical and structural directions.
???? Computation, Complexity, and Algorithms
4 P vs NP and the Nature of Efficient Proof
Resolve the P ≠ NP problem or determine whether efficient verifiability fundamentally differs from efficient solvability. Go further—define practical mathematics via complexity.
5 Mathematics of Explainable and Reliable AI
Develop a rigorous mathematical theory of explainability, alignment, and trust in learning systems; include interpretability theorems akin to Galois theory for neural nets.
6 Foundations of Quantum Algorithms and Complexity
Classify the true power of quantum computation: Does BQP ≠ P? Can we characterize quantum advantage in rigorous, structure-theoretic terms?
7Mathematics of Emergent Computation
Formalize models where computation emerges from dynamics, e.g., in biological or physical systems, beyond Turing models (e.g., morphogenesis, cellular automata, or chaotic attractors).
???? Number Theory and Arithmetic Geometry
8 General Langlands Program: Global Correspondences
Fully establish the Langlands correspondence for all reductive groups over number fields. Develop effective versions and connections with physics (e.g., via S-duality).
9 Uniformity and Effective Bounds in Diophantine Problems
Prove effective versions of theorems like Faltings’ or Mordell’s. Understand the computability of rational points on algebraic varieties.
10 Zeta Functions and Riemann Hypothesis in All Fields
Settle the Riemann Hypothesis for classical zeta and L-functions and generalize to global function fields, motives, and beyond.
???? Geometry, Topology, and Algebra
11 Geometric Langlands Program and Topological QFT
Establish the deep links between geometric representation theory, topological quantum field theory, and categorical sheaf theory.
12 Categorification of All Classical Invariants
Find higher categorical analogues of fundamental invariants (like the Jones polynomial, group cohomology), revealing deeper structures behind known theorems.
13 Moduli of Structures: Global Classification Theory
Fully describe the moduli spaces of key mathematical objects (e.g., Calabi–Yau manifolds, special holonomy metrics, stability conditions) in constructive and computable ways.
???? Physics and Mathematical Reality
14 Mathematics of Quantum Gravity
Find a rigorous and predictive framework for quantum gravity, possibly reconciling loop quantum gravity, string theory, and holography with categorical and geometric ideas.
15 Mathematics of Black Holes and Holography
Prove the AdS/CFT correspondence rigorously. Establish a dictionary between geometry in the bulk and quantum field theory on the boundary.
16 Mathematics of Time and Causality
Develop a mathematical theory of time that accounts for asymmetry, quantum causality, and emergent structure, possibly from thermodynamic or categorical principles.
???? Biology, Life, and Complexity
17 Mathematical Theory of Consciousness
Is it possible to formulate an axiomatic theory of conscious systems, their emergence, and their classification? What mathematical structures correspond to subjective experience?
18 Mathematics of Evolution and Open-Endedness
Formalize evolutionary processes as open-ended, creative systems. Characterize complexity growth, innovation, and transitions in mathematical terms.
19 Mathematics of the Cell: Internal Geometry of Life
Create a geometric or topological model of intracellular dynamics, explaining information flow, signaling, and metabolic regulation.
???? Unification and Meta-Mathematics
20 Theory of Mathematical Insight and Discovery
Develop a mathematical theory of creativity—what makes some concepts “deep”? Can AI models discover conjectures with Hilbertian value?
21 Universal Theory of Mathematical Structures
Develop a unifying theory that classifies all mathematical structures in a single language (e.g., via toposes, ∞-categories, or model theory), bridging logic, geometry, and algebra.
22 Self-Verifying and Evolvable Formal Systems
Build mathematical systems that evolve, extending their axioms to account for new phenomena (self-modifying formalism), while avoiding inconsistency.
???? Mathematics and the Universe
22 Ultimate Geometry of the Universe
Does the universe admit a canonical geometric model (e.g., algebraic variety, spectral triple, or topoi)? Can we mathematically derive the dimensionality, topology, and constants of physics?
Видно, что deepseek любит вставлять везде квантовую механику, а chatGPT топосы. |
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