human
数学の最も重要な未解決の問題の上位20をリストします。
gpt
数学における20の重要な未解決の問題は次のとおりです。

1.リーマン仮説
2.バーチとスウィンナートンダイヤーの推測
3.ホッジの推測
4. P対NP
5.ヤンミルの存在と大量のギャップ
6. Navier-Stokes方程式
7. 3次元でのノイマン問題の解決策の存在
8.原素数の境界ギャップの推測
9.ツインプライム推測
10.ゴールドバッハの推測
11.コラッツの推測
12. Hadamardの推測
13.オイラーの権力に対する推測
14. Fermatのより高い指数の最後の定理
15. Baum – Connesの推測
16. Oppenheimの推測
17.連続性仮説
18.ノビコフの推測
19. Hirschの推測
20.シャヌエルの推測


数学には多くの未解決の問題がありますが、重要性が主観的であり、数学の分野によって異なる可能性があるため、最も重要な上位20のリストの決定的なリストを決定することは困難です。ただし、ウィキペディアのページ「数学における未解決の問題のリスト」の数学のさまざまな未解決の問題に関する詳細情報を見つけることができます。
human
これらの問題がリストされている各問題で解決されていない理由をリストします。
gpt
Here are some reasons why the following 20 problems in mathematics are still unsolved:

1. Riemann Hypothesis: This conjecture deals with the distribution of prime numbers and their connection to the Riemann zeta function. It is unsolved because it requires a deep understanding of the properties of the zeta function, which remains a mystery in mathematics.

2. Birch and Swinnerton-Dyer Conjecture: This conjecture relates to the behavior of solutions to certain types of polynomial equations, known as elliptic curves. It is unsolved because of the complexity of the objects involved and the difficulty in analyzing their behavior.

3. Hodge Conjecture: This conjecture is related to the topology of algebraic varieties, which are geometric objects that arise in algebraic geometry. It is unsolved because it involves understanding the complex interplay between geometry, topology, and algebra.

4. P versus NP: This is a fundamental problem in computer science that asks whether every problem that can be checked by a computer in polynomial time can also be solved by a computer in polynomial time. It is unsolved because it is one of the most difficult problems in computer science and mathematics, with far-reaching implications for cryptography, algorithms, and more.

5. Yang-Mills Existence and Mass Gap: This is a problem in theoretical physics and mathematics that deals with the behavior of particles known as gauge bosons. It is unsolved because it involves the difficult problem of quantizing a theory with non-Abelian symmetries.

6. Navier-Stokes Equation: This is a set of partial differential equations that describe the behavior of fluids in motion. It is unsolved because it is a difficult problem in applied mathematics that involves understanding the behavior of turbulent fluids.

7. Existence of a Solution for the Neumann Problem in Three Dimensions: This is a problem in the theory of partial differential equations that deals with finding solutions to certain types of boundary value problems. It is unsolved because it is a difficult problem that requires a deep understanding of the properties of partial differential equations.

8. Bounded Gaps Conjecture for Prime Numbers: This conjecture is related to the distribution of prime numbers and asks whether there are infinitely many pairs of primes that are a bounded distance apart. It is unsolved because it is a difficult problem that requires a deep understanding of the distribution of prime numbers.

9. Twin Prime Conjecture: This conjecture is related to the distribution of prime numbers and asks whether there are infinitely many pairs of primes that differ by 2. It is unsolved because it is a difficult problem that requires a deep understanding of the properties of prime numbers.

10. Goldbach Conjecture: This conjecture asks whether every even integer greater than 2 can be expressed as the sum of two prime numbers. It is unsolved because it is a difficult problem that has resisted all attempts at proof for over 250 years.

11. Collatz Conjecture: This conjecture is related to the behavior of certain sequences of numbers and asks whether every starting value eventually reaches 1. It is unsolved because it is a difficult problem that has resisted all attempts at proof for over 80 years.

12. Hadamard Conjecture: This conjecture is related to the geometry of convex bodies and asks whether every convex body in n-dimensional Euclidean space can be covered by exponentially fewer translations of a symmetric convex body. It is unsolved because it is a difficult problem that requires a deep understanding of the geometry of convex bodies.

13. Euler's Conjecture on Sums of Powers: This conjecture asks whether there are any nontrivial integer solutions to the equation a^n + b^n = c^n for n > 2. It is unsolved because it is a difficult problem that has resisted all attempts at proof for over 250 years.

14. Fermat's Last Theorem for higher exponents: This conjecture is related to Fermat's Last Theorem and asks whether there are any nontrivial integer solutions to the equation a^n + b^n = c^n for n > 2. It is unsolved because it is a difficult problem that has resisted all attempts at proof for over a century.

15. Baum–Connes Conjecture: This conjecture is related to the theory of operator algebras and asks whether certain types of groups have the same K-theory as the integers. It is unsolved because it involves difficult algebraic and topological concepts.

16. Oppenheim Conjecture: This conjecture is related to the behavior of quadratic forms and asks whether every positive integer can be expressed as a sum of at most 3 squares. It is unsolved because it is a difficult problem that requires a deep understanding of number theory.

17. The Continuum Hypothesis: This is a problem in set theory that asks whether there is a set whose size is strictly between that of the integers and that of the real numbers. It is unsolved because it is a difficult problem that involves the structure of the real numbers and the nature of infinity.

18. Novikov Conjecture: This conjecture is related to the topology of manifolds and asks whether certain types of manifolds have a particular type of symmetry. It is unsolved because it involves difficult topological and geometric concepts.

19. The Hirsch Conjecture: This conjecture is related to the topology of high-dimensional spaces and asks whether the number of cells in a certain type of complex grows exponentially with the dimension. It is unsolved because it involves difficult topological concepts.

20. Schanuel's Conjecture: This conjecture is related to the behavior of transcendental numbers and asks whether certain types of exponential functions can take on algebraic values. It is unsolved because it involves difficult algebraic and transcendental number theory concepts.

These problems are unsolved due to their complexity, involving advanced mathematical concepts and techniques that are still being developed. While progress has been made in some cases, their solution remains elusive and continues to challenge mathematicians and researchers around the world.
human
非アベルの対称性とは何ですか、そしてなぜいくつかの粒子がこの対称性にあるのですか？
gpt
物理学では、対称性は粒子の挙動とその相互作用を記述する上で重要な役割を果たします。対称性とは、物理学の根底にある法則を変化させない変換です。たとえば、固定点の周りに粒子のシステムを回転させ、結果として生じる粒子の構成が同じように見える場合、物理学の法則は回転の前後に同じでなければなりません。

対称性の1つの重要なクラスは、アベルの対称性として知られています。これらは通勤する対称性です。つまり、変換を実行する順序は重要ではありません。アベルの対称性の例は、粒子物理学における電荷保存です。反対の電荷の2つの粒子を交換する場合、システムはまだ同じです。

一方、非アベルの対称性は通勤しません。変換を実行する順序が重要です。非アベル対称性の例は、量子染色体（QCD）の色電荷保存です。QCDは、強力な力を説明する理論であり、陽子と中性子にクォークとグルオンを一緒に保持します。クォークには異なる「色」（赤、緑、青）があり、強い力は色の電荷を運ぶ「グルオン」によって媒介されます。

一部の粒子は、対称グループの下で非文書的に変換される特定のタイプの電荷を運ぶため、非アベル対称性にあります。たとえば、QCDでは、Quarksは色の対称性である色の電荷を運びます。電子のような他の粒子は、色電荷を運ぶことなく、この対称性にはありません。

粒子物理学から凝縮物質物理学まで、物理学の多くの領域で重要な役割を果たしていない非アベルの対称性があります。彼らの特性と行動を理解することは、研究の積極的な分野です。