The Most Advanced Math Problem, Ranked

Voting rules: Choose the math problem you think is the most advanced!

Updated on Apr 29, 2024 06:16

In the realm of mathematics, the complexity and intricacy of problems can reach staggering heights, often requiring innovative solutions that push the boundaries of human knowledge. These challenging problems are not only a testament to our intellectual limits but also provide a gauge for understanding how far we've pushed the envelope in mathematical disciplines. Knowing which problems are currently deemed the most complex can inspire new generations of mathematicians to tackle these formidable challenges. By engaging with a community-driven ranking of such problems, both amateur and professional mathematicians gain insights into where collective efforts and interests are converging in the field. This interaction fosters a collaborative environment where ideas can be exchanged and strategies discussed, potentially leading to breakthroughs in solving these advanced problems. Consequently, your vote helps shape an understanding of global priorities and perceptions in the mathematical community.

What Is the Most Advanced Math Problem?

1

56
votes
The Riemann Hypothesis

Bernhard Riemann

It is considered one of the most significant unsolved problems in mathematics. Its solution would have profound implications for the distribution of prime numbers.
The Riemann Hypothesis is one of the most significant unsolved problems in mathematics. It was formulated by the German mathematician Bernhard Riemann in 1859 and is concerned with the distribution of prime numbers. This hypothesis states that all non-trivial zeros of the Riemann zeta function have a real part equal to 1/2.

Formulated: 1859

Subject: Distribution of prime numbers

Hypothesis: All non-trivial zeros have real part equal to 1/2

Zeta Function: Riemann zeta function

Number Theory: Important problem in number theory
The Riemann Hypothesis in other rankings
- Position 1 of 10 in Most difficult math question
- Position 2 of 10 in Most difficult Calculus problem
2

38
votes
P vs. NP

It is one of the most famous and important unsolved problems in computer science and mathematics. Its resolution would have major implications for cryptography and optimization.
P vs. NP is one of the most famous unsolved problems in computer science and mathematics. It deals with the question of whether every problem whose solution can be verified quickly (in polynomial time) can also be solved quickly (in polynomial time).

Problem Type: Computational complexity theory

Problem Statement: Is P (problems solvable in polynomial time) equal to NP (problems whose solutions can be verified in polynomial time)?

Importance: Considered one of the most important open problems in computer science and mathematics

Applications: Relevant to many fields including cryptography, optimization, artificial intelligence, and algorithm design

Difficulty: Listed as one of the Millennium Prize Problems by the Clay Mathematics Institute, with a $1 million prize for solving it
P vs. NP in other rankings
- Position 3 of 10 in Most difficult math equation
3

29
votes
Yang-Mills Existence and Mass Gap

It is one of the seven Millennium Prize Problems, and its solution would help in understanding the behavior of subatomic particles.
The Yang-Mills Existence and Mass Gap problem is a fundamental question in theoretical physics and mathematics. It is related to the behavior of quantum chromodynamics (QCD), which describes the strong force that binds quarks together in atomic nuclei and governs interactions between elementary particles.

Problem Type: Mathematical physics problem

Nature: Unsolved problem

Field: Theoretical physics and mathematics

Main Question: Does the Yang-Mills theory have a mass gap and exist in four dimensions?

Yang-Mills Theory: A gauge theory that describes interactions among elementary particles based on the symmetry group SU(N)
Yang-Mills Existence and Mass Gap in other rankings
- Position 5 of 10 in Most difficult unsolved mathematical problem
4

9
votes
Birch and Swinnerton-Dyer Conjecture

It is another Millennium Prize Problem that has implications for the arithmetic of elliptic curves. Its solution would help in understanding the behavior of prime numbers.
The Birch and Swinnerton-Dyer Conjecture is one of the unsolved problems in mathematics that relates to elliptic curves, which are a type of mathematical object with fascinating properties. It was proposed by Bryan Birch and Peter Swinnerton-Dyer in 1965. The conjecture posits that there is a fundamental connection between the arithmetic properties of elliptic curves and the behavior of their associated L-functions. If the L-function of an elliptic curve has a specific value at a certain critical point, then the curve has an infinite number of rational points. On the other hand, if the L-function has a specific value of zero at that point, then the curve has only finitely many rational points. The conjecture has wide-ranging implications in number theory and algebraic geometry.

Year Proposed: 1965

Type of Problem: Unsolved Problem

Mathematical Field: Number Theory and Algebraic Geometry

Subject Area: Elliptic Curves

Connection: Arithmetic properties of elliptic curves and their L-functions
Birch and Swinnerton-Dyer Conjecture in other rankings
- Position 2 of 10 in Most difficult unsolved mathematical problem
- Position 4 of 10 in Most difficult math question
- Position 7 of 10 in Most difficult equation to solve
5

12
votes
Navier-Stokes Equations

They describe the behavior of fluids and are essential in fields such as aerodynamics and weather prediction. However, their solutions are notoriously difficult to find.
The Navier-Stokes equations are a set of partial differential equations that describe the motion of fluid substances such as liquids and gases.

Equations: The equations consist of the continuity equation and the momentum equations.

Continuity equation: Describes the conservation of mass for an incompressible fluid.

Momentum equations: Describe the conservation of momentum for a fluid, taking into account pressure and viscous forces.

Nonlinear nature: The equations are nonlinear, making them difficult to solve analytically.

Important in fluid dynamics: They provide a mathematical description of the behavior of fluids in various scenarios.
Navier-Stokes Equations in other rankings
- Position 1 of 10 in Most difficult math equation
- Position 1 of 10 in Most difficult equation
- Position 3 of 10 in Most difficult math question
- Position 4 of 10 in Most difficult unsolved mathematical problem
- Position 8 of 10 in Most advanced mathematical proof used in reasoning
6

11
votes
Langlands Program

Robert P. Langlands

It is a vast and ambitious program that seeks to connect different areas of mathematics, such as number theory and representation theory. Its resolution would have far-reaching implications for mathematics as a whole.
The Langlands Program is a vast and ambitious mathematical endeavor that seeks to establish deep connections between number theory, algebraic geometry, and representation theory.

Bridging Fields: It aims to bridge the gap between different areas of mathematics, including number theory and harmonic analysis.

Dualities: It explores the concept of dualities, where ideas from one field can shed light on problems in another.

Automorphic Forms: It investigates the theory of automorphic forms and their associated L-functions.

Representations of Algebraic Groups: It studies the representations of algebraic groups over different fields.

Liftings: It explores the possibility of lifting objects from one field to another, such as lifting automorphic forms to Galois representations.
7

9
votes
Hodge Conjecture

William Vallance Douglas Hodge

It is a problem in algebraic geometry that concerns the topology of algebraic varieties. Its solution would have implications for the study of differential equations and physics.
The Hodge Conjecture is a famous conjecture in algebraic geometry. It deals with the relationship between two important mathematical structures, algebraic cycles and cohomology theory. The conjecture states that for a smooth projective algebraic variety, there exists a natural decomposition of the cohomology into pieces called Hodge classes, which are algebraic cycles. These algebraic cycles can be understood in terms of the geometry of the variety. The conjecture further asserts that any Hodge class can be represented by a rational linear combination of algebraic cycles.

Field: Algebraic geometry

Status: Unsolved

Conjectured: During the 1950s

Importance: One of the seven Millennium Prize Problems

Impact: Reveals deep connections between algebraic cycles and cohomology theory
Hodge Conjecture in other rankings
- Position 3 of 10 in Most difficult unsolved mathematical problem
- Position 7 of 10 in Most advanced mathematical proof used in reasoning
8

6
votes
Riemannian Geometry

Bernhard Riemann

It is a subfield of differential geometry that deals with the curvature and geometry of spaces. Its applications range from physics to computer vision.
Riemannian Geometry is a branch of mathematics that deals with the study of curved spaces and the properties of geometric objects on these spaces. It is named after the German mathematician Bernhard Riemann, who laid the foundations for this field in the mid-19th century.

Curvature: Riemannian geometry studies spaces with curvature, allowing for the exploration of non-Euclidean geometries.

Manifolds: Riemannian geometry focuses on studying smooth manifolds and their geometric properties.

Metric Tensor: It utilizes the metric tensor to define a notion of distance and angles on the curved spaces.

Geodesics: Riemannian geometry studies geodesics, which are the shortest paths between points on a curved surface.

Curvature Tensor: It introduces the concept of curvature tensors to measure the curvature properties of a geometric space.
9

7
votes
Hilbert's Problems

David Hilbert

They were a list of 23 problems proposed by David Hilbert in 1900, which set the agenda for 20th-century mathematics. Some of them have been solved, while others remain unsolved.
Hilbert's Problems refer to a collection of 23 mathematical problems presented by German mathematician David Hilbert in the year 1900. The problems were presented during a famous lecture at the International Congress of Mathematicians in Paris and were intended to highlight important unsolved questions in various branches of mathematics.

Number of problems: 23

Year of presentation: 1900

Purpose: Highlight unsolved questions in mathematics

Range of topics: Various branches of mathematics

Significance: Influenced the course of 20th-century mathematics
10

7
votes
Tlep · Public domain

Category Theory

It is a branch of mathematics that deals with abstract structures and their relationships. It has applications in fields such as computer science and physics.
Category Theory in other rankings
- Position 1 of 10 in Most advanced branch of mathematics
- Position 1 of 10 in Most advanced math subject

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Discussion

Ranking factors for advanced math problem

Complexity

Consider the mathematical concepts, techniques, and operations required to solve the problem. A more advanced problem will involve a higher level of mathematical concepts, such as calculus, number theory, topology, or abstract algebra.
Novelty

Evaluate the problem's originality, as more advanced problems tend to be new or relatively undiscovered. Problems that have been studied extensively or have known solutions may be less advanced than those that haven't been explored as extensively.
Applicability

Advanced problems can often be applied to multiple areas within mathematics or other disciplines, indicating that they have a broader impact on the understanding of various fields.
Difficulty

The level of difficulty in solving the problem is an important factor to consider. More advanced problems are typically more challenging and may require innovative thinking or advanced techniques.
Openness

Determine if the problem is open, meaning it doesn't have a known complete solution. Open problems are often considered to be more advanced since they still present ongoing challenges within the field.
Historical significance

Consider the problem's relevance and importance in the history of mathematics. A more advanced problem may have been a driving force behind significant developments within the field.
Interdisciplinary connections

Advanced math problems often connect to other areas of mathematics or branch out into other scientific disciplines, such as physics, computer science, and engineering.
Generalization potential

Advanced problems might lead to more general principles or theories, contributing to the development of a broader understanding within mathematics.
Depth

Assess the depth of the problem, meaning the layers of complexity that one needs to understand before being able to fully grasp and solve the problem. More advanced problems typically have greater depth.
Intellectual challenge

A more advanced problem will usually provide greater intellectual stimulation and demand higher levels of critical thinking, creativity, and persistence in order to find a solution.

About this ranking

This is a community-based ranking of the most advanced math problem. We do our best to provide fair voting, but it is not intended to be exhaustive. So if you notice something or Math problem is missing, feel free to help improve the ranking!

Statistics

1710 views
180 votes
10 ranked items

Voting Rules

A participant may cast an up or down vote for each Math problem once every 24 hours. The rank of each Math problem is then calculated from the weighted sum of all up and down votes.