Conjectures

A conjecture is an educated guess or a proposition that is put forward based on limited evidence, which has not yet been proven or disproven. In mathematics and science, conjectures arise from observations or patterns that suggest a certain conclusion, but they need formal proof or experimental validation to be accepted as a theorem or law.

In mathematics, a conjecture is a statement or hypothesis that is proposed to be true but has not yet been proven. When a conjecture has been proven true, it is no longer considered a conjecture; instead, it is termed a theorem.

Disproved conjectures refer to proposed statements or hypotheses in mathematics or science that were initially believed to be true but have been shown to be false through logical reasoning, counterexamples, or experimental evidence. In mathematics, a conjecture is an assertion that has not yet been proven or disproven. Once a conjecture is disproven, it is clear that it does not hold in all cases.

The Arnold conjecture, proposed by the mathematician Vladimir Arnold in the 1960s, is a statement in the field of symplectic geometry and dynamical systems. It relates to the fixed points of Hamiltonian systems, which arise in the study of physics and mechanics.

Blattner's conjecture is a conjecture in the field of algebraic topology and homotopy theory, specifically concerning the structure of topological groups and their associated homotopy groups. Proposed by the mathematician Robert Blattner, the conjecture suggests a connection between certain types of topological groups and the generation of their homotopy groups.

The Calogero conjecture, proposed by Salvatore Calogero in the early 1990s, is a conjecture in the field of mathematical physics, specifically in the study of integrable systems. It generally concerns certain mathematical structures known as "Calogero-Moser systems," which are defined on a set of particles interacting through a specific type of potential. The conjecture itself relates to the behavior of the eigenvalues of certain matrices that arise in the context of these systems.

The Chronology Protection Conjecture is a theoretical idea in physics that was proposed by physicist Stephen Hawking. It suggests that the laws of physics may prevent time travel to the past in order to avoid potential paradoxes and violations of causality.

The Coase Conjecture is a concept in economics proposed by economist Ronald Coase. It addresses the behavior of firms when they sell durable goods, particularly how they set prices over time. The conjecture suggests that if a firm sells a durable good (a product that lasts a long time, like cars or appliances) and has market power, it will face a challenge in setting prices optimally.

A conjecture is an educated guess or a proposition that is believed to be true based on preliminary evidence or reasoning, but has yet to be proven or substantiated. In mathematics, for example, a conjecture is a statement that appears to be true because of observed patterns or numerical evidence, but it requires a formal proof to be accepted as a theorem. Conjectures play a crucial role in the development of mathematical theories, as they often lead to further research and exploration.

The Conley Conjecture is a proposition in the field of dynamical systems, particularly related to the study of Hamiltonian systems and their behavior in the context of symplectic geometry. Formulated by Charles Conley in the early 1970s, the conjecture specifically concerns the existence of certain types of periodic orbits for Hamiltonian systems.

De Branges's theorem, often referred to in the context of de Branges spaces, is a significant result in the theory of entire functions, specifically related to the representation of certain types of entire functions through Hilbert spaces. The theorem addresses the existence of entire functions that can be represented in terms of their zeros and certain properties related to their growth and behavior. More formally, it provides conditions under which a function defined by its Taylor series can be expressed in terms of its zeros or certain integral representations.

The Duffin–Schaeffer theorem is a result in the field of number theory, specifically in the study of Diophantine approximation. It addresses the question of how well real numbers can be approximated by rational numbers under certain conditions.

"ER = EPR" is a conjecture in theoretical physics that connects two seemingly different concepts: wormholes (denoted by ER, after the physicists Einstein and Rosen) and quantum entanglement (denoted by EPR, after Einstein, Podolsky, and Rosen). The idea was proposed by the physicist Juan Maldacena in a paper published in 2013.

Ehrhart's volume conjecture is a conjecture in the field of combinatorial geometry and involves the study of convex polytopes and their integer lattice points. More specifically, it relates the number of integer points in dilates of a polytope to the volume of the polytope.

The Final State Conjecture is a concept in the context of black holes and quantum mechanics, specifically relating to the information paradox. It suggests that when matter falls into a black hole, and the black hole eventually evaporates due to Hawking radiation, the information about the initial state of the matter that formed the black hole is not lost—even after the black hole has completely evaporated.

Fuglede's conjecture, proposed by the mathematician Bjarne Fuglede in 1974, is a statement in the field of mathematics that relates to the concepts of spectral sets and tiling in Euclidean space. Specifically, the conjecture asserts that: A measurable subset \( S \) of \( \mathbb{R}^n \) can tile \( \mathbb{R}^n \) by translations if and only if it is a spectral set.

Hilbert's twelfth problem, proposed by the mathematician David Hilbert in 1900, is concerned with the theory of functions with respect to algebraic number fields and involves the study of the so-called "absolutely abelian extensions" of these fields. More specifically, the problem asks for a systematic method to construct all the abelian extensions of a given number field using explicit functions, particularly through the use of modular forms.

The Ibragimov–Iosifescu conjecture pertains to the behavior of certain types of stochastic processes, particularly concerning the convergence of $\phi$-mixing sequences. A sequence of random variables \((X_n)_{n \in \mathbb{N}}\) is said to be $\phi$-mixing if it satisfies a certain criterion that measures the dependence between random variables that are separated by a certain distance.

Khabibullin's conjecture, proposed by the mathematician Ildar Khabibullin, revolves around integral inequalities related to certain classes of functions, particularly focusing on the relationships that hold for integrals of products of functions and their transformations. The conjecture suggests specific bounds and properties for these integrals, often drawing upon known results in functional analysis, inequalities, and perhaps the theory of convex functions.

Lafforgue's theorem is a result in the field of mathematics, specifically in the area of number theory and the theory of automorphic forms. It is associated with Laurent Lafforgue and pertains to the Langlands program, which aims to connect number theory and representation theory.

A list of conjectures is generally a compilation of mathematical statements or propositions that are suspected to be true but have not yet been proven. Conjectures play a significant role in driving mathematical research, often guiding the development of theories and the exploration of new areas. They can be found across various fields within mathematics, including number theory, topology, geometry, and combinatorics.

Paul Erdős was a prolific Hungarian mathematician known for his work in number theory, combinatorics, and other areas of mathematics. He is also famous for posing numerous conjectures throughout his career, many of which remain unresolved. Here are some notable conjectures attributed to Erdős: 1. **Erdős–Ko–Rado Theorem**: Although this theorem was proven, Erdős contributed to formulating its limitations and extensions regarding intersecting families of sets.

The field of computer science encompasses various unsolved problems that challenge researchers and practitioners. Here are some notable unsolved problems in computer science: 1. **P vs NP Problem**: Perhaps the most famous problem in computer science, it asks whether every problem for which a solution can be verified quickly (in polynomial time) can also be solved quickly (in polynomial time). The Clay Mathematics Institute offers a $1 million prize for a correct solution.

Mahler's 3/2 problem is a question in the field of number theory, specifically related to the properties of real numbers and their representations. Named after the mathematician Kurt Mahler, the problem concerns the transcendental numbers and the approximation of real numbers by rational numbers. The essence of the problem deals with whether there exist sufficiently "nice" sequences of rational numbers that can approximate certain real algebraic numbers well, particularly those that satisfy specific linear forms.

The Main Conjecture of Iwasawa theory is a central result in the field of algebraic number theory, particularly in the study of the relationship between the arithmetic of modular forms and the theory of \( p \)-adic numbers. In simple terms, the conjecture relates the growth of certain \( p \)-adic \( L \)-functions to the ideal class group of an infinite abelian extension of a number field, particularly in the context of cyclotomic fields.

The Nagata–Biran conjecture is a conjecture in the field of symplectic geometry and Hamiltonian dynamics. It was formulated by the mathematicians Masahiro Nagata and Michael Biran. The conjecture relates to the properties of symplectic manifolds, particularly concerning the existence of certain types of Lagrangian submanifolds.

The Novikov self-consistency principle is a concept in the realm of theoretical physics, particularly in the context of time travel and general relativity. Proposed by the Russian physicist Igor Novikov in the 1980s, the principle addresses the paradoxes that arise when one considers scenarios involving time travel. At its core, the Novikov self-consistency principle asserts that any events that occur as a result of time travel must be self-consistent.

The Pacman conjecture, proposed by mathematicians in the context of topology and geometric analysis, deals primarily with the area of geometric shapes and their properties, particularly in relation to convex shapes. It essentially posits a relationship between the area of a certain shape, referred to as the "Pacman" shape, and various mathematical properties surrounding convex polygons. The conjecture gets its name from the resemblance of the shape to the well-known video game character Pac-Man.

The Ryu–Takayanagi conjecture is a theoretical proposal in the field of theoretical physics, particularly in the context of quantum gravity and the AdS/CFT correspondence, which relates gravitational theories in Anti-de Sitter (AdS) space to conformal field theories (CFT) defined on the boundary of that space.

Selberg's 1/4 conjecture, proposed by the Norwegian mathematician Atle Selberg, is a conjecture in the field of number theory and specifically related to the distribution of the zeros of the Riemann zeta function and other Dirichlet series.

"Space form" can refer to different concepts depending on the context in which it is used. Below are a few interpretations: 1. **Architectural Context**: In architecture and design, "space form" often refers to the relationship between the physical space and the forms (structures and shapes) that occupy it. This can involve the analysis of how different shapes and materials influence the perception and functionality of a space.

The Thomas–Yau conjecture is a conjecture in the field of algebraic geometry and differential geometry, particularly relating to the study of the geometry of certain types of spaces called "special Lagrangians" and the moduli space of stable sheaves. It was proposed by Thomas and Yau in the early 2000s.

The Weak Gravity Conjecture (WGC) is a principle proposed in the context of theoretical physics, particularly in string theory and quantum gravity. It was formulated primarily by peers in the field, including Nathan Seiberg, and is aimed at providing insights into the nature of gravity in scenarios involving compact extra dimensions, such as those found in many string theory models.