Unsolved problems in geometry cover a wide range of topics and questions that have yet to be resolved. Here are a few notable examples: 1. **The Poincaré Conjecture**: While this conjecture was solved by Grigori Perelman in 2003, its implications and related questions about the topology of higher-dimensional manifolds are still active areas of research.
The Abundance Conjecture is a concept within the field of number theory, focusing on the behavior of certain algebraic integers. Specifically, it deals with the distribution of prime numbers and the density of subsets of integers with specific properties. While the conjecture has been discussed in various contexts, it is often associated with the idea that among the integers, there exists a rich abundance of those that exhibit certain arithmetic properties, such as being prime or having a specific number of divisors.
The Atiyah conjecture on configurations is a mathematical statement concerning the representation theory of algebraic structures, specifically related to bundles of vector spaces over topological spaces. It is named after the British mathematician Michael Atiyah, who has made significant contributions to several areas of mathematics, including topology, geometry, and mathematical physics.
The Bass conjecture is a conjecture in algebraic K-theory, specifically concerning the K-theory of integral domains and, more generally, rings. It was proposed by Hyman Bass in the 1960s.
The Bombieri–Lang conjecture is a concept in number theory that relates to the distribution of rational points on certain types of algebraic varieties. Specifically, it deals with the behavior of rational points on algebraic varieties defined over number fields and has implications for understanding the ranks of abelian varieties and the distribution of solutions to Diophantine equations. The conjecture can be stated in a few steps for certain types of varieties, particularly for curves and higher-dimensional varieties.
The Carathéodory conjecture is a mathematical conjecture in the field of geometry that deals with the concept of convex polygons in three-dimensional space. Specifically, the conjecture states that for any simple closed convex surface in three-dimensional Euclidean space, the surface can be covered by at most five planes. This conjecture was proposed by the Greek mathematician Constantin Carathéodory in 1911.
Clifford's theorem is a significant result in algebraic geometry that deals with special divisors on a non-singular projective curve. It can be applied in the context of the study of linear systems and, more broadly, the theory of algebraic curves.
Dissection into orthoschemes is a concept in geometry, particularly in higher-dimensional spaces, that deals with the partitioning of a geometric object into pieces that can be individually described as orthoschemes. An orthoscheme is a generalization of a tetrahedron to higher dimensions where all faces meet at right angles (i.e., they are orthogonal).
The Fröberg conjecture, proposed by Anders Fröberg in 1981, is a conjecture in the field of algebraic geometry and commutative algebra. It deals with the study of the Betti numbers of a certain class of algebraic varieties, specifically focusing on the resolutions of certain graded modules.
The Fujita conjecture is a statement in the field of algebraic geometry, particularly concerning the minimal model program and the properties of algebraic varieties. Proposed by Takao Fujita in the 1980s, the conjecture pertains to the relationship between the ample divisor classes and the structure of the variety. Specifically, the Fujita conjecture relates to the growth of the dimension of the space of global sections of powers of an ample divisor.
Heesch's problem is a question in the field of geometry, specifically in relation to tiling and the properties of shapes. It asks whether a given shape can be extended into a larger shape by adding additional copies of itself, while maintaining a specific tiling condition—specifically, that the tiles fit together without gaps or overlaps.
Hilbert's fifteenth problem, presented by David Hilbert in 1900 as part of his famous list of 23 problems, concerns the nature of the solutions to certain types of polynomial equations. Specifically, it can be summarized as asking for the conditions under which a solution to a system of polynomial equations can be expressed in terms of elementary functions (such as addition, multiplication, and taking roots).
The Inscribed Square Problem refers to a geometric problem of finding the largest square that can be inscribed within a given shape, usually a convex polygon or a specific type of curve. The goal is to determine the dimensions and position of the square such that it fits entirely within the boundaries of the shape while maximizing its area.
The Nakai conjecture is a concept in the field of algebraic geometry, specifically related to the theory of ample and pseudoeffective line bundles.
The Pierce–Birkhoff conjecture is a conjecture in the field of lattice theory, specifically concerning finite distributive lattices and their Maximal Chains. It was proposed by the mathematicians Benjamin Pierce and George Birkhoff. The conjecture essentially deals with the nature of certain kinds of chains (series of elements) within these lattices and posits conditions under which certain structural properties hold.
Resolution of singularities is a mathematical process in algebraic geometry that aims to transform a variety (which can have singular points) into a smoother variety (which has no singularities) by replacing the singular points with more complex structures, often in a controlled way. This process is crucial for understanding geometric properties of algebraic varieties and for performing various calculations in algebraic geometry.
The Sato–Tate conjecture is a conjecture in number theory that describes the symmetry of the distribution of certain mathematical objects called elliptic curves over finite fields. Specifically, it relates to the number of points on an elliptic curve defined over a finite field and their distribution when examined from a statistical perspective.
The Section Conjecture is a significant hypothesis in the field of arithmetic geometry, particularly concerning the relationship between algebraic varieties and their associated functions or sections. It was formulated by mathematicians in the context of the study of abelian varieties and their rational points. More specifically, the conjecture relates to the *Neron models* of abelian varieties over a number field and their sections.
The Standard Conjectures on algebraic cycles are a set of conjectures in algebraic geometry that relate to the study of algebraic cycles and their properties, particularly in the context of algebraic varieties over a field. The conjectures were primarily formulated by Pierre Deligne, Alexander Grothendieck, and others in the mid-20th century.
The Virasoro conjecture is a fundamental result in the field of string theory and two-dimensional conformal field theory (CFT). It relates to the algebra of Virasoro operators, which are central to the study of CFTs, particularly in the context of two-dimensional quantum gravity and string theory. In essence, the conjecture asserts that there exists a certain relation between the partition functions of two-dimensional conformal field theories and the geometry of the underlying space.
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