The Oberwolfach problem is a problem in combinatorial design and graph theory that involves the arrangement of pairs (or "couples") of items, typically represented as graphs or edges. It is named after the Oberwolfach Institute for Mathematics in Germany, where the problem was first studied. The classical statement of the problem can be described as follows: You have a finite group of \( n \) people (or vertices) who need to meet in pairs over a series of days (or rounds).
The Lovász–Woodall conjecture is a conjecture in graph theory related to the concept of an ideal vertex cover and the independence number of graphs. Specifically, it provides a bound on the size of a minimum dominating set in terms of the independence number and the number of edges in a graph.
Harborth's conjecture is a hypothesis in the field of graph theory, particularly related to the properties of planar graphs. Specifically, it suggests that every planar graph can be colored using at most four colors such that no adjacent vertices share the same color. This assertion is closely related to the well-known Four Color Theorem, which states that four colors are sufficient to color the vertices of any planar graph.
"Erdős on Graphs" typically refers to the collection of works and contributions made by the famous Hungarian mathematician Paul Erdős in the field of graph theory. Erdős is known for his prolific output and collaborations, having published thousands of papers, many of which have shaped the development of various areas in mathematics, including combinatorics and graph theory.
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.
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 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 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.
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 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.
The Nakai conjecture is a concept in the field of algebraic geometry, specifically related to the theory of ample and pseudoeffective line bundles.
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.
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).
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.
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.
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.
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).
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.