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Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science.
Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object.
In combinatorics, an abstract simplicial complex (ASC), often called an abstract complex or just a complex, is a family of sets that is closed under taking subsets, i.e., every subset of a set in the family is also in the family. It is a purely combinatorial description of the geometric notion of a simplicial complex. For example, in a 2-dimensional simplicial complex, the sets in the family are the triangles, their edges, and their vertices.
Melvin Hochster is an American mathematician working in commutative algebra. He is currently the Jack E. McLaughlin Distinguished University Professor Emeritus of Mathematics at the University of Michigan.
In mathematics, specifically in combinatorial commutative algebra, a convex lattice polytope P is called normal if it has the following property: given any positive integer n, every lattice point of the dilation nP, obtained from P by scaling its vertices by the factor n and taking the convex hull of the resulting points, can be written as the sum of exactly n lattice points in P. This property plays an important role in the theory of toric varieties, where it corresponds to projective normality of the toric variety determined by P. Normal polytopes have popularity in algebraic combinatorics. These polytopes also represent the homogeneous case of the Hilbert bases of finite positive rational cones and the connection to algebraic geometry is that they define projectively normal embeddings of toric varieties.
In algebraic combinatorics, the h-vector of a simplicial polytope is a fundamental invariant of the polytope which encodes the number of faces of different dimensions and allows one to express the Dehn–Sommerville equations in a particularly simple form. A characterization of the set of h-vectors of simplicial polytopes was conjectured by Peter McMullen and proved by Lou Billera and Carl W. Lee and Richard Stanley (g-theorem). The definition of h-vector applies to arbitrary abstract simplicial complexes. The g-conjecture stated that for simplicial spheres, all possible h-vectors occur already among the h-vectors of the boundaries of convex simplicial polytopes. It was proven in December 2018 by Karim Adiprasito.
Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra.
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In geometry and combinatorics, a simpliciald-sphere is a simplicial complex homeomorphic to the d-dimensional sphere. Some simplicial spheres arise as the boundaries of convex polytopes, however, in higher dimensions most simplicial spheres cannot be obtained in this way.
Louis Joseph Billera is a Professor of Mathematics at Cornell University.
In mathematics, a Stanley–Reisner ring, or face ring, is a quotient of a polynomial algebra over a field by a square-free monomial ideal. Such ideals are described more geometrically in terms of finite simplicial complexes. The Stanley–Reisner ring construction is a basic tool within algebraic combinatorics and combinatorial commutative algebra. Its properties were investigated by Richard Stanley, Melvin Hochster, and Gerald Reisner in the early 1970s.
polymake is software for the algorithmic treatment of convex polyhedra.
In mathematics, a cyclic polytope, denoted C(n, d), is a convex polytope formed as a convex hull of n distinct points on a rational normal curve in Rd, where n is greater than d. These polytopes were studied by Constantin Carathéodory, David Gale, Theodore Motzkin, Victor Klee, and others. They play an important role in polyhedral combinatorics: according to the upper bound theorem, proved by Peter McMullen and Richard Stanley, the boundary Δ(n,d) of the cyclic polytope C(n,d) maximizes the number fi of i-dimensional faces among all simplicial spheres of dimension d − 1 with n vertices.
In mathematics, the upper bound theorem states that cyclic polytopes have the largest possible number of faces among all convex polytopes with a given dimension and number of vertices. It is one of the central results of polyhedral combinatorics.
Peter McMullen is a British mathematician, a professor emeritus of mathematics at University College London.
In polyhedral combinatorics, the hypersimplex is a convex polytope that generalizes the simplex. It is determined by two integers and , and is defined as the convex hull of the -dimensional vectors whose coefficients consist of ones and zeros. Equivalently, can be obtained by slicing the -dimensional unit hypercube with the hyperplane of equation and, for this reason, it is a -dimensional polytope when .
In mathematics, a polyhedral complex is a set of polyhedra in a real vector space that fit together in a specific way. Polyhedral complexes generalize simplicial complexes and arise in various areas of polyhedral geometry, such as tropical geometry, splines and hyperplane arrangements.
Jürgen Reinhard Gerhard Herzog was an Emeritus Professor of Mathematics at University of Duisburg-Essen, in Essen, Germany. From 1969 to 1975, he was Lecturer at University of Regensburg and from 1975 to 2009 a professor of Mathematics at University of Duisburg-Essen.
In the mathematical discipline of polyhedral combinatorics, the Gale transform turns the vertices of any convex polytope into a set of vectors or points in a space of a different dimension, the Gale diagram of the polytope. It can be used to describe high-dimensional polytopes with few vertices, by transforming them into sets with the same number of points, but in a space of a much lower dimension. The process can also be reversed, to construct polytopes with desired properties from their Gale diagrams. The Gale transform and Gale diagram are named after David Gale, who introduced these methods in a 1956 paper on neighborly polytopes.
Convex Polytopes is a graduate-level mathematics textbook about convex polytopes, higher-dimensional generalizations of three-dimensional convex polyhedra. It was written by Branko Grünbaum, with contributions from Victor Klee, Micha Perles, and G. C. Shephard, and published in 1967 by John Wiley & Sons. It went out of print in 1970. A second edition, prepared with the assistance of Volker Kaibel, Victor Klee, and Günter M. Ziegler, was published by Springer-Verlag in 2003, as volume 221 of their book series Graduate Texts in Mathematics.
References
A foundational paper on Stanley–Reisner complexes by one of the pioneers of the theory:
- Hochster, Melvin (1977). "Cohen–Macaulay rings, combinatorics, and simplicial complexes". Ring Theory II: Proceedings of the Second Oklahoma Conference. Lecture Notes in Pure and Applied Mathematics. Vol. 26. Dekker. pp. 171–223. ISBN 0-8247-6575-3. OCLC 610144046. Zbl 0351.13009.
The first book is a classic (first edition published in 1983):
- Stanley, Richard (1996). Combinatorics and commutative algebra. Progress in Mathematics. Vol. 41 (2nd ed.). Birkhäuser. ISBN 0-8176-3836-9. Zbl 0838.13008.
Very influential, and well written, textbook-monograph:
- Bruns, Winfried; Herzog, Jürgen (1993). Cohen–Macaulay rings. Vol. 39. Cambridge Studies in Advanced Mathematics: Cambridge University Press. ISBN 0-521-41068-1. OCLC 802912314. Zbl 0788.13005.
Additional reading:
- Villarreal, Rafael H. (2001). Monomial algebras. Monographs and Textbooks in Pure and Applied Mathematics. Vol. 238. Marcel Dekker. ISBN 0-8247-0524-6. Zbl 1002.13010.
- Hibi, Takayuki (1992). Algebraic combinatorics on convex polytopes. Glebe, Australia: Carslaw Publications. ISBN 1875399046. OCLC 29023080.
- Sturmfels, Bernd (1996). Gröbner bases and convex polytopes. University Lecture Series. Vol. 8. American Mathematical Society. ISBN 0-8218-0487-1. OCLC 907364245. Zbl 0856.13020.
- Bruns, Winfried; Gubeladze, Joseph (2009). Polytopes, Rings, and K-Theory. Springer Monographs in Mathematics. Springer. doi:10.1007/b105283. ISBN 978-0-387-76355-2. Zbl 1168.13001.
A recent addition to the growing literature in the field, contains exposition of current research topics:
- Miller, Ezra; Sturmfels, Bernd (2005). Combinatorial commutative algebra. Graduate Texts in Mathematics. Vol. 227. Springer. ISBN 0-387-22356-8. Zbl 1066.13001.
- Herzog, Jürgen; Hibi, Takayuki (2011). Monomial Ideals. Graduate Texts in Mathematics. Vol. 260. Springer. ISBN 978-0-85729-106-6. Zbl 1206.13001.