In mathematics and formal logic, a theorem is a statement that has been proven, or can be proven. The proof of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems.
Stephen Arthur Cook is an American-Canadian computer scientist and mathematician who has made significant contributions to the fields of complexity theory and proof complexity. He is a university professor emeritus at the University of Toronto, Department of Computer Science and Department of Mathematics.
In the philosophy of mathematics, ultrafinitism is a form of finitism and intuitionism. There are various philosophies of mathematics that are called ultrafinitism. A major identifying property common among most of these philosophies is their objections to totality of number theoretic functions like exponentiation over natural numbers.
The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global L-functions, which are formally similar to the Riemann zeta-function. One can then ask the same question about the zeros of these L-functions, yielding various generalizations of the Riemann hypothesis. Many mathematicians believe these generalizations of the Riemann hypothesis to be true. The only cases of these conjectures which have been proven occur in the algebraic function field case.
Experimental mathematics is an approach to mathematics in which computation is used to investigate mathematical objects and identify properties and patterns. It has been defined as "that branch of mathematics that concerns itself ultimately with the codification and transmission of insights within the mathematical community through the use of experimental exploration of conjectures and more informal beliefs and a careful analysis of the data acquired in this pursuit."
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures.
Doron Zeilberger is a mathematician, known for his work in combinatorics.
In mathematics, an alternating sign matrix is a square matrix of 0s, 1s, and −1s such that the sum of each row and column is 1 and the nonzero entries in each row and column alternate in sign. These matrices generalize permutation matrices and arise naturally when using Dodgson condensation to compute a determinant. They are also closely related to the six-vertex model with domain wall boundary conditions from statistical mechanics. They were first defined by William Mills, David Robbins, and Howard Rumsey in the former context.
Bruno Buchberger is Professor of Computer Mathematics at Johannes Kepler University in Linz, Austria. In his 1965 Ph.D. thesis, he created the theory of Gröbner bases, and has developed this theory throughout his career. He named these objects after his advisor Wolfgang Gröbner. Since 1995, he has been active in the Theorema project at the University of Linz.
A computer-assisted proof is a mathematical proof that has been at least partially generated by computer.
In mathematics and especially in combinatorics, a plane partition is a two-dimensional array of nonnegative integers that is nonincreasing in both indices. This means that
In mathematics, and more specifically in analysis, a holonomic function is a smooth function of several variables that is a solution of a system of linear homogeneous differential equations with polynomial coefficients and satisfies a suitable dimension condition in terms of D-modules theory. More precisely, a holonomic function is an element of a holonomic module of smooth functions. Holonomic functions can also be described as differentiably finite functions, also known as D-finite functions. When a power series in the variables is the Taylor expansion of a holonomic function, the sequence of its coefficients, in one or several indices, is also called holonomic. Holonomic sequences are also called P-recursive sequences: they are defined recursively by multivariate recurrences satisfied by the whole sequence and by suitable specializations of it. The situation simplifies in the univariate case: any univariate sequence that satisfies a linear homogeneous recurrence relation with polynomial coefficients, or equivalently a linear homogeneous difference equation with polynomial coefficients, is holonomic.
In mathematics, Richardson's theorem establishes the undecidability of the equality of real numbers defined by expressions involving integers, π, and exponential and sine functions. It was proved in 1968 by the mathematician and computer scientist Daniel Richardson of the University of Bath.
In mathematics, Gosper's algorithm, due to Bill Gosper, is a procedure for finding sums of hypergeometric terms that are themselves hypergeometric terms. That is: suppose one has a(1) + ... + a(n) = S(n) − S(0), where S(n) is a hypergeometric term (i.e., S(n + 1)/S(n) is a rational function of n); then necessarily a(n) is itself a hypergeometric term, and given the formula for a(n) Gosper's algorithm finds that for S(n).
In mathematics, the Dyson conjecture is a conjecture about the constant term of certain Laurent polynomials, proved independently in 1962 by Wilson and Gunson. Andrews generalized it to the q-Dyson conjecture, proved by Zeilberger and Bressoud and sometimes called the Zeilberger–Bressoud theorem. Macdonald generalized it further to more general root systems with the Macdonald constant term conjecture, proved by Cherednik.
The progression of both the nature of mathematics and individual mathematical problems into the future is a widely debated topic; many past predictions about modern mathematics have been misplaced or completely false, so there is reason to believe that many predictions today will follow a similar path. However, the subject still carries an important weight and has been written about by many notable mathematicians. Typically, they are motivated by a desire to set a research agenda to direct efforts to specific problems, or a wish to clarify, update and extrapolate the way that subdisciplines relate to the general discipline of mathematics and its possibilities. Examples of agendas pushing for progress in specific areas in the future, historical and recent, include Felix Klein's Erlangen program, Hilbert's problems, Langlands program, and the Millennium Prize Problems. In the Mathematics Subject Classification section 01Axx History of mathematics and mathematicians, subsection 01A67 is titled Future prospectives.
The David P. Robbins Prize for papers reporting novel research in algebra, combinatorics, or discrete mathematics is awarded both by the American Mathematical Society (AMS) and by the Mathematical Association of America (MAA). The AMS award recognizes papers with a significant experimental component on a topic which is broadly accessible which provide a simple statement of the problem and clear exposition of the work. Papers eligible for the MAA award are judged on quality of research, clarity of exposition, and accessibility to undergraduates. Both awards consist of $5000 and are awarded once every three years. They are named in the honor of David P. Robbins and were established in 2005 by the members of his family.
Christoph Koutschan is a German mathematician and computer scientist. He is currently with the Johann Radon Institute for Computational and Applied Mathematics (RICAM) of the Austrian Academy of Sciences.
Ira Martin Gessel is an American mathematician, known for his work in combinatorics. He is a long-time faculty member at Brandeis University and resides in Arlington, Massachusetts.
