James Haglund is an American mathematician who specializes in algebraic combinatorics and enumerative combinatorics, [1] and works as a professor of mathematics at the University of Pennsylvania. [2]
Haglund received his Ph.D. in 1993 from the University of Georgia, with the dissertation Compositions, Rook Placements, and Permutations of Vectors supervised by Earl Rodney Canfield. [3]
In 2005, together with M. Haiman and N. Loehr gave the first proof of a combinatorial interpretation of the Macdonald polynomials. In 2007, Haglund, Haiman and Loehr gave a combinatorial formula for the non-symmetric Macdonald polynomials.
Haglund is the author of The -Catalan Numbers and the Space of Diagonal Harmonics: With an Appendix on the Combinatorics of Macdonald Polynomials. [4]
In 2024, Haglund gave a talk at KAIST on Superization of Symmetric Functions. [5]
In 2015, together with Alexandre Kirillov and Ching-Li Chai, Haglund gave a talk at Penn Wharton China Center on Penn Math Day, sponsored by the University of Pennsylvania and Peking University. [6]
In 2006, he gave a Plenary Address at the 18th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC '06), San Diego (USA). [7]
Haglund was on the editorial boards of Transactions of the AMS, Journal of Combinatorics, and a few other academic journals.
Among the Ph.D. students supervised by Haglund are Mahir Bilen Can, Logan Crew, Sarah Katherine Mason, Anna Pun, Chunwei Song, and Meesue Yoo. [8]
In 2013, Haglund became an inaugural Fellow of the American Mathematical Society. [9]
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.
In combinatorics, bijective proof is a proof technique for proving that two sets have equally many elements, or that the sets in two combinatorial classes have equal size, by finding a bijective function that maps one set one-to-one onto the other. This technique can be useful as a way of finding a formula for the number of elements of certain sets, by corresponding them with other sets that are easier to count. Additionally, the nature of the bijection itself often provides powerful insights into each or both of the sets.
Ian Grant Macdonald was a British mathematician known for his contributions to symmetric functions, special functions, Lie algebra theory and other aspects of algebra, algebraic combinatorics, and combinatorics.
In mathematics, Macdonald polynomialsPλ(x; t,q) are a family of orthogonal symmetric polynomials in several variables, introduced by Macdonald in 1987. He later introduced a non-symmetric generalization in 1995. Macdonald originally associated his polynomials with weights λ of finite root systems and used just one variable t, but later realized that it is more natural to associate them with affine root systems rather than finite root systems, in which case the variable t can be replaced by several different variables t=(t1,...,tk), one for each of the k orbits of roots in the affine root system. The Macdonald polynomials are polynomials in n variables x=(x1,...,xn), where n is the rank of the affine root system. They generalize many other families of orthogonal polynomials, such as Jack polynomials and Hall–Littlewood polynomials and Askey–Wilson polynomials, which in turn include most of the named 1-variable orthogonal polynomials as special cases. Koornwinder polynomials are Macdonald polynomials of certain non-reduced root systems. They have deep relationships with affine Hecke algebras and Hilbert schemes, which were used to prove several conjectures made by Macdonald about them.
In mathematics, an LLT polynomial is one of a family of symmetric functions introduced as q-analogues of products of Schur functions.
In mathematics, the Littlewood–Richardson rule is a combinatorial description of the coefficients that arise when decomposing a product of two Schur functions as a linear combination of other Schur functions. These coefficients are natural numbers, which the Littlewood–Richardson rule describes as counting certain skew tableaux. They occur in many other mathematical contexts, for instance as multiplicity in the decomposition of tensor products of finite-dimensional representations of general linear groups, or in the decomposition of certain induced representations in the representation theory of the symmetric group, or in the area of algebraic combinatorics dealing with Young tableaux and symmetric polynomials.
Adriano Mario Garsia was a Tunisian-born Italian American mathematician who worked in analysis, combinatorics, representation theory, and algebraic geometry. He was a student of Charles Loewner and published work on representation theory, symmetric functions, and algebraic combinatorics. He and Mark Haiman made the n! conjecture. He is also the namesake of the Garsia–Wachs algorithm for optimal binary search trees, which he published with his student Michelle L. Wachs in 1977.
In mathematics, the n! conjecture is the conjecture that the dimension of a certain bi-graded module of diagonal harmonics is n!. It was made by A. M. Garsia and M. Haiman and later proved by M. Haiman. It implies Macdonald's positivity conjecture about the Macdonald polynomials.
In mathematics, Kostka polynomials, named after the mathematician Carl Kostka, are families of polynomials that generalize the Kostka numbers. They are studied primarily in algebraic combinatorics and representation theory.
Norman Linstead Biggs is a leading British mathematician focusing on discrete mathematics and in particular algebraic combinatorics.
In algebra and in particular in algebraic combinatorics, a quasisymmetric function is any element in the ring of quasisymmetric functions which is in turn a subring of the formal power series ring with a countable number of variables. This ring generalizes the ring of symmetric functions. This ring can be realized as a specific limit of the rings of quasisymmetric polynomials in n variables, as n goes to infinity. This ring serves as universal structure in which relations between quasisymmetric polynomials can be expressed in a way independent of the number n of variables.
In mathematics, Kronecker coefficientsgλμν describe the decomposition of the tensor product of two irreducible representations of a symmetric group into irreducible representations. They play an important role algebraic combinatorics and geometric complexity theory. They were introduced by Murnaghan in 1938.
In mathematics, a double affine Hecke algebra, or Cherednik algebra, is an algebra containing the Hecke algebra of an affine Weyl group, given as the quotient of the group ring of a double affine braid group. They were introduced by Cherednik, who used them to prove Macdonald's constant term conjecture for Macdonald polynomials. Infinitesimal Cherednik algebras have significant implications in representation theory, and therefore have important applications in particle physics and chemistry.
Sara Cosette Billey is an American mathematician working in algebraic combinatorics. She is known for her contributions on Schubert polynomials, singular loci of Schubert varieties, Kostant polynomials, and Kazhdan–Lusztig polynomials often using computer verified proofs. She is currently a professor of mathematics at the University of Washington.
Lauren Kiyomi Williams is an American mathematician known for her work on cluster algebras, tropical geometry, algebraic combinatorics, amplituhedra, and the positive Grassmannian. She is Dwight Parker Robinson Professor of Mathematics at Harvard University.
Stephanie van Willigenburg is a professor of mathematics at the University of British Columbia whose research is in the field of algebraic combinatorics and concerns quasisymmetric functions. Together with James Haglund, Kurt Luoto and Sarah Mason, she introduced the quasisymmetric Schur functions, which form a basis for quasisymmetric functions.
Sylvie Corteel is a French mathematician at the Centre national de la recherche scientifique and Paris Diderot University and the University of California, Berkeley, who was an editor-in-chief of the Journal of Combinatorial Theory, Series A. Her research concerns the enumerative combinatorics and algebraic combinatorics of permutations, Young tableaux, and integer partitions.
Jennifer Leigh Morse is a mathematician specializing in algebraic combinatorics. She is a professor of mathematics at the University of Virginia.
Julian Sahasrabudhe is a Canadian mathematician who is an assistant professor of mathematics at the University of Cambridge, in their Department of Pure Mathematics and Mathematical Statistics. His research interests are in extremal and probabilistic combinatorics, Ramsey theory, random polynomials and matrices, and combinatorial number theory.
Chunwei Song is a Chinese mathematician who specializes in combinatorics, graph theory and intellectual history. He has been working as a professor of mathematics at the Peking University. Prior to Peking University, Song held visiting faculty positions at Boston College, Massachusetts and the Tokyo Institute of Technology, Japan. In 2010, he was a visiting associate professor at the University of Delaware.
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