Francis G. Garvan (born March 9, 1955) is an Australian-born mathematician who specializes in

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777� ...

and

combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...

. He holds the position Professor of Mathematics at the

University of Florida The University of Florida (Florida or UF) is a public land-grant research university in Gainesville, Florida. It is a senior member of the State University System of Florida, traces its origins to 1853, and has operated continuously on its ...

. He received his Ph.D. from

Pennsylvania State University The Pennsylvania State University (Penn State or PSU) is a Public university, public Commonwealth System of Higher Education, state-related Land-grant university, land-grant research university with campuses and facilities throughout Pennsylvan ...

(January, 1986) with

George E. Andrews George Eyre Andrews (born December 4, 1938) is an American mathematician working in special functions, number theory, analysis and combinatorics. Education and career He is currently an Evan Pugh Professor of Mathematics at Pennsylvania State ...

as his thesis advisor. Garvan's thesis, ''Generalizations of Dyson's rank'', concerned the

rank of a partition In mathematics, particularly in the fields of number theory and combinatorics, the rank of a partition of a positive integer is a certain integer associated with the partition. In fact at least two different definitions of rank appear in the lite ...

and formed the groundwork for several of his later papers. Garvan is well-known for his work in the fields of ''q''-series and

integer partitions In number theory and combinatorics, a partition of a positive integer , also called an integer partition, is a way of writing as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same parti ...

. Most famously, in 1988, Garvan and Andrews discovered a definition of the

crank of a partition In number theory, the crank of a partition of an integer is a certain integer associated with the partition. The term was first introduced without a definition by Freeman Dyson in a 1944 paper published in Eureka, a journal published by the Mathem ...

. The crank of a partition is an elusive combinatorial statistic similar to the

which provides a key to the study of

Ramanujan congruences In mathematics, Ramanujan's congruences are some remarkable congruences for the partition function ''p''(''n''). The mathematician Srinivasa Ramanujan discovered the congruences : \begin p(5k+4) & \equiv 0 \pmod 5, \\ p(7k+5) & \equiv 0 \pmod 7 ...

in partition theory. It was first described by

Freeman Dyson Freeman John Dyson (15 December 1923 – 28 February 2020) was an English-American theoretical physicist and mathematician known for his works in quantum field theory, astrophysics, random matrices, mathematical formulation of quantum m ...

in a paper on ranks for the journal

Eureka Eureka (often abbreviated as E!, or Σ!) is an intergovernmental organisation for research and development funding and coordination. Eureka is an open platform for international cooperation in innovation. Organisations and companies applying th ...

in 1944. Andrews and Garvan's definition was the first definition of a crank to satisfy the properties hypothesized for it in Dyson's paper.

References

External links

*http://people.clas.ufl.edu/fgarvan

{{DEFAULTSORT:Garvan, Frank 1955 births Living people 20th-century American mathematicians 21st-century American mathematicians Australian mathematicians Number theorists University of Florida faculty