Randall Dougherty has made contributions in widely varying areas of mathematics, including set theory

Set theory

Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...

,
logic

Logic

In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...

, real analysis

Real analysis

Real analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real...

, discrete mathematics

Discrete mathematics

Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not...

, computational geometry

Computational geometry

Computational geometry is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems are also considered to be part of computational...

, information theory

Information theory

Information theory is a branch of applied mathematics and electrical engineering involving the quantification of information. Information theory was developed by Claude E. Shannon to find fundamental limits on signal processing operations such as compressing data and on reliably storing and...

 and coding theory

Coding theory

Coding theory is the study of the properties of codes and their fitness for a specific application. Codes are used for data compression, cryptography, error-correction and more recently also for network coding...

.

Dougherty is a three-time winner of the U.S.A. Mathematical Olympiad

International Mathematical Olympiad

The International Mathematical Olympiad is an annual six-problem, 42-point mathematical olympiad for pre-collegiate students and is the oldest of the International Science Olympiads. The first IMO was held in Romania in 1959. It has since been held annually, except in 1980...

, 1976, 1977, 1978
and a three time medalist in the International Mathematical Olympiad

International Mathematical Olympiad

The International Mathematical Olympiad is an annual six-problem, 42-point mathematical olympiad for pre-collegiate students and is the oldest of the International Science Olympiads. The first IMO was held in Romania in 1959. It has since been held annually, except in 1980...

. He is also a three-time Putnam Fellow

William Lowell Putnam Mathematical Competition

The William Lowell Putnam Mathematical Competition, often abbreviated to the Putnam Competition, is an annual mathematics competition for undergraduate college students of the United States and Canada, awarding scholarships and cash prizes ranging from $250 to $2,500 for the top students and $5,000...

 1978, 1979, 1980. Dougherty earned his Ph.D.

Ph.D.

A Ph.D. is a Doctor of Philosophy, an academic degree.Ph.D. may also refer to:* Ph.D. , a 1980s British group*Piled Higher and Deeper, a web comic strip*PhD: Phantasy Degree, a Korean comic series* PhD Docbook renderer, an XML renderer...

 in 1985 at University of California, Berkeley

University of California, Berkeley

The University of California, Berkeley , is a teaching and research university established in 1868 and located in Berkeley, California, USA...

 under the direction of Jack Silver

Jack Silver

Jack Howard Silver is a set theorist and logician at the University of California, Berkeley. He has made several deep contributions to set theory...

.

His notable accomplishments include the following.
With Matthew Foreman

Matthew Foreman

Matthew Dean Foreman is a set theorist at University of California, Irvine. He has made contributions in widely varying areas of set theory, including descriptive set theory, forcing, and infinitary combinatorics....

 he showed that the Banach-Tarski decomposition

Banach–Tarski paradox

The Banach–Tarski paradox is a theorem in set theoretic geometry which states the following: Given a solid ball in 3-dimensional space, there exists a decomposition of the ball into a finite number of non-overlapping pieces , which can then be put back together in a different way to yield two...

 is possible with pieces with the Baire property, solving a problem of Marczewski

Edward Marczewski

Edward Marczewski was a Polish mathematician. His surname until 1940 was Szpilrajn.Marczewski was a member of the Warsaw School of Mathematics...

 that remained unsolved for more than 60 years.
With Chris Freiling

Chris Freiling

Christopher F. Freiling is a set theorist responsible for Freiling's axiom of symmetry. He is currently a member of the faculty of the Department of Mathematics at California State University, San Bernardino.-External links:*...

 and Ken Zeger, he showed that linear codes are insufficient to gain the full advantages of network coding

Network coding

Network coding is a technique where, instead of simply relaying the packets of information they receive, the nodes of a network will take several packets and combine them together for transmission. This can be used to attain the maximum possible information flow in a network...

.