He held the John William and Helen Stubbs Potter Professorship in mathematics at Texas Christian University (TCU) from 1995 until his retirement in 2016.

[4] Doran is known for his research-level books, his award-winning teaching, and for his solution to a long-standing open problem due to Irving Kaplansky on a symmetric *-algebra.

He served as a Special Forces Instructor at the US Army Jungle Survival Center at Fort Sherman, Colón, Panama.

[11][12] Doran held the John William and Helen Stubbs Potter Professor of Mathematics at Texas Christian University from 1995 to 2016.

[20] The most recent conference was held at TCU, June 2011 with Phillip Griffiths, former Director of the Institute for Advanced Study as principal speaker.

Doran is known for his elegant solution[21] of a long-standing unsolved problem on a symmetric *-algebra left open by Irving Kaplansky in the Duke Mathematical Journal in 1949.

[22] Doran has published books, articles, and reviews on a variety of topics, often collaborating with other mathematicians, including J. M. G. Fell, Richard Kadison, Calvin Moore, Jonathan Rosenberg, Paul Sally, Robert Zimmer, and V. S. Varadarajan.