Gene Abrams

Gene Abrams
NationalityAmerican
EducationUniversity of Oregon, Ph.D. (1981)
Scientific career
FieldsMathematics, Algebra, Noncommutative Ring Theory, Morita Equivalence, Leavitt Path Algebras
InstitutionsUniversity of Colorado Colorado Springs
Thesis Rings with Local Units  (1981)
Doctoral advisorFrank Anderson
Websitehttps://academics.uccs.edu/gabrams

Gene Abrams is an American mathematician and Professor of Mathematics at University of Colorado Colorado Springs. He works in the area of Algebra, and he earned his Ph.D. in mathematics at the University of Oregon in 1981. Abrams' research interests are in noncommutative rings and their categories of modules, and he is known for his contributions to Morita equivalence, particularly Morita equivalence for nonunital rings.[1][2]

Leavitt path algebras

Abrams is credited as one of the founders of the subject of Leavitt path algebras.[3] Leavitt path algebras were simultaneously introduced in 2005 by Abrams and Gonzalo Aranda Pino[4] as well as by Ara, Moreno, and Pardo,[5] with neither of the two groups aware of the other's work.[6] Abrams has stated that his inspiration for Leavitt path algebras came after attending a CBMS Conference hosted by Paul Muhly, David Pask, and Mark Tomforde at the University of Iowa in 2004.[7] The topic of this CBMS conference was graph C*-algebras,[8] a particular class of C*-algebras studied in functional analysis, and the talks at the conference gave Abrams the idea to introduce Leavitt path algebras as algebraic analogues of the graph C*-algebras. The Leavitt path algebras are so-named because they are constructed from the path algebra of a graph and they also generalize Leavitt algebras.

Leavitt path algebras have been investigated by dozens of mathematicians since their introduction, and Abrams has been instrumental in the development of the theory.[9] The study of Leavitt path algebras has also promoted interactions between Analysis and Algebra, and there have been multiple conference on Leavitt path algebras aimed at bringing algebraists and analysts together to collaborate and share ideas.[10][11][12] Abrams is also one of the coauthors, with Pere Ara and Mercedes Siles Molina, of the book Leavitt path algebras,[13] published by Springer. In 2020 Leavitt path algebras were added to the Mathematics Subject Classification with code 16S88 under the general discipline of Associative Rings and Algebras.[14]

Outreach and popularization of mathematics

Abrams has been active in mathematics outreach and worked to popularize mathematical topics for general audiences. He is a member and organizer of the Colorado Math Circle.[15] He has spoken[16] at the Colorado Café Scientifique,[17] an organization based on the French Café Philosophique, where members of the general public receive an introduction to an interesting current scientific topic from an expert. Abrams also wrote an article for the MAA FOCUS describing his experiences at the Colorado Café Scientifique and encouraging more mathematicians to become involved in this kind of public outreach.[18]

Abrams was awarded the Carl B. Allendoerfer Award of the Mathematical Association of America in 2011 for his paper with Jessica Sklar, The Graph Menagerie: Abstract Algebra and the Mad Veterinarian.[19] The paper describes and provides a general solution to a topic in recreational math known as the "Mad veterinarian puzzles". One example of a Mad Veterinarian Puzzle is the following:

"Suppose a mad veterinarian creates a transmogrifier that can convert one cat into two dogs and five mice, or one dog into three cats and three mice, or a mouse into a cat and a dog. It can also do each of these operations in reverse. Can it, through any sequence of operations, convert two cats into a pack of dogs? How about one cat?"[20]

Honors and awards

  • Carl B. Allendoerfer Award, 2011
  • Rocky Mountain Section of the Mathematical Association of America Award for Distinguished University Teaching, 2002[21]
  • University of Colorado systemwide President's Teaching Scholar, 1996

References

  1. ^ Morita equivalence for rings with local units. Comm. Algebra 11 (1983), no. 8, 801–837.
  2. ^ Rings with Local Units. Thesis (Ph.D.), University of Oregon. 1981. 82 pp, ProQuest LLC
  3. ^ Sec. 1.7 of Leavitt path algebras. Lecture Notes in Mathematics, 2191. Springer, London, 2017. xiii+287 pp. ISBN 978-1-4471-7343-4; 978-1-4471-7344-1. Online Copy (PDF)
  4. ^ Abrams, Gene; Aranda Pino, Gonzalo; The Leavitt path algebra of a graph. J. Algebra 293 (2005), no. 2, 319--334.
  5. ^ Pere Ara, María A. Moreno, and Enrique Pardo. Nonstable K-theory for graph algebras. Algebr. Represent. Theory, 10(2):157–178, 2007.
  6. ^ Sec. 1.7 of Leavitt path algebras. Lecture Notes in Mathematics, 2191. Springer, London, 2017. xiii+287 pp. ISBN 978-1-4471-7343-4; 978-1-4471-7344-1. Online Copy (PDF)
  7. ^ Title slide and p.23 of "Leavitt path algebras: Entering Adulthood (d.o.b. May 31, 2004; Iowa City, Iowa)", slides from a talk given by Gene Abrams at the University of Iowa on March 30, 2018. Online Copy (PDF)
  8. ^ List of 2004 CBMS conferences
  9. ^ Leavitt path algebras: the first decade. Bull. Math. Sci. 5 (2015), no. 1, 59--120.
  10. ^ Workshop on Graph Algebras at Málaga, Spain in 2006 website
  11. ^ AMS Special Session on Graph Algebras in Analysis and Algebra held at the Joint Mathematics Meetings in 2010.
  12. ^ Graph algebras: Bridges between graph C*-algebras and Leavitt path algebras, a five-day workshop held at Banff International Research Station from April 21--April 26, 2013. Conference Website, Online Problem Page from the Workshop.
  13. ^ Leavitt path algebras: a primer and handbook. Lecture Notes in Mathematics, 2191. Springer, London, 2017. xiii+287 pp. ISBN 978-1-4471-7343-4; 978-1-4471-7344-1. Springer Book Page Online Copy (PDF)
  14. ^ 2020 Mathematics Subject Classification (PDF)
  15. ^ The Colorado Math Circle, members and organizers
  16. ^ Colorado Café Scientifique talk by Gene Abrams
  17. ^ Colorado Café Scientifique website
  18. ^ Café Scientifique Mathematics in the Microbrewery: Fermat Meets Fermentation, MAA FOCUS, Volume 27, Issue 6, August/September 2007 Online Copy (PDF)
  19. ^ The Graph Menagerie: Abstract Algebra and the Mad Veterinarian, Gene Abrams and Jessica K. Sklar, Math. Mag. 83 (2010), no. 3, 168--179.Online Copy (PDF)
  20. ^ 1 Plus 1 Makes Engaging Book: Mother and Daughter Bridge Generations and Disciplines by Dana Mackenzie (January 2013), Swarthmore College Bulletin. Online Copy
  21. ^ The Burton W. Jones Distinguished Teaching Award of the Rocky Mountain Section of the MAA, list of recipients