My research is in a field known as 'higher-dimensional algebra.' I wrote my dissertation Lie 2-Algebras under the guidance of John Baez.  My work blends Lie theory with elements of category theory and has connections to braid theory and Lie algebra cohomology.  I am also interested in the relationship between Lie algebras and algebraic structures known as quandles.  Thus, my interests lie in quantum algebra and quantum and geometric topology.  

In my dissertation, I defined and explored generalized Lie algebras, called 'Lie 2-algebras,' and classified them up to equivalence in terms of Lie algebra cohomology.  A Lie 2-algebra is a category equipped with algebraic structure much like that of a Lie algebra, but where the laws involving the bracket only hold up to isomorphism.  In my dissertation, I focused on semistrict Lie 2-algebras which are those where only the Jacobi identity fails to hold as an equation.   I showed that just as any Lie algebra gives a solution of the Yang-Baxter equation, a semistrct Lie 2-algebra gives a solution of the Zamolodchikov tetrahedron equation, which is the higher dimensional analog of the Yang-Baxter equation.  Furthermore, I explored the relationship between groups, Lie algebras, quandles, and braids, and described a novel means of passing from a Lie group to its Lie algebra. 

To learn more about my work, please read (pdf versions of) my:

• Research Summary

• Dissertation Abstract

And if you're feeling ambitious:  

• My Dissertation:  Lie 2-Algebras

• Carter, J.S.; Crans, A.; Elhamdadi, M.; and Saito, M. "Cocycle Deformations of Algebraic Identities and R-matrices."  Available as arXiv:0802.2294. 

• Carter, J.S.; Crans, A.; Elhamdadi, M.; Karadayi, E.; and Saito, M. "Cohomology of Frobenius Algebras and the Yang-Baxter Equation."  Available as arXiv:0801.2567. 

• Crans, A.; Fiore, T.; and Satyendra, R. "Musical Actions of Dihedral Groups." Accepted by The American Mathematical Monthly in February 2008. Available as arXiv:0711.1873. 

• Carter, J.S.; Crans, A.; Elhamdadi, M.; and Saito, M. "Cohomology of Categorical Self-Distributivity." Provisionally accepted by Journal of Homotopy and Related Structures in August 2007.  Available as arXiv:math/0607417. 

• Carter, J.S.; Crans, A.; Elhamdadi, M.; and Saito, M. "Cohomology of the adjoint of Hopf algebras." Journal of Generalized Lie Theory and Applications Vol. 2 (2008), No. 1: 19 - 34.  Also available as arXiv:0705.3231.

• Baez, J; Crans, A.; and Wise, D. "Exotic Statistics for Loops in 4d BF Theory." Advances in Theoretical and Mathematical Physics Vol. 11 (2007), No 11: 707 - 749. Also available as arXiv:gr-qc/0603085.

• Baez, J.; Crans, A.; Schrieber, U.; and Stevenson, D. "Loop Groups to 2-Groups."  Homology, Homotopy, and its Applications Vol. 9 (2007), No. 2: 101- 135. Also available as arXiv:math/0504123.  

• Baez, J.; and Crans, A. "Higher Dimensional Algebra VI:  Lie 2-Algebras."  Theory and Applications of Categories.  12(2004), 492-538.  Also available as arXiv:math/0307263. 

• Crans, A.; Fallat, S.; and Johnson, C. "The Hadamard Core of the Totally Nonnegative Matrices."  Linear Algebra and its Applications. 328 (2001), 203—222.

• Crans, A.; and Weinhold, R. `"What to Do on Your Summer Vacation." Math Horizons. February 2001, 23-26

• Cameron, N.; Crans, A.; and Killpatrick, K. Pacific Coast Undergraduate Mathematics Conference 2008 National Security Agency Conference/Workshop Grant $14,650; MAA Undergraduate Conferences NSF Grant DMS-0241090 $3,000; Raytheon $2,000

• Crans, A.; DeCoste, R.; and Stewart, S.A. Career Mentoring Workshop for Women MAA TENSOR Grant $6,000

• Cameron, N.; Crans, A.; and Killpatrick, K. Pacific Coast Undergraduate Mathematics Conference 2007 MAA Undergraduate Conferences NSF Grant DMS-0241090 $3,000

• Cameron, N.; Crans, A.; and Killpatrick, K. Pacific Coast Undergraduate Mathematics Conference 2006 MAA Undergraduate Conferences NSF Grant DMS-0241090 $2,500

• Crans, A. AWM Travel Grant to attend Categories in Algebra, Geometry, and Mathematical Physics Conference at Macquarie University, July 2005, $1,257

• "Musical Actions of Dihedral Groups," Knotting Mathematics and Art: Conference in Low Dimensional Topology and Mathematical Art, University of South Florida, November 2007   

• "R,C,H,O," Summer Program for Women in Mathematics, The George Washington University, July 2007   

• "Meet the 3-sphere," with Liz Stanhope, Summer Mathematics Program for Women Undergraduates, Carleton College, June 2007   

• "A Survey of Higher Lie Theory," Higher Categories and Their Applications, Fields Institute, Toronto, Canada, January 2007   

• "Categorical Self-Distributivity," AWM Workshop, Joint Mathematics Meetings, New Orleans, January 2007  

• "Quandle Queries," Knots in Washington XXIII, George Washington University, Washington, DC, November 2006 

• "Lie 2-Groups, Lie 2-Algebras, and Loop Groups," Midwest Topology Seminar/Category Theory and its Applications Conference in memory of Saunders Mac Lane, University of Chicago, April 2006

"2-Groups:  An introduction to higher-dimensional groups," University of South Florida, January 2006 -->

• "Higher-Dimensional Algebra:  Weakening the Notion of Equality," Invitation to Research Seminar, The Ohio State University, January 2006

• "Loop Groups and Lie 2-Algebras,"   Categories in Algebra, Geometry and Mathematical Physics Conference, Macquarie University, July 2005
  
  This talk concluded a series of three talks on Higher Gauge Theory in which John Baez, Danny Stevenson, and I presented the results of our paper, "Loop Groups to 2-Groups."  Our abstracts and slides are available here.

• "Making Distinctions:  Interpreting the Notion of Sameness," SMPosium for the Summer Mathematics Program for Women Undergraduates, Carleton College, July 2005

• "2-Braids, 2-Groups, and Lie 2-Algebras," Quantum Topology--Contemporary Issues and Perspectives Conference, Snowbird, UT, June 2005

• "Matrix Groups:  Where Symmetry meets Geometry," Minicourse for EDGE (Enhancing Diversity in Graduate Education) Program for women, North Carolina A & T State, June 2005

• "Solutions of the Yang-Baxter and Zamolodchikov Tetrahedron Equations," Braid Groups and Applications Conference, Banff Research Center (BIRS), October 2004

• "Higher Linear Algebra," n-Categories:  Foundations and Applications Conference, Institute for Mathematics and its Applications, June 2004
  

• "Lie 2-Algebras," AWM Workshop Poster Presentation, Joint Mathematics Meetings, Phoenix, January 2004

• "Braids and the Yang-Baxter Equation:  An Application of Knot Theory to Physics," MAA Invited Paper Session on Applications of Topology to Biology, Chemistry, and Physics, Joint Mathematics Meetings, Phoenix, January 2004

• My other mathematical interests include algebraic coding theory, knot theory, the history of mathematics, and the relationship between music and mathematics.  Here's an excellent site for math history.   

• My advisor's advisor's advisor's advisor's advisor's advisor's advisor's advisor's advisor was Gauss!  Of course you can trace my mathematical lineage using The Mathematics Genealogy Project, but it's more fun to read the whole story as told by my advisor, John Baez, in Week 166 of his column This Week's Finds in Mathematical Physics.

• My Erdos number is 3, which I determined using the Erdos Number Project.   My Bacon number is 2.  Thus, my Erdos-Bacon number is 5.

• Did you see that my advisor is running for president in 2024?  Not as exciting, but still fun:  he was honored with the title of Geek O' the Week in April 2005.  

• And...who doesn't love a fun math comic?
• Did you hear? I was Time's 2006 Person of the Year!