# Mathematics

__Research Interests__

__Research Interests__My research is in a field known as 'higher-dimensional algebra'. My dissertation, *Lie 2-Algebras,* was written 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, *Lie 2-Algebras,* 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 `categorified' Lie algebra, meaning that is consists of 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. I focused on *semistrict* Lie 2-algebras which are those where only the Jacobi identity fails to hold as an equation and I showed that just as any Lie algebra gives a solution of the Yang-Baxter equation, a semistrict 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.

My current interests, supported by the *Collaboration Grants for Mathematicians* program through the Simons Foundation (2015 - 2020), focus on self-distributive structures such as quandles and racks. A quandle is a set equipped with two binary operations satisfying axioms that capture the essential properties of group conjugation and algebraically encode the Reidemeister moves from classical knot theory. I am interested in the relationships between self-distributive structures and their (co)homology and crossed modules, group and Lie algebra (co)homology theories, and knot and knotted surface invariants.

__Selected Publications__

__Selected Publications__(for a complete list, please see my Curriculum Vitae)

- Bonatto, M.; Crans, A.; and Whitney, G. "On the Structure of Hom Quandles." Available as arXiv:1808.01738.
- Anders, K.; Crans, A.; Foster-Greenwood, B.; Mellor, B.; and Tymoczko, J. "Graphs Admitting only Constant Splines." Available as arXiv:1807.11515.
- Crans, A; Hoste, J.; Mellor, B.; and Shanahan, P. "Finite n-quandles of torus and two-bridge links." Available as arXiv:1806.05727.
- Crans, A; Mukherjee, S.; and Przytycki, J. "On Homology of Associative Shelves."
*Journal of Homotopy and Related Structures,*(2016) DOI:10.1007/s40062-016-0164-9. Also available as arXiv:1603.08590. - Crans, A.; Ganzell, S.; and Mellor, B. "The Forbidden Number of a Knot."
*Kyungpook Math Journal*. Vol. 55 (2015), No. 2: 485 - 506. Also available as arXiv:1305.5200. - Crans, A.; Rovetti, R.; and Vega, J. "Solving the
*KO Labyrinth*."*Mathematics Magazine*. Vol. 88 (2015), No. 1: 27 - 36. - Crans, A.; Wagemann, F.; "Crossed modules of racks."
*Homology, Homotopy and Applications*. Vol. 16 (2014), No. 2: 85 - 106. Also available as arXiv:1310.4705. - Crans, A.; Nelson, S.; "Hom Quandles."
*Journal of Knot Theory and its Ramifications*. Vol. 23 (2014), No. 2. Also available as arXiv:1310.0852. - Crans, A.; Przytycki, J.; and Putyra, K. "Torsion in one-term distributive homology."
*Fundamenta Mathematicae*. Vol. 225 (2014): 75 - 94. Also available as arXiv:1305.5200. - Crans, A.; and Rovetti, R. "Beyond Formulas: A Collaboration between Liberal Arts Honors Underclassmen and Senior Math Majors".
*Honors in Practice*Vol. 7 (2011), 115 - 126. - Crans, A.; Fiore, T.; and Satyendra, R. "Musical Actions of Dihedral Groups."
*The American Mathematical Monthly*, Vol. 116 (2009), No. 6: 479 - 495. Also available as arXiv:0711.1873. - Carter, J.S.; Crans, A.; Elhamdadi, M.; and Saito, M. "Cohomology of Categorical Self-Distributivity."
*Journal of Homotopy and Related Structures*Vol 3. (2008), No. 1: 13 - 63. Also available as arXiv:math/0607417. - 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 5: 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.

*Examples of final student projects resulting from this collaboration include recreations of photographs of Sacred Heart Chapel and St. Robert's Hall. Original music (Copyright 2011) for each of these was composed by Mike Hyziak and Eleni Maureas, EMMPH. *

*This paper was awarded the 2011 Merten M. Hasse Prize by the Mathematical Association of America as a "noteworthy expository paper appearing in an Association publication."*

__Selected Grants__

__Selected Grants__(for a complete list, please see my Curriculum Vitae)

- Crans, A.
*Collaboration Grants for Mathematicians,*Simons Foundation, 2015 - 2020; $35,000 - Crans, A.; and Killpatrick, K.
*Pacific Coast Undergraduate Mathematics Conference 2015,*MAA Undergraduate Conferences; $6,000 - Crans, A.; and Killpatrick, K.
*Pacific Coast Undergraduate Mathematics Conference 2014,*MAA Undergraduate Conferences; $3,000 - Crans, A.; Frey, S.; and Killpatrick, K.
*Pacific Coast Undergraduate Mathematics Conferences 2013,*NSF Mathematical Sciences Infrastructure Program; $19,011 - Cameron, N.; Crans, A.; and Killpatrick, K.
*Pacific Coast Undergraduate Mathematics Conferences 2012,*NSF Mathematical Sciences Infrastructure Program; $23,056 - Cameron, N.; Crans, A.; and Killpatrick, K.
*Pacific Coast Undergraduate Mathematics Conference 2011,*National Security Agency Conference/Workshop Grant; $10,000 - Crans, A.
*Categorified Quandles,*National Security Agency Young Investigator's Grant, Summers 2010 and 2011; $29,709 - Cameron, N.; Crans, A.; and Killpatrick, K.
*Pacific Coast Undergraduate Mathematics Conference 2010,*MAA undergraduate conferences NSF Grant DMS-0241090; $3,500

__Selected Presentations__

__Selected Presentations__(for a complete list, please see my Curriculum Vitae)

- "Quintessential Quandle Queries," MAA Invited Address,
*Joint Mathematics Meetings,*San Diego, January 2018 - "Frosting Fairness, Finally!," Invited Speaker,
*MAA DC-MD-VA Section Spring Meeting,*Frostburg University, April 2017 - "A Magical Number," Invited Speaker,
*MAA DC-MD-VA Section Spring Meeting,*AWM Workshop, Frostburg University, April 2017 - "A Surreptitious Sequence: the Catalan Numbers,"
*National Math Festival*, April 2017 - "Patterns + Women = Figures in Mathematics,"
*National Math Festival*, April 2017 - "A Surreptitious Sequence: the Catalan Numbers,"
*Doris G. Gordon Lecture Series,*Denison University, March 2017 - "Musical Actions of Dihedral Groups," Plenary Speaker,
*MAA Michigan Section Spring Meeting,*Hillsdale College, April 2016 - "A Surreptitious Sequence: the Catalan Numbers,"
*George Kitchen Memorial Lecture,*Kalamazoo College, March 2016 - "Unital Shelves,"
*AMS Special Session on Knots in Washington State,*Joint Mathematics Meetings, Seattle, January 2016 - "A Surreptitious Sequence: the Catalan Numbers,"
*National Math Festival,*April 2015 - "Patterns + Women = Figures in Mathematics,"
*National Math Festival,*April 2015 - "Cracking the Cubic: Cardano, Controversy, and Creasing," Keynote Speaker,
*Southeastern Conference for Undergraduate Women in Mathematics,*Clemson University, November 2014 - "A Surreptitious Sequence: the Catalan Numbers,"
*MAA Distinguished Lecture Series,*MAA Carriage House, May 2014 - "Crossed Modules of Racks,"
*AMS Special Session on Algebraic Structures Motivated by Knot Theory,*Joint Mathematics Meetings, Baltimore, January 2014 - "Patterns + Women = Figures in Mathematics,"
*Sonya Kovalevsky Mathematics Day,*Keynote Speaker, California State University at Fresno, November 2013 - "A Surreptitious Sequence: the Catalan Numbers,"
*Math Encounters,*Museum of Mathematics, September 2013 - "Musical Mathematics,"
*Bay Area Mathematical Adventures,*Santa Clara University, November 2012 - "Cracking the Cubic: Cardano, Controversy, and Creasing," Invited Speaker,
*MAA DC-MD-VA Spring Section Meeting,*Stevenson University, April 2012 - "Patterns + Women = Figures in Mathematics," Keynote Speaker,
*Expanding Your Horizons,*James Madison University, March 2012 - "A Surreptitious Sequence: the Catalan Numbers," Invited Speaker,
*MAA EPaDel Fall Section Meeting,*Bryn Mawr College, November 2011 - "Count Me In!," Alder Award Presentation,
*Mathfest 2011,*Lexington, Kentucky, August 2011 - "Million Dollar Mathematics," Keynote Speaker,
*Southern California Undergraduate Math Day,*University of California, San Diego, April 2011 - "Pascal's Patterns," Keynote Speaker,
*University of Oklahoma Math Day,*November 2010 - "Musical Actions of Dihedral Groups," Invited Speaker,
*MAA So Cal-Nev Spring Section Meeting,*Harvey Mudd College, April 2010

*A (much!) shorter version of this talk appears on YouTube.*

*The above talk was presented after being honored with the Henry L. Alder Award for "Distinguished Teaching by a Beginning College/University Mathematics Faculty Member" by the Mathematical Association of America.*

(Please e-mail me if you would like the musical tracks to accompany the above talk.)

__Fun Stuff__

__Fun Stuff__- Here's an essay about me from 2016 written for the AWM Essay Contest: Biographies of Contemporary Women in Mathematics.
- Here's an interview with me from 2013 by the Girls' Angle: Math Club for Girls.
- 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.
- Did you hear? I was
*Time's*2006 Person of the Year!