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Mathematics

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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.

My current interests, supported by the Collaboration Grants for Mathematicians program through the Simons Foundation (2015 - 2023), 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.

Recent Publications

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

• Crans, A. and Whitney, G. eds. "The Mathematical Playgorund: People and Problems from 31 Years of Math Horizons." MAA | AMS Press Problems Books (2024).
• Crans, A. and Whitney, G. "Integral Tiling Pentagons." Mathematics Magazine. Vol. 96 (2023), No. 2. DOI 10.1080/0025570X.2023.2176101.
• Bonatto, M.; Crans, A.; Nasybullov, T.; and Whitney, G. "Quandles with Orbit Series Conditions." Journal of Algebra. Vol. 567 (2021): 284 – 309. DOI 10.1016/j.algebra.2020.09.026. Also available as arXiv:2004.10227.
• Horvat, E. and Crans, A. "From biquandle structures to Hom-biquandles." Journal of Knot Theory and its Ramifications. Vol. 29 (2020), No. 2. DOI 10.1142/S0218216520400088. Also available as arXiv:1907.10259.
• Anders, K.; Crans, A.; Foster-Greenwood, B.; Mellor, B.; and Tymoczko, J. "Graphs Admitting only Constant Splines." Pacific Journal of Mathematics Vol. 304 - 2 (2020): 385 -- 400. DOI 10.2140/pjm.2020.304. Also available as arXiv:1807.11515.
• Bonatto, M.; Crans, A.; and Whitney, G. "On the Structure of Hom Quandles." Journal of Pure and Applied Algebra Vol. 223 (2019), No. 11. Also available as arXiv:1808.01738.
• Crans, A; Hoste, J.; Mellor, B.; and Shanahan, P. "Finite n-quandles of torus and two-bridge links." Journal of Knot Theory and its Ramifications Vol. 28 (2019), No. 3. Also available as arXiv:1806.05727.

Recent Grants

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

• Crans, A. Collaboration Grants for Mathematicians, Simons Foundation, 2015 - 2023; $35,000

Recent Presentations

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

• "What, No Repeat?," Eaves Lecture, University of Kentucky, April 2023
• "Playing with Patterns," Sonia Kovalevsky Day, Cal Poly Pomona, April 2021
• "Cracking the Cubic: Cardano, Controversy, and Creasing," Mathematical Association of America LA/MS Spring Section Meeting, Anderson Lecture, Loyola University, February 2020
• "An Introduction to Quandles," "The Topology of Quandles," and "The Algebra of Quandles," Knots and Braids in Norway NCTU, Trondheim, May 2019
• "Why Pi? Humanity's unnatural obsession with computing, memorizing, reciting, and celebrating the value of pi to an excessive level of accuracy," An Evening of Unnecessary Detail, London, May 2019
• "Cracking the Cubic: Cardano, Controversy, and Creasing," Mathematical Association of America Indiana Spring Section Meeting, Invited Speaker, University of Indianapolis, April 2019
• "Frosting Fairness, Finally!," McDougal Lecture, Lawrence University, April 2019
• "Frosting Fairness, Finally!," Shenandoah Undergraduate Mathematics and Statistics Conference, Opening Address, James Madison University, October 2018
• "More than Meets the Pi," Pi Day Keynote Speaker, Los Angeles Valley College, March 2018
• "Quintessential Quandle Queries," MAA Invited Address, Joint Mathematics Meetings, San Diego, January 2018
• "Quandles and Links with Finite n-Quandles," Mile High Conference on Nonassociative Mathematics, Plenary Speaker, University of Denver, August 2017
• "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

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!