image

Research

My background is in application-oriented mathematics, and I am especially interested in applications to medicine, biology, and biochemistry. My current projects involve modeling (1) pumping performance valveless tubular hearts (supported by NSF awards DMS-2152110 and DMS-2152040); (2) emergence of novel genes; (3) intracellular phototroph:heterotroph symbioses; and (4) mitigation of mosquito-borne illnesses using Wolbachia. For a list of some of my research articles, either refer to my curriculum vitae or see below.

Research with undergraduates

I love to mentor qualified undergraduates, usually (but not always!) on research projects involving applications of mathematics. If you are a Harvard undergraduate interested in mathematics research and/or exploring senior thesis options, please e-mail me at:

Selected publications and presentations

62. J.W. Cain and Y. He, Exploring Mechanisms for Reversal of Flow in Tunicate Hearts, submitted, 2026.

61. I. Chavarri, Y. He, M. Chen and J.W. Cain, Mixed Types of Uncertainty Quantification Using Fuzzy Set and Dempster-Shafer Theories, submitted, 2025.

60. J.W. Cain, I.M. Raymond, and N.C. Källersjö, Towards Resolving Cubic Tic-Tac-Toe: A Matching-Based Approach , arXiv, 2025.

59. D. Dye and J.W. Cain, Efficacy of Wolbachia-Based Mosquito Control: Predictions of a Spatially-Discrete Mathematical Model, PLoS One, 19.3 (2024), e0297964.

58. N. Ranc, J.W. Cain, F. Cagnacci, and P.R. Moorcroft, The Role of Memory-Based Movements in the Formation of Animal Home Ranges, Journal of Mathematical Biology, Vol. 88 (2024), 59.

57. M. Hill, B. Lawson, J.W. Cain et al., Sustained Beneficial Infections: Priority Effects, Competition and Specialization Drive Patterns of Association in Phototroph:Heterotroph Mutualisms, Frontiers in Ecology and Evolution, Vol. 11 (2023), 1221012.

56. F. Li, R. Rane, V. Luria et al., Phylogenomic Analyses of the Genus Drosophila Reveals Genomic Signals of Climate Adaptation, Molecular Ecology Resources, Vol. 22.4 (2022), pp. 1559--1581.

55. S. Menon and J. W. Cain, Approaches Toward Understanding Delay-Induced Stability and Instability, SIAM Review, Vol. 63 (2021), pp. 625--637.

54. J. A. Weber, S. G. Park, V. Luria et al., The Whale Shark Genome Reveals How Genomic and Physiological Properties Scale with Body Size, Proceedings of the National Academy of Sciences of the USA, August 2020, DOI: 10.1073/pnas.1922576117

53. A. Vonica, N. Bhat, K. Phan et al., Apcdd1 is a Dual BMP/Wnt Inhibitor in the Developing Nervous System and Skin, Developmental Biology, Vol. 464 (2020), pp. 71--87.

52. J.W. Cain, Luran He, and Lindsay Waldrop, Modeling Action Potential Reversals in Tunicate Hearts, Physical Review E, Vol. 102 (2020), pp. 062421.

51. J.W. Cain, Review of "Biological Clocks, Rhythms, and Oscillations, by D. Forger", SIAM Review, Vol. 60 (2018), pp. 1013--1016.

50. M. Lindsay and J.W. Cain, Improved Lower Bounds on the Classical Ramsey Numbers R(4,22) and R(4,25). http://arxiv.org/abs/1510.06102

49. D.G. Schaeffer and J.W. Cain, Ordinary Differential Equations: Basics and Beyond. Springer, 2016.

48. J.W. Cain, Chemical Reaction Kinetics: Mathematical Underpinnings, In: Molecular Life Sciences: An Encyclopedic Reference, Springer, 2014.

47. J.W. Cain, Equilibria and Bifurcations in the Molecular Biosciences, In: Molecular Life Sciences: An Encyclopedic Reference, Springer, 2014.

46. J.W. Cain, Mathematical Models in the Sciences, In: Molecular Life Sciences: An Encyclopedic Reference, Springer, 2014.

45. J.W. Cain, Mathematics of Fitting Scientific Data, In: Molecular Life Sciences: An Encyclopedic Reference, Springer, 2014.

44. K.G. Workman, S. Zhao, and J.W. Cain, Restricted Feedback Control in Discrete-Time Dynamical Systems with Memory, Physical Review E, Vol. 89, 2014, 042903.

43. J.W. Cain, Stability of Impulsively-Forced Excitable Fibers to Perturbations of the Forcing Period, SIAM Journal on Applied Mathematics, Vol. 74, 2014, pp. 191--207.

42. J.W. Cain, Chaos Control in Cardiac Tissue: Some Mathematics Behind Terminating Cardiac Arrhythmias, Springer Proceedings in Mathematics and Statistics (B. Toni, ed.), Vol. 37, 2014.

41. J.W. Cain, Using Local Control to Stabilize Global Behavior in Excitable Media, Proceedings of the Fifth International Conference on Applied Mathematics and Informatics, Cambridge, MA, January 2014.

40. J.W. Cain, To Make Free or Not to Make Free?---That is the Question, Joint Mathematics Meetings, special session on open source publishing, Baltimore, MD, January 2014.

39. J.W. Cain, Restricted Feedback Control and Discrete-Time Dynamical Systems, Joint Mathematics Meetings, special session on difference equations and applications, Baltimore, MD, January 2014.

38. J.W. Cain, Chaos Control in Cardiac Tissue: Local Stimulation versus Far-Field Pacing, Joint Mathematics Meetings, San Diego, CA, January 9--12, 2013.

37. J.W. Cain, Controlling Cardiac Alternans via Point Stimulation versus Far-Field Pacing, Proceedings of the 13th International conference on Mathematics and Computers in Biology and Chemistry (MCBC 2012), Iasi, Romania, June 2012.

36. J.W. Cain, Controlling Nonlinear Systems with Memory: An Example from Cardiac Dynamics, Dynamics Days 2012, Baltimore, MD, January 4--7, 2012.

35. J.W. Cain, CardioMathematics, Notes of the Canadian Mathematical Society, Vol. 44, No. 1, Feb. 2012.

34. "Keeping the Beat", a podcast interview with the American Mathematical Society, edited by Mike Breen, 2011. See also the Math Moments poster

33. J.W. Cain, Mathematics, Cardiac Phenomena, and the Diagnosis of Pathology, Future Cardiology, Vol. 7, No. 5, Sept. 2011, pp. 587--590.

32. Q&A: Taking Mathematics to Heart , an interview with Science Careers (from the journal Science ), edited by Kate Travis, April 29, 2011.

31. J.W. Cain and A.M. Reynolds, Ordinary and Partial Differential Equations: An Introduction to Dynamical Systems , ISBN#: 978-0-9824062-3-6, published August 2010, 418 pages, hardback.

30. J.W. Cain, Feedback Control of Cardiac Alternans, SIAM Annual Meeting/SIAM Conference on the Life Sciences, Pittsburgh, PA, July 12, 2010.

29. J.W. Cain, Taking Math to Heart: Mathematical Challenges in Cardiac Electrophysiology, Notices of the American Mathematical Society, Vol. 58., No. 4 (April 2011), pp. 542--549.

28. N.B. Menke, J.W. Cain, A.M. Reynolds, D.M. Chan, R.A. Segal, T.M. Witten, D.G. Bonchev, R.F. Diegelmann, and K.R. Ward, An In Silico Approach to the Analysis of Acute Wound Healing , Wound Repair and Regeneration, Vol. 18 (2010), pp. 105--113.

27. D.T. Taylor, J.W. Cain, D.G. Bonchev, S.S. Fong, A.A. Apte, and L.E. Pace, Toward a Classification of Isodynamic Feed-Forward Motifs , Journal of Biological Dynamics, Vol. 4 (2010), pp. 196--211.

26. J.W. Cain, Local Asymptotic Stability and Difference Equations of Arbitrary Finite Order, Joint Mathematics Meetings, Washington, DC, January 5--8, 2009.

25. J.W. Cain, A Kinematic Model of Wave Propagation , International Journal of Pure and Applied Mathematics, Vol. 49 (2008), pp. 599--605.

24. J.W. Cain, A Kinematic Model of Wave Propagation, invited address, International Conference of Applied Mathematics and Computing, Plovdiv, Bulgaria, August 12--18, 2008.

23. J.W. Cain, A Kinematic Model of Wave Propagation in an Excitable Medium, AMS Spring Eastern Meeting, New York, NY, March 15--16, 2008.

22. S.V. Narayanan and J.W. Cain, Mathematical Investigation of a Feedback Control Algorithm for Suppressing Electrical Alternans in Cardiac Tissue, TIMBER 2007: Undergraduate Research Conference in Quantitative Biology, November 2-3, 2007, Appalachian State University, Boone, NC.

21. A.A. Apte, J.W. Cain, D.G. Bonchev and S.S. Fong, Cellular Automata Simulation of Topological Effects on the Dynamics of Feed-Forward Motifs , Journal of Biological Engineering, 2:2 (2008)

20. H. Sedaghat, M.A. Wood, J.W. Cain, C.K. Cheng, C.M. Baumgarten and D.M. Chan, Complex Temporal Patterns of Spontaneous Initiation and Termination of Reentry in a Loop of Cardiac Tissue , Journal of Theoretical Biology, Vol. 254 (2008), pp. 14--26.

19. J.W. Cain and D.G. Schaeffer, Shortening of Cardiac Action Potential Duration Near an Insulating Boundary , Mathematical Medicine and Biology, Vol. 25 (2008), pp. 21--36.

18. J.W. Cain, Analysis of Transient Alternans in a Paced Cardiac Fiber, Applications of Analysis to Mathematical Biology, Durham, NC, May 21--23, 2007.

17. C.M. Berger, J.W. Cain, J.E.S. Socolar, and D.J. Gauthier, Control of Electrical Alternans in Paced Myocardium Using Extended Time-Delay Autosynchronization , Physical Review E, Vol. 76 (2007), pp. 041917. (Selected to appear in the Virtual Journal of Biological Physics Research, Volume 14, Issue 9, November 1, 2007).

16. S.V. Narayanan and J.W. Cain, Analysis of a Modified Feedback Control Technique for Suppressing Electrical Alternans in Cardiac Tissue , MEJAM undergraduate research journal, Vol. 5 (2007), pp. 1--17.

15. J.W. Cain, Criterion for Stable Reentry in a Ring of Cardiac Tissue , Journal of Mathematical Biology, Vol. 55 (2007), pp. 433--448.

14. D.G. Schaeffer, J.W. Cain, D.J. Gauthier, S.S. Kalb, W. Krassowska, R.A. Oliver, E.G. Tolkacheva and W. Ying, An ionically based mapping model with memory for cardiac restitution , Bulletin of Mathematical Biology, Vol. 69 (2007), pp. 459--482.

13. H. Sedaghat, C.M. Baumgarten, J.W. Cain, D.M. Chan, C. Cheng, C.M. Kent and M.A. Wood, Modeling Spontaneous Initiation and Termination of Reentry in Cardiac Tissue, Dynamics Days 2007: International Conference on Chaos and Nonlinear Dynamics, Boston, MA, January 2007.

12. J.W. Cain, C.M. Berger and D.J. Gauthier, Controlling Electrical Alternans in Paced Myocardium Using Extended Time-Delay Autosynchronization, Dynamics Days 2007: International Conference on Chaos and Nonlinear Dynamics, Boston, MA, January 2007.

11. J.W. Cain and D.G. Schaeffer, Shortening of Cardiac Action Potential Duration Near a Non-Conducting Boundary, Dynamics Days 2007: International Conference on Chaos and Nonlinear Dynamics, Boston, MA, January 2007.

10. J.W. Cain. Review of R.C. Robinson's, "An Introduction to Dynamical Systems: Continuous and Discrete." J. Difference Equations and Applications, Vol. 12 (2006), No. 9, pp. 957--958.

9. J.W. Cain, C.M. Berger and D.J. Gauthier, Adaptive Control of Abnormal Rhythms in a Patch of Cardiac Tissue, Joint SIAM/SMB Conference on the Life Sciences, Raleigh, NC, August 2, 2006.

8. J.W. Cain, Criterion for Stable Reentrant Excitation in a Ring of Cardiac Tissue, Dynamics Days 2006: International Conference on Chaos and Nonlinear Dynamics, Bethesda, MD, January 2006.

7. J.W. Cain, Dynamic behavior of a paced cardiac fiber , SIAM J. Applied Mathematics, Vol. 66 (2006), No. 5, pp. 1776--1792.

6. J.W. Cain and D.G. Schaeffer, Two-term asymptotic approximation of a cardiac restitution curve , SIAM Review, Vol. 48 (2006), No. 3, pp. 537--546.

5. J.W. Cain, Issues in the one-dimensional dynamics of a paced cardiac fiber, Ph.D. Dissertation, Duke University (2005) .

4. D.G. Schaeffer, J.W. Cain, D.J. Gauthier, S.S. Kalb, W. Krassowska, R.A. Oliver, E.G. Tolkacheva and W. Ying, An ionically based mapping model with memory for cardiac restitution, Gordon Research Conference on Cardiac Arrhythmia Mechanisms, Buellton, CA, February 2005.

3. J.W. Cain, E.G. Tolkacheva, D.G. Schaeffer, and D.J. Gauthier, Rate-dependent propagation of cardiac action potentials in a one-dimensional fiber , Phys. Rev. E., vol. 70 no. 6 (2004), pp. 061906 (Selected to appear in the Virtual Journal of Biological Physics Research, Volume 8, Issue 12, December 15, 2004) .

2. J.W. Cain, E.G. Tolkacheva, D.G. Schaeffer, and D.J. Gauthier, Rate-dependent waveback velocity of cardiac action potentials in a one-dimensional cable, International Conference for Mathematics in Biology and Medicine: Annual Meeting for the Society for Mathematical Biology, July, 2004.

1. J.W. Cain, Simulating discordant alternans with a two-current model, International School on Biomathematics, Bioengineering and Clinical Aspects of Blood Flow, MSRI, August, 2002.