Over the past several decades, a collection of conceptual mathematical approaches have been developed with considerable public attention due to the expectation at the time that the new approach would revolutionize research. These approaches involved complex mathematics that offered new insights for describing and analyzing patterns in systems at multiple scales of space, time and organization. These very general mathematical approaches were applied to hosts of physical, biological and social science questions, sometimes providing novel results. As with many innovations, the hopes for each as a general approach to deal with “everything” have not been met, yet each has enhanced the toolbox of researchers. There are connections between these, as well as to other approaches such as cellular automata, agent-based methods and network science.

The objective of this 1-credit seminar is to outline the key ideas of each of these efforts to create a mathematical theory of everything and discuss the connections between them. These approaches were often developed with biological examples in mind, and our discussions will focus on life science applications, though each of these have been applied in numerous contexts. Participants will not be expected to develop detailed understanding of all of these approaches but will be expected to choose a particular one, read with care several detailed articles or books associated with the approach that could connect to their own research interests, inform the instructor regularly about what they are reading, and be prepared as the semester progresses to comment in class about what they have learned about their chosen topic. The instructor will provide an extensive list of papers and other reference material for each of the approaches, provide a conceptual overview of each of the approaches with a bit of mathematical detail, and guide discussions in collaboration with course participants.

In addition to attending class each week, participants are expected to share their understanding of the topic they have chosen. At the end of the semester, each participant is expected to produce a short video (2 minutes) that describes, for a literate audience who are not trained in mathematics, the key concepts in the approach they have chosen to read about, how this approach has been applied and what has been learned about some area of biology from its development.

This seminar will meet once a week on Mondays from 10:10-11:00 in the NIMBioS suite of rooms (primarily Claxton 105). Attendees should sign up for one credit hour (unless prior arrangement is made with the Dr. Gross).

Possible Video Project: The National Academies of Sciences and Engineering have created a project called Elevating Mathematics that "invites early career professionals and students who use mathematics in their work to submit short video elevator speeches describing how their work in mathematics is important and relevant to our everyday lives." You may wish to submit to this as part of the expectation for registered class participants to create a video on a topic relevant to the course.

Regarding suggestions on how to prepare appropriate short videos, general links about communicating science are on the NIMBioS Resource Library page , some of which contain info about talking on camera. Another link with multiple resources is SciFund Challenge Solutions for Creating Impact with Science Videos .

Some Game Theory papers to consider:

Karl Sigmund and Christian Hilbe. 2012. "Game Theory" Encyclopedia of Theoretical Ecology (online library link)

For a recent pop-culture connection to game theory and anonymous giving see
Larry David and the Game Theory of Anonymous Donations which notes the connection to a recent paper "The signal-burying game can explain why we obscure positive traits and good deeds" by Moshe Hoffman, Christian Hilbe and Martin Nowak as well as the connection to a heavily-cited paper "Five Rules for the Evolution of Cooperation" by Martin Nowak

The papers mentioned on September 16 regarding the use of cooperative game approaches in evolution were Joan Roughgarden, Meeko Oishi and Erol Akcay: Reproductive Social Behavior: Cooperative Games to Replace Sexual Selection published in Science 311:965-969 (2006) which generated a huge number of responses Letters in response to Roughgarden et al., Debating Sexual Selection and mating Strategies published in Science 312:689-697 (2006) . The other paper mentioned regarding cooperative game theory was

Some introductory papers about bifurcations and stability:

Michio Kondoh. 2012. "Resilience and Stability" Encyclopedia of Theoretical Ecology (online library link)

Fabio Dercole and Sergio Rinaldi. 2012. "Bifurcations" Encyclopedia of Theoretical Ecology (online library link)

Some review papers about catastrophe theory and ecology:

Catastrophe Theory and Ecology. Chapter 9 in: Integration of Ecosystem Theories: A Pattern. Springer Netherlands (online library link)

Loehle, C. 1989, Catastrophe theory in ecology: a critical review and an example of the butterfly catastrophe. Ecological Modelling 49:125-152 (online library link)

Some papers about chaos and ecology:

Constantino, Robert and Robert Desharnais. 2012. Chaos. Encyclopedia of Theoretical Ecology (online library link)

Beckage, B., L. J. Gross and S. Kauffman. 2011. The limits to prediction in ecological systems. Ecosphere 2:1-12 https://doi.org/10.1890/ES11-00211.1 (online library link)

Bjornstad, O. N. 2015. Nonlinearity and chaos in ecological dynamics revisited. PNAS 112:6252-6253 https://doi.org/10.1073/pnas.1507708112 (online library link)

Sugihara1, G., R. May, H. Ye1, C-h. Hsieh, E. Deyle, M. Fogarty, Stephan Munch. Detecting Causality in Complex Ecosystems DOI: 10.1126/science.1227079 (online library link)

Slides from presentations:

Slides from Presentation on September 4, 2019 - Introduction to Seminar and Intro to game Theory (.pptx file)

Slides from Presentation on September 9, 2019 - Introduction to Game Theory (.pptx file)

Slides from Presentation on September 16, 2019 - Introduction to Game Theory - expanded slides from previous week (.pptx file)

Slides from Presentation on September 23, 2019 - Introduction to Bifurcation Theory (.pptx file)

Slides from Presentation on October 14, 2019 - Introduction to Catastrophe Theory (.pptx file)

Slides from Presentation on October 28, 2019 - Introduction to Chaos Theory (.pptx file)

Slides from Presentation on November 11, 2019 - Introduction to Complexity Theory (.pptx file)

Slides from Presentation on November 18, 2019 - More on Complexity Theory (.pptx file)