From March 10-23, 2004, we were in residence at the Rockefeller Foundation’s Bellagio Study and Conference Center located in Bellagio on Lake Como in northern Italy. The Bellagio Center opened in 1959 to allow scholars, scientists, artists, writers, as well as policy-makers and practitioners from around the globe to pursue their research and creative work. Its setting is idyllic with views of Lake Como and the Alps, coupled with magnificent gardens and parks. The beauty of the setting,  the support that we received, the rewarding interactions with other scholars outside our disciplines and the musicians, artists, and writers that were in residence during our stay, provided us with a truly unique and exceptional environment in which to pursue our research project.

The title of our team project was: Dynamics of Complex Networks in an Environment of Risk and Uncertainty: Theoretical Foundations and Applications to Global Supply Chains and International Financial Networks. We were only the 152nd research team in the Center’s history since research teams were instituted at the Center in the late 1990s and the first operations research team. Our proposal had been submitted over a year earlier in response to a letter of invitation that Anna Nagurney received following her Distinguished Chaired Fulbright in Innsbruck, Austria in Spring 2002.

The research team consisted of Patrizia Daniele, Monica Gabriela Cojocaru, and Anna Nagurney. Patrizia Daniele was born in Sicily and received her Bachelor’s degree at the University of Catania and her doctorate in Applied Mathematics and Computer Science from the University of Naples in 2000.  In 1996, she was awarded the Gioacchino Iapichino Award by President Oscar Luigi Scalfaro of Italy, after a national competition, for one of her papers on variational inequalities (cf. Daniele (1994)). Patrizia is an Associate Professor of Operational Research at the University of Catania.

Monica Cojocaru was born in Romania and received her Bachelor’s and Master’s degrees in Mathematics at the University of Bucharest. She received her doctorate in Mathematics from Queen’s University in Kingston, Canada in 2002 and was an NSERC postdoctoral fellow at the Centre des Recherches Mathematiques in Montreal. In 2003, Monica won the NSERC University Faculty Award competition for the Assistant Professor position that she presently holds at the University of Guelph.

Anna Nagurney is the John F. Smith Memorial Professor at the University of Massachusetts. She received her doctorate in Applied Mathematics with a specialty in Operations Research from Brown University.  Among the awards that she has received are: the Kempe Prize from the University of Umea, Sweden, an NSF Visiting Professorship for Women and a Faculty Award for Women, an Eisenhower Faculty Fellowship, and two AT&T Industrial Ecology Fellowships.

The three research team members were greeted at the Milan airport on March 10 and were driven to the Bellagio  Center (an approximately 2 hour drive). While at Bellagio, we stayed at our own minivilla, which was set up for our use and provided us with access to laptops, Internet connections, as well as a white board and printer. Our research interests and backgrounds created an incredible synergy from which we could conduct our research project. Although we had interacted through email, only Patrizia and Anna had ever met face to face, but we knew each other through our research interests and publications.  Monica Cojocaru  had established results critical to the success of the project in her thesis and subsequent publications (see Cojocaru (2002) and Cojocaru and Jonker (2003)).  In particular, she had generalized some of the results of Dupuis and Nagurney (1993) (see also Nagurney and Zhang (1996)) in projected dynamical systems to Hilbert spaces. Patrizia Daniele, in turn, had been working on evolutionary variational inequalities. She had developed time-dependent analogues of some of the traffic network, spatial price equilibrium, and financial equilibrium models (cf. Daniele, Maugeri, and Oettli  (1998),  Daniele (2002, 2003)) that Nagurney had formulated and studied (with students and co-authors) as finite-dimensional variational inequality problems (see Nagurney (1993, 2000), Nagurney and Siokos (1997), and the references therein).

From the first day of our residency, we established a rhythm that enabled us to work intensively from early in the morning until very late at night. In addition, we took part in some incredible and very memorable intellectual conversations and discussions during the meals as well as the evening seminars. For the most part, we worked together in the conference room of our villa, brainstorming, questioning, connecting, researching the existing literature, marking up the white board with equations and diagrams, and establishing our results. Our primary goal was to build the theoretical foundations through the connections (and unification) of projected dynamical systems and evolutionary variational inequalities on Hilbert spaces. From the former, we could then gain powerful computational procedures, whereas from the latter we could gain a new richness in terms of model development and analysis in  operations research/management science, as well as in economics, finance, as well as engineering (notably, transportation science),  and environmental sciences. The theoretical results that we obtained while at the Bellagio Center, the applications that can now be explored include some that can be interpreted in a new way, as well as entirely new and very exciting ones (notably, to global  supply chain networks and international financial networks) will be written up in a series of papers.

When the intensity of our research and work necessitated a respite, we would take some time for a meal, a walk, or a song; laughter would soon follow. We got to share with artists, musicians, writers, human rights activists, public health experts, and historians what we consider to be the beauty of mathematics, its relevance, and its special nature in terms of research, creativity, and scientific discovery. We found that many of the residents eagerly anticipated our seminar presentation which was unique not only since it was being presented by three females but by three female operations researchers!

Our presentation took place on Friday night, March 19, 2004.  We discussed the context of our work, with Anna providing the background and foundations, and focusing on a spectrum of network-based applications that had been studied as finite-dimensional variational inequality problems and as projected dynamical systems. Some applications that were described included supply chain networks, international financial networks with intermediation as well as recycling networks (for relevant papers, see: http://supernet.som.umass.edu). Patrizia followed with an overview of her modeling work on time-dependent spatial price and financial network models using evolutionary variational inequalities, coupled with her contributions to the existence and uniqueness of the solutions. Monica provided not only a discussion of infinite-dimensional projected dynamical systems but even showed  dynamic trajectories using MAPLE. She also discussed why math is important, what is special about math, and listed some of the unsolved problems dating to the 19th century. The three of us also summarized the work that we had accomplished during the residency and highlighted the impact that it might have. The discussions that followed our seminar were incredibly animated

The research team residency was an experience of a life-time. It demonstrated the rewards of a sustained  collaboration in an environment of great beauty and without the obligations of daily life. It was a privilege to be able to partake in the activities of the Bellagio Center during our two week tenure and we hope to return to our disciplines what we learned and discovered.  Finally, we hope that we helped to get across the idea that operations researchers can be very dynamic, social, and engaging individuals, with interests that transcend different areas.

For background on the Rockefeller Foundation’s Bellagio Center Program, see: http://www.rockfound.org
 

Acknowledgments

We are deeply grateful to the Rockefeller Foundation’s Bellagio Center program and to its staff for giving us this incredible opportunity.
 

References

Cojocaru, M. G., Projected Dynamical Systems on Hilbert Spaces, Ph. D. Thesis, Queen’s University, Kingston, Ontario, Canada (2002).

Cojocaru, M. G. and Jonker, L. B., Existence of Solutions to Projected Differential Equations in Hilbert Spaces, Proc. Amer. Math Soc. 132 (2004), 183-193, online May 2003.

Daniele, P., Dual Variational Inequality and Applications to Asymmetric Traffic Equilibrium Problem with Capacity Constraints, Le Matematiche 49 –Fasc. II (1994), 211-222.

Daniele, P., Time-Dependent Spatial Price Equilibrium Problem: Existence and Stability Results for the Quantity Formulation Model, to appear in Journal of Global Optimization (2002).

Daniele, P., Evolutionary Variational Inequalities and Economic Models for Demand Supply Markets, M3AS: Mathematical Models and Methods in Applied Sciences 4 (13) (2003), 471-489.

Daniele, P., Maugeri, A. and Oettli, W., Time-Dependent Variational Inequalities, Journal of Optimization Theory and its Applications 103 (1999), 543-555.

Dupuis, P. and Nagurney, A., Dynamical Systems and Variational Inequalities, Annals of Operations Research 44 (1993), 9-42.

Nagurney, A.,  Network Economics: A Variational Inequality Approach, Kluwer Academic Publishers, Dordrecht, The Netherlands (1993).

Nagurney, A., Sustainable Transportation Networks, Edward Elgar Publishing, Cheltenham, England (2000).

Nagurney, A. and Siokos, S., Financial Networks: Statics and Dynamics, Springer-Verlag, Heidelberg, Germany (1997).

Nagurney, A. and Zhang, D., Projected Dynamical Systems and Variational Inequalities with Applications, Kluwer Academic Publishers, Boston, Massachusetts (1996).
 

Last Update: March 25, 2004