Friday, May 27, 2016

Reflection on John Forbes Nash Jr. and Connections

The past week or so I have been delighted, every time that I logon from my office at All Souls College at Oxford University, where I am now a Visiting Fellow,  to see the beautiful photo of Sir Andrew Wiles receiving the 2016 Abel Prize in Oslo, Norway from Crown Price Hakon. The Abel prize  is the most prestigious international prize awarded in mathematics.  Sir Wiles proved Fermat's Last Theorem, an achievement which has brought him great and much-deserved renown. The interview itself is an inspiration and speaks of his years of studying the problem before all the pieces for the proof fit together.

I could not help thinking back to one year ago and recalling that, one of my scholarly giants, whom I cite in numerous papers (probably about 100), John F. Nash Jr.,  was awarded the Abel Prize in May 2015. He was awarded it with Louis Nirenberg.

My husband brought me the latest issue of the Notices of the AMS to my daughter's graduation last weekend saying that I had to bring the issue back with me to Oxford, and I did. On the cover was a photo of John Forbes Nash Jr. and in the issue were both tributes and a fascinating interview with Nash, his last.
This morning I had a chance to read the interview conducted by Professors Martin Raussen and Christian Skau with Nash on May 18, 2015, one day before the Abel Prize ceremony in Oslo, Norway and only 5 days before he died tragically with his wife following a return flight and being driven in a taxi from the Newark Airport.

I had been at that airport both last Thursday and Sunday, flying in from Heathrow London to attend my daughter's college graduation from Lafayette College. My experiences with drivers to and from Newark Airport merits a separate blogpost.

John Nash received his PhD from Princeton University at age 21 and his dissertation was on Noncooperative Games. His contributions included results for manifolds and even partial differential equations and for decades he suffered from mental illness but in the 1990s became well enough and was awarded the Nobel Prize in Economic Sciences in 1994. Interestingly, his advisor at Princeton was Al Tucker (of course of Kuhn-Tucker nonlinear programming fame) and he actually shortened his dissertation. Nash had had some results on cooperative game theory at that point as well. All Tucker's son, Al Tucker Jr., had interviewed me for a tenure track faculty position at SUNY Stony Brook and I had gotten an offer while I was on the job market while finishing my PhD, but I chose to come to UMass Amherst.

I have utilized Nash's results in game theory for modeling a plethora of competitive supply chain network problems from those arising in food applications to even fast fashion (some Fellows at dinner last night were asking me about what is fast fashion and there are both H&M and Zara outlets 2 blocks away). It was a fun discussion. And the Fellow who raised the question is a Harvard-educated economic historian who after speaking with me said - "You are an Operations Researcher!"

I have also with students and collaborators been doing alot of work on cybersecurity and cybercrime and, again, Nash's results on Nash Equilibria are crucial to my model formulation and subsequent analysis and computations. 

This past year we have been tackling some very challenging problems. These have required extensions of Nash Equilibrium to Generalized Nash Equilibrium in the case of a disaster relief model that we developed with a brilliant undergraduate at the Isenberg School of Management, and now alum, and an Economics Professor, as well as the development and solution of Nash Bargaining models from cooperative game theory for information sharing among firms that are subject to cyberattacks with applications to financial service firms, retailers, and energy companies.

And speaking of other connections, our paper: A Supply Chain Game Theory Framework for Cybersecurity Investments Under Network Vulnerability, Anna Nagurney, Ladimer S. Nagurney, and Shivani Shukla, in Computation, Cryptography, and Network Security, N.J. Daras and M.T. Rassias, Editors, Springer International Publishing Switzerland (2015) pp 381-398, which was co-edited as you can see by Michael T. Rassias, is mentioned in the last paragraph of the Nash interview. The paragraph reads:  After the end of the interview proper, there was an informal chat about John Nash’s main current interests. He mentioned again his reflections about cosmology. Concerning publications, Nash told us about a book entitled “Open Problems in Mathematics” that he was editing with the young Greek mathematician Michael Th. Rassias, who was conducting postdoctoral research at Princeton University during that academic year.

This book is scheduled to be published by Springer sometime this summer.

And I know Rassias's father, who is another brilliant mathematician. He co-edited with my great colleague Panos Pardalos the volume that another paper of ours appears in: A Dynamic Network Economic Model of a Service-Oriented Internet with Price and Quality Competition, Anna Nagurney, Dong Li, Sara Saberi, and Tilman Wolf, in Network Models in Economics and Finance, V.A. Kalyagin, P.M. Pardalos, and T. M. Rassias, Editors, Springer International Publishing Switzerland (2014) pp 239-264. That volume was based on a conference in Athens, Greece, and I posted some photos from it on my blog.