Abstract:
A bijective proof shows that two objects are naturally equivalent by exhibiting a natural bijection. This talk will describe a few familiar bijective proofs. It will then exhibit an interesting bijection in a context involving connected graphs and biconnected graphs. The fundamental combinatorial objects are (1) the set of connected graphs on a vertex set and (2) the set of all connected graphs on a vertex set with a distinguished vertex. These play a role in equilibrium statistical mechanics (of lattice particle systems). They correspond to (1) the potential as a function of activity and to (2) the density as a function of activity. The combinatorial identity reflects a very general way in which the potential may be regarded as a function of density. All this is on the level of formal expansions, but some convergence results are available.
Biography:
William G. Faris is Visiting Professor of Mathematics at NYU Shanghai. He is also Professor Emeritus at the University of Arizona. Prior to that, he was a Fulbright Lecturer at the Independent University of Moscow. He has also been a Visiting Professor at the Institut des Hautes Etudes Scientifiques, a Visiting Member at the Courant Institute of Mathematical Sciences at NYU, a Visiting Fellow at the Newton Institute, and a Visiting Scholar at the University of British Columbia. He holds a Ph.D. from Princeton University and a BA from the University of Washington. Professor Faris's research interests are mathematical physics, applied probability, and combinatorics. His books include Self-Adjoint Operators (Springer, 1975), Martingale Methods in Elementary Probability(Independent University of Moscow, 1996), and Diffusion, Quantum Theory, and Radically Elementary Mathematics (Princeton University Press, 2006). He has published over fifty articles in mathematics and mathematical physics.
Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai