Mathematics in the 21st Century

PIERRE CARTIER, member of Institut des Hautes Études Scientifiques (IHES) in Bures sur Yvette, France, is emeritus professor in the University Paris 7. Prof. Cartier was a student of the École Normale Supérieure where he obtained his PhD under the supervision of Henri Cartan. He has been a member of the Bourbaki group for over 30 years. In algebraic geometry, a number of objects bear his name: the Cartier duality between Abelian varieties or formal groups, Cartier divisors. He has been a closed collaborator from Grothendieck.A.D.R. CHOUDARY is director general at the Abdus Salam School of Mathematical Sciences (ASSMS), GC University Lahore, Pakistan. Prof. Choudary received his PhD in mathematics at the University of Bucharest. In 2002, he received the award of “Most Inspirational Educator” and in 2003, received the award of “Distinguished Professor of Research” at Central Washington University, USA. Earlier, he served as a coordinator of the Faculty of Mathematics at the University of Guadalajara, Mexico, and professor of mathematics at Central Washington University, USA. An author of three books, Prof. Choudary has been a visiting scholar at a number of universities and institutes in Europe, USA, Asia and Australia. He received several grants from US Department of Education and other similar agencies during his stay in the USA.MICHEL WALDSCHMIDT is emeritus professor at Université Pierre et Marie Curie (Paris 6), Paris, France. Prof. Waldschmidt studied at the University of Nancy. Then he went to Bordeaux, where he obtained his PhD in 1972 under the supervision of Jean Fresnel. He was immediately offered a temporary position in the University of Orsay by Henri Cartan (1972-73) and in 1973, he was appointed as a professor at the University Paris VI, where he taught until the end of 2011. He is an expert in number theory, especially Diophantine problems, including transcendence methods.

• Chapter 1. About phase transition and zero temperature.- Chapter 2. Approximation on Curves.- Chapter 3. Discriminants, polytopes, and toric geometry.- Chapter 4. Lecture on the abc conjecture and some of its consequences.- Chapter 5. Some results on chromaticity of quasi-linear hypergraphs.- Chapter 6. Basic properties of the non-abelian global reciprocity map.- Chapter 7. Poisson Smooth Structures on Stratified Symplectic Spaces.- Chapter 8. Some classical problems in number theory via the theory of K3 surfaces.- Chapter 9. On Ramsey (2K2;K4) - minimal graphs.- Chapter 10. On the motion induced by a flat plate that applies oscillating shear stresses to an Oldroyd-B fluid. Applications.- Chapter 11. Equilibrium in choice of generalized games.- Chapter 12. Hamiltonian connectedness of Toeplitz graphs.- Chapter 13. Cosmos and its furniture.- Chapter 14. On super edge-antimagic total labeling of Toeplitz graphs.