Topics in Mathematical Physics in View of Differential Equations
2018-01-01 12:13  华东师范大学

Differential equations play important roles in fundamental science and present great challenges and opportunities to research mathematicians. In theoretical physics, many profound concepts, predictions, and advances were pioneered through mathematical insights gained from the study of the equations governing basic physical laws. Why do some materials demonstrate zero electric resistance when cooled (superconductivity)? How do the two strands in an entangled DNA double helix become separated when heated (DNA denaturation)? Why does the universe have a ﬁnite past (big bang cosmology)? Why are all the electric charges integer multiples of a minimal unit (charge quantization)? Why is that the basic constituents of matter known as quarks can never be found in isolation (quark conﬁnement)? These are some of the exemplary situations where mathematical investigation is essential. The purpose of this series of lectures is to provide a vista-type overview of a broad range of topics in mathematical physics which may well be illustrated through differential equations. Emphasis will be given to the mathematical structure and physical descriptions of various basic problems and to the appreciation of the power of functional analysis, although close attention will also be given to the links with other areas of mathematics, including geometry, algebra, and topology.
List of units of the course
1. Hamiltonian systems and applications
Motion of a massive particle
The vortex model of Kirchhoﬀ
The N-body problem
Hamiltonian function and thermodynamics
Hamiltonian modeling of DNA denaturation
2. The Schrodinger equation and quantum mechanics
Path to quantum mechanics
The Schrodinger equation
Quantum N-body problem
The Hartree–Fock method
The Thomas–Fermi approach
3. The Maxwell equations, Dirac monopole, etc
The Maxwell equations and electromagnetic duality
The Dirac monopole and Dirac strings
Charged particle in an electromagnetic ﬁeld
Removal of Dirac strings, existence of monopole, and charge quantization
4. Abelian gauge ﬁeld equations
Spacetime, Lorentz invariance and covariance, etc
Relativistic ﬁeld equations
Coupled nonlinear hyperbolic and elliptic equations
Abelian Higgs model
5. The Ginzburg–Landau equations for superconductivity
Heuristic proof of the Meissner eﬀect
Energy partition, ﬂux quantization, and topological properties
Vortex-lines, solitons, and particles
From monopole conﬁnement to quark conﬁnement
6. Non-Abelian gauge ﬁeld equations
The Yang–Mills theory
The Schwinger dyons
The ’t Hooft–Polyakov monopole and Julia–Zee dyon
The Glashow–Weinberg–Salam electroweak model
7. The Einstein equations and related topics
Einstein ﬁeld equations
Cosmological consequences
Static solution of Schwarzschild
8. Charged vortices and the Chern–Simons equations
The Julia–Zee theorem
The Chern–Simons term and dually charged vortices
The Rubakov–Tavkhelidze problem
9. The Skyrme model and related topics
The Derrick theorem and consequences
The Skyrme model
Comments on fractional-exponent growth laws and knot energies
10. Strings and branes
Relativistic motion of a free particle
The Nambu–Goto strings
p-branes
The Polyakov string, conformal anomaly, and critical dimension
11. The Born–Infeld geometric theory of electromagnetism
Formalism
The Born–Infeld theory and a generalized Bernstein problem
Charge conﬁnement and nonlinear electrostatics

Yisong Yang is Professor of Mathematics at Polytechnic School of Engineering of New York University. Prior to joining Polytechnic, he held academic appointments at University of Minnesota (postdoctoral fellow), University of New Mexico (assistant professor), Carnegie Mellon University (visiting assistant professor), and Institute for Advanced Study (member). He was also a Professor of Mathematics at Yeshiva University. He received his Ph. D. in Mathematics from University of Massachusetts at Amherst in 1988.
His research is in the fields of nonlinear partial differential equations and global analysis, with main interest focused on mathematical problems arising in theoretical physics. His contributions include construction of vortices in supersymmetric field theory, which are essential for the quark confinement mechanism, derivation of universal growth laws relating energy and topology for knots arising in a quantum field theory model in arbitrary dimensions, and existence theory for cosmic strings, which are responsible for matter accretion and galaxy formation in the early universe.
He is a Fellow of American Mathematical Society.
He is the author of two books: Solitons in Field Theory and Nonlinear Analysis, Springer-Verlag, 2001, and A Concise Text on Advanced Linear Algebra, Cambridge University Press, 2014.