教学内容：

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

ADM mass and related topics

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

Knots in the Faddeev 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

授课安排：

第1-2周：Hamiltonian 系统及其应用

第3-4周：Schrodinger方程与量子力学

第5周：Maxwell方程与Dirac磁单极

第6-7周：规范场方程,超导与Ginzburg-Landau方程

第8-9周：非交换规范场方与Chern-Simons方程

第10-11周：Einstein方程

第12-13周：Skyrme模型，Faddeev 扭节

第14周：弦与膜，期末考试

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.