1. MATH 488-588
Time and place: MWF 10:00 am – 10:50 am Gould Simpson 701
Office hours:
MWF: 3 pm – 4 pm
Final Exam: Wednesday, May 9, 10:30 am – 12:30 pm
Course Content
Solitons, the nonlinear localized objects, play a very important role in different areas: nonlinear optics, hydrodynamics, plasma theory, superfluidity, and magnetism. Also, they are important for the theory of general relativity – the black holes are solitons. It is remarkable that this broad variety of physical phenomena, from microscopic to astronomic scale, can be described by unified mathematical apparatus that was intensively developed during last four decades. The mathematical theory of solitons, known as the Inverse scattering method, is closely connected to the spectral theory of differential operators and to the classical theory of integrable Hamiltonian systems.
In this course we will discuss the basic elements of both physics and mathematics of solitons. We will make accent on pure algebraic method for construction of solitonic solutions via local and non-local delta-problem. Then we will develop the Inverse scattering method for the Schrodinger and Dirac operators and for their discrete versions. The course will be organized as follow:
The course will available for undergraduate students with basic knowledge of ODE, linear algebra and complex analysis. Lecture notes will be posted on internet.
2. MATH 456-556
Time and place: MWF 1:00 pm – 1:50 pm PAS 414
Office hours: MWF: 11:30 am – 12:45 pm
Textbook: Richard Haberman, Applied Partial Differential Equations
Prerequisites: Math 223, 254
Final Exam: Monday, May 7, 1:00 pm – 3:00 pm
Withdrawal: Students may withdraw from this course, or change to audit until February 7 if they desire complete deletion of the course enrollment from their records. February 8 – March 6 withdrawal is possible with the consent of the instructor. In this case the grade of W is given if the student is passing; the grade of E is given if the student is failing at the time of withdrawal. Also, a change to audit is possible in this time interval if the student is passing. March 6 is the last day for administrative drops for Spring 2012. Form must be received in Administration room 210 by 5:00 pm.
Homework: Homework problems will be assigned every two weeks. Totally 6 homeworks will be assigned.
Examinations: Two midterm exams will take place (March 21 and April 27) and there will be a two-hour final exam. The final examination will be based upon the entire course. Books and notes are allowed for these examinations. The use of electronic calculators is permitted. Your final grade for this course will be determined by the scores of all examinations and the homework grades. The final exam counts 40%, each one-hour test counts 15%, and the homeworks will contribute 30% to the total score.
Syllabus
Week 1. Heat equation. 1.1 – 1. 5.
Week 2. Fourier series. Periodic extension. Convergence theorem. Fourier sine and cosine series. 3.1 – 3.3.
Week 3. Term by term differentiation and integration of Fourier series. 3.4 – 3.5. Complex form of Fourier series. 3.6. Bessel unequality and Parseval’s identity.
Week 4. Method of separation of variables. Heat equation with zero temperature at finite ends. Insulated ends. Heat conduction in a thin circular ring. 2.3, 2.4.
Week 5.
Week 6. Vibrating string and membranes. 4.1 – 4.5
Week 7. Sturm-Liouville Eugenvalue problem. 5.1 – 5.7
Week 8. Heat and string equations with boundary conditions of the third kind. Asymptotic properties of large eigenvalues. 5.8 – 5.10
Week 9. Nonhomogenous problems. 8.1 – 8.6
Week 10. The Dirac delta-function. Green equation for time-independent problems. One-dimensional steady-state heat equation. 9.1 – 9.4.3
Week 11. Green function for Poisson equation. 9.5.1 – 9.5.9
Week 12. Fourier transform. Laplace and Poisson equation in infinite domain. Chapter 10.
Week 13. Green’s function for wave equation in infinite domain. 11.2, 4.6.
Green’s function for heat equation in infinite domain. Self-similar solution. 11.3
Week 14. The method of characteristics for linear and quasilinear wave equation. Shock waves. Chapter 12.
Week 15-16. Dispersive waves. Oscillating of stressed and rotating beam. Stability. Solitons. Chapter 14 (essentially modified)
Due February 6
Due February 22
Due March 19
Due April 4
Due April 20
Due May 2
1. MATH 488-588
Differential
Geometry, General Relativity, and Cosmology
Course Content
The course will cover the following topics: 1. Tensor algebra; 2. Special theory of relativity; 3. Calculus of differential forms on manifolds; 4. Tensor analysis on manifolds. Affine connection and covariant differentiation; 5. Riemannian geometry, Curvature and Ricci tensors. Geodesics; 6. Riemannian spaces of diagonal curvature and their integrability. Spaces of constant curvature and flat connection. N-orthogonal coordinate systems; 7. Basic principles of general relativity. Einstein equations. The simplest solution of Einstein equations (Kasner metrics); 8. Gravitational waves; 9. Spherically-symmetric gravitational field. Black holes. Horizon. Volkov-Oppenheimer metrics; 10. Geodesics in the field of a black hole. Deflection of light in the gravitational field. Gravitational lenses; 11. Charged black holes. Naked singularity; 12. Rotating black hole: Kerr’s solution. Black hole as a source of energy;
13. Gravitational solitons: the charged rotating black holes. Partial integrability of Einstein equations;
14. Are the travels in time possible? 15. Basic cosmological models. Red shift. Gravitational collapse;
16. Black matter and black energy. The predictable fate of the Universe; 17. Origin of the Universe. Inflation. Why our Universe is spatially flat?
Motivation
The goal of this course is to give students the mathematical apparatus making it possible to deal with such urgent physical problems as basic cosmological models, the origin and predictable fate of Universe, and to explain why the concept of dark matter and dark energy is necessary to be in agreement with observable astronomic data. After taking this course, the student will get a solid knowledge of black holes, the resting ones as well as charged and rotating, and will be able to answer the question: why are travels in time not possible? More mathematically oriented students will probably concentrate on exact solutions of the Einstein equations and on other applications of the theory of integrable systems to differential geometry. We will climb on these peaks of human intellectual achievements starting from pretty low and modest levels. First, we will study the tensor algebra in affine, Euclidean, and pseudo-Euclidean spaces, the calculus of differential forms on manifolds, and then the basic points of Riemannian geometry. Then we will spend most part of time on study of general relativity.
Prerequisites
The course will be within reach for both graduate and undergraduate students from Mathematical, Physical, and Astronomy Departments. The prerequisites are modest: linear algebra (Math 415) and ordinary differential equations (Math 252). The knowledge in PDE is desirable but not necessary.
Textbooks and other course materials
The main text for the course will be a set of lecture notes developed by the instructor. I will not follow strictly any particular textbook but will use extensively three of them: 1. David Lovelock and Hanno Rund, Tensors, Differential Forms and Variational Principles; 2. Landau L.D. and Lifshitz E.M. Course of Theoretical Physics. Vol. 2. The classical theory of fields; 3. B.A. Dubrovin, S.P. Novikov and A.T. Fomenko. The Modern Geometry. Vol. 1.
Some supplementary materials will be provided in electronic form.
Due February 9
Due March 28
Collapse of charged dust by Bole Yang
Geodesics in homogeneous isotropic space by A. Shah
and P. Whalen
2. MATH 456-556
Time and place: MWF 1:00 pm – 1:50 pm PAS 418
Office hours: MWF: 11:00 am – 12:00 am
Textbook: Richard Haberman, Applied Partial Differential Equations
Prerequisites: Math 223, 254
Final Examination: Tuesday, May 10, 1:00 pm – 3:00 pm
Withdrawal: Students may withdraw from this course, or change to ``audit'' until Wednesday, February 9 if they desire complete deletion of the course enrollment from their records. After February 9 but before Wednesday, March 9, withdrawal is possible with the consent of the instructor. In this case the grade of W is given if the student is passing; the grade of E is given if the student is failing at the time of withdrawal. Also, a change to ``audit'' is possible in this time interval if the student is passing. After March 9 withdrawal is only possible in the most extraordinary circumstances. Approval of the Dean is required, and the same regulations as for withdrawal after February 9 apply with regard to the grades of W and E.
Homework: Homework problems will be assigned every two weeks. Totally 6 homeworks will be assigned.
Examinations: Two midterm exams will take place (March 11 and May 4) and there will be a two-hour final examination. The final examination will be based upon the entire course. Books and notes are allowed for these examinations. The use of electronic calculators is permitted. Your final grade for this course will be determined by the scores of all examinations and the homework grades. The dates of each test will announced at least one week before it is scheduled. The final exam counts 40%, each one-hour test counts 15%, the homeworks will contribute 30% to the total score.
Syllabus
Week 1. Heat equation. 1.1 – 1. 5.
Week 2. Fourier series. Periodic extension. Convergence theorem. Fourier sine and cosine series. 3.1 – 3.3.
Week 3. Term by term differentiation and integration of Fourier series. 3.4 – 3.5. Complex form of Fourier series. 3.6. Bessel unequality and Parseval’s identity.
Week 4. Method of separation of variables. Heat equation with zero temperature at finite ends. Insulated ends. Heat conduction in a thin circular ring. 2.3, 2.4.
Week 5.
Week 6. Vibrating string and membranes. 4.1 – 4.5
Week 7. Sturm-Liouville Eugenvalue problem. 5.1 – 5.7
Week 8. Heat and string equations with boundary conditions of the third kind. Asymptotic properties of large eigenvalues. 5.8 – 5.10
Week 9. Nonhomogenous problems. 8.1 – 8.6
Week 10. The Dirac delta-function. Green equation for time-independent problems. One-dimensional steady-state heat equation. 9.1 – 9.4.3
Week 11. Green function for Poisson equation. 9.5.1 – 9.5.9
Week 12. Fourier transform. Laplace and Poisson equation in infinite domain. Chapter 10.
Week 13. Green’s function for wave equation in infinite domain. 11.2, 4.6.
Green’s function for heat equation in infinite domain. Self-similar solution. 11.3
Week 14. The method of characteristics for linear and quasilinear wave equation. Shock waves. Chapter 12.
Week 15-16. Dispersive waves. Oscillating of stressed and rotating beam. Stability. Solitons. Chapter 14 (essentially modified)
Due
Due February 21, 2011
Due March 25
Due April 11
Due April 22
Due May 4
2. MATH 456-556
1. MATH 421-521
2. MATH 488-588
Draft of lecture notes (checked up to Lecture
13th for now).
V.E.Zakharov/Department of Mathematics/Program in Applied
Mathematics/University of Arizona/Tucson, AZ 85721