Delaware State University




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Delaware State University

Department of Applied Mathematics and Theoretical Physics

Dover, DE 19901

Topology and Geometry in Physics


60-711-00, 3 cr.

Text:

M. Göckler & T. Schücker: Differential Geometry, Gauge Theories, and Gravity

recommended:

C. Nash & S. Sen: Topology and Geometry for Physicists

S. Goldberg: Curvature and Cohomology

C. von Westenholz: Differential Forms in Mathematical Physics

The aim of the course is to generalize vector and tensor calculus and introduce the general theory of cohomology on curved and topologically non-trivial spaces, including fiber bundles and quotient spaces. This will include applications in classical mechanics, classical field theory, statistical mechanics and quantum theory. The emphasis in this course is on the adaptation of these mathematical methods and techniques to their swift and effective application in solving advanced problems in theoretical physics.

Prerequisite: Mathematical methods of Physics IV (26-667) or equivalent (vector calculus, linear algebra, tensors, real and complex analysis, multivariate calculus). A successful student is expected to gain a working knowledge of the covered material, so as to be able to (1) follow the applications in the literature, (2) solve typical problems in the field, and (3) discuss adequately the term-paper subject.

Topical schedule:

  • Basic Topology, Differentiable and Complex Manifolds

Application: Electromagnetism, Thermodynamics

  • Differential Geometry on Manifolds

Application: Tensor Algebra on Curved Manifolds

  • Homotopy and Homology

Application: Magnetic Monopoles, Cosmic Strings and Domain Walls

  • De Rham and Dolbeault Cohomology

Application: The Lichnerowitz Equation & Field Fluctuations

  • Fiber Bundles, Quotient Spaces and Algebraic Varieties

Application: Constrained and Gauge-Symmetric Systems

  • Curvature, Connections and Characteristic Classes

Application: General Relativity & Gauss-Bonet Theorem

  • Singularities, Deformations and Resolutions

Application: Spacetime- and Configuration Space-Changing Phase Transitions

  • Holonomy and Monodromy

Application: Spacetime and Configuration Space Curvature and Dynamical Invariants

  • Morse Theory

Application: Spacetime and Configuration Space Topological Defects

1. Course Title/Number: Topology and Geometry in Physics / 60-711-00

2. Number of Credits: 3

3. Curriculum Program Title: Ph.D. in Applied Mathematics and Theoretical Physics

4. Curriculum/Course is: [ X ] New [     ] Revised

[     ] Required Course [ X ] Elective Course

5. List Prerequisites:

26-667 (Mathematical methods of Physics IV), or equivalent (vector calculus, linear algebra, tensors, real and complex analysis, multivariate calculus)

6. List Courses Being Replaced or Changed:

This is a new course.

7. List Courses Being Deleted:

No courses are being deleted.

8. Needs Statement:

This course is needed for students pursuing a Ph.D. in all areas of theoretical physics and especially for those interested in fundamental physics. The frameworks of general relativity and Yang-Mills gauge theories, describing respectively gravity and all other known fundamental interactions of matter, all involve the geometric structure called fiber bundles and admit a unified description by means of exterior calculus. Also, nontrivial boundary conditions (e.g., anti-periodicity for fermions), certain aspects of quantum interaction of matter and electromagnetic fields (e.g., Bohm-Aharonov effect, Berry’s phase) all have a topological origin. This course lays the foundations to understanding these phenomena in fundamental and applied physics.

9. Catalog Description of the Course:

This course generalizes vector calculus to curved and topologically non-trivial spaces and introduces fiber bundles, quotient spaces and homological algebra. The emphasis is on applications in classical mechanics, statistical mechanics, classical field theory and quantum mechanics.

10. List of Objectives of the Course:

(1) To provide an introduction to the body knowledge and techniques of topology and geometry as applied in theoretical physics. (2) To see how these techniques apply to the analysis of phenomena in classical mechanics, statistical mechanics, classical field theory and quantum theory. (3) To learn how to identify those phenomena throughout theoretical physics, which are best described using these methods. (4) To develop the problem-solving skills associated with the application of these methods in theoretical physics, and learn how to extract experimentally verifiable information from such application.

11. Course Outline:

See the “Topical schedule” section in the attached brief syllabus.

12. Show how the proposed course fits into the curriculum or course sequence:

This course is an elective within the curriculum of the Ph.D. program in theoretical physics, and is indispensable for students focusing on fundamental physics. For an overview of pre-requisite dependences and the course’s relation to other courses proposed herein, please see the attached “Proposed Course Dependencies” chart.

13. Are there comparable courses in other departments?

No.

14. How will the students be affected by this course change?

This course provides the students an opportunity to increase their integration with the research program of the Department of Applied Mathematics and Theoretical Physics, by understanding the mathematical underpinnings of the techniques that are used in contemporary theoretical physics. This course will improve students’ professional competence, employability in technical fields and ability to pass professional examinations. Neither this course nor its prerequisites increase the total number of semester hours in this curriculum or the number of credit hours required for graduation.

15. What effect will this new course have on College resource?

None: this course will not require new or additional resources or staffing.

16. How will the course benefit the College?

This course will address applications of topology and geometry in various areas of fundamental and applied physics, some of which lie at the foundation of numerous other disciplines in science: engineering (e.g., spin networks as memory devices), nanotechnology (e.g., topological (in)stability of macro-molecules), biotechnology (e.g., braiding of DNA and RNA molecules and protein folding), cosmology (e.g., cosmic topological defects as accretion sources), etc.

17. How will the change affect the program?

This course will introduce students to a few select topics in “higher” mathematics and their application in various branches of physics. This course will be one of the electives specific to the Ph.D. program (concentration in theoretical physics) in this department. Besides providing such a cross-disciplinary broadening of knowledge for the students in this program, it also serves as a prerequisite to Superstrings and Beyond, also proposed herein.

18. Evaluation of Student Performance:

Homework Assignments 40 %

Term-paper (take-home final) 60 %

Sample homework assignments, in-term and final examination question-sheets, work sheets, course notes, review sheets and term papers will be accessible on-line.

Course Structure: Three (3) 50-minute lectures per week.

References

  1. M. Göckler and T. Schücker: Differential Geometry, Gauge Theories, and Gravity (Cambridge University Press, 1989; ISBN = 0521378214)

  2. C. Nash and S. Sen: Topology and Geometry for Physicists (Academic Press, 1988; ISBN = 0125140819)

  3. S. Goldberg: Curvature and Cohomology (Dover Publications, 1998; ISBN = 048640207X)

  4. C. von Westenholz: Differential Forms in Mathematical Physics (North Holland, 2nd ed.. 1978; ISBN = 0444854371)

Submitted to Department of Applied Mathematics and Theoretical Physics

by: Tristan Hubsch, on 27th of November, 2007.



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