DEPARTMENT OF PHYSICS AND ASTRONOMY -- (202) 806-6245 (main office), -5830 (fax)

Advanced Mathematical Methods in Physics 1
PHYS-266 [CRN: 82756]: MWF, 3:30 - 4:20, TKH #103
[Topics ][ Daily Schedule ][ e-Gear ][Welcome ]

Instructor: Tristan Hübsch

TKH 213, 806-6267 thubsch@mac.com
Office hrs.: MW 11-12 & 4:30-5:30, and by appointment (at least one day ahead, confirmed)

Required Textbooks:

1: Ch.Nash & S.Sen, Topology and Geometry for Physicists

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

 

Recommended reading:

S. Goldberg, Curvature and Cohomology
C. von Westenholz, Differential Forms in Mathematics
T. Hubsch , Calabi-Yau Manifolds—A Bestiary for Physicists

and other sources, as given in class

Component	Time	Remark	% of Grade
Homework	As assigned in class	Late HW = 0 credit !!!	40%
Term paper	By the last day of classes	Topical	60%

The aim of the course is to introduce, develop and discuss various methods of determining global characteristic properties of spaces which occur in theoretical and mathematical physics. Throughout the course, the emphasis is on the applications of these results rather than their proofs. The course is to be regarded as an statistically significant sampler rather than a definitive compendium, and students are strongly encouraged to study particular topics mentioned in the course, in detail and depth surpassing the discussions in class; this is the purpose of the term paper.

“Success = 1% inspiration + 99% perspiration”--T.A. Edison
But, learning is still 100% learnig + 0% teaching.

• Topology vs. Geometry: §1
• Differential Geometry: §2 and §§ 1-3
• Metric and Other Structures on Manifolds: §§ 4-7
• Lie groups: §§8
• The Fundamental Group, Homotopy and Homology Groups: §3-5
• De Rham Cohomology vs. Homology: §6
  
• Fibre Bundles, Topological Defects: §7-10,  §§9-10
• Spin and Anomalies: §§11-13
  

Day-to-day schedule: Students are expected to read ahead

08/26: Introductory Matters
08/28: Motivations and introduction to topology: §1.1-2
08/30: Basic notions in topology: §1.2-5
09/02: Observed Holiday: Labor Day
09/04: Exterior algebra: §§1.
09/06: Vectors, frames, differential forms, pullbacks and exterior derivatives : §§2.1-6
09/09: Poincaré lemma, de Rham cohomology, integration and stokes' theorem: §§2.7-10
09/11: Metric structures: §§3.
09/13: Maxwell's equations, connection/potential, curvature/field strength: §§4.1-3
09/16: Exterior covariant derivative, Yang-Mills and gauge theories: §§4.4-7
09/18: The equivalence principle, metric connection, action and field equations: §§5.1-5
09/20: Energy-momentum of matter, Einstein gauge and the geometry of curvature and torsion: §§5.6-9
09/23: The Lie derivative: §§6.
09/25: Manifolds, submanifolds, tangent space, and frames: §§7.1-7
09/27: Differential forms on manifolds, integration, metric and gauge theories: §§7.8-14
09/30: Lie groups, their representations and algebras: §§ 8.1-3
10/02: Relations between Lie groups and Lie algebras, the Maurer-Cartan form: §§8.4-6
10/04: The fundamental group, §3.1-2
10/07: Simplexes, triangulation and calculations: §3.3-5
10/09: Homology groups and simplexes, abelian groups, §4.1-3
10/11: Relative homology groups, exact sequences, and torsion, Künneth and Euler-Poincaré formulas: §4.4-6
10/14: Observed Holiday: Columbus Day
10/16: Higher homotopy groups: §5
10/18: De Rham cohomology vs. homology: §6
10/21: Fibre bundles: §7.1-4
10/23: Sections, singularities, reduction and contraction: §7.5-6
10/25: Hamiltonian, almost complex and G-structures: §7.7-8
10/28: Connection, curvature, parallel transport, and Binachi identities: §7.9-14
10/30: Tangent bundle, three connections: Levi-Civita, Yang-Mills and Maxwell: §7.5-20
11/01: Characteristic classes: §7.21-26
11/04: Calculation of and with characteristic classes: § 7.27-30
11/06: Morse inequalities and lemma, §8.1-2
11/08: Symmetry breaking and equilibira, §8.3-4
11/11: Topological defects and homotopy theory: §9
11/13: Instantons I: §10.1-5
11/15: Instantons II: §10.6-10
11/18: Instantons and twistors, §10.11-15
11/20: Instantons and holomorphic vector bundles, monopoles and the Aharonov-Bohm effect, §10.16-20
11/22: Clifford algebras, spinors and the Dirac operator: §§11.1-5
11/25: The Dirac action, spin structures and Kähler fermions, §§11.6-8
11/27: Anomalies and an algebraic approach to them§§12 .
11/29: Observed Holiday: Thanksgiving Recess
12/02: Anomalies from perturbative calculations; §§13.
Algebraic geometry crash course:
12/04: Spacetime in string theory, compactification and Calabi-Yau manifolds.
12/06: Spectral sequences, diagram chasing and group actions as calculational tools.

Collaboration policy

Collaboration -- but not blind copying -- on the homework assignments is strongly encouraged; students should use this to learn from each other. However, no collaboration is permitted on the term papers: by signing them, the students implicitly agree to abide by this policy. Violation of this policy is covered under University regulations on academic dishonesty and cheating.

Presentation and organization

While a neat presentation of home,- quiz- and exam-work is not required for full credit, it certainly makes it easier to assess the quality of the work and give the proper credit due. In all cases, include a simple sketch if it might help conveying the approach or the calculations. Where necessary, include all units and symbols such as the measure of an integral, arrow on a vector, vertical bars for the absolute value of a quantity, for the magnitude of a vector or for the determinant of a matrix, etc.

• Homework: Submitted answer sheets should be stapled (preferably in the top-left corner). The Student's full name (and ID number) should be indicated on the first (cover) page. Solutions should be labeled and ordered consecutively by the problem numbers. Include all calculation/work and give full reference to any result quoted from any source other than the text, and page/equation number for results quoted from the text. Homework assignments will be graded by a grader with whom unsatisfactory credit should be disputed prior to appealing to the instructor; the instructor may override the grader's assessment.
• Term paper: All students are required to write a term paper on a subject from the topical schedule and submit it by the last day of classes. A list of tentative topics will be presented in class, by the end of the first month of the course. The term papers must be typed or typeset in a format of a research paper, complete with references and any appropriate auxiliary material (tables, diagrams, figures, etc.), and are expected to be of a 15-page length

ADA disclaimer

Howard University is committed to providing an educational environment that is accessible to all students.  In accordance with this policy [details], students in need of accommodations due to a disability should contact the Office of the Dean for Special Student Services at 202-238-2420, for verification and determination of reasonable accommodations as soon as possible after admission to the Law School, or at the beginning of each semester.

© Tristan Hübsch, 2002