File Name: an introduction to differential geometry and topology in mathematical physics .zip
Differential Geometry is the study of smooth manifolds. Manifolds are multi-dimensional spaces that locally on a small scale look like Euclidean n -dimensional space R n , but globally on a large scale may have an interesting shape topology.
- Introduction to Geometry and Topology
- Topology and Geometry for Physics
- DIFFERENTIAL GEOMETRIC METHODS IN PHYSICS: AN INTRODUCTION TO YANG-MILLS THEORIES
It seems that you're in Germany. We have a dedicated site for Germany. A concise but self-contained introduction of the central concepts of modern topology and differential geometry on a mathematical level is given specifically with applications in physics in mind.
Introduction to Geometry and Topology
Phys Spring Tuesday, Thursday am to am. Online only: Zoom invites will be sent to registered students. Covariant and contravariant vectors and tensors. Vector fields: Lie bracket and integrability.
Differential forms: exterior differentiation, Poincare's theorem, integration of p-forms, Stokes' theorem. Introduction to topology: homology and cohomology. Groups and Group Representations. Elementary group theory: subgroups, cosets and conjugacy classes. Representations of finite groups: characters and orthogonality. Physical applications: molecular vibrations, quantum mechanics. Lie groups and Lie algebras: semisimiple and compact groups, representations, roots and weights.
Complex Analysis. Complex differentiability. Conformal mapping and its physical applications. Cauchy, Taylor, and Laurent theorems, smooth vs. Applications to contour integration, solution of differential equations and asymptotics. Below are links to various course-related documents.
Homework number 0, due 4pm Feb 3rd. Homework number 1, due 4pm Feb 10th. Homework number 2, due 4pm Feb 17th, Porter's solutions. Homework number 3, due 4pm Feb 24th. Yun's solutions. Homework number 4, due 4pm March 3rd. Homework number 5, due 4pm March 10th. Homework number 6, due 4. Homework number 7, due 4pm April 7. Homework number 8, due 4pm April Homework number 9, due 4pm April Homework number 10, due 4pm April Lecture notes.
The lecture notes are available to the general public. Please give me feedback - tell me of typos, sign errors, obscurities, or plain conceptual errors. My notes on torsion can be found near the bottom of this page. I recommend but do not require that you purchase Mathematics for Physics: A guided tour for graduate students by myself and Paul Goldbart.
Cambridge University Press I do not yet know what the UI bookstore is selling it for. The grades will be recorded via gradescope, but the traditional gradebook can accessed though "my. The gradescope grades will eventually be transferred there.
You may choose your own term paper topic, but I will wish to check its suitability. Here is short list of papers that might provide some inspiration. Please e-mail me with your suggested topics by the end of the spring break. The papers should be in the order of five pages long although there will be no penalty for longer works if you need the space and should be typeset in TeX or LaTex.
You may find this this essay on the art of writing mathematics to be of use. Another useful article is this article by N. David Mermin. The term papers will be due at the end of the scheduled time for the final exam: 10pm on Monday May 11th People who figure in our story:. Hopf W. Office: ESB. Phone:
Topology and Geometry for Physics
Phys Spring Tuesday, Thursday am to am. Online only: Zoom invites will be sent to registered students. Covariant and contravariant vectors and tensors. Vector fields: Lie bracket and integrability. Differential forms: exterior differentiation, Poincare's theorem, integration of p-forms, Stokes' theorem. Introduction to topology: homology and cohomology.
Differential geometry is a mathematical discipline that uses the techniques of differential calculus , integral calculus , linear algebra and multilinear algebra to study problems in geometry. The theory of plane and space curves and surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century. Since the late 19th century, differential geometry has grown into a field concerned more generally with the geometric structures on differentiable manifolds. Differential geometry is closely related to differential topology and the geometric aspects of the theory of differential equations. The differential geometry of surfaces captures many of the key ideas and techniques endemic to this field.
DIFFERENTIAL GEOMETRIC METHODS IN PHYSICS: AN INTRODUCTION TO YANG-MILLS THEORIES
Szczyrba 6. Scanned Lecture notes. Rather than giving all the basic information or touching upon every topic in the field, this work treats various selected topics in differential geometry. The presentation assumes knowledge of the elements of modern algebra groups, vector spaces, etc.
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It seems that you're in Germany. We have a dedicated site for Germany. This book provides an introduction to topology, differential topology, and differential geometry. It is based on manuscripts refined through use in a variety of lecture courses. The first chapter covers elementary results and concepts from point-set topology. An exception is the Jordan Curve Theorem, which is proved for polygonal paths and is intended to give students a first glimpse into the nature of deeper topological problems. The second chapter of the book introduces manifolds and Lie groups, and examines a wide assortment of examples.
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