Curriculum: Mathematical Physics

Concept for a five year program (undergraduate plus graduate) for Mathematical Physics

Mission Statement:

Spend five years of your life to learn in a unique atmosphere with great people everything you need to know to become an excellent physicist and mathematician.

Overview:

This exposé describes a concept for a five year program in Mathematical Physics of the highest standard - both in the terms of the content and the way it is taught. From the beginning mathematics and physics are on the same footage and after the first two year the students should have an equally well-founded background in mathematics and physics - not until the third year the students can specialise on a variety of different topics between the fields. The fourth year can be used to enjoy an exchange program with another university or to specialise further in advanced fields, while the fifth year is for an elaborated research project. During the whole program the students have additional tutorials in small groups, supervisions alone or for two and discussion hours with present lecturers. Furthermore, additional two or three week schools at different places around the world will help to prepare the students for new topics and make them curious for the connection between the courses.

Concept of Principal Dean:

A very important institution is the Principal Dean who is in charge for a year and responsible for the fine tuning between the different lectures and their content. He informs all the lectures about the progress of the classes, so that every lecture exactly knows what the students learn in the different classes. Furthermore, the Principal Dean is together with the fellowss responsible for correcting the Lecture Notes, optimizing the Problem Sheets and providing optimal Solutions.

Concept of the three Terms per year:

Each year consists of three terms with 9 weeks per term. Courses can go over the whole term or just over some weeks. The main part of the evaluation is done in supervisions - however, written or oral exams can be scheduled at the end of the third term per year.

Concept of the three Schools per year:

There will be summer (winter and spring) schools between the terms as part of the curriculum. From beginning the function of the schools is to prepare the students for the following term: They should get a rough overview over the different topics, the most important mathematical methods, some applications and the underlying structure. Therefore, the schools are quite challenging, but it is not expected to provide a complete treatment, but rather to point out the most important, central points of the topics covered. The students shall get a good overview what is going on in the next term and a rough idea how the different topics are related. This is an optimal preparation for the lectures, because they can really concentrate on understanding the subtleties and do not waste time by getting into the new topics. Furthermore, they will get good feeling for individual difficulties and can concentrate on particular aspects over the following term. For the students of the later years the schools become optional and more free: They can organize their own lectures (including finding lecturers) and can mix between the years. They can do also an internship or attend other international schools instead.

There are schools between all terms which go over 2 or 3 weeks and take place at different places in the world (nice places like in the mountains, near the sea, on an island, a little bit isolated). The students have different lectures, additional challenging tutorials or projects they work on in groups. Students of different years will attend the same schools. While the first years have their own lectures, tutorials and projects, students of the later years (for whom the schools are optional) can mix and organize their time more flexible. In particular, they can organize their own seminars and invite lecturers they would like to learn from.

Concept of Lectures:

The purpose of lectures is to tell the students what they need to know. Every lecturer is different, has his own style and should be free to shape his lecture completely free. However, it is very important to introduce efficient methods to ensure a permanent dialogue between lecturer and the students. This includes periodic evaluation, but also regular meetings with the different supervisors and teaching fellows. Lecturers are expected to be experts in the particular field and to give innovative lectures. Hereby we do not define what an innovative lecture has to be, but describe an idea of what it could be: thorough knowledge of a field often comes with hard work and with developing an individual perspective on what the most important current problems are, what key points should be understood and where conceptual difficulties lie. Therefore, we hope that a true expert of his field will almost automatically give an innovative lecture incorporating his current up-to-date perspective on the field.

Concept of Lecture Notes:

For every course there are particular prepared lecture notes. In the first years these lecture notes are prepared in cooperation of the lecturer and fellows. Later on the lecturers are encouraged to improve the existing lecture notes to have a convergent process of continuously improving lecture notes. Students are also encouraged to give additional comments and material. This can be realised in form of a wiki where students and lecturers can edit, improve and comment the notes, but at the end of a course this wiki should provide the opportunity to print out well structured lecture notes in LaTeX standard while the future students can still see all comments and corrections in the wiki version. All notes are free accessible for all Internet users (but without the right to edit or to comment).

[This point may lead to problems and should be thought through: different lecturers will have very different teaching styles and it might not be possible to use much from former lecturers. However, it is assumed that well-written comprehensive lecture notes that are continuously improved are very useful to support the students.]

Concept of Applications:

Wednesday is reserved for applications of the current material: Most of the time people will be part of small groups (for example 8 persons per group with pairs of 2). In these small groups people can work on different projects connected to the material. During the first years these applications contain introductory seminars (with presentations), LaTeX, experimental labs, programming, advanced experimental lab and sophisticated seminars (research presentation, blackboard talk). Each group is  supported by a fellow and can organize its time very flexible. Detailed explanations of the project task are available in advance, so that the students can prepare. The projects shall be designed in a way that they help to gain a better understanding of the material.

Concept of Tutorials:

The students work together on problems which encourage to discuss and deepen the understanding the material. Concrete calculations shall help to develop an intuition for the material and the underlying structures. The environment for the tutorials should be an open space with different opportunities for discussion and collaboration (tables, cosy sofas, nice corners with chairs, enough blackboards around) - fellows are available to ask questions. The fellows should ensure that everybody can cope with the problems and if not they can support individual students.

Concept of Supervisions:

Supervisions are intense discussions between a teaching fellow and small groups of maximal four students. The content of the supervisions can be freely chosen by the teaching fellow based on the student’s need.

Concept of Evaluation

There are two different types of evaluation: Assessment of the personal learn progress and grading with marks relative to other students (also compared to students of other universities). While the former concept is of real importance to make the individual learn experience as good as possible, the latter way of grading is a necessary evil due to the fact that in our current education system marks and gradings are important for applications.

Assessment: For all courses there is a detailed “List of personal progress’, reporting skills, knowledge and deeper understanding the students are aimed to gain by taking them. These lists contain both particular contents of a course as “understanding of the proof for the implicit function theorem” and general concepts as “understanding and ability to apply integration by parts”. Every week the supervisors update these lists (and comment if necessary), so that the student knows exactly where he has to work harder (of course beside this the supervisor advises the students how to proceed). All kind of problems and misunderstanding of the current material can instantaneously lead to additional discussions, reading recommendations or training problems to ensure an almost complete understanding. The “List of personal progress” is online accessible by the students and all lecturer/fellows/supervisors - an online tool enables them to get detailed statistics of the whole class, so that general difficulties can be easily noticed.

Grading: There are two different ways of grading, namely oral and written exams. While supervisions and problem sheets give already a detailed picture of the student’s abilities, additional exams can complete this picture in a seemingly objective way. The meaning of the different marks has to be defined in a proper way and such that it is comparable to the marks of comparable universities. Because the “List of personal progress” should be in agreement with the grading, discrepancies between them can cause additional supervisions or the obligation to retake exams or parts of it. The students have to show a complete understanding of the rough course content to pass - a failing causes immediately additional support. If a student is over-strained by the program and additional support does not help the responsible fellows/supervisors assist to find a proper solution as retaking classes or moving to another program which is less intense.


Content:

Year 1: CLASSICAL PHYSICS

Courses:

Mathematics: Calculus I, Calculus II, Calculus III, Linear Algebra I, Linear Algebra II, Differential Geometry I, Algebraic Structures I (Group theory)

Physics: Classical Mechanics, Thermodynamics, Geometrical Optics, Electrodynamics

Application: Overview over Mathematics and Physics, LaTeX, Experimental Lab, Numerical Methods,

Additional: All students have to attend the three offered schools to give an overview over the next semester: Basic Mathematics and theoretical Physics, Symplectic Geometry, Electrodynamics

Year 2: MODERN PHYSICS

Courses:

Mathematics: Complex Analysis, Calculus IV, Differential Equations,

Physics: Electrodynamics, Special Relativity, General Relativity, Quantum Mechanics, Statistical Physics,

Applications: Programming,

Additional: All students have to attend the three offered schools to give an overview over the next semester: Relativity, Quantum Mechanics, Statistical Physics

Year 3: APPLICATIONS

Available Courses (partly free choice):

Mathematics: Algebra, Topology (Characteristic classes), Number theory, Partial differential equations, Stochastic, ...

Physics: Relativistic Quantum Mechanics, Introduction to Quantum Field Theory, Elementary Particles, Condensed Matter, ...

Applications: Advanced experimental lab, seminar (topic presentation), seminar (blackboard talk)

Additional: Students are encouraged to organize their own schools (convincing lecturers, combining topics, working also with other sciences). Furthermore, all students are expected to do a summer internship before the third year begins.

Year 4: ADVANCED APPLICATIONS - can be used for an international exchange program

Available Courses (free choice):

Mathematics: Algebraic Geometry, Algebraic Topology, Functional analysis, ...

Physics: Advanced General Relativity, Standard Model, Quantum Field Theory, Cosmology, String Theory, Algebraic Topology, Conformal Field Theory, String Theory, Condensed Matter, Quantum Gravity, Quantum Information, Quantum Foundation, ...

Applications: Group seminars, Problem classes, self-organized seminars, ...

Additional: Summer research can be used to write a paper, but also to do an internship or to travel around. A part of the students is expected to help organizing the summer schools for the first years.

Year 5: RESEARCH PROJECT

Available Courses (free choice):

Mathematics: All courses from Year 4 can be chosen.

Physics: All courses from Year 4 can be chosen.

Application: All courses from Year 4 can be chosen.

Additional: One-year research project with thesis and the aim to publish. A part of the students is expected to help organizing the summer schools for the first years.

Weekly Schedule for the Terms:

Monday

Tuesday

Wednesday

Thursday

Friday

08:00-09:00

Breakfast

Breakfast

Breakfast

Breakfast

Breakfast

09:00-10:30

Lecture

Lecture

Application

Lecture

Lecture

11:00-12:30

Lecture

Lecture

Application

Lecture

Lecture

12:30-14:00

Lunch

Lunch

Lunch

Lunch

Lunch

14:00-16:00

Tutorial

Tutorial

Application

Tutorial

Discussion

16:00-18:00

Supervision

Supervision

Application

Supervision

Discussion

18:00-19:00

Dinner

Dinner

Dinner

Dinner

Dinner

Weekly Schedule for the Schools:

Monday

Tuesday

Wednesday

Thursday

Friday

Saturday

Sunday

08:00-9:00

Breakfast

Breakfast

Breakfast

Breakfast

Breakfast

Breakfast

Breakfast

09:00-10:30

Lecture

Lecture

Lecture

Lecture

Discussion

11:00-12:30

Seminar

Seminar

Seminar

Seminar

Discussion

12:30-14:00

Lunch

Lunch

Lunch

Lunch

Lunch

Lunch

Lunch

14:00-17:00

Tutorial

Tutorial

Tutorial

18:00-19:00

Dinner

Dinner

Dinner

Dinner

Dinner

Dinner

Dinner

20:00-21:30

Talk

Talk

Talk

Talk

Party

Talk