Mathematical Physics  

Coördinator: Prof. dr. Erik Koelink

From the time of Newton (1642-1727) until about 1930, mathematics and theoretical physics were inseparable. Breakthroughs typically took place simultaneously in both areas, and progress at both fronts often even resulted from the work of a single scientist, such as Newton himself, Huygens, Euler, Lagrange, Laplace, Fourier, Gauss, Poisson, Cauchy, Jacobi, Hamilton, Riemann, and Poincaré. Their work provided the foundations of 'classical' mathematical physics (as well as of large areas of mathematics), which culminates in the field of partial differential equations (the Maxwell equations are a case in point). A second stage in the development of mathematical physics is connected with some of the greatest names in 20th century science, like Einstein, Born, Dirac and Wigner on the physics side, and Hilbert, Weyl, von Neumann and Kolmogorov on the maths side. The cross-fertilization of mathematics and physics led by these people was instrumental in establishing key areas of modern physics like general relativity and quantum mechanics as well as parts of mathematics like differential geometry, Lie groups and functional analysis.

This typical cross-fertilization subsided between about 1930-1975, when research at the frontiers of physics felt no need for advanced or new mathematics (whose relevance to physics was even openly derided by Feynman), whilst simultaneously mathematics began to be developed according to its own internal criteria established by Hilbert and others (notably the French Bourbaki group). (In addition, after 1945 some of the greatest mathematicians like Grothendieck refused to make use of insights from modern physics because of its connection to nuclear
weapons.) This has led, for example, to the creation of modern algebraic geometry and algebraic number theory by Weil, Grothendieck, Serre, Deligne, and others. This development may be said to have culminated in the extremely deep and abstract proof of Fermat's Last Theorem by Wiles in the mid-1990s. From about 1975, however, mathematical physics has began to regain the élan it used to have.

First, mathematicians like Atiyah, Singer and Penrose, and physicists like 't Hooft and Witten recognized the connection between differential geometry and gauge theories. This connection goes via the notion of index theory (originating in analysis) and is crucial, for example, in the technical implementation of Sacharov's scenario for baryogenesis mentioned earlier. This recognition has led to very important progress in both physics (magnetic monopoles, instantons, anomalies, and other topological phenomena in classical and quantum field theory) and mathematics (e.g. Donaldson theory and Floer homology). In its immediate wake, deep relationships between algebraic geometry and quantum field theory and string theory were discovered and developed by Witten, Kontsevich and others. The work of Dijkgraaf, Verlinde and Verlinde also played an important role here.

Second, Connes began to develop an entire new domain of mathematics called noncommutative geometry on the basis of ideas from quantum physics (e.g. the Dirac equation), operator algebras (an area of mathematics created by von Neumann in the 1930's) and index theory (the field launched by Atiyah and Singer just mentioned). This body of work has led to breakthroughs in a number of areas in pure mathematics (like index theory and foliation theory). Furthermore, as might have been expected, the subject was successfully applied to physics within a decade after its inception, for example to the quantum Hall effect, the theory of quasicrystals, and the Standard Model of elementary particle physics. More recently, noncommutative geometry has also been related to renormalization theory in perturbative quantum field theory, and to string theory.

Third, the classical area of integrable systems (going back to Lagrange, Jacobi and others, with important later contributions by Lax) underwent a complete rejuvenation in that it got related to the Langlands program (originally an area of pure mathematics in which number theory and representation theory interacted). Also, the notion of a Frobenius manifolds emerged from the work of Witten, Manin, Dijkgraaf and others as a new setting for integrable systems. These three areas together have culminated in an independent field of research called the geometric Langlands program.

The above developments have been widely recognized by the mathematical community. Atiyah has been awarded both of the two most prestigious prizes in mathematics, viz. the Abel prize (with Singer) and the Fields Medal. Connes and Kontsevich won the Fields Medal as well, as did Witten (although he is a physicist). Lax and Serre were  awarded the Abel Prize. Penrose has won the Wolf Prize and numerous other awards, as did 't Hooft, who won the Nobel prize. And so on and so forth.

Research in Nijmegen
Each of the three current research directions in mathematical physics just mentioned is well represented at Nijmegen, and there are other themes as well, so students interested in research in mathematical physics are offered a rich choice. Clauwens' research is in algebraic topology. Heckman's research lies in the interaction between Lie theory, integrable systems, and geometry, and is closely related to the third topic above. It is currently centered around the link between the geometric Langlands program and Hitchin's integrable system.
Koelink's research is on the interplay of representation theory of Lie and quantum groups and special functions as well as the interaction of quantum groups and operator algebras and non-commutative geometry.
  Landsman's research combines noncommutative geometry with quantization theory (i.e. the theory that tries to establish the precise mathematical relationship between classical and quantum physics). One goal is the quantization of singular spaces, in the hope of eventually developing a quantum theory of the Big Bang (a purely classical notion which according to Stephen Hawking and others is probably smoothened out in quantum theory). Maassen works in quantum probability, including the application of stochastic calculus to the interaction of molecules with light, and the functional analysis of quantum noise. In addition he works in the modern theory of quantum information and quantum computing.
Van Suijlekom's main interest is in the mathematical structure of quantum gauge theories: the building blocks of the Standard Model of elementary particle physics. Again, noncommutative geometry provides an ideal mathematical setup for studying these theories.  Müger specializes in category theory, operator algebras and their applications to quantum field theory. He is also interested in constructive quantum field theory. Steenbrink's work is mainly in algebraic geometry. Over the last few years it has focused on two streams: the study of discriminant complements and moduli spaces, and the study of threefolds which are double covers of projective space (double solids).

Preparation
When coming from the BSc Program in mathematics, the student is recommended to prepare for the MSc Program by filling in the free space in the BSc Program with (a slection of) the following courses:

1st semester:
Mechanics 1B (3 ec)
Mechanics 2B (3 ec),

2nd semester:
Mechanical waves(3 ec),
Introduction to quantum mechanics (3 ec)

3d semester:
Analytical mechanics (3 ec),
Special relativity (3 ec)
Vibrations and waves (3 ec)

4th semester:
Quantum mechanics 1a & 1b (6 ec),
Electricity and magnetism 1 & 2 (6 ec),
Introduction Fourier theory (3 ec),
Curves and surfaces (3 ec)

5th semester:
Introduction to Functional Analysis (6 ec)
Quantum mechanics 2 (6 ec)

6th semester:
Introduction to partial differential equations (6 ec)
Quantum mechanics 3 (6 ec)

Up to 18 ec of courses from the 5th and 6th semester may be taken in the master programme.

Master programme
The master programme is tailor made for each student taking the research variant in Mathematical Physics. We advise the to contact as soon as possible a supervisor in order to compose a programme for the research master Mathematical Physics. In case you don't know who should act as your supervisor, please contact prof.dr. E. Koelink as soon as possible.

In 2010-2011 the following courses fit well into the research master Mathematical Physics:

Furthermore the following courses are offered in the National Master Program or by the MRI that fit well with the research master Mathematical Physics:

A selection of courses form the Physics MSc programme can be taken, such as Group Theory, Classical Electrodynamics, Statistical Mechanics and Quantum Mechanics. Moreover related courses from Applied Stochastics, such as Spin Glasses and Neural Networks, Information Theory can also be taken. .