Stefan Müller

Research Fellow (Postdoc)
School of Mathematics
Korea Institute for Advanced Study
Mailing address
office 1426
phone: +82-2-958-3722
fax: +82-2-958-3786 (shared)
email: mueller{at}

Research and Interests

Mathematics can only be understood backwards; but it must be lived forwards.
— freely adapted from Søren Kierkegaard, Journals IV A 164 (1843)

My primary interests and areas of research are

  • Topological Hamiltonian and contact dynamics
  • Symplectic and contact geometry and topology
  • Topological dynamics

and the deeper topological structure underlying symplectic and contact topology. Topological Hamiltonian dynamics and topological contact dynamics are natural and genuine extensions of smooth Hamiltonian and contact dynamics to topological dynamics, and of symplectic and contact diffeomorphisms to homeomorphisms as transformations preserving the additional geometric structure. These new theories have numerous applications to their smooth counterparts, as well as to other areas of mathematics, such as topological dynamics, in particular in low dimensions, and to geodesic flows on Riemannian manifolds. Applications are also expected to billiard dynamics. I also have an interest in action selectors in Hamiltonian Floer theory and related subjects.

A topological Hamiltonian or contact dynamical system consists of a topological Hamiltonian or contact isotopy (a continuous isotopy of homeomorphisms), together with a possibly non-smooth topological Hamiltonian function on the underlying symplectic or contact manifold, and a topological conformal factor (a continuous function) in the contact case. By definition, this Hamiltonian function is the limit of a sequence of smooth Hamiltonian functions with respect to the usual Hofer metric. Moreover, the corresponding sequence of smooth Hamiltonian or contact isotopies converges uniformly to the above continuous isotopy, and their smooth conformal factors converge uniformly to the above continuous function associated to the limit isotopy. In other words, we consider limits of smooth Hamiltonian and contact dynamical systems with respect to a metric that combines topological and dynamical information. Similarly, topological automorphisms of a symplectic or contact structure or form are C^0-limits of symplectic or contact diffeomorphisms (together with their conformal factors in the contact case).

A topological Hamiltonian or contact isotopy is not generated by a vector field, and may not even be Lipschitz continuous; nonetheless, it is uniquely determined by its associated topological Hamiltonian function, and in turn, in the contact case it determines uniquely its topological conformal factor. Composition and inversion can therefore be defined as in the smooth case, and the usual transformation law continues to hold. Conversely, every topological Hamiltonian or contact isotopy possesses a unique topological Hamiltonian function. The topological automorphism groups of a contact structure and a contact form exhibit surprising rigidity properties analogous to the well-known Eliashberg-Gromov rigidity in the case of a symplectic structure.

The uniqueness theorems for topological Hamiltonian and contact dynamical systems, rigidity, the group structures, and the transformation law, provide ample evidence that Hamiltonian and contact dynamics are in fact topological theories, and a priori smooth invariants can be extended to topological Hamiltonian and contact dynamical systems. The study of these topological theories is to a large extend motivated by physical considerations and by various continuity phenomena already present in their smooth counterparts, and in turn, they have applications to smooth Hamiltonian and contact dynamics as well as to topological dynamics and low-dimensional topology. For example, one can answer important questions of V. I. Arnold regarding the continuity of the helicity of a volume preserving isotopy, and its behavior under conjugation by volume preserving homeomorphisms, provided the isotopy and the transformation can be described as lifts from a manifold one dimension lower. The relation between topological Hamiltonian dynamics and topological contact dynamics is … [More]

Publications and Preprints

The truth is... to overcome the terrible inhibitions that almost every writer faces when
he has to stop vaguely talking and dreaming about his work; when the time comes to sit
down and write definite text whose short-comings stare bleakly back at their creator...

— Hans Selye, From Dream to Discovery


Ph.D. thesis

  • The group of Hamiltonian homeomorphisms and C^0-symplectic topology, Ph.D. thesis, the University of Wisconsin - Madison, 2008, 104+v pages

ArXiv Preprints

In preparation

To view this list of publications and preprints with a brief summary of each item or as a pdf file click [here].


When [Erwin Schrödinger] went to the Solvay conferences in Brussels, he would walk from the station
to the hotel where the delegates stayed, carrying all his luggage in a rucksack and looking so like a
tramp that it needed a great deal of argument at the reception desk before he could claim a room.

— Paul A. M. Dirac, quoted in Robert L. Weber, Pioneers of Science: Nobel Prize Winners in Physics (1980)

A comprehensive list of conferences I attended or intend to attend in 2011 and 2012. Conferences marked with an * are those where I gave or will give a research talk.


I hear and I forget. I see and I remember. I do and I understand.
— Chinese Proverb

My present position at Korea Institute for Advanced Study is a research-only fellowship. In the past, I have been a teaching assistant at the University of Wisconsin-Madison during my years as a graduate student. I have taught many classes as a lecturer at various college levels, while other classes were in the lecture-discussion format with divided teaching responsibilities. In addition I have experience in teaching summer classes for underprivileged high school students entering college, as well as for graduate students preparing for qualifying exams in geometry and topology. In the fall of 2008, I taught a seminar lecture series at Seoul National University on action selectors in Hamiltonian Floer theory. Details can be found in my curriculum vitae and my teaching statement.


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