Professor Nicolai Reshetikhin from
Department of Mathematics, University of
California, Berkeley, is a world scholar of Mathematics and
Mathematical Physics. The affiliation of Professor Reshetikhin as
a Danish National Research Foundation Niels Bohr Visiting Professor
during the coming 5 years will help to consolidate and
considerably further extend CTQM's position among the leading
research centers. He has previous collaborations with the director
of CTQM, Associate
Professor Jørgen Ellegard Andersen, and the
continuation of this work on quantization of moduli spaces will be
the focal point of the interaction and integration of Professor
Reshetikhin's vast expertise. Professor Reshetikhin has
considerable research interest in common with several of the other
researchers at the center as well as a number of researchers in
the *Quantum algebra groups* at the Department of Mathematical
Sciences.

On the mathematical
physics side, his presence will give a boost to quantum field
theory and string theory in Denmark and
enhance the development of this area between mathematics and physics
planned at the newly created Center for Theory in the Natural
Sciences at the Faculty of
Science at Aarhus University.

Professor Nicolai Reshetikhin was born on the 10'th of October 1958 in St. Petersburg, where he, in 1984, received his Ph.D from the Steklov Mathematical Institute. After a two-year prize fellowship from Harvard in 1989--91 and the distinction of giving an address at the age of 32 at the International Congress of Mathematicians in 1990 in Kyoto, he was hired at University of California, Berkeley and has been a full professor of mathematics there since 1993. He is further the winner of a Sloan Fellowship, a Humboldt award and he is on the editorial board of a number of international mathematics and theoretical physics journals.

Professor Nicolai Reshetikhin's research area is naturally split into
three fields: * Quantum topology and moduli spaces, Quantum
algebras and their representations* and *Statistical mechanics
related to quantum gauge theory and string theory*.
He is one of
the two founders of the field "Quantum topology" and he belongs
to the absolute elite in all three fields, in which he has
authored over a 100 research papers all published in international journals.
A short version of Professor Reshetikhin's CV is
available.

See also:

- University of Aarhus press release.
- Danish National Research Foundation press release.
- Jyllands-posten: "Verdensklasse-forskere til Aarhus Universitet"

#### Scientific context

One of the long term challenges of science is to understand and describe the universe around us, and the fundamental forces which govern it.

In the modern picture of matter, fundamental forces such as electromagnetic, weak, and strong interactions are described by the Grand Unified Theory (also known as the Standard model). It is a complex local quantum field theory and despite the fact that everything that is known about it is confirmed by experiments, many mathematical problems in the theory remain unresolved. One such is the confinement problem. The Clay Mathematics Institute at Harvard identified this problem as one out of the seven Millennium problems with the prize of one million dollars attached.

Yet one of the most fundamental forces in nature, gravity, is not included in the standard model. The reason is simple: up until now nobody knows how to develop a quantum theory of gravity. On the other hand the existing paradigm of a quantized nature of fundamental forces requires it. The most realistic theory which may unify under one paradigm both the classical gravity and the local quantum theory is string theory (Green, Schwarz and Witten). This is why so much effort went into this subject during the last two decades. But the mathematical problems involved in string theory are even more formidable than those for local quantum field theory.

Because of the scales of energies involved, cosmology is basically the only way to verify string theory experimentally. This is one of the goals of the satellite mission Planck, due to be launched in 2007. This connects string theory with cosmology on a very conceptual level.

In quantum field theory there are models that can be "solved". They include: integrable models (describing such important dynamical phenomena as asymptotical freedom and dynamical mass generation), conformal field theories (describing critical phenomena), and topological field theories (capturing the complexity of space-time). Similarly, in string theory, topological strings can be regarded as a model of string theory that focuses on the interaction between the structure of the world-sheet and the target space.

All of these models involve the most advanced mathematical structures, and in many cases, mathematicians has developed them further (such as it was the case with electromagnetism and quantum mechanics half a century ago). For example, integrable, conformal, and topological theories have "hidden" representation theory of quantum groups as a reflection of an "inner symmetry", an amazing beautiful discovery to which N. Reshetikhin has contributed most profoundly. That conformal field theories, topological field theories, and topological string are all based on the theory of moduli spaces and their quantization is largely due to E. Witten (who will visit CTQM this comming summer). The fascinating interaction and interplay between these two viewpoints are described below in the case of quantum Chern-Simons theory.

This brings the theory of moduli spaces and their quantization into the forefront of modern mathematical physics (the term which we reserve for the mathematical wing of theoretical physics).

On the mathematical side the study of moduli spaces goes at least a century back. These ideal and simple geometric objects appeared in mathematics long before they appeared in physics. A vast mathematical knowledge of these spaces including their dimensions, geometric shapes and complicated compactifications has been developed and today the theory of moduli spaces involves a substantial amount of modern abstract mathematics, including large parts of topology, geometry, and algebra.

Many related fundamental problems in the theory of moduli spaces are still outstanding, but recently there has been significant progress on several fronts which includes important advances by N. Reshetikhin, J.E. Andersen and I. Madsen. The aim is to continue this challenging research and make further advances on the topology and quantization of moduli spaces and its relation to quantum invariants of 3-manifolds as discussed below.

#### General Mathematical setting

N. Reshetikhin is one of the two founders of the area of pure mathematics now called "Quantum Topology". - In the late 80'ties, mathematical physicist E. Witten was studying quantum gauge theories, in particular quantum Chern-Simons theory, based on Feynman path integral formulations. He argued that Chern-Simons theory was reproducing V. Jones' famous knot polynomial, a construction from pure mathematics. Shortly thereafter N. Reshetikhin and V. Turaev came up with a mathematical construction of this theory using the concept of Topological Quantum Field Theory (TQFT) -- a concept introduced by M. Atiyah, G. Segal and E. Witten. N. Reshetikhin's ICM talk in 1990 in Kyoto was devoted to this new discovery, which became the beginning of Quantum Topology.

Quantum Topology grew and developed rapidly up through the 90'ties, and is now a well established branch of mathematics. In the last years it became increasingly clear that in order to deepen our knowledge of these quantum invariants of 3-dimensional manifolds one should understand better the corresponding moduli spaces. The fact that these invariants are related to the geometric quantization of the moduli spaces goes all the way back to Witten's work. But many things remain to be done. For example, a full geometric construction of these invariants is still lacking.

Despite this, tremendous progress has been achieved in this direction during the last few years. For example, the asymptotical faithfulness of the natural action of the mapping class groups on the quantized moduli spaces was successfully proven by J.E. Andersen. New invariants of knots and 3-manifolds were constructed where the moduli spaces play an even more essential role. A substantial part of it has been led by N. Reshetikhin and is closely related to the work of V. Fock (who will visit CTQM summer 2006 - Summer 2007) and A. Goncharov on quantum moduli spaces.

Geometric quantization of moduli spaces; representations of corresponding quantum algebras; the relations between quantum invariants, the topology of moduli spaces, and topological strings; will be the main focus of the envisioned cooperation. Within the last few years researchers at CTQM have contributed significantly to the study of the topology and geometric quantization of moduli space and therefore the attachment of N. Reshetikhin and the prospect of this collaboration come at a very opportune moment.

Before N. Reshetikhin's ground-breaking joint work with V. Turaev, his work was chiefly concerned with the integrability of classical and quantum systems and with the corresponding models in statistical mechanics. This led him to pioneering results in the theory of quantum groups and their representations. Among his major results are the development of the hierarchical "Bethe ansatz", the discovery of the quantum algebra analog of SL(2,C) and analysis of its category of representations, exact solutions of a number of important models in 2-dimensional quantum field theory and statistical mechanics. He continues to be an innovating researcher at the forefront in these fields. It is therefore also planed that N. Reshetikhin will cooperate with the mathematicians associated to quantum algebra group at IMF and with the mathematicians and physicist attached CTN.

Revised 19.12.2006

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