The Scientific Context:

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1 The Scientific Context: The role of mathematics in our understanding of nature has been recognized for millennia. Its importance is especially poignant in modern theoretical physics as the cost of experiment escalates and the mathematical complexity of physical theories increases. Major challenge: Discover the mathematical mechanisms to define quantum field theory as a mathematical entity, to justify mathematically the recipes used by physicists, and to unify quantum theory with gravity. Classical and quantum mechanics rests on solid mathematical foundations, which were developed over the last two centuries. In contrast, quantum field theory, which plays a central role in modern theoretical physics, lacks a mathematical foundation. The mathematical difficulties involve averaging over infinite-dimensional spaces, which have defied rigorous interpretation and yet are effective in practice according to explicit recipes in physics. For example, the Standard Model of particle physics successfully predicts to a dozen decimals experimental results from high-energy colliders such as CERN or SLAC. Quantum field theory has enjoyed several important successes in the last two decades where a number of models were constructed despite the difficulties described above. In all these cases, the theory admits a large family of symmetries, and the problematic infinite-dimensional averages reduce to averages over smaller ensembles, called moduli spaces, which are in many cases finite-dimensional and permit mathematically rigorous calculations. Also, low-dimensional models unifying gravity with quantum theory have been developed successfully, and the key geometrical objects are again moduli spaces. Quantum geometry, so-named since the quantization arises from geometric methods designed to make a passage from classical geometry to quantum theory, plays the fundamental role in all these constructions. Quantum geometry of Moduli Spaces combines the many ideas and techniques developed over the last decades and provides the current most promising resolution of the major challenge for a large class of quantum field theories.

2 Research Mission: In response to this need for a systematic mathematical approach to quantum field theory and string theory, we propose to establish the Center for the Quantum Geometry of Moduli Spaces at the University of Aarhus, with the aspiration to become the world-leading center at this crucial interface between mathematics and theoretical physics. The conceptual goals of the proposed Center are: to significantly advance the understanding of moduli spaces and their quantization in order to provide the mathematical underpinnings of contemporary and future physical theories; to train the next generation of scholars and researchers at the Center to be the world-leading experts on this important confluence of mathematics and theoretical physics; to build the Center into a world-renowned catalyst of collaborative cutting-edge research and training in the quantum geometry of moduli spaces. The specific research mission of the proposed Center is to develop the quantum geometry of moduli spaces so as to provide complete mathematical models for a number of quantum field theories. In particular four specific deep research topics will receive our primary attention: The Geometric Langlands Program and Higgsbundle Moduli Spaces Toeplitz operators and geometric quantization of moduli spaces Combinatorial models for moduli spaces and quantum moduli spaces Quantum representation theory and perturbative invariants Topologically twisted quantum Yang-Mills theories.the standard model of particle physics is a quantum Yang- Mills theory. Quantum Chern-Simons theory. Two spatial and one time dimension. Quantum gravity so far only in one and two spatial dimensions plus one time dimension. Relevant for our understanding all quantum field theories.

3 The proposed Center: To meet these objectives, the Center will build on the existing Niels Bohr Professorship initiative around Professor Nicolai Reshetikhin, the Aarhus based researchers involved in this initiative and the regularly recurring long-term visitors it has attracted to Aarhus, in particular: Professor Robert Penner, recurring visitor from University of Southern California; Professor Vladimir Fock. recurring visitor from Strasbourg University; In addition, Professor Maxim Kontsevich, IHES Paris, recipient of the 2000 Fields Medal (the mathematical equivalent of the Nobel Prize) and the 2008 Crawford Prize; Savilian Professor of Geometry Nigel Hitchin, University of Oxford, recipient of the 2000 Sylvester Medal and 2002 Polya Prize; are enthusiastic partners in the Center and its activities contributing their time, effort, and institutional funds and resources. These are outstanding scholars whose commitment ensures the success of the Center. The research programs of these personnel are intrinsically and deeply inter-connected forming a natural base for their collaborative research. The expected advances and milestones detailed in our joint research mission will constitute major progress in our mathematical understanding of the world around us and the forces which govern it. Further unexpected practical spin-off to the protein folding problem in Biology and Quantum Computing in Computer Science/Physics are under development. The scientific strength and visibility of the team and their existing collaborations with one another and around the globe warrant the success of the Center in achieving its ambitious intellectual goals, in attracting the highest level international collaborators and affiliates, and in educating and stimulating the next generation of Danish mathematicians and mathematical physicists. The Human Ressource Mission includes the opportunity for Danish students and post docs to participate at the apex of this exciting field and take part in breakthroughs among this elite team.

4 The proposed center director Jørgen Ellegaard Andersen born 16 th of June D.Phil in Mathematics from Oxford University in C.B. Morrey Jr. Assistant Professor at Department of Mathematics at U. C. Berkeley, where he participated in and co-directed several NSF grants during 1992 to Associate Professor of Mathematics at in The Danish Bank Scholarship Prize in 2000 was spent at the Mathematical Sciences Research Institute in Berkeley. Clay Professor at UC Berkeley funded by the Clay Mathematics Institute 2001 to Returned to in 2002 and in 2007 he was appointed professor of mathematics. International leader in the field of quantization of moduli spaces, Hitchin's connection and its relation to Toeplitz operators and the Reshetikhin-Turaev quantum field theories, where he has produced a number of breakthrough results written up in 34 research papers published in international journals. The main research results include: The asymptotic faithfulness of the Reshetikhin-Turaev quantum field theory representations of the mapping class groups. Resolution of the Property T question for mapping class groups using quantum geometry of moduli spaces, a long-standing problem in geometric group theory. The Reshtikhin-Turaev quantum field theory determines the unknot. He currently directs: CTQM, initiated in 2006 based on a grant from FNU; The Niels Bohr Professorship (Nicolai Reshetikhin) funded by Danish National Research Foundation; GEOMAPS, a PhD-school coordinating Danish research groups on the interface of geometry and theoretical physics; ITGP, a European Science Foundation network which involves the top 170 researcher within the area of Interactions of Low-Dimensional Topology and Geometry with Mathematical Physics from 21 countries. Editor of two top international mathematics journals. Supervisor of 9 PhD-students, five of whom currently work on Quantum Geometry of Moduli Spaces. Plenary speaker at numerous international conferences, master classes and distinguished lecture series.

5 The Principal Researchers: Professor Jørgen Ellegaard Andersen Professor Henning Haahr Andersen Assoc. professor Alexei Venkov Assoc. Professor Anna Beliakova Niels Bohr Professor Nicolai Reshetikhin Recurring visitor from U.C. Berkeley Professor Robert Penner Recurring visitor from U. Southern California Professor Vladimir Fock Recurring visitor from Strasbourg University Professor Maxim Kontsevich, Fieldsmedalist Institut des Hautes Études Scientifiques Savilian Professor Nigel Hitchin Oxford University