Erwin Schrödinger International Institute for Mathematical Physics Contents

2013 Erwin Schrödinger International Institute for Mathematical Physics contents

Erwin Schrödinger International Institute Preface—Turning Over the Page ...... 2 for Mathematical Physics Boltzmanngasse 9 Erwin Schrödinger International Institute

1090 Vienna / Austria for Mathematical Physics—Structure and Activities ...... 5 T +43-1-4277-28301 F +43-1-4277-9283 [email protected] At the Interface of Mathematics and Physics www.esi.ac.at Some Thoughts on the Fruitful and Ongoing Interaction Between Mathematics and Physics

by Wendelin Werner ...... 10

Algebras, Groups and Strings

by Peter Goddard ...... 12

A Deeper Union? Poincaré on Mathematics and Mathematical Physics

by Jeremy Gray ...... 16

A Physicist Involved in Purely Algebraic Research: The Case of H. B. G. Casimir

by Martina R. Schneider ...... 21

Changing Views—Images and Figures of Thought in the Mathematical Sciences

by Joachim Schwermer ...... 25

Scientific Activities

Thematic Programmes...... 32

Research in Teams Programme ...... 33

Junior Research Fellowship Programme ...... 34

Summer Schools ...... 35

Impressions ...... 36

Senior Research Fellowship Programme ...... 38

Gathering Evidence: My Time as a Senior Research Fellow

by James W. Cogdell ...... 39

Data and Statistics ...... 44

1 / esi 2013 preface preface

Preface―Turning Over the Page of these activities add an important personal funding for the Erwin Schrödinger International by Joachim Schwermer perspective to this catalogue of possible scholarly Institute for Mathematical Physics to support the experiences. five-year appointment of a Simons Junior Professor In view of the strategic exchanges between in Mathematics or Mathematical Physics. mathematics and mathematical physics that have It is the Institute’s foremost objective to ad- occurred over the course of the last decades and vance scientific knowledge ranging over a broad even beyond, one finds personal reflections of band of fields and themes in mathematics and scientists on their experiences as scholars whose theoretical physics. Creating a space where fruitful work is rooted in mathematics and physics. They collaborations and the exchange of ideas between Restart. In October 2010, when the Erwin the new governing board of the esi, called the found a creative way to capture their thoughts re- scientists can unfold is decisive. With the esi now Schrödinger International Institute for Math- “Kollegium”, to carry on the mission of the esi garding the interaction between mathematics and being a research centre within the University of ematical Physics (esi) had been in existence as an and to retain its international reputation. These mathematical physics and specific advances in their Vienna, the best way of achieving this is to ensure independent research institute since 1993, the aims include, in particular, to support research at work. In the same vein, historical case studies deal- that the esi continues to interweave leading inter- scientific directorate and the international commu- the University of Vienna, to contribute to its inter- ing with fundamental facets or specific develop- national scholars and the local scientific community. nity of scholars had to learn with great distress of national visibility and appeal, and to stimulate ments in which the interrelations of mathematics The research and the interactions that take place at the intention of the government of Austria to cease the scientific environment in Austria. Setting aside and physics have unfolded extend the discussion. the Institute will have a lasting impact on those funding for the esi. Due to budgetary measures all the technical issues the transition process of Another note follows the interplay between thought who get their scientific education in Vienna. affecting a large number of independent research the esi involved, and which had to be taken care and images in the mathematical sciences and how institutions in Austria, funding of the esi would be of, the esi could begin restoring its fundamental visual thinking may function in the creative process Joachim Schwermer terminated as of January 1st, 2011. Since its start it scientific activities. This was and still is the prevalent in mathematics, physics and their crossroads. Director was the mission of the esi to advance research in task: striving to be excellent and thereby keeping its Finally, a concluding section in this booklet Erwin Schrödinger International Institute mathematics, physics and mathematical physics at position within the international scientific commu- documents to some extent the scientific activities of for Mathematical Physics the highest international level through fruitful nity of scholars as a research institute with a specific the esi, starting from its beginnings. The Institute interaction between scientists from these disciplines. unique character. The esi is a place that is very con- has always remained true to its commitment: pro- An abrupt end for the scientific activities of the ducive to research and, at the same time, integrates moting and supporting original scholarship in the Institute and the closure of the esi appeared on the scientific education and research in mathematics mathematical sciences. horizon. Weeks of trembling uncertainty followed, and mathematical physics. mixed with signs of a solution in which the Univer- In retrospect, though the planning horizon Securing the future. The Erwin Schrödinger Inter­ sity of Vienna would be involved. In the wake of was very short, the Thematic Programmes, scheduled national Institute for Mathematical Physics fills a a protest action by renowned scholars and academic already far ahead for 2011 and 2012, turned out in unique role in (post-) graduate education and sci- institutions worldwide, an agreement was achieved the end to be successful scientific events. Addition- entific research in mathematics and mathematical in January 2011 that the esi could continue to exist ally, various workshops and other activities could be physics, and in Austria, in particular. However, one but now as a research centre (“Forschungsplatt­ solicited for the year 2012 on short notice, involving, has to give careful consideration to ways in which form”) at the University of Vienna. As a partner in in particular, young researchers who came to the the esi arrangements might be made more stable this agreement the Ministry of Science and Research Institute for the first time as organizers. In addition, as it goes into a period of various challenges re- (bmwf) guaranteed to fund the “new” esi through by January 1, 2012, the Erwin Schrödinger Institute garding its financial resources. As mentioned, the University yearly with a reduced budget until had established the Research in Teams Programme the Ministry has only guaranteed to fund the esi 2015. At a time when pure research and scholarly as a new component in its spectrum of scientific through the University until 2015. Thus, in the near activities are undervalued, the opportunities for activities. future a fundamental decision has to be reached scholars and young researchers that the Institute as to whether the government of Austria and the provides have never been more necessary. The This booklet. With the 20th anniversary of the esi University of Vienna are prepared to host and University of Vienna took the chance and created occurring precisely at this critical time, it is neces- fund an institution like the esi, engaged as it is in a home for “one of the world’s leading research sary to utilize this moment, as it were, not only to fundamental research in mathematics and physics. institutes in mathematics and theoretical physics”, celebrate the rich heritage of the Institute but also Due to the lack of sufficient funding schemes in as Peter Goddard, the chair of the international to position us for the future. Specifically, this booklet Austria in such cases, the usual mechanisms of review committee for the Institute, commissioned serves as an overview of the institutional structure third party funding are not feasible in this case. by the bmwf, and its members put it in 2010 in of the Erwin Schrödinger International Institute for The esi is, of course, firmly bound to seeking funds a letter to the Ministry. Mathematical Physics, turned into a research centre from other sources but it is only in the position to at the University of Vienna in June 2011, and the complement adequate basic financial support in Stabilization. With this new institutional framework various programmatic pillars of its scientific activ- this way. With regard to the latter point, as a first in place since June 2011, it was the main task of ities. Recollections of scientists involved in some step, the Simons Foundation recently approved

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ESI―Structure and Activities by Joachim Schwermer

1. The Institute and its Mission • By inviting individual scientists who col­laborate with members of the local scientific community. The Erwin Schrödinger International Institute for Even through the transition period the esi had to Mathematical Physics (esi) was founded in Vienna, go through in the years of 2010 and 2011, the esi Austria, in 1992, and became fully operational in has remained a leading international centre for re- April 1993. On June 1, 2011, the esi assumed its role search in the mathematical sciences. This position as a research centre within the University of Vienna. has been achieved with a minimal deployment of The mission of the Institute is: resources, financial and human, especially when • to advance research in mathematics, physics and compared with similar institutes in other countries. mathematical physics at the highest international level through fruitful interaction between scien- tists from these disciplines; 2. The Institute’s Scientific Management and • to support research at the University of Vienna its Resources and surrounding universities and to stimulate the scientific environment in Austria. The arrangements that provide for the scientific The transition of the Erwin Schrödinger Institute direction and administration of the Institute are from an independent research institute to a “For- perhaps among the noteworthy features of the esi. schungsplattform” at the University of Vienna was Indeed, the Institute is run in a quite minimalist a complicated process. There are far-reaching dif- fashion. ferences in operation as a consequence of the uni- The organizational structure of the esi is as versity’s involvement in the running of the Institute. follows: The esi is governed by a board (“Kolle- This includes issues concerning payments to par- gium”) of six scholars, necessarily faculty members ticipants, modifications to the premises and future of the University of Vienna. These members of the funding prospects. However, the Institute has con- board are appointed by the President (Rektor) of the tinued to function, even flourish, during the radical University after consultations with the Deans of the changes of its status. Faculties of Physics and Mathematics. It currently The Institute currently pursues its mission consists of Goulnara Arzhantseva (Mathematics), in a number of ways: Adrian Constantin (Mathematics), Piotr T. Chruściel • Primarily, by running four to six thematic pro- (Physics), Joachim Schwermer (Mathematics), Frank grammes each year, selected about two years in Verstraete (Physics), and Jakob Yngvason (Physics). advance on the basis of the advice of the Interna- All members of the Kollegium still act as Professors tional esi Scientific Advisory Board. at the University. • By organizing workshops and summer schools at In addition, the Scientific Advisory Board of shorter notice. the esi plays a crucial role in keeping this Institute • By a programme of Senior Research Fellows (srf), alive scientifically. The members of the Scientific who give lecture courses at the esi for graduate Advisory Board of the esi, which currently consists students and postdocs. of seven international scholars, have a variety of • By a programme of Research in Teams, which tasks: they assess the programme proposals sub- offers teams of two to four Erwin Schrödinger mitted to the esi, they point out interesting scien- Institute Scholars the opportunity to work at the tific developments in the area of mathematics and Institute for periods of one to four months, in mathematical physics, and suggest topics and pos- order to concentrate on new collaborative research sible organizers for future activities of the Insti- in mathematics and mathematical physics. tute, and, most importantly, during the yearly

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meeting, they review—and criticize—the scientific scientific budget are used for these activities each 4. Senior Research Fellowship Programme successful and internationally held in high esteem performance of the Institute during the past year year. In addition, smaller programmes, workshops but unfortunately it came to an end because funding and make suggestions for possible improvements. and conferences are organized at shorter notice, as In order to stimulate the interaction of the Institute’s by the bmwf was terminated with the end of 2010. Though the name of the Institute only contains well as visits of individual scholars who collaborate activities with the local community, the Institute The presence of the Junior Research Fellows at the mathematical physics as the subject of concern, with scientists of the University of Vienna and the initiated a Senior Research Fellowship Programme Institute, together with the Fellows of the European mathematics plays an equally important role in its local community. in 2000. Its main aim is attracting internationally Post-Doc Institute, had a very positive impact on the scientific activities. The list (see page 44) of research areas in math- renown scientists to Vienna for longer visits. These esi’s scientific atmosphere through their interaction On June 1, 2011, the Scientific Advisory ematics, physics and mathematical physics covered scholars would interact with graduate students and with participants of the thematic programmes, Board of the esi was restructured. Only scholars by the scientific activities of the Erwin Schrödinger postdocs in Vienna, in particular, by offering lecture through lively discussions with other postdocs and who are not affiliated with a scientific institution in Institute in the years 1993 to 2012 shows a remark- courses on an advanced graduate level. This pro- also through the series of jrf seminars. In conjunc- Austria can be appointed as members. Thus, at the able variety. gramme enables Ph.D. students and young post- tion with the jrf Programme, the esi had regularly same time, its composition changed. For the sake The pages of the annual esi report, available doctoral fellows at the surrounding universities to offered Summer Schools which combined series of of continuity, John Cardy (Oxford), Horst Knörrer on its web page, provide ample evidence that the communicate with leading scientists in their field introductory lectures by international scholars with (eth Zürich), Vincent Rivasseau (Paris) and Herbert high quality of the scientific programmes was sus- of expertise. more advanced seminars in specific research areas. Spohn (München) were reappointed. As new tained and, in particular, undiminished during and However, even though the jrf programme had to members Isabelle Gallagher (Paris), Helge Holden shortly after the radical changes the Institute had to be discarded, the Institute continues its long term (Trondheim) and Daniel Huybrechts (Bonn) joined face. Longer thematic programmes and the open 5. ESI Scholars policy of vertical integration of scientific education the Board, starting January 2012. approach to research they offer and encourage form and research. Summer Schools are still essential The day-to-day functioning of the esi is a fundamental pillar of the work of the esi. The By January 1, 2012, the Erwin Schrödinger Institute components of the scientific activities of the esi. overseen by the Director. The Director is appointed Institute provides a place for focused collaborative had established the Research in Teams Programme In 2010, a “May Seminar in Number theory” by and accountable to the Rektor of the University. research and tries to create the fertile ground for as a new component in its spectrum of scientific took place to introduce young researchers to excit- Besides the ongoing oversight of the esi, the Di- new ideas. activities. The programme offers teams of two to four ing recent developments of current research at the rector chairs the Kollegium, represents the esi at It is generally noted, as already the Review Erwin Schrödinger Institute Scholars the opportunity crossroads of arithmetic and other fields. During meetings of the European Institutes and has re- Panel of the esi pointed out in its report in 2008, to work at the Institute in Vienna for periods of one the summer 2011 a school dealt with recent devel- sponsibility for the budget of the esi. The Director that over the last years the esi has widened the to four months, in order to concentrate on new opments in mathematical physics. Jointly with the makes sure the esi functions in a way manner with range of its thematic programmes and other sci- collaborative research in mathematics and math- European Mathematical Society (ems) and the its mission. entific activities from being originally more nar- ematical physics. The interaction between the team International Association of Mathematical Physics The administrative staff of the Institute, cur- rowly focused within mathematical physics. The members is a central component of this programme. (iamp) the esi organized in 2012 the “Summer rently consisting of three people, two of them work- Scientific Directorate has increased the scope The number of proposals, on themes of topical in- School on Quantum Chaos”. This Instructional ing on part time basis, is also extremely lean but very of the activities mounted by the Institute into areas terest, was high. However, due to limited resources, Workshop attracted more than 45 graduate students, efficient in handling the approximately 450 visitors of mathematics more remote at present from theo- the Kollegium could only accept four of these ap- postdocs and young researchers from all over the per year. retical physics. This process will continue in the plications for the year 2012. The first scholars within world. A poster session accompanied this event. Situated at Boltzmanngasse 9 in Vienna, the same fashion, with special emphasis on the fruitful this programme were at the esi in June 2012. Other esi is housed in the upper floor of a two hundred- interactions between mathematics and mathemati- teams are already accepted for the year 2013. year-old Catholic seminary. This building provides a cal physics. Of course, by invitation only, the esi continues 7. Conclusion quiet and secluded environment. By its distinctive The themes of the programmes which are in to have individual scientists as visitors who pursue character, the esi is a place that is very conducive to place in 2013 range from “The Geometry of Topo- joint work with local scientists. In some cases these With the esi now being a research centre within research. logical d-Branes”, “Jets and Quantum Fields for collaborations originate from previous thematic pro- the University of Vienna, the best way of benefiting The Institute is still funded by the Austrian lhc and Future Colliders” over “Forcing, Large grammes which took place at the esi. the local community is to ensure that the esi con- Federal Ministry for Science and Research, via the Cardinals and Descriptive Set Theory” to “Heights tinues to function at the highest level internation- University of Vienna, but it works on the basis of in Diophantine Geometry, Group Theory and ally and thus attract the world’s leading scholars much smaller resources financially than in the years Additive Combinatorics”. 6. Junior Fellows and Summer Schools to Vienna where their presence will enhance and before 2011. In 2014 the esi will host four thematic stimulate research further. This has been and still is programmes, the first one dealing with “Modern Funding from the Austrian Federal Ministry for the approach successfully followed by the esi. ■ Trends in Topological Quantum Field Theory”, Science and Research (bmwf) enabled the Institute 3. Thematic Programmes and Workshops followed by one centred around “Combinatorics, to establish a Junior Research Fellowship Pro- Geometry and Physics”. The programmes “Topo­ gramme (jrf). Its purpose was to provide support The Institute’s scientific activities are centred around logical Phases and Quantum Matter” and “Minimal for advanced Ph.D. students and postdoctoral four to six larger thematic programmes per year. Energy Point Sets, Lattices and Designs” cover the fellows to allow them to participate in the activities Planning for these programmes typically begins second half of the year, supplemented by various of the esi. Grants were given for periods between two years in advance. About three quarters of the workshops. two and six months. This programme was very

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Some Thoughts on the Fruitful starts to walk and sense gravity, when one starts to boundaries between different randomly created look and feel shapes, textures, colors, occurs simulta- domains, or as level lines of randomly created and Ongoing Interaction Between neously with some sort of intuitive conceptualization mountains. of these experiments in our brain, and one could This interaction between mathematical and therefore even argue that the interaction between physical academic communities is not such an easy Mathematics and Physics physics and mathematics is a natural continuation one. To start with, the vocabulary used, the back- of this very natural dual discovery of the physical ground material that is assumed to be known, the by Wendelin Werner world and its interpretation/understanding. rules of what information a paper should contain, This relation can be very direct (like for in- are very different, so that it is extremely difficult Photo: Antoine Taveneaux stance trying to understand mathematical aspects for a mathematician to extract the useful needed of relativity, which is still an ongoing hot topic in information for his research out of a physics paper, Wendelin Werner is a mathematics) or indirect, more based on analogies and vice-versa. We all spent quite frustrating and German-born French or on abstract generalizations. It is very rare to find long hours looking at a physics paper without mathematician working in a recent important mathematical result that is not understanding its logic. The direct face-to-face the area of self-avoiding random walks, Schramm- related in some way or the other to physics. A look interaction can however be much more productive, Loewner evolution, and related at the list of prize-winners (for instance for the Abel but it requires time, and repetition. Just one subjects in probability and Prize, Shaw Prize, Wolff Prize etc.) backs this conversation is often not enough… In that respect, mathematical physics. In assertion very convincingly. mathematicians in our area have been particularly 2006, at the 25th International My own personal scientific life has in fact not fortunate to be able to benefit from such timely Congress of Mathematicians only been shaped by the motivation that comes from programs that did take place at places such as the in Madrid, he received the the relation to physics of the type that I have just esi or the Isaac Newton Institute for Mathematical Fields Medal. Werner became described, but also more directly from the contact Sciences in Cambridge, where one could meet our a member of the French with physics as an academic field. Indeed, a number friends from theoretical physics, and interact with Academy of Sciences in 2008. of the questions that I have been studying have been them, in the lecture room as well as in office dis- He is professor at the Uni- initially either studied or raised by physicists, with cussions or around a glass of beer. This has certainly versity Paris-Sud and part- time at the École Normale motivations that range from the experimental study helped a lot in the exchange of ideas. Supérieure in Paris. of phase transitions (how spontaneous magnetiza- When one tries to think about scientific tion of iron depends on the temperature, say) to strategy, organization and funding of the academic theoretical questions about the nature of interac- life, the main motivation is to see how to help and tions (related to field theory). In fact, in the par- stimulate novel creative scientific ideas. It is impor- ticular topic that I have been working on, we were tant to pay a salary to academics and more generally Sample of a large critical he mutual cross-fertilization between math- very fortunate that theoretical physicists such as to provide them with good working conditions, but percolation cluster: under- ematics and physics is a well-documented fact, Bertrand Duplantier or John Cardy actually “came to it is also essential to put them in the right “environ- standing such random shapes T and has been a constant feature of the history us” with precise mathematical challenges, i.e. precise ment”, where they can interact with those other in the fine-mesh limit has of science. In today’s fast-changing world, where statements (“conjectures”) that were calling not only scientists that would bring them some fresh, new been the topic of numerous spectacular progress is been made in the under- structures, that seem to have little direct appli- for a rigorous proof, but also for an intuitive under- and maybe unsettling approaches to the questions works in the physics and standing of Life Science, and where the computing cations or even little concrete meaning in the real standing that the physical theories were not able to that they are looking at, to feed their curiosity. This mathematics communities. power provided by technology opens new doors, it outside world. Of course, each mathematician has fully provide. In fact, together with Greg Lawler and is why such places like the esi have played, play and can be questioned whether the progress driven by his/her personal history and motivations, but most Oded Schramm, we also worked on a prediction of will continue to play an essential part in the future this privileged interaction between mathematics and of the time, they are to a large extent emotionally Benoît Mandelbrot, who can also be viewed as a developments of our disciplines. ■ physics belongs to the past or whether it is still, and driven (the recent exhibition by the Fondation mediator between natural sciences and mathematics maybe even more than ever, a major driving force in Cartier “Mathématiques, un dépaysement soudain” (see his book “The fractal geometry of nature”). the evolution of these two disciplines. In these few illustrates this very well). Mathematics is, for many of In this area, key mathematical objects that have led lines, I would like to explain why, based on my us, a way to express or capture some aspects of our to new insight have been the Schramm-Loewner personal mathematical experience, I believe that the interaction with the outside world, and the questions Evolutions (invented by Oded Schramm), that pro- latter case holds, and why some institutions such as we study are often in some sense a continuation of vide random ways to draw curves in the plane, the Erwin Schrödinger Institute have a pivotal role those questions that we faced as children or teen- making use of conformal transformations of the to play in this direction. agers, when learning about how our physical world plane (these are the angle-preserving distortions A first preliminary question is to wonder what is structured and functions. After all, the experimen- of portions of the plane), as they turn out to be the reasons are that lead some minds to choose to tal exploration of the outside world as a baby, when the only natural candidates to describe curves that devote their life to study abstract mathematical one learns how to move arms in space, where one appear in various parts of physics, for instance as

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Algebras, Groups and Strings to add resonance pole contributions to Regge essential in the string theory context, is now References by Peter Goddard contributions. However, it became apparent that, named the Virasoro algebra.⁽³⁾ The proposition that (1) G. Veneziano, Nuovo Cimento more ambitiously, it might be considered as the Virasoro’s conditions did indeed eliminate the ghost 57A (1968) 190–197. starting point (or ‘Born term’) for a perturbative states was then a mathematically precise conjec- (2) J. H. Conway, Proc. Nat. Acad. ture that could be formulated just in terms of the Sci. USA 61 (1968) 398–400; Bull. series expansion for the scattering amplitude in a London Math. Soc. 1 (1969) fundamental theory of the strong interactions, rather Virasoro algebra: it was the requirement that a 79–88. than just a provisional phenomenological descrip- certain space defined by the algebra was positive (3) M. Virasoro, Phys. Rev. D1 Emerging and interweaving connections between finite group theory, infinite-dimensional tion. For this purpose, in addition to generalizations semi-definite. With the efforts of several theoretical (1970) 2933–2936. symmetries, and string theory are related from a personal perspective, illustrating the symbi- to the scattering of arbitrary number of particles, physicists⁽⁴⁾, within two and a half years, the result, (4) R. Brower and C. B. Thorn, we also need higher order, or loop, contributions known as the No-Ghost Theorem, was proved. Nucl. Phys. B31 (1971) 163–182; otic relationship between mathematics and theoretical physics. E. Del Giudice, P. Di Vecchia and to ensure consistency with unitarity, i.e. conserva- A second severe challenge in the development S. Fubini, Ann. Phys. 70 (1972) tion of probability. of the theory of dual models was the consistent 378–398; he birth of string theory, or more accurately ematicians of exciting developments in a very Some clear obstacles were present to devel­ construction of loop contributions. Typically, there R. C. Brower and P. Goddard, Nucl. Phys. B40 (1972) 437–444; its conception, was announced in the Hofburg different area, namely, finite group theory. John oping dual models into a consistent fundamental are divergences in loops that have to be removed by R. C. Brower, Phys. Rev. D6 (1972) T in Vienna forty-five years ago, in the late sum- Conway had become intrigued with the Leech lat- theory. In particular, quantum theories consistent renormalization (unless, in very special theories, 1655–1662; P. Goddard and C. B. Thorn, Phys.

Photo: Cliff Moore mer of 1968. Gabriele Veneziano proposed to the tice, a lattice of points in 24-dimensional Euclidean with special relativity are likely to generate negative they cancel as the result of supersymmetry). In Lett. 40B (1972) 235–238. 14th International Conference on High Energy space describing an extremely close packing of answers for some of the quantum mechanical prob- certain dual model loops, a new problem arose: an (5) C. Lovelace, Pomeron Form Peter Goddard is a math- Physics an explicit formula for a ‘scattering ampli- (hyper-)spheres, and its symmetries. He had calcu- abilities specifying the results of scattering processes, unwanted singularity, a branch cut in an energy ematical physicist who has Factors and Dual Regge Cuts, tude’ describing the interaction of strongly interact- lated precisely how many there were of these sym- that thus would make no sense. These negative variable, which would violate the axiom of unitarity Phys. Lett. 34B (1971) 500–506. made contributions to quan- ing subnuclear particles, such as pions.⁽¹⁾ His for- metries, more than 8.3 × 10¹⁸. With any lattice probabilities are associated with certain potential unless it could be transformed into a pole, in which tum field theory, string theory (6) Y. Nambu, in Proceedings of the International Conference on and conformal field theory. mula was simply expressed in terms of the Beta there is always the obvious symmetry of reflection states of the system, known as ghost states. To have a case it would signal a new particle in the theory. function introduced by Euler in the 18th century, through the origin and identifying symmetries that satisfactory interpretation, such a theory needs to Late in 1970, Claud Lovelace, desperate to make Symmetries and Quark Models His achievements have been held at Wayne State University, recognized by the award, whose properties are set out in the standard text differ by such a reflection gives a group of half the have a consistent way of excluding ghosts states; the theory consistent, considered varying the dimen- June 18–20, 1969, ed. R. Chand jointly with David Olive, of books on the methods of mathematical physics. size, now known as the Conway group, C0₁. that is, we should be able to define a subspace of the sion of space-time, away from the familiar four, and (Gordon and Breach, New York, the Dirac Medal of the Inter- Veneziano’s proposal resolved a controversy Conway quickly showed that his group, C0₁, potential states of the theory, which would be re- even fantasized about varying the number of sets 1970) 269–277; T. Goto, Prog. Th. Phys. 46 (1971) national Centre for Theoretical about how the description of a scattering amplitude is simple, and so as near as one can get to a basic garded as the physical states, free of ghost states of Virasoro-like conditions. Relaxing reality in this 1560–1569. Physics in 1997, and appoint- in terms of resonances, as a sum of poles in the en- building block of finite group theory. It is now producing negative probabilities, and generating way, he found that the dual model would be saved if (7) L. N. Chang and F. Mansouri, ment as a Commander of the ergy variable, s, could be reconciled with the high- known that the finite simple groups come in 18 only other physical states, and no ghosts, on the dimension, D, of space-time were 26 (25 space Phys. Rev. D5 (1972) 2535–2542; Order of the British Empire in energy description of the amplitude, as an asymp- infinite series and 26 exceptional sporadic cases. scattering. and one time) and the Virasoro conditions were F. Mansouri and Y. Nambu, Phys. 2002. He is a Fellow of the Lett. 39B (1972) 375–378; totic series of terms which are powers of s, with Conway’s group, C0₁, turned out to be the fifth Typically, the space of physical states is doubled in number. Royal Society of London and P. Goddard, J. Goldstone, C. Rebbi exponents dependent on the angle of scattering, largest of the sporadic simple groups, the largest characterized by the vanishing of a set of (linear) At the time, to many physicists, Lovelace’s and C. B. Thorn, Nucl. Phys. B56 an Honorary Fellow of Trinity (1973) 109–135. College, Cambridge. From called Regge pole contributions. Some physicists had one that had been discovered at that time. Further- conditions, and the consistency of these conditions speculations seemed at best irrelevant, even a joke, 2004 to 2012, he served as argued that the resonance terms should be added more, other new simple groups were found inside with scattering amounts to a symmetry of the theory. because considering extra dimensions to space-time the eighth Director of the to the Regge pole contributions, so that there could it and it provided a crucial step towards the overall In quantum electrodynamics (qed), it is gauge sym- was viewed as just science fiction. But his Delphic Institute for Advanced Study be interference between these features of the ampli- classification of finite simple groups, which took metry that plays this role in eliminating the ghosts. observation was rehabilitated, if not totally ex- in Princeton. Before that tude, but Veneziano’s amplitude showed that the sum over 150 years to complete. As you may know, the In the theory of dual models, there are, in a sense, plained, by the proof of the No-Ghost Theorem, he was Master of St John’s over resonances and the asymptotic series of Regge number 26 also plays a particular role in string infinitely more ghost states than in qed, and, in re- because it showed that ghosts were absent if and College and Professor of pole contributions could be alternative, equivalent theory, as a special dimension of space-time. Our sponse, an infinite-dimensional symmetry is needed only if D is no greater than 26, and that, when Theoretical Physics in the descriptions of the amplitude. The resonance and story is full of coincidences that turn out to be clues to eliminate them. Early on, it was realized that the D = 26, the Virasoro conditions indeed do effec- University of Cambridge, Regge pole descriptions of the amplitude were said of much deeper and unexpected connections, but fundamental constituents scattering in dual models tively behave as though they were two sets of con- where he played a leading to be ‘dual’; the extensions and generalizations of the occurrence of 26 in these two places really is a were more analogous in some sense to a sort of vi- ditions rather than one (mimicking what happens in role in establishing the the Veneziano amplitude that were made by many coincidence (I suppose!); the 24-dimensionality brating medium, a ‘rubber band’ or a ‘string’, rather qed). Thus, Lovelace’s consistency requirements are Newton Institute and the Centre for Mathematical theoretical physicists over the next few years were of the Leech lattice is quite another matter. than to conventional particles, and there are ghost precisely met, as if by miraculous coincidence. But Sciences. called dual models. Dual models proved to be the Having moved to cern, Geneva, in 1970, as modes associated with each of the infinitely many these results encourage another, geometrical inter- embryo (or perhaps, it would be more apposite to a postdoctoral fellow, I became engaged in the de- vibrational modes of the string. pretation. A string moving in four-dimensional say, the larval form) of what emerged within a few velopment of the theory of dual models. Veneziano’s Towards the end of 1969, at the cost of an space-time can vibrate in two dimensions trans- years to become string theory. original formula, proposed for processes involving assumption that necessitated massless spin 1 par- versely and one longitudinally; a string moving in In the summer of 1968, I was a research stu- the scattering of two particles, had been generalized ticles (like the photon), Miguel Virasoro identified D-dimensional space-time can vibrate in D – 2 dent in Cambridge, working on other aspects of to processes involving any number of particles. Ini- an infinite-dimensional symmetry, which had the transverse dimensions and one longitudinal dimen- strong interaction scattering amplitudes and hearing tially, it might have been regarded as an interesting potential to eliminate the ghost states. The algebra sion. When D = 26, the extra effectiveness of the gossip from my friends amongst the pure math­ example, resolving the controversy over whether generating this symmetry, with the central term Virasoro conditions removes the longitudinal

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oscillations and the only significant vibrations are because the dimension of the smallest space, in with the group of rotations in a Euclidean space, of research. Vertex operator constructions from References those in the D – 2 = 24 transverse dimensions. And, which M can be thought of acting as rotations, is not and some of the beautiful and important results string theory were rediscovered or imported into (8) J. G. Thompson, Bull. London over the next two decades, it would emerge that the 24 but 196,883. from the theory of such algebras generalize to Kac- the theory of Kac-Moody algebras, and then used to Math. Soc. 11 (1979) 352–353; J. H. Conway and S. Norton, Bull. occurrences of the special dimension 24 in finite In 1974, even before the Monster had been Moody algebras. provide a natural setting for the Monster group, and London. Math. Soc. 11 (1979) group theory and as the number of transverse di- constructed, while the evidence for its existence, It was realized about 1980 that the basic ways a framework within which the Moonshine conjec- 308–339. mensions in what was becoming string theory are although extremely strong, remained circumstantial, of constructing or representing Kac-Moody algebras tures could be proved using the No-Ghost theorem (9) V. Kac, Func. Anal. Appl. 1 essentially related. John McKay spotted that this dimension differed by involve the same mathematical objects that describe from string theory; and ideas from Kac-Moody (1967) 328–329; R.V. Moody, Bull. Amer. Math. Soc. 73 (1967) In retrospect, that the longitudinal oscillations one from the coefficient 196,884 in the expansion of the interaction of strings, called vertex operators. algebras were imported back into string theory to 217–221. of the string should be absent when the passage to the elliptic modular function j(τ), a function of This established a two-way flow of information, provide new methods of realizing gauge symmetry (10) P. Goddard and D. Olive, quantum theory leaves the symmetries of theory central importance in complex variable theory, an enabling physicists, in particular, to understand there. in Vertex Operators in Math- intact, when the theory is free from anomalies, is apparently very different branch of mathematics. He that Kac-Moody algebras played a role in physical The story provides a fine illustration of the ematics and Physics: Proceed- clear from the description of the mechanics of the sent his observation to John Thompson, then visiting theories and could be used to provide new ways of symbiotic relationship between mathematics and ings of a Conference November 10–17, 1983 ed. J. Lepowsky, string originally proposed by Nambu and Goto.⁽⁶⁾ In Princeton from Cambridge, who generalized it to realizing gauge symmetries.⁽¹⁰⁾ In string theory, theoretical physics. Its plot could not have been S. Mandelstam and I. Singer their action principle, longitudinal motions of the observe that the next four coefficients of j(τ) could vertex operators are part of what is called a confor- guessed in advance; it is more wonderful than any- (Springer-Verlag, 1984) 51–96; string have no dynamical significance; they are only also be expressed as simple sums of a few of the mal field theory which describes the structure of one could have imagined. Few of the chapters could Int. J. Mod. Phys. A1 (1986) 303–414; reparametrizations. When D ≠ 26, the passage to dimensions of spaces in which the putative Monster the string. Igor Frenkel, James Lepowsky and Arne have been written as predetermined deliverables on D. J. Gross, J. A. Harvey, E. Martinec quantum theory is corrupted by anomalies, and group might be represented as acting as rotations. Meurman⁽¹¹⁾ found the natural setting for the multi-year research programs. Lovelace’s suggestion and R. Rohm, Nucl. Phys. B256 longitudinal modes, which seem impossible to treat Since both the coefficients of j(τ) and the dimension Monster group by constructing a special conformal of a 26-dimensional space-time and Thompson’s (1985) 253–284. consistently, are introduced by quantization. This of these representation spaces get big rather quickly, field theory, called a vertex operator algebra in this conjecture of a detailed relationship between the (11) I. Frenkel, J. Lepowsky and ♮ A. Meurman, Proc. Nat. Acad. Sci. motivated an understanding⁽⁷⁾ of how the quantum this would be an extreme coincidence, unless there context, acting in V . Their conformal field theory elliptic modular functions and representations of USA 81 (1984) 3256–3260; Vertex mechanics of a relativistic string would lead to the was a deeper reason. He conjectured that this would was special amongst conformal field theories in the Monster group led others to doubt their reason. operator algebras and the monster spectrum of states found in the dual model, pro- be true for all the coefficients of j(τ), and that this much the same way that the Leech lattice is special And, of course, the story is not yet over. Physicists (Academic Press, New York, 1988) vided that D = 26, and to a precise relationship phenomenon reflected the existence of an infinite- amongst 24-dimensional lattice. They showed that Tohru Eguchi, Hirosi Ooguri, and Yuji Tachikawa (12) R. E. Borcherds, Invent. Math. 109 (1992) 405–444. between the geometrical string picture and the dimensional space, sliced or graded into finite-di- the group of symmetries of this vertex operator have found a new form of ‘Moonshine’ relating the calculational framework of dual models. Gradually mensional layers, in which M could be represented algebra was indeed M, and in this way proved the representations of the Mathieu sporadic simple (13) P. Goddard, Proceedings of the International Congress of it came to be accepted that string theory required naturally acting on each layer. Conway together with conjectures of Thompson. group, M₂₄, to a weak Jacobi form, the elliptic genus Mathematicians, Berlin 1998. more than the familiar four dimensions of space- Simon Norton, another Cambridge group theorist, To go further and prove the Moonshine con- associated to a K3 surface.⁽¹⁴⁾ To what new ques- (Deutsche Mathematiker-Vereini- time, a total of 26 for the original string theory or generalized Thompson’s work to conjecture a modu- jectures of Conway and Norton, Richard Borcherds, tions, connections and concepts this will lead, we gung, 1998) 99–108. 10 for the more elaborate and realistic superstring lar function, and a series connected with the hypo- originally a student of Conway, defined the notion have yet to imagine… ■ (14) T. Eguchi, H. Ooguri and Y. Tachikawa, Experimental theory, and these extra dimensions should be curled thetical natural infinite-dimensional representation of a Generalized Kac-Moody algebra. He enlarged Math. 20 (2011) 91–96. up on a scale of 10¯³⁵m, with the theory being inter- space, for each of the elements 8.1 × 10⁵³ elements V♮, so that it became something like the transverse I am grateful to Richard Borcherds, Matthias Gaberdiel and Siobhan Roberts for helpful comments on a draft of this text. preted as a unified theory of all the fundamental of the Monster, conjectures which they called vibration states of a string and defined a Generalized interactions, rather than one describing strong Monstrous Moonshine.⁽⁸⁾ Kac-Moody algebra associated with the space de- interaction physics on the scale of the nucleus, its This posed the challenge of finding the natural fined by Virasoro-like physical state conditions original context. infinite-dimensional representation space, later imposed in this larger space. In a denouement which While the attention of theoretical physicists denoted V♮, and a structure within it of which M brings together the strands of our plot, by applying shifted away from string theory, from the mid 1970s was the symmetries, so ‘explaining’ the existence of a generalized form of the Weyl character formula, to the mid 1980s, there were a number of significant M in the same way that the Leech lattice ‘explains’ a central result from the theory of compact Lie al- developments in mathematics which were or would the existence of C0₁, with the important historical gebras which extends to Generalized Kac-Moody become intimately related to string theory. In 1981, difference that, whereas the Leech lattice led to the algebras, and using the No-Ghost theorem of string Robert Griess announced the construction of the discovery of Conway’s group, it would be the Mon- theory, Borcherds was able to prove the Moonshine largest sporadic simple group, usually known as ster group that was leading to the discovery of V♮ conjectures.⁽¹²⁾ For this tour de force, he was award- the Monster group, M. It has about 8.1 × 10⁵³ el- and whatever lived inside. The construction of V♮ ed a Fields medal in 1998.⁽¹³⁾ ements, more than 36 orders of magnitude bigger and the structure inside it which could be regarded In this story, extending over thirty years, I than Conway’s group. As well as being the biggest, as providing the raison d’être for the Monster, as its have tried to outline briefly how three seemingly it contains all but 6 of the other sporadic simple symmetry group, brings together concepts from disparate theories or themes, which began about groups within it in some sense. Just as C0₁ is the string theory and another new development from 1968—string theory, Kac-Moody algebras, and the group of symmetries of the 24-dimensional Leech the late 1960s: the infinite-dimensional algebras developments in finite group theory following from lattice, up to reflections, so it is natural to seek to an introduced independently by Victor Kac and Robert the discovery of Conway’s group, C0₁—became in- object, albeit an abstract one, of which M is the Moody⁽⁹⁾. These Kac-Moody algebras generalized tertwined and cross-fertilized, revealing profound group of symmetries. This is challenging not least the compact Lie algebras, such as those associated connections between previously unrelated areas

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A Deeper Union? Poincaré on Mathematics and Mathematical Physics by Jeremy Gray

his topic invites the historian to speculate: usually too slow, and the approximations involved are mathematics and mathematical physics two too complicated for such approaches to yield ef- T subjects, or two aspects of the same subject? fective numerical procedures (see Poincaré 1890). Even if we grant that in the modern world both Poincaré did not regard the lack of rigour in subjects have become so large that no-one can be mathematical physics as acceptable. He could see no expected to know all of even one of them, we find clear place to draw the line between mathematics surprises such as string theory, with a strong claim and mathematical physics, no way to be sure that a to being fundamental in both physics and topology. less than rigorous proof was not in fact misleading, If we set aside disciplinary issues, is there a and argued that in any case a rigorous proof teaches Photo: Susan Laurence single domain containing both mathematics and something. As we have just seen, the more important Jeremy Gray is a British mathematical physics, or are there two domains with distinction for him was between ‘right’ and ‘wrong’ historian of mathematics perhaps some bridges between them? The view that proofs—proofs that capture the essence of the working on topics in algebra, there are two domains was implied as long ago as problem and those that do not. geometry, analysis, and the Greek times. Archimedes in his Method regarded For Poincaré, the problem was always: How philosophy of mathematics some arguments drawn from considerations that to proceed? Isolated facts had no appeal for him, in the 19th and 20th centuries. In 2009 he was awarded would be convincing in physics as having only a he said, but a class of facts held together by analogy the Albert Leon Whiteman heuristic character when used in a strictly math- was valuable because it brings us into the presence Memorial Prize of the ematical setting. In other words, there might be of a law. He openly echoed Ernst Mach in his 1908 American Mathematical arguments in mathematical physics that draw their address to the International Congress of Mathemati- Society for his work on the certainty from an understanding of physics that has cians in Rome when remarking that “The impor- history of mathematics. He is not been captured in precise mathematical terms. tance of a fact is measured by the return it gives— presently a Professor of the On this view, the world displays a limited range that is, by the amount of thought it enables us to History of Mathematics at of possibilities, which it presents for reasons that economise”. The elegance of a good proof reflects an the Open University, and an may not be understood. 19th century arguments in underlying harmony that in turn introduces order Honorary Professor at the potential theory had a similar character. Mathemati- and unity and “enables us to obtain a clear compre- University of Warwick. cal physicists such as Helmholtz and Maxwell could hension of the whole as well as its parts. But that is not see the point of the cumbersome and restricted also precisely what causes it to give a large return”. proofs offered by Dirichlet, Riemann, and Schwarz. The aesthetic response to mathematics was chiefly Henri Poincaré reflected at length on these appreciated by Poincaré as a sign of its efficacy—he issues. He argued that more-or-less intuitive proofs set less store by unconscious feelings of certainty of theorems in potential theory that are based on which could, he admitted, prove on rational analysis an appeal to Dirichlet’s principle are without value to be worthless. for the mathematician, although they are of the right In downplaying the importance of isolated sort to satisfy a physicist because they leave the me- facts, Poincaré was seeking to diminish the inexpert chanism of the phenomena apparent. On the other view of physics, or any science, that it serves up hand, the rigorous proofs he knew, including his strange things we would never have known other- own, he regarded as wrong in kind, because they wise. Indeed, there are many strange—and many do not mimic the physical process. Moreover, they familiar—facts about the physical world that to this depended on convergence arguments that were day lack good mathematical explanations. His point

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Footnotes was that, as a leading expert in mathematical phys- and because they played a central role in an effec- the laws of nature as propose theories—those are References (1) For more about Poincaré, see ics, with a particular interest in the theories of tive theory. the lessons to take from Poincaré’s philosophising. Gray, J. J. 2012. Henri Poincaré: Gray, J. J. 2012 and the references magnetism, electricity, and optics, the only way to Theories could change. Poincaré did wonder Two generations later, the physicist Eugene Wigner a scientific biography, Princeton cited there. University Press, Princeton. work was theoretically. And for Poincaré, a theo­ if the mathematical physics he knew, which he called speculated on the unreasonable effectiveness of Heath, Sir T. L. 1897. Works of retical understanding was a mathematical one. the physics of principles, was not indeed coming to mathematics in physics. Poincaré could not have Archimedes. Dover Publications. In one crucial respect, he argued, a fact had an end, and would have to be replaced by a more found it unexpected at all. If mathematical physics Wigner, E. 1960. The Unreasonable travelled the opposite way, from mathematical probabilistic form of physics if and when thermody- is conducted in the language of mathematics, mat- Effectiveness of Mathematics in physics to mathematics. This, he said, was the gift namics was better understood. He knew very well ters could not be otherwise. We might raise the the Natural Sciences. Communica- of the continuum, the structure being defined in that the branch of physics he knew best and upon possibility that nature might be too unruly to admit tions on Pure and Applied Mathematics 13 (1): 1–14. his lifetime as the real numbers, but upon which which he worked most extensively, was in trouble in detectable regularities at all, but that invites the reply th Poincaré, H. 1890. Sur les the calculus had been based for at least a century. the early years of the 20 century. Hertz’s theory of that then there would be no sentient beings. Could équations aux dérivees partielles Poincaré knew that there were other mathematical magnetism, electricity, and optics could not account other beings understand the universe with radically de la physique mathématique. continua, but the one we believe to be appropriate for Fizeau’s experiments on the speed of light in different mathematics to ours? Poincaré never ad- American Journal of Mathematics 12, 211–294. Oeuvres 9, 28–113. for mathematical physics was responsible, in his water; its only rival, Lorentz’s, could only do so by dressed that question directly. He did speculate on Poincaré, H. 1908. L’avenir des view, for all of mathematics except some algebra abandoning Newton’s third law (to every action whether other minds could perceive the universe as mathématiques. Revue générale (group theory and combinatorics). there is an equal and opposite reaction). non-Euclidean, and his answer was a straight-for- des sciences pures et appliquées What had flowed from mathematics to math- Lorentz was not much bothered by this prob- ward yes. His message was that mathematics and 19, 930–939. Also in Atti del IV congresso internazionale dei ematical physics was language. As he put it, “Math- lem. He was exploring the novel idea that all matter mathematical physics are two aspects of the same matematici, 1909, 167–182. Only ematics is the only language the physicist speaks.” might be electro-magnetic in nature. Were that to be enterprise. Each advances the other. And whether partially in Science et Methode, Poincaré was entirely familiar with the passage from true, might it not be that a law about the behaviour string theory succeeds or fails, there can be little and in the English trl. “The Future th of Mathematics”, Monist, 20, theoretical to experimental physics and back, but of matter would have to be rewritten when matter doubt that the best physics of the 20 century was 1910, 76–92 and Science and th his touch was less sure. What he knew about in was no longer a fundamental concept? But Poincaré written in the language of 20 century mathematics, method. A complete English physics, what he sought to increase our understand- found this approach unsatisfactory. To him it was and one must suppose the new physics of the 21st translation is given in Gray, J. J. 2012. Poincaré replies to Hilbert: ing of, was mathematical physics, in which the ob- ad hoc, a hypothesis invented to get people out of a century will follow the same pattern. ■ on the future of mathematics, ca. jects under discussion should have good definitions difficulty, but one that did not meet the criteria of 1908, Mathematical Intelligencer, and be treated according to agreed mathematical leading to new discoveries. 34, 15–29. rules. The best definitions, the deepest insights, He felt the same way about the hypothesis of would be those that made the theories work most the supposed Lorentz contraction as Lorentz had smoothly and produce new discoveries most ef- proposed it. To Poincaré, this was as if nature was fectively. giving one a nudge in the ribs (a “coup de pouce”)— The rules were not exclusively mathematical. something it never otherwise did. He preferred to He attached particular significance to Newton’s reformulate Lorentz’s idea of local time and make it laws of motion, the conservation of energy, and the part of a theory of space and time in which measure- principle of least action. In his view, these rules ments in different moving frames of reference were had passed beyond the stage where they were to be handled by the transformations of what we now call disputed. He gave the example of a satellite in orbit the Lorentz group, and which Poincaré described round a planet and which displayed behaviour shortly before Einstein presented his theory of spe- inconsistent with Newtonian gravity. We would not, cial relativity. he said, any longer try out alternatives to the inverse This is not the place to analyse why, having square law. We would instead look for other forces done so much, Poincaré in the end chose to keep to acting on the satellite that were distorting its orbit. a separation of space and time and not move over to Certain results, formerly a matter of empirical test- space-time, or, if you prefer, to keep to the Galilean ing and discussion, had been elevated to the status group and not switch to the Lorentz group. Famous- of axioms in the relevant theory. They could not ly, he and Einstein did not understand what each be contested within the theory. other had done, and Poincaré was technically right: Nonetheless, they were what he called con- one can do everything Einstein described in the ventions. They were not laws of nature or absolute special theory with the Galilean group and different truths (in his opinion we have no access to such behaviours of rods and clocks.⁽¹⁾ things), but had been raised to the status of incon- That there was a choice to make, that it is made testable truths within a theory because they had on grounds of efficacy or convenience, as Poincaré achieved a high degree of empirical confirmation sometimes put it, that we do not so much discover

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A Physicist Involved in Purely Algebraic Research: The Case of H. B. G. Casimir by Martina R. Schneider

he legacy of Paul Ehrenfest (1881–1933), an and some physicists, e.g. John Clarke Slater⁽⁴⁾, tried Martina R. Schneider is a Austrian physicist, is kept at the Dutch Mu- to find ways to avoid it. But Ehrenfest was keen to historian of mathematics at T seum Boerhaave in Leiden and is indeed a learn about it. He invited leading experts (Wolfgang the Institute of Mathematics treasury of knowledge. Here one can gain insight Pauli, Eugene Paul Wigner, Walter Heitler) as well at Mainz University. She has published several mono- into Ehrenfest’s scientific motivations and problems, as the mathematician van der Waerden to Leiden to graphs on the interrelation- learn about his personal opinions on more general clarify matters. Casimir who was present during ship between mathematics (scientific) issues, about the research communities these guest lectures considered van der Waerden’s and physics in the 19th and and the research processes he was involved in during lecture to be brilliant. 20th centuries. Her Ph.D. th the first third of the 20 century. This treasure trove Van der Waerden knew Ehrenfest from his thesis on van der Waerden’s also contains letters to Ehrenfest written by the student days. In fact, van der Waerden’s first publi- group-theoretical contribu- Dutch theoretical physicist Hendrik Brugt Gerhard cation ever was a popular science account of a lec- tions to quantum mechanics Casimir (1909–2000). Casimir was one of Ehren- ture on relativity theory given by Ehrenfest at an was awarded with the Nach- fest’s students who kept Ehrenfest informed about institute for workers’ education⁽⁵⁾. When van der wuchsförderpreis­ (young his subsequent scientific activities. In these letters Waerden became professor at Groningen in October researchers’ prize) of the one can gain insight into the development of the first 1928, Ehrenfest frequently wrote him, asking him Saxon Academy of Sciences in Leipzig. Recently she has purely algebraic proof of the complete reducibility of questions on group theory, especially on certain turned to studying the finite-dimensional representations of semi-simple sections of Weyl’s monograph dealing with represen- historiography of mathemat- Lie-groups (see⁽¹⁾, especially chapter 16). This proof tations of the rotation group. It was Ehrenfest who ics by analyzing the develop- was published in a joint paper by Casimir and the urged van der Waerden to develop a calculus which ments in the history of mathematician Bartel Leendert van der Waerden should be analogous to the tensor calculus in special ancient Greek mathematics (1903–1996) in 1935⁽²⁾. relativity⁽⁶⁾. With the help of this calculus, named during the 20th century. In 1926 Casimir started studying physics, spinor-calculus upon Ehrenfest’s instigation, the mathematics and astronomy at Leiden university, relativistic wave equation of the electron could be where Ehrenfest had been teaching since 1912. The handled more easily and conveniently. bright young student was just in his third semester In his Ph.D. thesis, completed in November when he was allowed to participate in Ehrenfest’s 1931, Casimir gave an elegant mathematical deduc- research colloquium. After passing the “Doctoraal- tion of the quantum mechanical equations of motion examen” in June 1928, Casimir turned to theoretical of the spinning top and sketched how the formalism physics: relativity theory and quantum mechanics. could be applied to describe the (external) rotation Casimir and van der Waerden probably met of molecules⁽⁷⁾. In this context, he constructed an in Leiden in 1928/29 during a series of guest lectures operator which commuted with any representation organized by Ehrenfest around group theoretical of a semi-simple Lie-group⁽⁸⁾—the Casimir-operator methods in quantum mechanics. At that time a was born. number of articles on the topic by diverse authors In September 1931 Casimir became assistant as well as the first monograph by the mathematician to Pauli at the eth Zurich. Pauli was one of the first Page 20, 22, 23: Letter Hermann Weyl had appeared⁽³⁾. Like most physi- to use group theory in quantum mechanics in his from Casimir to Ehrenfest; cists, Ehrenfest was unfamiliar with the mathemati- attempt to describe the spin of an electron as an ESC 2, S. 9, 201 (date: cal theory, i.e. representation theory of groups. The intrinsic two-valuedness of the wave function of an November 7, 1932); Archive, term “group plague” (Gruppenpest) was coined electron. During his time as assistant in Hamburg, Museum Boerhaave, Leiden

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References probably in the winter term 1926/27, Pauli attend- with the help of the Casimir-operator, but was out that way, but I am very happy about it never- van der Waerden. On the one hand, some of the References (1) Schneider, M. R.: Zwischen ed a series of lectures by Emil Artin on hypercom- unable to generalize his proof to arbitrary semi- theless. And Pauli was absolutely delighted: mathematicians, like von Neumann and van der (6) van der Waerden, B. L.: zwei Disziplinen. B. L. van der plex systems (algebras). According to Pauli, Artin simple Lie-groups. However, in November 1932, He had always said that it could not be anything Waerden, were open to the physical theory and Spinoranalyse. Nachrichten von Waerden und die Entwicklung der der Gesellschaft der Wissenschaf- Quantenmechanik. Springer- started the lecture with a remark on continuous Casimir informed Ehrenfest that a solution had been profound.”⁽⁹⁾ to its special problems. Others like Weyl became ten zu Göttingen. Math.-Phys. Verlag, Berlin, Heidelberg 2011 groups, i.e. Lie-groups. Artin could not discuss them found. Casimir had written to van der Waerden, The development of the Casimir-operator is even deeper involved in quantum mechanics. Klasse (1929), pp. 100–109. (Mathematik im Kontext). because no algebraic proof existed for the complete who was professor in Leipzig at the time. Van der not only an example of a two-way transfer of knowl- On the other hand, physicists, like Pauli, Casimir, (7) Casimir, H. B. G.: Rotation of a (2) Casimir, H. B. G.; van der reducibility of the representations of semi-simple Waerden had succeeded in proving complete re- edge and techniques between mathematics and Wigner and Guilio Racah, took up mathematical rigid body in quantum mechanics. Wolters, Groningen 1931 (Ph.D. Waerden, B. L.: Algebraischer continuous groups. The only proof known was one ducibility for arbitrary semi-simple Lie-groups along physics, but also represents a methodological shift questions and problems. Thus for them group theory Beweis der vollständigen thesis, University of Leiden). Reduzibilität der Darstellungen presented by Weyl in 1925/26. Although Weyl’s the lines of Casimir’s proof for the rotation group. in the status of group theory: at first physicists had also became an object of research—not of physics, (8) Casimir, H. B. G.: Ueber die halbeinfacher Liescher Gruppen. proof was an important break-through, it unfortu- Van der Waerden was not satisfied because his proof made group theory and quantum mechanics com- but of mathematics, of algebra. Casimir’s applica- Konstruktion einer zu den irre- Mathematische Annalen 111 duziblen Darstellungen halb- (1935), pp. 1–12. nately used integrals and analytic means, instead of was based on a study of three cases. Only one of the patible and used group theory mainly as a tool, but tion of the Casimir-operator to the problem of algebraic ones. When describing to Ehrenfest how cases could be solved quickly with the help of the then group theory itself became an object of their complete reducibility, however, stands out as an einfacher kontinuierlicher Gruppen (3) Weyl, H.: Gruppentheorie und gehörigen Differentialgleichung. Quantenmechanik. Hirzel, Leipzig unsatisfied mathematicians, including Weyl himself, Casimir-operator⁽²⁾. The delay in the publication of research in the field of mathematics. In my opinion, example of research done by a physicist who was Proceedings Koninklijke 1928. were with Weyl’s proof, Casimir quoted Pauli as the proof could have been due to van der Waerden’s what was vital for this transition from being a tool to not motivated by problems of physics, but of pure Nederlandse Akademie van Wetenschappen 34 (1931), (4) Slater, J. C.: The theory of having said: “The mathematicians wandered around attempt to improve the proof. Casimir was rather becoming an object of research was the network of mathematics. ■ pp. 844–846. complex spectra. The Physical in tears.”⁽⁹⁾ proud that van der Waerden could not do without scientists which evolved around group theoretical Review 34 (1929), pp. 1293–1322. (9) Casimir, H. B. G.: Letter to When Casimir started working in Zurich, his operator: methods in quantum mechanics. Since group theory Ehrenfest, Zurich, November 7, (5) van der Waerden, B. L.: Pauli set him the task of proving the full reducibility “V. d. Waerden is now trying to simplify the was new to the physicists, it needed to be explored 1932. Museum Boerhaave, Over Einsteins relativiteitstheorie. Leiden, ESC 2, S. 9, 201. De socialistische gids: theorem for semi-simple Lie-groups with algebraic thing; he would like to avoid the quadratic form. To by them. Some physicists turned to mathematicians: (translation MS) maandschrift der Sociaal-Democ- means only. This was a purely mathematical prob- tell you the truth, I have hopes that this won’t work. Pauli went to Artin’s lectures, Wigner asked John ratische Arbeiderspartij 6 (1921), pp. 54–73, 185–204. lem. Within one year Casimir had solved the Of course I am aware that it was more luck von Neumann for help, Ehrenfest invited mathema- problem for the group of 3-dimensional rotations than intellectual power that everything turned ticians to give lectures. Casimir wrote to Weyl and

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Changing Views―Images and Figures of Thought in the Mathematical Sciences by Joachim Schwermer

— Part I —

Patterns of scientific discovery. Some sixty years tional structures, and the place of insight in an ago thinking about science meant contemplating evolving structure of ideas.”⁽²⁾ scientific knowledge. In the case of the mathemati- Gruber certainly puts forth a perspective that cal sciences this scholarly work tended to be done stands in stark contrast to the single “moment” of from the technical point of view of the history of discovery. ideas. This view has changed considerably. To make sense of science, one has to think about both sci- Images and figures of thought as part of the process entific knowledge and the practice with which of abstraction. One of the patterns that play an it engages. important role in this intellectual process is the Photo: Philipp Steinkellner Scientific thought and work is not a single structure of analogies upon which a scientist may process but a complex pattern of activities aimed have drawn in his or her system of thought. To Joachim Schwermer is towards certain ends. There has been a strong ten- explore the ongoing mechanisms of innovation and Professor of Mathematics dency to view scientific discoveries as the outcomes to shed light on the edges of conception it is nec- at the University of Vienna and Director of the Erwin of sudden insights. However, this view of scientific essary to reflect on the ways in which “Gedanken- Schrödinger International creativity has turned out to be too simple to grasp bilder”, i.e. figures of thought, metaphors and Institute for Mathematical the many facets and complexity of the scientific images function in the scientific work. Figures of Physics, Vienna. His research enterprise. To some extent this is the inevitable thought are particularly interesting because they focuses on questions arising result of the more detailed attention to case studies are characterized by both a specific certainty in in arithmetic algebraic geo- in the history of sciences. By an analysis of the capturing the invisible hidden behind the appear- metry and the theory of auto- investigative pathways of a thought process over ances and an intuitive uncertainty (or vagueness) morphic forms. He takes a extended periods of time, one can see the multitude to open further explanation. It is the aim of this keen interest in the math- of themes, the gradual growth of certain points of short note to follow the interplay of thought and ematical sciences in the 19th th view, repeated encounters with certain structural image in the mathematical sciences by examining and 20 centuries in their ideas, their development, their rejection involved certain cases. historical context. in the scientific process. It has become ever more widely recognized Perception of reality, or, what is an image? This is an that scientific creativity might be viewed as a process apparently simple question, and some people might of growth. As a step towards the eventual construc- have an easy answer but I do not. We live in a culture tion of a “theory of creative thinking“ the psycholo- where images are produced at a rate that is nothing gist Howard E. Gruber called for an “evolving sys- short of bewildering, where photographs, films, tems approach” in the study of scientific creativity, advertisements, computer pictures are churned out Page 24: Sketch of the transparency for the i.e. a “conceptualization of scientific thought as so quickly that our heads start to spin. In addition, lecture “Raum und Zeit”, protracted, purposeful, constructive work”.⁽¹⁾ we have the treasures of art, the contemporary works September 21, 1908, Cologne. Gruber wrote: of artists on one hand, and we have the scientific Cod. Ms. Math. Arch. 60: 2, “Issues to be dealt with include: intentionality, images, simple drawings in mathematics or the Bl. 25. SUB Göttingen, the relations between emotions and thought, sci- physical sciences, we have images generated out of Spezialsammlungen und entific thinking as a series of structural transforma- large sets of data gained by scanning tunnelling Bestandserhaltung,­ Abt. tions, metaphoric thought as part of the process microscope or representing events in high energy Handschriften und Seltene of abstraction, differential uptake of complex idea- physics. We are in the middle of a still unfolding Drucke

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debate⁽³⁾ which focuses on the role of images in can help us to develop the intuition needed in later published under the title “Raum und Zeit” hour on, space by itself and time by itself are to art and sciences, focuses on the ways in which the process of conceptualization.⁽⁶⁾ This is not an in 1909. sink fully into the shadows and only a kind of union images help to shape our perception of reality. At accident. Our brains seem to be organized in such a An analysis of the content of this talk and other of the two should preserve their autonomy. the same time, there is the urgent need to have way that they are extremely concerned with vision. sources reveals substantial evidence that Minkows- First of all I would like to indicate how, [start- precise distinctions in dealing with images in dif- We are highly capable of recognizing, understand- ki’s visual-geometric insight had a significant in- ing] from mechanics at present, one could arrive ferent contexts—historical, cultural, psychological, ing, and making sense of the world that we see upon fluence on the formation of his concept of space through purely mathematical considerations at philosophical, aesthetical. What is the relation visual patterns. Therefore, spatial intuition, or spatial and time. Basic geometric notions as symmetry, changed ideas about space and time”.⁽⁷⁾ between “picturing science” and “producing art”, perception, (räumliche Anschauung, as one says in invariance, familiar to him as a mathematician, play An important point in Minkowki’s exposition what are the differences?⁽⁴⁾ Moreover, we have to German) is an enormously powerful tool. Pattern key roles in his thinking on the confusing world of was the following where he drew attention to the explore an old distrust of images, questioning both recognition, the capacity to take in a large amount of the electron theory at that time. Minkowski’s views physical space encoded in three orthogonal coor- the truth of imagination and means of represen­ information by an instantaneous visual action, is a of the nature of physical reality are directed by his dinates and, simultaneously, an arbitrarily-directed tation. fundamental virtue of the human mind. Images can capability of “Räumliche Anschauung”, i.e. his temporal axis. He introduced the equation, given capture a richness of relations within a given setting extraordinary spatial intuition. Beyond the separa- in terms of a quadratic form of signature (1, 3), Visual perception—its ambiguity. In his book “Art in a way that even accurate verbal description can tion of space and time, which is imposed on us by and Visual Perception”⁽⁵⁾ Rudolf Arnheim discus- never unfold. We have to acknowledge the dynamic given experience, the postulate of relativity presents c²t² – x² – y² – z² = 1 sed some of the virtues of vision. In particular, he and polysemic signifying power from images. physical reality in its frame as a union of space described how the foundations of our present and time. where c was an unspecified weight assigned to knowledge of visual perception are rooted in the Minkowski started his lecture with a quite the time coordinate. Suppressing two of the space studies of the Gestalt psychology. Vision is more — Part II — radical statement: axes in y and z, he derived his decisive space-time than a mechanical recording of elements. It is a “Gentlemen, the conceptions of space and diagram. creative grasp of significant patterns in reality. The Visualization. Text books in geometry covering time, which I would like to develop before you, arise This mathematical formulation resembled the simple act of looking at a given object is character- Euclidean geometry, passing through various stages from the soil of experimental physics. Therein lies approach Minkowski was familiar with in his studies ized by a complex interaction between properties of non-Euclidean geometry to Riemannian geometry their strength. Their direction is radical. From this in the arithmetic theory of quadratic forms. In his supplied by the object and the intellectual perception provide a menagerie of drawings and images to en- of the object. Over the centuries, artists, scientists rich our understanding of the geometrical objects and each of us have struggled to explore the nature we are interested in. Sketches or drawings, given in of this relationship of tension and to come up with personal discussions or seminar talks, motivate the Transparency for the some satisfying visual concepts to represent objects. ideas for rigorous proofs of mathematical results. lecture “Raum und Zeit”, To be more explicit: the elementary task Even more, in a dynamic process, teaching math- September 21, 1908, Cologne. of depicting on a two-dimensional plane a three- ematics at the blackboard intertwines writing and Cod. Ms. Math. Arch. 60: 2, Bl. 26. SUB Göttingen, dimensional object is a difficult one. Just try to talking, symbols play the role of an object, diagrams Spezialsammlungen und represent a chair on a piece of paper. Different are decisive components in the interplay of thinking Bestandserhaltung, Abt. people will get different drawings. and writing. In any case, the visualization of geo- Handschriften und Seltene The situation gets even more complicated or metrical configurations, the figurative marking of Drucke interesting if the notion of space is involved. One analogies between different mathematical structures might discuss how distortions create space, how or fields, the change of the point of view, are used as objects create space, one might trace back how vari- a device to transmit mathematical knowledge and ous spatial constructions have developed over the to inspire an intuitive understanding. generations and cultures, one might focus on the discovery of central perspective, of the “punta di fuga”. Central perspective is a product of visual im- — Part III — agination, a key to solving the problem of spatial organization. Spatial intuition in Minkowski’s work. A case study of one scientist, Hermann Minkowski, may help us Räumliche Anschauung—spatial intuition. On the one to understand the role of visual thinking in the hand, numerous examples make it evident that there evolving structures of knowledge. is one decisive side of the endless intellectual strug- Minkowski presented his views and his theory gle scientists have to face in their work: images of an absolute world (= Postulat der absoluten Welt) deceive, they lead us to argue on ambiguous prem- to a wider public on September 21, 1908, when he ises. On the other hand, we need scientific images delivered a lecture to the meeting of the Assembly because only images as visual elements of thought of Natural Scientists and Physicians in Cologne,

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Footnotes “Probevorlesung” for his Habilitation in Bonn, consciously directed towards a certain perception, Results of the request (1) cf. Howard E. Gruber, The given on March 15, 1887, Minkowski had successfully all these elements constitute essential steps in the to draw a chair. evolving systems approach to laid out his geometric approach to questions in the artistic creative work. But artists have the tendency creative scientific work: Charles Darwin’s early thought. In: theory of quadratic forms. Following a suggestion to hide their sketches, they view them as their T. Nickles (ed.), Scientific Discovery: of Carl Friedrich Gauss, made in 1840 and pursued most private sources of inspiration. They view a Case Studies, 113–130, 1980. by Gustav L. Dirichlet in 1848, Minkowski inter- sketch book as the battlefield where their artistic (2) Howard. E. Gruber, ibid. p. 114. preted quadratic forms as lattices in space. This imagination has to find its formal expression, where (3) cf. e.g. the exhibition (and geometric point of view was already increasingly compository or aesthetic demands have to match the accompanying catalogue) apparent in Minkowski’s early papers when he re- their artistic “Willen”, i.e. their conviction and desire. “Iconoclash—beyond the image wars in science, religion and art”, interpreted results obtained in an arithmetical mode But this is the difficult pathway along which an ZKM, Karlsruhe, 2002, or, Bettina as results about lattices in space. But now, as outlined artwork comes to fruition. Heintz, Jörg Huber (ed.), Mit in his Probevorlesung, the approach evolved into On the other side, mathematics asks for a dem Auge denken – Strategien der Sichtbarmachung in wissen- a direct treatment of the arithmetically defined strict logical-analytical exposition of its results, schaftlichen und virtuellen Welten, objects [that is, the quadratic forms] in the geomet- a clear deduction of arguments in giving a proof. Wien – New York 2001. ric framework. This was quite a decisive step, a turn It is difficult to trace the true motivations, the idea (4) This pair of notions is taken from arithmetic to geometry. At the same time, behind an argument or a specific point of view, from the title of the book: Caroline A. Jones, Peter Galison (ed.), Minkowski introduced the concept of volume into because the publication is convoluted with tech- Picturing Science, Producing Art. his analysis. This simple idea led to a fundamental nical details, condensed to the mathematical subject New York, London 1998. result, the so-called lattice point theorem. The matter. But the mathematical practice is very dif- (5) Rudolf Arnheim, Art and method of investigation was definitely led by some ferent. Mathematicians at work need sketches, they Visual Perception—a Psychology of the Creative Eye, Berkeley- kind of spatial intuition. This capability of visual- sketch an idea by drawing an image in a discussion Los Angeles, 1957. geometric thinking is a unique aspect of Minkows- with a colleague, they use this device, hopefully, (6) As one testimony one can refer ki’s approach to questions in number theory. It is in the classroom, sketches show up in talks to the to A. Einstein’s “Autobiographical steadily directed through geometric concepts and scientific community. Sketches or images in mathe- Notes” where he explicitly stated notions. In this way his “Geometry of Numbers” matics capture patterns of thought, they can bring to how images become an ordering element of thought and, finally, unfolded. It is viewed as one of Minkowski’s funda- light some underlying structures by suppressing the a concept. Another example is mental contributions to mathematics. However, this unimportant related to a specific question. But the discussed by Peter Galison in “The visual thinking also forms one pillar in Minkowski’s way that sketches are used and how open they are to Surpressed Drawing: Paul Dirac’s Hidden Geometry”, Representa- later investigations in the theory of special relativity. be perceived depends on the personal and social tions 72 (2000), 145–166. context. (7) „M. H.! Die Anschauungen über Raum und Zeit, die ich Ihnen — Part IV — Rigorous concepts matter. A work of art can convey entwickeln möchte, sind auf experimentell-physikalischem some of the essential mysteries of life, beyond the Boden erwachsen. Darin liegt ihre By examining the case of Minkowski, I indicated frames of ideas. It can unfold the depth of our feel- Stärke. Ihre Tendenz ist eine in some detail how images and visual thinking may ings, it can make us rethink the truth of imagina- radikale. Von Stund’ an sollen Raum für sich und Zeit für sich function in the creative process in mathematics or tion and the means of representation. At its best völlig zu Schatten herabsinken physics. However, images form the core of art. By its meaning is open ended, it reveals mystery. In und nur noch eine Art Union der creating their own reality, images enable the scientist contrast, in its best form, a mathematical work beiden soll Selbständigkeit bewahren. Ich möchte zunächst as well as the artist to capture the invisible and help represents an idea, and its value lies in the rigorous ausführen, wie man von der to shape our perception of reality. In light of this foundation of the inherent concepts. Revealing the gegenwärtig angenommenen connection I would like to conclude with some re- precise nature of an object of our mathematical Mechanik wohl durch eine rein mathematische Überlegung zu marks on similarities, dissimilarities between art and inquiry within a given framework, making clear the veränderten Ideen über Raum mathematics, remarks towards certain distinctions constraints of a given representation or description, und Zeit kommen könnte.“ which have to be made in dealing with images in capturing beyond the necessary verification the In: H. Minkowski, Gesammelte Abhandlungen [ed. D. Hilbert], different contexts. essence of a correspondence between objects in Leipzig 1911, vol. II, p. 431–444. apparently different fields, these tasks mark the heart Sketches matter. Initial sketches to capture the idea of mathematical work. The role of mathematics is of a prospective painting, drawings to bring out the to understand the mystery and communicate that formal organization of a composition, scattered understanding to others. ■ configurations of lines to fix distortions of the pic- torial elements so that, e.g. the viewers eye is un-

28 / esi 2013 29 / esi 2013 Scientific Activities thematic programmes research in teams programme

Thematic Thematic programmes in 2013: Research Teams at the ESI in 2012: Teichmüller Theory Disordered Oscillator Systems (organized by L. Funar, Y. Neretin, Bruno Nachtergaele (UC Davis), Programmes A. Papadopoulos, R. Penner), in Teams Robert Sims (U Arizona), Günter Stolz January 28–April 21, 2013 (U Alabama, Birmingham), June 18–August 5, 2012 The Geometry of Topological D-Branes, Programme Categories and Applications Whittaker Periods of Automorphic Forms (organized by S. Gukov, A. Kapustin, Erez Lapid (Hebrew U), L. Katzarkov and Y. Soibelman), Zhengyu Mao (Rutgers U Newark), April 22–July 6, 2013 July 1–July 31, 2012

Jets and Quantum Fields for LHC and Future Colliders Twisted Conjugacy Classes in Discrete Groups Thematic programmes are (organized by A. H. Hoang and I.W. Stewart), The Erwin Schrödinger Institute Alexander Fel’shtyn (Szczecin U), the main scientific activities July 1–July 31, 2013 Research in Teams Programme is Evgenij Troitsky (Moscow State U), taking place at the ESI. These in place since the beginning of July 23–August 23, 2012 Forcing, Large Cardinals and Descriptive Set Theory programmes are extending over a (organized by S. Friedman, M. Goldstern, 2012. It offers teams of two to Resolution of Surface Singularities longer period of time, including A. Kechris and W. H. Woodin), four ESI scholars the opportunity in Positive Characteristic several workshop periods and September 2–October 25, 2013 to work at the Institute in Vienna Dale Cutkosky (U Missouri), Herwig Hauser (U Vienna), Hiraku Kawanoue research stays of individual Heights in Diophantine Geometry, Group Theory for periods of one to four months, (Kyoto U), Stefan Perlega (U Vienna), scientists. There are usually four and Additive Combinatorics in order to concentrate on new Bernd Schober (U Regensburg), to six thematic programmes per (organized by R. Tichy, J. Vaaler, collaborative research in math- November 5–December 21, 2012 M. Widmer and U. Zannier), year that have to be proposed October 21–December 20, 2013 ematics and mathematical phys- about two years in advance and ics. The interaction between the Teams at the ESI in 2013:

are chosen by the Scientific Thematic Programmes in 2014: team members is a central com- Degenerate Eisenstein Series for GL(n) Advisory Board. ponent of this programme. Marcela Hanzer (U Zagreb), Modern Trends in Topological Quantum Field Theory Goran Muic (U Zagreb), (organized by J. Fuchs, L. Katzarkov, January 7–Feburary 7, 2013 N. Reshitikin and C. Schweigert), February 2–March 29, 2014 On the First-order Theories of Free Pro-p Groups, Group Extensions and Free Products of Groups Don Zagier, Combinatorics, Geometry and Physics Montserrat Casals-Ruiz (U Oxford), Collège de France, Paris, France (organized by A. Abdesselam, C. Krattenthaler, Ilya Kazachkov (U Oxford), and Vladimir and MPI Mathematik, Bonn, A. Tanasa, F. Vignes-Tourneret), N. Remeslennikov (Russian Academy of Science), Germany June 2–July 31, 2014 February 18–March 16, 2013

Topological Phases and Quantum Matter Non-Commutative Geometry and Spectral Invariants (organized by N. Read, J. Yngvason, Alan Carey (Australian National U), and M. Zirnbauer), Harald Grosse (U Vienna), Jens Kaad (U Bonn), August 4–September 12, 2014 May 27–June 23, 2013

Minimal Energy Point Sets, Lattices and Designs Nuclear Dimension and Coarse Geometry (organized by C. Bachoc, P. Grabner, E. Saff Erik Guentner (U Hawaii), Jan Spakula and A. Schürmann), (U Münster), Rufus Willett (U Hawaii), September 29–November 22, 2014 May 27–June 23, 2013

33 / esi 2013 junior research fellowship programme summer schools

am a number theorist a little over halfway EMS IAMP Summer School on Quantum Chaos. From Junior Research through a six month stay in Vienna, working at Summer July 30 to August 3 2012 a Summer School on I the esi as a Junior Research Fellow, and what Quantum Chaos took place at the esi. Organized by Nalini Anantharaman, Stéphane Nonnenmacher, follows are a few thoughts on my time here so far. Fellowship Despite being called an Institute for Math­ Schools Zeév Rudnick and Steve Zelditch, it was the first ematical Physics, the esi has a diverse range of one of a series of schools on various topics of math- mathematical activities, and has certainly kept this ematical physics supported by the iamp and the Programme particular number theorist stimulated and occupied ems. Young researchers from more than 10 different with organised events—there have been number countries spent a week in Vienna to learn about the theoretic seminars and lecture courses running research field of “Quantum Chaos”. non-stop for over three months. All of these activ- This research field aims at understanding the ities have been conducted in a relaxed atmosphere, dynamics of quantum (or wave) systems admitting with all those in attendance happy to talk to a young a chaotic classical counterpart and is situated at the researcher like myself. The Institute’s support for interface of mathematics and physics. As the sum- From 2004 to 2010, the ESI those of us just beginning our research careers is Summer Schools are short mer school was mainly adressed to graduate students Junior Research Fellowship Pro- admirable. programmes of one to two weeks and postdocs, the week started with several basic courses that introduced the subject and the basic gramme was in place. Its purpose The esi, occupying the top floor of a priests’ that are aimed at young re- seminary, is a wonderfully tranquil place to sit and methods and concepts. These introductory lectures was providing support for ad- think—it is close to the centre of the city and easily searchers such as graduate were held from Monday to Wednesday and covered vanced Ph.D. students and accessible by public transport, yet feels peaceful and students and postdocs. These topics such as the elementary theory of dynamical systems, semiclassical analysis, and the theory postdoctoral fellows to allow calm. It is too easy to spend all of one’s time at the schools usually consist of a Institute! Nothing gets in the way of doing math- of random matices, amongst others. The second them to participate in the activ- ematics. The staff at the esi, both academic and series of basic introductory part of the week was devoted to a series of more ad- ities of the ESI. Grants were given administrative, have been unfailingly friendly. The lectures in the beginning, vanced talks, presenting some recent work and new developments. for periods between two and mathematicians, though busy, have always involved complemented by some more me in the local events and found time for a math- A special event of the summer school was six months. ematical chat. I was invited to and took part in sev- advanced talks on ongoing the poster session on Tuesday evening, in which the When the programme had to eral of the seminars taking place at the University research, and are organized participants presented their own research. Especially for younger researchers at the beginning of their be terminated due to lack of of Vienna. by renown scientists in math- One of the great benefits of a visit to the esi academic career this served as a good opportunity funding in 2010, there had been is the chance it provides to spend some time in the ematics and mathematical to present and discuss their research interests and more than 150 Junior Research city of Vienna. Of course, I have been to Schön­ physics. problems with experts of the field in an informal setting. The poster session turned out to be well- Fellows at the ESI, coming from brunn, seen ‘The Kiss’ and been to the opera—like all good tourists should—but I have been fortunate attended and very lively and it also showed the many over 30 different countries. to be shown a few things not in the guide books. different research interests and backgrounds that The friends I have made amongst the staff at the were represented at the summer school. ■ esi and students of the University have been kind enough to show me some of their own favourite places in Vienna—be they tucked-away restaurants, coffee houses or nightclubs. I am not even sure the six months of my stay will be long enough to sample all of the astonishing array of coffee houses. Apart from sating my worrying new addiction to coffee and cake, I have found them a pleasant setting for doing mathematics (though not as pleasant as the esi, of course!). Now, if I can only finish this tome of Musil whilst I am here…

Adam Joyce Spring 2006

Most of the ESI Junior Research Fellows, May 2006

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Senior Research Gathering Evidence: Fellowship My Time as a Senior Research Fellow Programme by James W. Cogdell

The Erwin Schrödinger Institute The origins of a colony (I). I was a Senior Research Artin to motivate the global and local Langlands regularly offers lecture courses Fellow (srf) at the esi during the winter term of correspondences, which roughly state that n-dimen- and research seminars at an 2011–2012. As part of my “duties’’ I gave a course on sional local or global Galois representations should “L-functions and Functoriality’’. The principle of be locally or globally modular, i.e., attached to ap- advanced graduate level. These functoriality is one of the central tenets of the propriate representations of GL(n) in such a way courses and seminars are taught Langlands program; it is a purely automorphic that preserves their L-functions. This is Langlands’ by Senior Research Fellows of avatar of Langlands vision of a non-abelian class formulation of a “non-abelian class field theory’’. field theory. There are two main approaches to We discussed how one would formulate such a the ESI who stay in Vienna for functoriality. The one envisioned by Langlands is Langlands correspondence for groups G other than Photo: Patty Merry a period of several months. through the Arthur-Selberg trace formula, and with GL(n). Putting all this together, we formulated the James W. Cogdell was a In bringing together young re- the recent work of Ngô, Arthur, and others this is Langlands Functoriality Conjecture as a process of now becoming available. The second method is that transferring local and automorphic representations student of Piatetski-Shapiro searchers working in Vienna and of L-functions as envisioned by Piatetski-Shapiro of G to GL(n) mediated by the local and global at Yale, finishing in 1981. He has held faculty positions at and is based on the converse theorem for GL(n). Langlands correspondences, that is, by L-functions. senior scientists from abroad, Rutgers University, Oklahoma The overall purpose of the series of lectures was to Finally we discussed how one would then use the the ESI Senior Research Fellows State University, and The Ohio develop and explain the L-function approach to Converse Theorem on GL(n) as a tool for establish- State University, as well as programme contributes in an functoriality. The course consisted of 12 lectures of ing cases of this Functoriality Conjecture. several visiting positions effective way to the scientific 90 minutes each together with the accompanying In attendance I had 5–6 graduate students through the years. He has training of graduate students question periods of 45 minutes each. The first lec- plus 2 faculty from the University of Vienna. The spent all of his professional ture covered the theory of modular forms and their graduate students ranged from early career to fin- life thinking about L-func- and postdocs of Vienna’s L-functions, the classical GL(2) theory, as developed ishing. Besides the lectures, every week I gave out tions, but makes no claims universities. by Hecke in the 1930’s. The second lecture began the problem sets and a list of historical references for the of truly understanding them adelic theory of automorphic representations for week’s lectures. Even though the problem sessions (and doubts the claims of GL(n), covering the basic definitions of cuspidal were designed for student solutions of the exercis- anyone who says they do). automorphic representations and their decomposi- es—and some were indeed used for that purpose— tion into local representations. Lectures 3 through 9 we also used them as open Q & A sessions where were spent on the theory of L-functions for GL(n) anyone could ask questions. If I asked, they were and the twisted L-functions for GL(n)×GL(m). The related to the exercises-detail questions. But when tenth lecture was devoted to the Converse Theorem they asked, while there was the occasional detail for GL(n). The eleventh lecture was devoted to de- question about lecture material, they were more scribing an arithmetic family of Euler products that often philosophical or historical or questions about are conjecturally nice, the Artin L-functions at- relations with other ideas in the area or surrounding tached to an n-dimensional representation of a areas. For example, while the lectures were centered Galois group. In this lecture we surveyed the results around integral representations of L-functions, the of class field theory and then the contents of Artin’s Q & A sessions prompted me to give an impromptu three papers on his non-abelian L-functions from outline of the Langlands-Shahidi method, relating 1923, 1930, and 1931. In the final lecture we applied analytic properties of L-functions to Fourier coef- Page 38 and 40: Clippings the “moral theorem’’ coming from the Converse ficients of Eisenstein series. These impromptu from the course notebook that Theorem to the conjecturally “nice’’ L-functions of expositions are equally challenging and ultimately Cogdell used for his lectures.

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satisfying when things go well. It forces an interest- pation in the meeting “Emil Artin—His Work and gramme. (Both Shahidi and Tsai were in residence Viennese composers, and Cerha was in attendance. ing concentration and perspective. Giving the stu- His Life’’ held at the esi in January of 2006. For the at the esi in the month of January 2012 as part of I attended the premier of an organ and electronics dents such opportunity for open ended questions accompanying volume on Artin I had prepared a the “Automorphic Forms: Arithmetic and Geom- piece written and performed by Wolfgang Mitterer, and discussions gives them a sense of control over paper On Artin L-functions based on my 2006 lec- etry’’ programme.) In this work, we show that the another contemporary Viennese composer, as part the direction and makes them an active participant ture. I took this time at esi to revise this article local Langlands correspondence for GL(n) preserves of the Wien Modern festival. On the improvised side in the course. and in particular I have now included much of the both the exterior square and symmetric square I heard the pianist Craig Taiborn and then a duo material that I discussed in the lecture on Artin L- and ε-factors. This is a measure of robustness of consisting of Theo Beckmann (voice) and Michael The origins of a colony (II). The srf program should L-functions in my course. So for my own intellectual the local Langlands correspondence and hopefully Wollny (piano). I also made my first foray into the be very rewarding for both the students and the development, the chance to give the srf course was will help us understand the local Artin ε-factors Wiener Staatsoper to hear a performance of Janáček’s lecturer. In my case, my srf tenure coincided with of significant importance. better. As part of this work, we established the local “Aus einem Totenhaus’’. One of the advantages of an esi Programme that I co-organized on “Auto­ The Senior Research Fellow programme, par- analytic stability of the exterior square γ-factor for an interest in contemporary music is that tickets morphic Forms: Arithmetic and Geometry’’ held in ticularly when coupled with an esi Thematic Pro- supercuspidal representations of GL(n), a result of seemed to be readily available and I could decide January and February of 2012. My course was on gramme, is one important way in which the esi independent interest in the local theory of automor- with very little notice what to attend. (For the more L-functions for GL(n) and was intended as a good impacts graduate education at the University of phic forms. This paper will eventually find a home classical fare at the Musikverein or the Staatsoper preparatory course for the topic of the Programme. Vienna. As part of its mission to further the math- in the esi Preprint archive. I understand one must plan and purchase tickets Indeed, when the Programme came around, all of ematical and physical sciences, an impact on the before one even boards the plane to Vienna.) There my students were regular participants, coming to next generation of scientists is paramount. Contact, On the other shore. I also took advantage of the cen- was even music at the esi! As part of the Programme most of the lectures (when they didn’t have other both formal and informal, with the Senior Research tral location of Vienna and the esi to take scientific on Automorphic Forms, there was a piano recital by duties) and some of the junior participants in the Fellow can be very important for a students devel- trips both to the west and to the east. Before the start Gülsin Onay, a Turkish State Artist, with a program Programme attended a few of my later course lec- opment. I still feel the influence of a summer of my srf duties I traveled to Oberwolfach to take of Beethoven, Bartok, and Saygun. It was attended tures. Two of the more advanced students from the workshop on Complex Varieties that I attended part in the meeting on “Emigration of Mathemati- by the workshop participants, other members of the course were invited to speak in the Programme. in Montreal in the summer of 1979, where I had the cians and Transmission of Mathematics: Historical esi community and two of the Turkish ambassadors Their participation in the Programme was very re- good fortune to interact with and attend lectures Lessons and Consequences of the Third Reich’’ dur- in Vienna. Vienna also has wonderful museums warding to me as their instructor in the course. On by Douady, Hirzebruch, Mumford,… as well as ing late October and early November of 2011. This throughout the city. Probably the museum highlight the other hand, the srf lectures and their prepara- making connections with the algebraic geometers was related to my interest in Artin as mentioned of this stay was the opening of the Claes Oldenburg tion impacted my own work. As part of the course, of my generation. I was very pleased to be able to above. In addition, I took a flight to Zurich to speak exhibit at mumok. I also finally made it to see “The in order to help motivate the Langlands Program repay a bit of this mathematical debt by participa- on my work with Shahidi and Tsai at eth and then Last Judgement’’ triptych by Hieronymus Bosch at and the connections with arithmetic, I decided to ting in the srf programme of the esi. took the train to Lausanne to speak on the same the Akademie für die Bildenden Künste. In a more devote one lecture to a survey of Artin L-functions. work at epfl. During the height of the frigid cold classical vein, I toured the Lichtenstein Museum, Artin wrote three papers on these L-functions in On the open road. While the preparing and giving spell of the winter of 2011 I boarded the train at near the esi, sadly during its last week as a public 1923, 1930 and 1931. It was quite a challenge to try of the srf lectures did take time and effort, there was Wien Meidling for Budapest to give two talks at museum. Vienna has a very eclectic food scene, and to distill these papers down to a single lecture, but plenty of time to devote to my own research agenda. the Rényi Institute. While the trips to Oberwolfach, I did eat most of my meals out. I will admit I have in doing this I had several insights into Artin’s mo- During my tenure as a srf I also worked on a joint Zurich and Lausanne were planned in conjunction a proclivity for both traditional and contemporary tivation and sense of structure that he built into project with F. Shahidi of Purdue and T-L. Tsai of with my trip to esi, the trip to Budapest was some- Viennese fare, and Austrian wines, and there are these L-functions. Not only were these revelations the National University of Taiwan on the “Local what serendipitous, only arising when a colleague at many fine restaurants and cafes both near and far intellectually satisfying, they were necessary for me Langlands correspondence for GL(n) and the exte- Rényi learned through e-mail that I was but a train from the esi to partake from (as well as some very to be able to give a coherent presentation of Artin’s rior and symmetric square ε-factors’’. This project ride away. His reaction was an immediate “when are interesting local “hole in the walls”). But when I work in my class and I subsequently expanded an was begun before my residence at esi, but the final you coming to give a talk?’’, and so I made my first think back on it, I realize that I also had fine Chi- article I had written on Artin L-functions to include work and the writing were done at esi, both during trip to Budapest. nese, Greek, Italian, Japanese, Persian, and Vegetar- these insights. This article dates from my partici­ my srf tenure and during the subsequent Pro- ian meals as well. But, in spite of all the wondrous At the Blue Unicorn. Alas, the life of a Senior Research things one can do in Vienna, I think I still find that Fellow is not all fun and games… there are serious what I enjoy most is just wandering the streets, par- cultural duties to perform in Vienna. Vienna is a ticularly in the first district. I prefer them at night, musical city: Mozart, Beethoven, Mahler, … But my when they are more likely to be empty, and covered tastes run more post-Darmstadt to improvised, and with snow if possible. (Unfortunately, there was Vienna is a perfect city for that as well. Where the little snow during this visit.) My only regrets are the more classically inclined might favor the Musik­ places I didn’t get to visit: the Wittgenstein House, verein, I frequented the Konzerthaus. I attended a the Anselm Kiefer exhibit, Roithamer’s Cone, … Klangforum Wien concert which featured pieces by but there will be future visits. ■ Boulez, Cerha, Feldman, Furrer, and Neuwirth. Cerha, Furrer and Neuwirth are all contemporary

41 / esi 2013 Data and Statistics data and statistics—thematic programmes 1993–2013 data and statistics—thematic programmes 1993–2013

The Foundational Period ESI Thematic Programmes Condensed Matter Physics—Dynamics, — 1998 — — 2000 — Aspects of Foliation Theory in Geometry, Geometry and Spectral Theory Topology and Physics (J. Glazebrook, Spectral Geometry and Its Applications Duality, String Theory and Theory 1990–1993 1993–2013 (V. Bach, R. Seiler), M- F. Kamber, K. Richardson), August 6, 1995–February 24, 1996 (L. Friedlander, V. Guillemin), (H. Grosse, M. Kreuzer, S. Theisen), July 15–November 30, 2002 February 1–June 30, 1998 March 15–July 15, 2000 The years 1990–1993, preceding the official — 1993 — Noncommutative Geometry and Quantum Field foundation of the Erwin Schrödinger International Schrödinger Operators With Magnetic Fields Representation Theory Theory, Feynman Diagrams in Mathematics and Two-Dimensional Quantum Field Theory — 1996 — Institute for Mathematical Physics in Vienna, had (I. Herbst, T. Hoffmann-Ostenhof, (V. Kac, A. Kirillov), April 1–July 31, 2000 Physics (H. Grosse, J. Madore, D. Kreimer, been a time of intense preparations involving (H. Grosse), February 15–June 30, 1993 Condensed Matter Physics—Dynamics, J. Yngvason), March 1–June 30, 1998 Confinement J. Mickelsson, I. Todorov), many mathematicians and physicists. This phase Geometry and Spectral Theory August 26–November 22, 2002 Schrödinger Operators Number Theory and Physics I. Convexity (W. Lucha, A. Martin, F. Schöberl), is very well documented and described in the (V. Bach, R. Seiler), (T. Hoffmann-Ostenhof), 1993 (P. M. Gruber), September–December, 1998 May 1–June 30, 2000 Appendix A to the Scientific Report for the years August 6, 1995–February 24, 1996 1993–2002 which can be found on the web page Differential Geometry (P. W. Michor), 1993 Number Theory and Physics II. Quantum Field Algebraic Groups, Invariant Theory, and — 2003 — of the ESI under www.esi.ac.at/about/reports.html. Topological, Conformal and Integrable Theory and the Statistical Distribution of Prime Applications (B. Kostant, P. W. Michor, Walter Thirring, Peter Michor and Heide Narnhofer, Field Theory (K. Gawedzki, H. Grosse), Mathematical Population Genetics and Numbers (I. Todorov), F. Pauer, V. Popov), acting on behalf of the scientific community, February 15–May 14, 1996 Statistical Physics (E. Baake, M. Baake — 1994 — September 1–November 30, 1998 August 1–December 29, 2000 played a decisive role in the foundational period of and R. Bürger), Representation Theory With Applications to the ESI and beyond. Their initiative was well taken Ergodicity in Non-Commutative Algebras Quantization, Generalized BRS Cohomology Quantum Measurement and Information December 1, 2002–February 28, 2003 Mathematical Physics (I. Penkov, J. A. Wolf), up by the Ministry of Science and Research. (H. Narnhofer), spring 1994 and Anomalies (R. A. Bertlmann, M. Kreuzer, (A. Zeilinger, A. Eckert, P. Zoller), April 1–June 30, 1996 Kakeya-Related Problems in Analysis W. Kummer, A. Rebhan, M. Schweda), September 3–December 20, 2000 Three workshops, organized in the years Mathematical Relativity (P. C. Aichelburg, (A. Iosevich, I. Laba, D. Müller), Mathematical Problems of Quantum Gravity September 28–December 31, 1998 1991–1992, underlined the importance of the R. Beig), July 1–September 15, 1994 February 15–April 15, 2003 (A. Ashtekhar, P. C. Aichelburg), enterprise to found a research institute at the Charged Particle Kinetics Gibbsian Random Fields July 1–August 31, 1996 — 2001 — Penrose Inequalities (R. Beig, P. Chruściel, interface of mathematics and physics in Vienna (R. Dobrushin), (C. Schmeiser, P. Markowich), and resulted in considerable momentum. They August 1–December 31, 1994 Scattering Theory W. Simon), June 2–July 29, 2003 Hyperbolic Dynamical Systems With Singularities October 5, 1998–January 31, 1999 (V. Petkov, A. Vasy, form the first scientific activities related to M. Zworski), March 1–July 31, 2001 Spinors, Twistors and Conformal Invariants (D. Szász), September 1–December 31, 1996 Poisson Geometry and Moment Maps the ESI: (A. Alekseev, T. Ratiu, S. Haller, (A. Trautman, V. Souek, H. Urbantke), Random Walks (V. Kaimanovich, K. Schmidt, — 1999 — P. W. Michor), August 1–October 15, 2003 Interfaces Between Mathematics and Physics I September 1–October 31, 1994 W. Woess), February 19–July 13, 2001 — 1997 — Functional Analysis (P. W. Michor, H. Narnhofer, W. Thirring), Quanternionic and Hyper-Kähler Manifolds (J. B. Cooper), Gravity in Two Dimensions (W. Kummer, Mathematical Cosmology (P.C. Aichelburg, May 22–23, 1991 Ergodic Theory and Dynamical Systems January 1–July 31, 1999 H. Nicolai, D. V. Vassilevich), (D. Alekseevsky, S. Salamon), G. F. R. Ellis, V. Moncrief, J. Wainwright), (A. Katok, K. Schmidt, G. Margulis), September 8–October 31, 2003 Interfaces Between Mathematics and Physics II September 1–December 31, 1994 Nonequilibrium Statistical Mechanics June 15–August 15, 2001 January 1–August 30, 1997 (P. W. Michor, H. Narnhofer, W. Thirring), (G. Gallavotti, H. Posch, H. Spohn), Mathematical Aspects of String Theory March 2-6, 1992 Mathematic Relativity (R. Beig), February 1–March 31, 1999 — 1995 — (M. Blau, J. Figueroa O’Farrill, A. Schwarz), — 2004 — January 1–June 30, 1997 75 years of Radon Transform Holonomy Groups in Differential Geometry September 3–November 16, 2001 Complex Analysis (F. Haslinger), Geometric and Analytic Problems Related to (S. Gindikin, P. W. Michor), Spaces of Geodesics and Complex Structures in (D. Alekseevsky, K. Galicki, C. LeBrun), January 1–March 31, 1995 Nonlinear Schrödinger and Quantum Boltzmann Cartan Connections (T. Branson, A. Cap, August 31–September 4, 1992 General Relativity and Differential Geometry September 6–December 31, 1999 Equations (P. Gérard, P. Markowich, J. Slovak), January 2–April 20, 2004 Noncommutative Differential Geometry (L. Mason, P. Nurowski, H. Urbantke), Complex Analysis (F. Haslinger, H. Upmeier), N. J. Mauser, G. Papanicolau), fall 2001 March 1–July 31, 1997 String Theory in Curved Backgrounds and (A. Connes, M. Dubois-Voilette, August 1–November 15, 1999 P. W. Michor), spring 1995 Boundary Conformal Field Theory Local Quantum Physics (D. Buchholz, Applications of Integrability (A. Alekseev, (H. Grosse, A. Recknagel, V. Schomerus), Field Theory and Differential Geometry H. Narnhofer, J. Yngvason), — 2002 — L. Faddeev, H. Grosse), March 1–June 30, 2004 September 1–December 31, 1997 (G. Marmo, P. W. Michor), August 15–October 31, 1999 Developed Turbulence (K. Gawedzki, May 15–July 31, 1995 Tensor Categories in Mathematics and Physics Nonlinear Therory of Generalized Functions A. Kupiainen, M. Vergassola), (J. Fuchs, Y.-Z. Huang, A. Kirillov, May 15–July 14, 2002 Geometry of Nonlinear Partial Differential (M. Oberguggenberger), M. Kreuzer, J. Lepowsky, C. Schweigert), Equations September–December, 1997 (A. Vinogradov), spring 1995 Arithmetic, Automata, and Asymptotics May 31–July 9, 2004 (R. Tichy, P. Grabner), Gibbs Random Fields and Phase Transitions Singularity Formation in Non-linear Evolution March 22–July 5, 2002 (R. Dobrushin, R. Kotecký), fall 1995 Equations (P. C. Aichelburg, P. Bizoń), Quantum Field Theory on Curved Space Time July 7–August 15, 2004 Reaction-Diffusion Equations in Biological (K. Fredenhagen, R. Wald, J. Yngvason), Context (K. Sigmund, R. Bürger, Many-Body Quantum Theory July 1–August 31, 2002 J. Hofbauer), (M. Salmhofer, J. Yngvason), September 1–November 15, 1995 September 1–December 31, 2004

44 / esi 2013 45 / esi 2013 data and statistics—thematic programmes 1993–2013 data and statistics—thematic programmes 1993–2013

— 2005 — — 2007 — — 2009 — — 2011 — — 2013 — Summer and

Open Quantum Systems (J. Derezinski, Automorphic Forms, Geometry and Arithmetic Representation Theory of Reductive Groups— Bialgebras in Free Probability (M. Aguiar, Teichmüller Theory (L. Funar, Y. Neretin, Winter Schools G. M. Graf, J. Yngvason), (S. S. Kudla, M. Rapoport, J. Schwermer), Local and Global Aspects (G. Henniart, F. Lehner, R. Speicher, D. Voiculescu), A. Papadopoulos, R. Penner), January 20–March 31, 2005 February 11–February 24, 2007 G. Muic and J. Schwermer), February 1–April 22, 2011 January 28–April 21, 2013 January 2–February 28, 2009 Summer School on Nonlinear Wave Equations Modern Methods of Time-Frequency Analysis Amenability (A. Erschler, V. Kaimanovich, Nonlinear Waves (A. Constantin, The Geometry of Topological D-Branes, (Y. Brenier, S. Klainerman, N. Mauser, (J. J. Benedetto, H. G. Feichtinger, K. Schmidt), February 26–July 31, 2007 Number Theory and Physics (A. Carey, J. Escher, D. Lannes, W. Strauss), Categories and Applications (S. Gukov, A. Selberg), July 7–11, 2004 K. Gröchenig), April 4–July 8, 2005 H. Grosse, D. Kreimer, S. Paycha, April 4–June 30, 2011 M. Herbst, A. Kapustin, L. Katzarkov, Mathematical and Physical Aspects of S. Rosenberg and N. Yui), Y. Soibelman), April 22–July 6, 2013 Summer School on Vertex Algebras and Related Geometric Methods in Analysis and Probability Perturbative Approaches to Quantum Field Dynamics of General Relativity: Numerical March 2–April 18, 2009 Topics (E. Frenkel, V. Kac, J. Schwermer), (J. Cooper, P. W. Jones, V. Milman, P. Müller, Theory (R. Brunetti, K. Fredenhagen, and Analytical Approaches (L. Andersson, Jets and Quantum Fields for LHC and Future June 12–July 2, 2005 A. Pajor, D. Preiss, C. Schütt, C. Stegall), D. Kreimer, J. Yngvason), Selected Topics in Spectral Theory (B. Helffer, R. Beig, M. Heinzle, S. Husa), Colliders (A. H. Hoang, I. W. Stewart), May 25–August 5, 2005 March 1–April 30, 2007 T. Hoffmann-Ostenhof and A. Laptev), July 4–September 22, 2011 July 1–31, 2013 Winter School: Langlands Duality and Physics May 5–July 25, 2009 (E. Frenkel, N. Hitchin, J. Schwermer, Complex Analysis, Operator Theory and Poisson Sigma Models, Lie Algebroids, Combinatorics, Number Theory, and Dynamical Forcing, Large Cardinals and Descriptive Set K. Vilonen), January 9–20, 2007 Applications to Mathematical Physics Deformations, and Higher Analogues Large Cardinals and Descriptive Set Theory Systems (M. Einsiedler, P. Grabner, Theory (S. D. Friedman, M. Goldstern, (F. Haslinger, E. Straube, H. Upmeier), (H. Bursztyn, H. Grosse, T. Strobl), (S. Friedman, M. Goldstern, R. Jensen, C. Krattenthaler, T. Ziegler), A. Kechris, J. Kellner, W. H. Woodin), Summer School on Combinatorics and Statistical September 5–November 11, 2005 August 1–September 20, 2007 A. Kechris and W. H. Woodin), October 1–November 30, 2011 September 2–October 25, 2013 Physics (M. Drmota, C. Krattenthaler, June 14–27, 2009 B. Nienhuis, M. Bousquet-Mélou), Geometry of Pseudo-Riemannian Manifolds Applications of the Renormalization Group Heights in Diophantine Geometry, Group Theory July 7–18, 2008 With Applications to Physics (D. Alekseevsky, (G. Gentile, H. Grosse, G. Huisken, Entanglement and Correlations in Many-Body — 2012 — and Additive Combinatorics (R. Tichy, J. Vaaler, H. Baum, J. Konderak), V. Mastropietro), Quantum Mechanics (B. Nachtergaele, M. Widmer, U. Zannier), Summer School on Current Topics in September 1–December 31, 2005 October 15–November 23, 2007 F. Verstraete and R. Werner), Automorphic Forms: Arithmetic and Geometry October 21–December 20, 2013 Mathematical Physics (C. Hainzl, R. Seiringer, August 10–October 17, 2009 (J. W. Cogdell, C. Moeglin, G. Muic, J. Yngvason), July 21–31, 2008 J. Schwermer), January 3–February 28, 2012 — 2006 — — 2008 — The d-bar-Neumann Problem: Analysis, Winter School: Mathematics at the Turn Geometry and Potential Theory (F. Haslinger, K-Theory and Quantum Fields (M. Ando, of the 20th Century: Explorations and Beyond Arithmetic Algebraic Geometry (S. S. Kudla, Combinatorics and Statistical Physics B. Lamel, E. Straube), A. Carey, H. Grosse, J. Mickelsson), (D. D. Fenster, J. Schwermer), M. Rapoport, J. Schwermer), (M. Bousquet-Mélou, M. Drmota, October 27–December 23, 2009 May 21–July 27, 2012 January 7–12, 2009 January 2–February 18, 2006 C. Krattenthaler, B. Nienhuis), The Interaction of Geometry and Representation ESI May Seminar 2010 in Number Theory February 1–June 15, 2008 Diophantine Approximation and Heights — 2010 — Theory. Exploring New Frontiers (A. Cap, (J. Schwermer), May 2–9, 2010 (D. Masser, H. P. Schlickewei, Metastability and Rare Events in Complex A. L. Carey, A. R. Gover, C. R. Graham, Summer School on Cartan Connections, Geometry W. M. Schmidt), February 27–May 12, 2006 Systems (P. Bolhuis, C. Dellago, E. van den Quantitative Studies of Nonlinear Wave J. Slovak), September 3–14, 2012 of Homogeneous Spaces, and Dynamics Eijnden), February 1–April 30, 2008 Phenomena (P. C. Aichelburg, P. Bizoń, (A. Cap, Rigidity and Flexibility Modern Methods of Time-Frequency Analysis II (V. Alexandrov, W. Schlag), January 7–February 28, 2010 C. Frances, K. Melnick), July 10–23, 2011 I. Sabitov, H. Stachel), April 23–May 6, 2006 Hyperbolic Dynamical Systems (L.-S. Young, (H. G. Feichtinger, K. Gröchenig), Summer School in Mathematical Physics H. Posch, D. Szász), May 25–July 5, 2008 Quantum Field Theory on Curved Space-Times September 10–December 15, 2012 Gerbes, Groupoids, and Quantum Field Theory and Curved Target-Spaces (M. Gaberdiel, (C. Hainzl, R. Seiringer), Operator Algebras and Conformal Field (P. Aschieri, H. Grosse, B. Jurco, S. Hollands, V. Schomerus, J. Yngvason), August 16–24, 2011 J. Mickelsson, P. Xu), May 8–July 31, 2006 Theory (Y. Kawahigashi, R. Longo, March 1–April 30, 2010 EMS-IAMP Summer School Quantum Chaos K.-H. Rehren, J. Yngvason), Complex Quantum and Classical Systems August 25–December 14, 2008 Matter and Radiation (V. Bach, J. Fröhlich, (N. Anantharaman, S. Morris Zeltditch, and Effective Equations (E. Carlen, J. Yngvason), May 5–July 30, 2010 St. Nonnenmacher, Z. Rudnick), L. Erdős, M. Loss), July 30–August 3, 2012 May 15–August 15, 2006 Topological String Theory, Modularity and Non-Perturbative Physics (L. Katzarkov, Homological Mirror Symmetry (A. Kapustin, A. Klemm, M. Kreuzer, D. Zagier), M. Kreuzer, A. Polishchuk, June 6–August 15, 2010 K.-G. Schlesinger), June 12–28, 2006 Anti-de Sitter Holography and the Quark-Gluon Global Optimization, Integrating Convexity, Plasma: Analytical and Numerical Aspects Optimization, Logic Programming, and (A. Rebhan, K. Landsteiner, S. Husa), Computational Algebraic Geometry (I. Bomze, August 2–October 29, 2010 I. Emiris, A. Neumaier, L. Wolsey), October 1–December 23, 2006 Higher Structures in Mathematics and Physics (A. Alekseev, H. Bursztyn, T. Strobl), September 1–November 11, 2010

46 / esi 2013 47 / esi 2013 data and statistics—junior research fellows 2004–2011 data and statistics—junior research fellows 2004–2011

Junior Research Fellows Alexandre Stefanov, Bulgaria Number of fellowships granted: 15 Henning Bostelmann, Germany Radu Saghin, Romania — 2010 — Jesper Tidblom, Sweden Number of months granted: 41 for 2007 Pierluigi Falco, Italy Maria Schimpf, Austria 2004–2011 Christian Tutschka, Austria Josh Garretson, usa Josef Silhan, Czech Republic Call for proposals: February 12, 2010 Matteo Viale, Italy Junior Research Fellows present at Gerald Gotsbacher, Austria Rafal Suszek, Poland Number of applications: 21 Vojtěch Žadník, Czech Republic the esi in 2006 (7 female, 21 male ): Peggy Kao, Taiwan Balint Vetö, Hungary From 2004 to 2011 the ESI Junior Research Number of fellowships granted: 9 Aleksey Kostenko, Ukraine Le Anh Vinh, Vietnam Fellowships Program was in place. There was a Katie Bloor, Great Britain Number of months granted: 27 for 2010, Gandalf Lechner, Germany Mihaly Weiner, Hungary total of 153 fellowships granted out of 515 appli- Francesco D’Andrea, Italy 8 for 2011 — 2005 — Christian Lübbe, Germany Lenka Zalabova, Czech Republic cations. The total number of months granted was Spyridon Dendrinos, Greece Sean Murray, Ireland Junior Research Fellows present at 510. The Junior research fellows came from 38 st Martyn De Vries, Netherlands 1 Call for proposals: June 15, 2005 Tomasz Paterek, Poland different countries and stayed for a period of Karla Diaz-Ordaz, Mexico the esi in 2010 (6 female, 20 male): Number of applications: 36 Pietro Polesello, Italy — 2009 — 3 months and 10 days on average. Number of fellowships granted: 9 Pierluigi Falco, Italy Julia Reffy, Hungary Camilo Arias Abad, Colombia Number of months granted: 5 for 2005, Anton Galaev, Russia 1st Call for proposals: April 17, 2009 Below you find a list of all Junior Research Florian Schätz, Austria Matteo Cardella, Italy 25 for 2006 Victor Junwei Guo, China Number of applications: 44 Fellows present at esi between April 26th 2004 Evelina Shamarova, Russia Claudio Dappiaggi, Italy Eman Hamza, Egypt Number of fellowships granted: 13 th Wonmin Son, Korea and February 28 2011. 2nd Call for proposals: October 15, 2005 Nataliya Ivanova, Ukraine Zywomir Dinew, Poland/Bulgaria Mihaly Weiner, Hungary Number of months granted: 32 for 2009, Number of applications: 34 Adam Joyce, Great Britain Slawomir Dinew, Poland/Bulgaria Michael Wohlgenannt, Austria 14 for 2010 Number of fellowships granted: 11 Wolfgang Lechner, Austria Alexander Fish, Israel — 2004 — Number of months granted: 36 for 2006 Richard Miles, Great Britain 2nd Call for proposals: October 11, 2009 Richard Green, Australia Thierry Monteil, France Number of applications: 20 Rika Hagihara, Japan st — 2008 — 1 Call for proposals: February 15, 2004 Junior Research Fellows present at Ian Morris, Great Britain Number of fellowships granted: 7 Myrto Kallipoliti, Greece Number of applications: 40 the esi in 2005 (5 female, 17 male): Angelika Kroner, Austria Milan Mosonyi, Hungary 1st Call for proposals: April 11, 2008 Number of months granted: 24 for 2010 Number of fellowships granted: 18 Tomasz Paterek, Poland Helge Krüger, Germany Sarah Bailey, United States Number of applications: 34 Number of months granted: 48 for 2004, Evangelia Petrou, Greece Junior Research Fellows present at Wojciech Krynski, Poland Christian Böhmer, Germany Number of fellowships granted: 10 19 for 2005 Michail Pevzner, Russia the esi in 2009 (9 female, 18 male): Christiane Losert, Austria Jessica Barrett, Great Britain Number of months granted: 16 for 2008, Peter Pickl, Germany Rongmin Lu, Singapore 2nd Call for proposals: May 31, 2004 Matthias Birkner, Germany 14 for 2009 Lior Alexandra Aermark, Israel Pietro Polesello, Italy Kostyantyn Medynets, Ukraine Number of applications: 38 Elena Cordero, Italy Jose Aliste, Chile Catherine Richard, France 2nd Call for proposals: November 14, 2008 Wolfgang Moens, Belgium Number of fellowships granted: 9 Anton Galaev, Russia Emanuela Bianchi, Italy Hisham Sati, Lebanon Number of applications: 36 Vladimir N. Salnikov, Russia Number of months granted: 8 for 2004, Sebastian Guttenberg, Germany Francis Brown, Great Britain Jean Savinien, France Number of fellowships granted: 9 Christian Ortiz, Chile 18 for 2005, 2 for 2006 Marcela Hanzer, Croatia Claudio Dappiaggi, Italy Emanuel Scheidegger, Switzerland Number of months granted: 34 for 2009 Piotr Przytyczki, Poland Anne-Katrin Herbig, Germany Slawomir Dinew, Poland Chris Rogers, usa 3rd Call for proposals: November 15, 2004 Evelina Shamarova, Russia Felipe Leitner, Germany Junior Research Fellows present at Zywomir Dinew, Poland Florian Schätz, Austria Number of applications: 65 Mathieu Stienon, Belgium Thomas Neukirchner, Germany the esi in 2008 (7 female, 16 male): Anastasia Jivulescu, Romania Susanne Schimpf, Germany Number of fellowships granted: 7 Alexandr Usnich, Belarus Kasso Okoudjou, Benin Lukasz Kosinski, Poland Nora Seeliger, Germany Number of months granted: 20 for 2005 Alexander Powell, United States Hendrik Adorf, Germany Rongmin Lu, Singapore Marcel Vonk, The Netherlands Mikhail Pevzner, Russia Vasiliki Anagnostopoulou, Greece Anca Matioc, Romania Junior Research Fellows present at — 2007 — Matthias Westrich, Germany Nenad Teofanov, Serbia Caterina Cusulin, Italy Kostyantyn Medynets, Ukraine the esi in 2004 (4 female, 14 male): Mark Williamson, Great Britain Jan Tichavsky, Czech Republic 1st Call for proposals: April 30, 2007 Philipp Geiger, Austria Karin Melnick, usa Wolfgang Angerer, Austria Alexandre Stefanov, Bulgaria Number of applications: 37 Neven Grbac, Croatia Wolfgang Moens, Belgium Jessica Barrett, Great Britain Stefan Wenger, Switzerland Number of fellowships granted: 9 Harald Grobner, Austria Mathieu Molitor, France — 2011 — Matthias Birkner, Germany Marcin Wiesniak, Poland Number of months granted: 19 for 2007, Minh Ha Quang, Vietnam Milan Mosonyi, Hungary Jeremy Clark, usa Michael Wohlgenannt, Austria 16 for 2008 Eman Hamza, Egypt Carolina Neira, Colombia Junior Research Fellows present at Ionas Erb, Germany Vojtěch Žadník, Czech Republic Matthieu Josuat-Verges, France Nicolas Raymond, France the esi in 2011 (2 female, 2 male): Borislav Gajic, Serbia Roland Zweimüller, Austria 2nd Call for proposals: November 10, 2007 Peggy Kao, Australia Jean Ruppenthal, Germany Angelika Kroner, Austria Alessandro Giuliani, Italy Number of applications: 41 Aleksey Kostenko, Ukraine Josef Silhan, Czech Republic Wojciech Krynski, Poland Marcela Hanzer, Croatia Number of fellowships granted: 15 Christian Lübbe, Germany Jean-Charles Sunye, France Nora Seeliger, Germany Bianca Mladek, Austria — 2006 — Number of months granted: 57 for 2008 Mate Matolcsi, Hungary Kirsten Vogeler, Germany Mark Williamson, Great Britain Ari Pakman, Argentina Philippe Nadeau, France Zhituo Wang, China 1st Call for proposals: March 31, 2006 Junior Research Fellows present at Milena Radnovic, Serbia Maryna Nesterenko, Ukraine Jiangyang You, China Number of applications: 27 the esi in 2007 (3 female, 17 male): Karl Georg Schlesinger, Germany Lenka Zalabova, Czech Republic Number of fellowships granted: 12 Lei Zhang, China Jeff Selden, usa Stuart Armstrong, Canada Number of months granted: 22 for 2006, Magdalena Zych, Poland 9 for 2007 Christoph Bergbauer, Germany Olivier Bernardi, France 2nd Call for proposals: October 31, 2006 Number of applications: 42

48 / esi 2013 49 / esi 2013 data and statistics—senior research fellows 2002–2012 data and statistics—senior research fellows 2002–2012

Senior Research Fellows — 2003/04 — — 2005/06 — — 2007/08 — — 2009/10 — — 2011/12 — 2002–2012 fall term 2003: fall term 2005: fall term 2007: fall term 2009: fall term 2011:

Vladimir Mazya (Linköping U, Sweden): Emil Straube (Texas A&M U, College Christos N. Likos (U Düsseldorf, Germany): Peter West (King’s College, London, uk): James W. Cogdell (Ohio State U, Columbus, From 2002 to 2012, there have been 48 lectures Sobolev Spaces With Applications to PDE Station, usa): The L²-Sobolev Theory of the Introduction to Theoretical Soft Matter Physics Supergravity Theories usa): L-functions and Functoriality held by Senior Research Fellows that were at the d-bar-Neumann Problem, continuation of the spring term 2012: ESI for a longer research period. Jürgen Rohlfs (U Eichstätt, Germany): spring term lecture course Radoslav Rashkov (Sofia U, Bulgaria): Jeff McNeal (Ohio State U, Columbus, usa): Algebraic Groups Over Number Fields and Dualities Between Gauge Theories and Strings L²-Methods in Complex Analysis Related Geometric Questions Bernard Helffer (U Paris Sud-Orsay, France): Detlev Buchholz (U Göttingen, Germany): Introduction to the Spectral Theory for Schrödinger John Barrett (U Nottingham, uk) and spring term 2010: Fundamentals and Highlights of Algebraic — 2002/03 — Peter van Nieuwenhuizen (C. N. Yang Operators With Magnetic Fields and Application Richard Szabo (U of Edinburgh, uk): Quantum Fields Institute for Theoretical Physics, Stony Two short courses on Theoretical Physics Neven Grbac (U Rijeka, Croatia): fall term 2002: Eisenstein Series Brook U, New York, usa): N=1 and N=2 Boban Velickovic (U Paris Diderot, France): Eduard Feireisl (Academy of Sciences, Prague, Czech Republic): Mathematics and Arkadi Onishchik (Yaroslavl State U, Russia): Supersymmetry and Supergravity, continuation Introduction to Descriptive Set Theory Stefan Hollands (Cardiff U, uk): Quantum Complete Fluid Systems Real Representation Theory of Lie Algebras of the spring term lecture course — 2008/09 — spring term 2006: Field Theory on Curved Spacetimes and Lie Groups spring term 2004: fall term 2008: David Masser (U Basel, Switzerland): Peter West (King’s College, London, uk): Anatoly Vershik (Steklov Institute, St. Theory Werner Ballmann (U Bonn, Germany): Heights in Diophantine Geometry Nigel Higson (Penn State U, usa): E- Petersburg, Russia): Measure Theoretic Über die Geometrie der Gebäude Index Theory, Groupoids and Noncommutative Constructions and Their Applications in Ergodic Mathai Varghese (U Adelaide, Australia): Geometry Theory, Asymptotics, Combinatorics, Jürgen Fuchs (Karlstad U, Sweden): K-Theory Applied to Physics — 2010/11 — and Geometry Conformal Field Theory Goran Muic (U Zagreb, Croatia): Selected Topics in the Theory of Automorphic Forms for fall term 2010: spring term 2003: — 2006/07 — Reductive Groups Tykal Venkataramana (Tata Institute, Michael Lacey (Georgia Institute of — 2004/05 — Representations Contributing fall term 2006: Feng Xu (UC Riverside, usa): Operator Mumbai, India): Technology, Atlanta, usa): Recent Trends fall term 2004: Algebras and Conformal Field Theory to Cohomology of Arithmetic Groups in Fourier Analysis Ioan Badulescu (U Poitiers, France): spring term 2011: Manfred Salmhofer (U Leipzig, Germany): Representation Theory of the General Linear spring term 2009: Peter van Nieuwenhuizen (C. N. Yang Renormalization Theory—Analysis and Group Over a Division Algebra Institute for Theoretical Physics, Stony Bruno Nachtergaele (UC Davis, usa): Applications Michael Loss (Georgia Institute of and Quantum Spin Systems. An Introduction to the Brook U, New York, usa): N=1 N=2 Thomas Mohaupt (U Liverpool, uk): Technology, Atlanta, usa): Spectral Inequal- Supersymmetry and Supergravity General Theory and Discussion of Recent Anton Wakolbinger (U Frankfurt, Black Holes, Supersymmetry and Strings ities and Their Applications to Variational Developments Germany): Stochastische Prozesse aus der Problems and Evolution Equations Populationsgenetik Miroslav Englis (Academy of Sciences, Prague, Czech Republic): Analysis on Raimar Wulkenhaar (U Münster, Germany): Michael Baake (U Bielefeld, Germany): Spektraltheorie dynamischer Systeme und Vlatko Vedral (Imperial College, London, Complex Symmetric Spaces Spektrale Tripel in nichtkommutativer Geometrie aperiodische Ordnung uk): Foundations of Quantum Information und Quantenfeldtheorie spring term 2007: spring term 2005: Kenneth Dykema (Texas A&M U, College Miroslav Englis (Academy of Sciences, Station, usa) and Roland Speicher Werner Ballmann (U Bonn, Germany): Prague, Czech Republic): Analysis on Complex (U Saarland, Germany): Free Probability Theory Kählergeometrie Symmetric Spaces, continuation of the fall term 2006 lecture course Peter West (King’s College, London, uk): Jan Derezinski (Warsaw U, Poland): Symmetries of Strings and Branes, continuation Operator Algebras and Their Applications Vadim Kaimanovich (International U of the spring term lecture course in Physics Bremen, Germany): Boundaries of Groups: Geometric and Probabilistic Aspects Anatoly Vershik (Steklov Institute, St. Petersburg, Russia): Representation Theory Thomas Mohaupt (U Liverpool, uk): of Symmetric Groups, Graphs, Universality Black Holes, Supersymmetry and Strings, continuation of the fall term 2006 lecture Emil Straube (Texas A&M U, College course Station, usa): The L²-Sobolev Theory of the d-bar-Neumann Problem

50 / esi 2013 51 / esi 2013 data and statistics—people at the esi

Kollegium (Scientific Governing Board) Contributors: Imprint: June 1, 2011–December 31, 2013: James W. Cogdell Publisher: Joachim Schwermer, Department of Mathematics Joachim Schwermer Professor of Mathematics, U Vienna, Ohio State University Director Director 231 W. 18th Ave. Erwin Schrödinger Columbus, oh 43210 International Institute for Goulnara Arzhantseva, usa Mathematical Physics Professor of Mathematics, U Vienna, [email protected] Boltzmanngasse 9 Deputy Director 1090 Vienna / Austria Peter Goddard Jakob Yngvason, School of Natural Sciences Editors: Professor of Theoretical Physics, U Vienna, Institute for Advanced Study Joachim Schwermer Deputy Director Einstein Drive Susanne Schimpf Princeton, nj 08540 Piotr Chruściel, Photos: usa Professor of Physics, U Vienna ESI, Österreichische [email protected] Zentralbibliothek für Physik, Adrian Constantin, Jeremy Gray U Wien, Philipp Steinkellner Professor of Mathematics, U Vienna Faculty of Mathematics, Computing Design & Production: Frank Verstraete, and Technology Philipp Steinkellner Professor of Physics, U Vienna The Open University www.steinkellner.com Walton Hall Milton Keynes Printing: International Scientific Advisory Board MK7 6AA uk Grasl FairPrint as of January 1, 2013: [email protected] 2540 Bad Vöslau www.grasl.eu John Cardy, Martina R. Schneider Professor of Theoretical Physics, U Oxford Institut für Mathematik © 2013 Erwin Schrödinger Johannes Gutenberg-Universität International Institute for Isabelle Gallagher, Staudingerweg 9 Mathematical Physics Professor of Mathematics, U Paris-Diderot 55099 Mainz Deutschland Helge Holden, [email protected] Professor of Mathematics, Norwegian U of Science and Technology, Trondheim Joachim Schwermer Erwin Schrödinger International Institute Daniel Huybrechts, for Mathematical Physics Professor of Mathematics, U Bonn Boltzmanngasse 9 Horst Knörrer, 1090 Wien Professor of Mathematics, eth Zürich Österreich [email protected] Herbert Spohn, Professor of Mathematics, Wendelin Werner Technical U Munich Laboratoire de Mathématiques Université Paris-Sud Vincent Rivasseau, Bâtiment 425 Professor of Theoretical Physics, 91405 Orsay Cedex U Paris-Sud, Orsay France [email protected]

Administration:

Alexandra Katzer, Head

Maria Marouschek

Beatrix Wolf

52 / esi 2013 2013 Erwin Schrödinger International Institute for Mathematical Physics