Annual Report 2005 - Fields Institute - University of Toronto

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found that it leads to opportunities to cooperate with other institutes on related programs. We also coordinate a list of institutes worldwide, and we find this an attractive way to share information and to mentor new institutes. Research institutes have changed the face of mathematics research, and have drawn attention to the role of communication and of interdisciplinary research in stimulating research mathematics. In Canada, the mathematics institutes have played an important part in focusing international attention on Canadian mathematics, as well as on giving Canadian mathematicians a window on inter- M e s s a g e f r o m t h e D i r e c t o r national research directions. Fields is proud to be a part of this enterprise. As another busy year draws to a close, the nature of what it means to be a successful institute becomes clearer. In this report, you will discover how we have contributed to the effort. Barbara Lee Keyfitz, Director Fields Institute 2005 ANNUAL REPORT 8
The Geometry of String Theory: September 2004–June 2005 Retrospective on the Thematic Program Organizing Committee: Kentaro Hori and Lisa Jeffrey (Toronto), Mikhail Kapranov (Yale and Toronto), Boris Khesin (Toronto), Rob Myers (Perimeter), and Amanda Peet (Toronto) Scientific Committee: Alexander Beilinson (Chicago), Jim Bryan (UBC), D. Freed (Austin), Kentaro Hori (Toronto), Jacques Hurtubise (McGill), Lisa Jeffrey (Toronto), Mikhail Kapranov (Yale and Toronto), Sheldon Katz (Urbana- Champaign), Boris Khesin (Toronto), Robert Myers (Perimeter Institute), Amanda Peet (Toronto), Edward Witten (I.A.S.) and Noriko Yui (Queen’s) This year’s thematic program on the Geometry of String Theory was hosted jointly by the Fields Institute and Perimeter Institute for Theoretical Physics (in nearby Waterloo). The activities were essentially divided equally between Fields and Perimeter. The central idea defining string theory is that, when viewed with sufficiently high resolution, all elementary particles will appear to be extended one-dimensional objects, i.e., strings. From this relatively simple starting point emerges an extraordinarily rich mathematical structure. For example, the internal consistency of the theories requires that the strings propagate in a ten-dimensional spacetime. So, six of the dimensions must be curled up on a compact geometry in order to reproduce the four-dimensional physics which we observe. In fact it is the intricacies of this internal geometry which are responsible for the complex physical interactions which emerge in the four-dimensional world. This is a simple example which illustrates the central role which geometry plays in string theory. Further, string theory incorporates ‘supersymmetry’, a symmetry which changes the spins of elementary particles Organizers: Lisa Jeffrey, Mikhail Kapranov, Boris Khesin, Robert Myers, and Amanda Peet T h e m a t i c P r o g r a m s pairing each fermion with a boson – this is the origin of the ‘super’ in superstrings. On the physical side, supersymmetry plays an important role in taming high energy divergences which appear in theories of point particles. On the mathematical side, this symmetry is fundamental in constructing new topological invariants, e.g., in Seiberg- Witten theory. More recently, it has been realized that string theory is more that just a theory of strings. That is, there are other kinds of extended objects, known as Dirichlet-branes or D-branes, which play an important role. In fact in the past ten years, D-branes have been the source of a profusion of new ideas and remarkable progress in our understanding of string theory. In particular, Dbranes played a crucial role in constructing new relations or ‘dualities’ between what were previously seen as five different consistent superstring theories. Instead it is now believed that they represent different phases of a single unified framework, commonly known as M-theory. Of course, mathematics played an important role in this progress as well. For example, the proper classification of D-branes comes from K-theory. String theory originated from a ‘failed’ attempt in the 1960’s to describe the nuclear interactions. Soon after, it evolved towards a very much more ambitious goal of being ‘the theory of everything’– that is a providing a unified framework to describe all of the elementary particles and fundamental forces in nature. Combining Einstein’s theory of gravity with the standard quantum theory used to describe physics at subatomic scales has lead to perplexing inconsistencies which have mystified physicists for over fifty years. Hence finding a quantum theory of gravity is often seen as the holy grail of theoretical physics. String theory is seen by many as the leading contender in this quest since as well as containing the appropriate structures Fields Institute 2005 ANNUAL REPORT 9
Page 1 and 2: Fields Institute 2005 ANNUAL REPORT
Page 3 and 4: Institute Profile 2 Message from th
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Career: Some Remarks. On August 7,
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Jeremy Quastel and Tom Salisbury Ne
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the property (T) in 2001.) The work
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This conference brought together ma
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Many speakers reported on recent pr
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and former president of the Interna
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S. Mitchell (UBC) An asymptotic fra
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Combinatorics workshop participants
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Workshop participants Each of the i
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Workshop participants Speakers: Ran
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Bruce Allison their work, discuss n
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Poster participants talks, each of
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Workshop on Mathematical Programmin
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University) presented recent result
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volume are Rajendra Bhatia (India),
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SEMINARS Algebraic Combinatorics Se
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Geometry and Model Theory Seminar S
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ments. All of the results obtained
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Vladimir Vinogradov (Ohio) On “co
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parameters - for example, in the ca
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Winter 2004 Meeting of the Canadian
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Special Sessions: Automatic Sequenc
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meeting introduced an Industry/Acad
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The workshop was attended by approx
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tri-annually. MOPTA 05 will be held
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Lowell Aronoff (Cannex Financial Ex
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Kuznetsov, he models each firm usin
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ment returns. The object of the res
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Camp participants Math Camp Program
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This Fields-sponsored symposium was
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Fields Institute Fellows The honour
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Fields Institute Communications Ser
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BARBARA LEE KEYFITZ Director TOM SA
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Scientific Advisory Panel 2004-2005
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and of the Board of Directors of th
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Industrial Advisory Board The Indus
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students at the University of Ottaw
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and graduate students are involved
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F i n a n c i a l S t a t e m e n t
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Fields Institute 2005 ANNUAL REPORT
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