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Banff International Research Station Proceedings 2004 Contents Five-day Workshop Reports 1 1 Interactions Between Model Theory and Geometry (04w5534) 3 2 Topology of Manifolds and Homotopy Theory (04w5533) 12 3 Orthogonal Polynomials; Interdisciplinary Aspects (04w5530) 14 4 Model Reduction Problems and Matrix Methods (04w5513) 27 5 Analytic and Geometric Aspects of Stochastic Processes (04w5023) 29 6 Celestial Mechanics (04w5012) 37 7 BIRS Workshop on Mathematics and Creative Writing (04w5555) 45 8 Microeconometrics of Spatial and Grouped Data (04w5036) 49 9 Singular Cardinal Combinatorics (04w5523) 51 10 Mathematical Structures in Economic Theory and Econometrics (04w5536) 63 11 Knots and Their Manifold Stories (04w5037) 66 12 New Developments on Variational Methods and Their Applications (04w5033) 69 13 Mathematical Foundations of Scientific Visualization, Computer Graphics, and Massive Data Exploration (04w5043) 80 14 Aperiodic Order: Dynamical Systems, Combinatorics, and Operators (04w5001) 83 15 Semimartingale Theory and Practice in Finance (04w5032) 94 16 New Horizons in String Cosmology (04w5021) 96 17 Advances in Complexity Theory (04w5100) 99 18 Convex Geometric Analysis (04w5014) 108 19 Modelling Protein Flexibility and Motions (04w5017) 115 20 Geometric Evolution Equations (04w5008) 123 21 Conformal Geometry (04w5006) 125 22 Stochastic Processes in Evolutionary and Disease Genetics (04w5015) 135 i ii 23 Statistical Science for Genome Biology (04w5519) 144 24 Dynamics, Control and Computation in Biochemical Networks (04w5550) 146 25 Combinatorial Hopf Algebras (04w5011) 150 26 Pluripotential Theory and its Applications (04w5035) 153 27 Commutative Algebra: Homological and Birational Theory (04w5027) 155 28 Quantum Computation and Information Theory (04w5041) 159 29 Self-Stabilizing Distributed Systems (04w5531) 164 30 Free Probability Theory (04w5028) 167 31 Braid Groups and Applications (04w5526) 178 32 Mathematical Image Analysis and Processing (04w5512) 187 33 The Structure of Amenable Systems (04w5045) 197 34 Functional Differential Equations (04w5026) 205 35 New Techniques in Lorentz Manifolds (04w5521) 207 36 Explicit Methods in Number Theory (04w5502) 218 37 Diophantine Approximation and Analytic Number Theory (04w5507) 220 38 Mathematical Models for Biological Invasions (04w5539) 224 39 Generalizations of de Bruijn Cycles and Gray Codes (04w5039) 236 40 Numeracy and Beyond (04w5044) 245 41 Resolution of Singularities, Factorization of Birational Mappings, and Toroidal Geometry (04w5510) 247 Two-day Workshop Reports 261 42 Human Infant Speech Perception and Language Acquisition: Rules vs. Statistics (04w2552) 263 43 2-day Retreat on Mathematical Ecology and Evolution (04w2540) 266 44 PIMS PDF Meeting (04w2542) 269 45 Mathfair Workshop (04w2600) 270 46 Directions in Combinatorial Matrix Theory (04w2525) 273 47 Decentralized Discrete Event Systems: Structure, Communication, and Control (04w2040) 277 48 Adaptive Wavelet and Multiscale Methods for Partial Differential Equations (04w2055) 280 49 The Design and Analysis of Computer Experiments for Complex Systems (04w2056) 282 iii 50 Combinatorial and Algorithmic Aspects of Networking and the Internet (04w2059) 284 51 Linear Operators: Theory, Applications and Computations (04w2063) 286 52 Alberta Topology Seminar (04w2064) 287 53 Theoretical Physics Institute (TPI) Symposium 2004 (04w2544) 289 54 PIMS Executive Retreat (04w2545) 291 55 Pacific Northwest Numerical Analysis Seminar (04w2053) 292 56 Data Mining MITACS Industry Session (04w2065) 294 57 Canadian Mathematical Leadership Retreat (04w2067) 296 58 MITACS Theme Meeting: Communication Networks and Security (04w2069) 297 59 MITACS Project Meeting: Modelling Trading and Risk in the Market (04w2070) 299 60 Number Theorists Weekend (04w2505) 301 61 MITACS Environment and Natural Resources Theme Meeting (04w2066) 304 Focused Research Group Reports 307 62 Robust Analysis of Large Data Sets (04frg501) 309 63 String Field Theory Camp (04frg538) 312 64 Kinetic Models for Multiscale Problems (04frg049) 315 Research in Teams Reports 321 65 Cohomogeneity One Manifolds With Positive Sectional Curvature (04rit525) 323 66 Modular Invariants and NIM-reps (04rit048) 326 67 Pi in the Sky Meeting (04rit300) 329 68 Maximal Functions in Non-commutative Analysis (04rit004) 331 69 Geometrical Analysis in One and Several Complex Variables (04rit524) 334 70 Geometry and Deformation Theory of Hyperbolic 3-manifolds (04rit057) 337 71 Stability and Computations for Stochastic Delay-Differential Equations (04rit047) 341 72 Study of Affine Surfaces with Self-maps of Degree > 1 and the Jacobian Problem (04rit553) 343 73 Competing Species, Predator-Prey Models and Measured-valued Diffusions (04rit050) 345 74 Geometry and Analysis on Cauchy Riemann Manifolds (04rit554) 348 75 Research on Stochastic Models for the Web Graph and Other Scalefree Networks (04rit060) 352 iv Summer School Reports 355 76 MITACS–MSRI–PIMS Special Program on Infectious Diseases, Summer School & Workshop (04ss101 & 04ss100) 357 Five­day Workshop Reports 2 Five-day Workshop Reports Chapter 1 Interactions Between Model Theory and Geometry (04w5534) March 14–18, 2004 Organizer(s): Jan Denef (Katholieke Universitat¨ Leuven), Deirdre Haskell (McMaster University), Ehud Hrushovski (Hebrew University), Angus Macintyre (University of Ed- inburgh), Anand Pillay (University of Illinois, Urbana-Champaign), Patrick Speissegger (McMaster University) Model theory is in a period of rich activity. Advances in pure model theory are finding immediate ap- plications, in particular to the model theory of fields, while the applications are themselves motivating the abstract developments. Applied model theory is using ideas and methods from other parts of mathematics, ranging from homology theory to complex analytic geometry. These two strands of research were exhibited at the BIRS workshop. The workshop was used as an opportunity to exhibit and elucidate two large pieces of technical work which have been in the process of development for several years. The first of these, “Homol- ogy in o-minimal theories” is a prime example of the way in which a mathematical tool can be developed to apply in a wider model-theoretic context. The second tutorial, “Imaginaries in valued fields” illustrates the way in which an applied context (in this case, an algebraically closed valued field) has motivated a theoret- ical development (the notion of stable domination). The three afternoon sessions presented recent research developments in three different active areas of research in applied model theory. A. Berarducci: Tutorial on o-minimality The principal aim of the tutorial was to describe the “transfer approach” in the study of definable groups and definable manifolds in an o-minimal structure, where by “transfer approach” I essentially mean a reduction to the classical (i.e. locally compact) case. The 5 talks of the tutorial focused in particular on work in collaboration with M. Otero in this direction, as well as on related results and conjectures by other authors, which I describe in the sequel. The definable sets in any o-minimal structure M admit a cell decomposition analogous to the cylindric cell decomposition for semialgebraic subsets of Rn with the difference that cells are bounded by graphs of de- finable continuous functions rather than semialgebraic functions. The analogy with semialgebraic geometry may however be misleading, since in general an o-minimal structures may not admit an easy description of its definable continuous functions (hence of its definable sets). The problem is the lack of quantifier elimination theorems or model completeness theorems. We are interested in definable groups in an o-minimal structure M, namely groups whose underlying set is definable and with a definable group operation. (We will assume in the sequel that M expands a field, although for many results this assumption is not necessary.) Such groups have been studied by many authors and the results so far obtained suggest a close analogy between definable groups and classical Lie groups. 3 4 Five-day Workshop Reports In particular it follows from the results of A. Pillay [26] that if the underlying set of the o-minimal structure is the real line, then every definable group is indeed a Lie group, while Y. Peterzil, A. Pillay and S. Starchenko [24] obtained matrix representation theorems for definable groups which confirm the analogy with Lie groups. Despite these results, many fundamental questions on definable groups remain open. The difficulty is that many tools available in the study of classical Lie groups are not available in the o-minimal context, mainly due to the fact that definable groups and definable manifolds may fail to be locally compact (if the structure M is not locally compact). This motivates the search for “transfer theorems” which allow, for various problems, a reduction to the case when the underlying set of M is the real line. A considerable amount of work in this direction was motivated by the following problem. It is known that every compact Lie group has torsion elements (namely elements of finite order) and Peterzil and Steinhorn [25] asked whether the same holds for every “definably compact” group (such groups need not be compact in the classical sense). It was known (Strzebonski [31]) that if a prime p divides the “o-minimal Euler charac- teristic” E(G) of a definable group G, then G has an element of order p. (The o-minimal Euler characteristic of a set is defined as the number of its even dimensional cells
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