Fieldsnotes fallwinter 1112

<div>SUMMER 2011 Thematic Program on the Mathematics of Constraint Satisfaction LAST SUMMER, THE FIELDS INSTITUTE HOSTED the Thematic Program on the Mathematics of Constraint Satisfaction. It was organized by Venkatesan Guruswami (Carnegie Mellon University), Pavol Hell (Simon Fraser University), Matt Valeriote (McMaster University), and Ross Willard (University of Waterloo). A great many combinatorial problems may be cast as finite-domain constraint satisfaction problems (CSPs), or equivalently as homomorphism problems about finite relational structures. A simple example is graph 4-colourability. Planar graphs are famously known to be 4-colourable; however, most graphs that arise in applications are not planar, and it is reasonable to ask of a given graph whether it is 4-colourable or not. Cast as a CSP, this question amounts to being given a finite graph G = ( V , E) and asking whether there exists an assignment χ from V to the target set { 0, 1, 2, 3 } that satisfies the constraints χ( u) /= χ( v) for ( u, v) ∈ E. Cast as a homomorphism problem, 4-colourability amounts to being given G and asking whether there exists a graph homomorphism from G to K 4 the complete graph on 4 vertices. 4-colourability is hard (NP-complete). If one changes the target set and/or the nature of the constraints, or equivalently, if one replaces K 4 with another graph, hyper-graph, etc., one gets a new CSP or homomorphism problem. One area of theoretical computer science attempts to classify which CSPs or homomorphism problems are tractable, indeed to prove a dichotomy (either tractable or NP-complete). A sub-area seeks to further classify the tractable CSPs. Another area seeks in the hard case to find tractable approximate answers or to prove that even these are hard. Finite relational structures generalize graphs and for years have played a central role in disparate disciplines including graph theory itself, multiple-valued logic, database theory, computational complexity theory, and finite model theory. They provide the natural, appropriate level of generality for CSPs and homomorphism problems. The summer Thematic Program focused on algorithmic, complexity, and approximability issues of combinatorial problems on finite relational structures, including and particularly focused on CSPs, homomorphisms and related problems. The three main, interrelated strands of the program were: the “algebraic approach” to CSPs, graph homomorphism problems, and approximate solutions to hard CSPs. For each of these strands an associated workshop was held during the program. The workshops gave focus to each of the strands and served to highlight the connections between them. Graph Homomorphism Problems Graph homomorphisms are central to graph theory and have been studied in their own right for years. They are viewed in graph theory as a common generalization of various versions of graph colourings, and find applications in many parts of graph theory, in scheduling and operations research in general, and even in statistical physics. One of the most popular problems, that of detecting the existence of a homomorphism, is a special constraint satisfaction problem. The Workshop on Graph Homomorphisms was held at the Fields Institute during the week of July 11 and was organized by Pavol Hell, Claude Tardif (Royal Military College), and Xuding Zhu (Zhejiang Normal University). The workshop focused on all aspects of graph homomorphisms, from those directly related to CSPs, such as minimum cost and list homomorphisms, versions of projectivity, and polymorphisms, to those related to basic graph theoretic notions such as colourings, tree-width and tree-depth, and those related to notions from category theory, such as adjoint functors, to those related to statistical physics such as counting homomorphisms, and the study of connection matrices. The workshop goal of bringing together the main players in all aspects of graph homomorphisms was amply met. The “Algebraic Approach” Spectacular advances in the dichotomy problem have been achieved recently via the so-called “algebraic approach” pioneered by Jeavons. As in descriptive complexity, where complexity classes are studied through the properties of associated logics, under the algebraic approach one associates universal algebras, or collections of such algebras, to classes of CSPs. It turns out that some well studied properties of universal algebras perfectly capture the complexity of the corresponding problems. In particular, the existence of homomorphisms from finite powers of a relational structure to itself (called polymorphisms) which satisfy certain axiomatic conditions serve to organize all known results regarding the classification of CSPs by their complexity. The axiomatic conditions that appear to capture various levels of complexity of CSPs, align perfectly with the classification theory for finite universal algebras called tame congruence theory developed in the 1980s and 90s. Moreover, the same axiomatic conditions can be used to design efficient algorithms and to study the combinatorial structure of CSPs. During the week of August 2, 2011 the Workshop on Algebra and CSPs was held at Fields. It was organized by Libor Barto (Charles University and McMaster University), Andrei Krokhin (Durham University), and Ross Willard. The workshop highlighted the recent advances on the CSP Dichotomy Conjecture arising from the algebraic approach and focused on the various algebraic notions and results 6 FIELDSNOTES | THE FIELDS INSTITUTE for Research in Mathematical Sciences</div> <div>THEMATIC PROGRAMS WINTER 2012: PROSPECTIVE Thematic Program on Galois Represent IN 1972, ALL THE EXPERTS IN THE arithmetic of automorphic forms could gather together in Antwerp for a single conference. Soon thereafter, simultaneous developments (Deligne’s proof of the Weil conjectures and work on Langlands’ functoriality conjectures) split the field along an arithmetic-automorphic divide. Only now has this divide begun to narrow. The recent proof of the Sato-Tate conjecture has made it clear that future progress in arithmetic requires a thorough understanding of the general representation theory of automorphic representations. This has happened simultaneously with the nascent development of the p-adic local Langlands program, which also draws on intuition and techniques from representation theory, and was an essential ingredient in recent progress on the Fontaine-Mazur conjecture. These results may well be the first fruits of an abundant harvest. In particular, new results in the theory of automorphic representations, such as the proof of the Fundamental lemma and the construction of good models for Shimura varieties, have only begun to be explored. Thus, significant new developments can be expected in the next few years. The blossoming of new ideas presents both a formidable challenge and an opportunity for current and recent graduate students in number theory, as well as the number theory community as a whole, since the technical tools required for progress in the field are now so diverse. The Thematic Program on Galois Representations and Automorphic Forms aims to bridge the divide between researchers in disparate fields that nevertheless share a connection to the Langlands program. A key goal of the program is to bring together leading experts in the fields of Galois representations, automorphic forms, and related subjects, with the aim of communicating recent discoveries to a variety of audiences, including graduate students and beginning investigators, and experts from other fields, ultimately with the intention of furthering progress in the Langlands program. The study of Galois representations and automorphic representations formed a pair of threads woven throughout the history of number theory, and thus, the divergence between the two fields over the past forty years has been somewhat anomalous. The recent convergence 2 FIELDSNOTES | THE FIELDS INSTITUTE for Research in Mathematical Sciences Illustration by Keith Yeomans</div> <div>12 FIELDSNOTES | THE FIELDS INSTITUTE for Research in Mathematical Sciences</div> <div>Électricité de France, with the overarching goals of building a quantitative history of financial crises, and developing early warning indicators for various types of crises. The undergraduate group was assigned to read This Time is Different, a study done by Reinhart and Rogoff. The group gathered and analyzed a broad range of data sets using indicators chosen by the supervisors. The indicators were tested on annotated crisis data and official market data from 1960 onwards. Consequences of this type of work include reducing the risk of future crises and gaining a better overall understanding of global and systemic risk. The data used by the group was compiled from over 125 countries and covers some countries back more than 800 years ago. The project is only in its infancy but has already received considerable attention. Model Theory of Operator Algebras Participants: Ferenc Bencs, Nigel Sequeira, Louis-Phillippe Thibault Supervisors: Ilijas Farah (York), Bradd Hart (McMaster) This group worked on a problem at the intersection of model theory and operator algebra theory. To find results in the model theoretic setting, the group established an Ehrenfeucht-Fraïssé game. The purpose of building the game was to translate a winning strategy in the context of the game to criteria for the convergence of limits in the model theoretic context. The supervisors are regular participants in the Toronto Set Theory Seminar and Operator Algebra seminar, and Geometry and Model Theory Seminar at the Fields Institute, respectively. Study of the Development of Glaucoma Participants: Luke Broemeling, Linda Liu, Vishal Siewnarine Supervisors: Irwin Pressman (Carleton), Sivabal Sivaloganathan (Waterloo) Together with the Centre for Mathematical Medicine and researchers in England, this group studied glaucoma, an incurable disease. They modeled the flow of the fluid in the eye, in hope of “unlocking the mystery behind what causes glaucoma.” Within a few weeks, the group produced mathematical models to study the underlying mechanisms that have been postulated as the cause. When fluid pressure increases in the eye, strain is put on the optic nerve and sight can be lost as a result. The group used non-dimensionalization techniques as well as tools from fluid dynamics. In particular, they derived a set of partial differential equations from a version of lubrication theory, based on the Navier-Stokes equations. In building and analyzing their model, they studied a number of recent papers dealing with fluid mechanics in the anterior chamber, the posterior chamber, and Schlemm’s canal. Combinatorial Rigidity and Graph Constructions Participant: Rebecca Tessier Supervisors: Anthony Nixon (Fields), Elissa Ross (York) The 2011 Fall Thematic Program on Discrete Geometry and Applications played a significant role in this project, as both supervisors were primarily at the Institute for that Program. Tessier investigated (2,l)-tight graphs for l = 0,1. These are classes of graphs defined by simple counting inequalities on the number of edges in any vertex induced subgraph. Specifically if X is a subset of the vertex set V then the subgraph induced by X contains at most 2|X|-l edges with equality when X=V. Particularly she investigated constructive characterisations of these graphs. That is, building procedures for generating all such graphs from the minimal example. The problem is well understood when the graphs are allowed to have loops and multiple edges. However in some applications to rigidity theory loops and multiple edges can have no meaning. Here such constructions are required to stay within the class of loopless or simple graphs. To attack this problem Tessier considered the usual Henneberg type moves, other existing operations such as vertex splitting as well as some new operations and proved exactly when these operations stay within the classes of (2,l)- tight graphs. In July, students participated in the Fields Undergraduate Network’s (FUN) Workshop on Discrete Mathematics, which took place at Carleton University. Towards the end of the Program, they presented their research progress at a mini-conference coorganized by the University of Toronto Math Union and FUN. The Fields community has seen this program undergo rapid growth. It was put together by the staff, the Director, and Deputy Director of the Institute along with financial support from the MITACS Globalink program. The summer Program fosters the interest in research among undergraduates as well as in the unique collaborative and pedagogical opportunities it has to offer. This year’s Program was a success due to its wide diversity in culture and mathematical beauty. As participant Lucas Bentivenha put it: “Math is international.” Richard Cerezo THE FIELDS INSTITUTE for Research in Mathematical Sciences | FIELDSNOTES 15</div> <div>CALL FOR PROPOSALS, NOMINATIONS, AND APPLICATIONS For more information about Fields Institute proposals, nominations, or applications, please visit our website: www.fields.utoronto.ca/proposals Mathematics of Planet Earth 2013 The Fields Institute invites proposals for activities related to the 2013 year of emphasis on the Mathematics of Planet Earth. Mathematics plays a key role in many of the processes affecting Planet Earth, both as a fundamental discipline and as an essential component of multidisciplinary and interdisciplinary research. We encourage proposals for events connecting mathematics with areas such as natural disasters, resource management, transportation, energy production and utilization, the World Wide Web, health care delivery, climate change, sustainability, and control of disease and epidemics. For more information on the Mathematics of Planet Earth 2013, please visit: www.mpe2013.org THEMATIC AND FOCUS PROGRAMS The Fields Institute solicits proposals for a variety of programs in areas of current research interest in the mathematical sciences: (1) Major thematic programs, six months in length. (2) Thematic or focus programs, from one to two months in length to run concurrently with our major thematic programs; in particular, twomonth summer programs of an interdisciplinary nature. Proposals or letters of intent should be submitted by March 15 or September 15, with a lead time of at least two years recommended for six-month programs. GENERAL SCIENTIFIC ACTIVITIES Proposals for short scientific events in the mathematical sciences should be submitted by October 15, February 15, or June 15 of each year, with a lead time of at least one year recommended. Activities supported include workshops, conferences, seminars, and summer schools. POSTDOCTORAL FELLOWSHIPS The Fields Institute’s Postdoctoral Fellowships provide for a period of research activity at the Institute and participation in our programs. We are currently soliciting applications for Fields Postdoctoral Fellowships and Jerrold E. Marsden Postdoctoral Fellowships. Qualified candidates who will have a recent PhD (awarded normally not more than five years before tenure of the Fellowship) are encouraged to apply. OUTREACH PROPOSALS The Fields Institute provides support for projects whose goal is to promote mathematical culture at all levels and bring mathematics to a wider audience. Faculty at Fields sponsoring universities or affiliates are invited to submit a proposal to the Fields Outreach Competition. There are two submission deadlines each year, June 1 and December 1. Proposals should include a detailed description of the proposed activity and the target audience. A budget indicating other sources of support is also required. CRM-FIELDS-PIMS PRIZE NOMINATIONS The CRM-Fields-PIMS Prize is the premier Canadian award in recognition of exceptional research achievement in the mathematical sciences. The candidate’s research should have been conducted primarily in Canada or in affiliation with a Canadian university. Nominations for the 2013 CRM-Fields-PIMS Prize should be sent to PIMS. Please send nominations no later than November 1, 2012. THANKS to our SPONSORS MAJOR SPONSORS Government of Ontario— Ministry of Training, Colleges, and Universities; Government of Canada—Natural Sciences and Engineering Research Council (NSERC) PRINCIPAL SPONSORING UNIVERSITIES Carleton University, McMaster University, University of Ottawa, University of Toronto, University of Waterloo, University of Western Ontario, York University AFFILIATED UNIVERSITIES Brock University, University of Guelph, University of Houston, Iowa State University, Lakehead University, University of Manitoba, University of Maryland, Nipissing University, University of Ontario Institute of Technology, Queen’s University, Royal Military College, Ryerson University, University of Saskatchewan, Trent University, Wilfrid Laurier University, University of Windsor CORPORATE SPONSORS Algorithmics, General Motors, QWeMA, R 2 Financial Technologies, Sigma Analysis and Management The Fields Institute receives and welcomes donations and sponsorships from individuals, corporations, or foundations, and is a registered charity. The Fields Institute is grateful to all its sponsors for their support. THE FIELDS INSTITUTE for Research in Mathematical Sciences | FIELDSNOTES 21</div> <div>Fields of Gold Speeches often start with “I would like to thank...”, and except that this note is not a speech, it will be no different. I would like to thank Ed Bierstone — in reverse chronological order, for having offered me to write a few words reflecting on my time at Fields, at the place where the accustomed reader of the Fields Notes would expect the Director’s Message. In correct chronological order, for the opportunity to serve as Deputy Director of the Fields Institute, a truly unique and extremely rich experience, and for too many hours and days I had the pleasure to work with him between both these points in time, than this page would allow me to describe. I would also like to thank to Juris Steprans with whom I started my time at Fields, as Interim Deputy Director in the first half of 2009, while he was serving as Acting Director, before beginning my term as Deputy Director (after a year sabbatical) in July 2010. The number of discussions we had about Institute matters as well as topics that matter most for the spirit of the Institute — such as mathematical problems of mutual interest we then started to solve! — seems uncountable. Last, but in no way least, I would like to express my thanks to every staff member at Fields. The friendliness, openness and dedication they bring to the Institute, make it a pleasure for the several thousand participants of our scientific activities to visit Fields, and for the directorate to work with them. It has been an extremely exciting time at Fields, and I am happy for having had the opportunity to contribute to the many initiatives the Institute has recently undertaken. Among the most intriguing ones is obviously the series of Fields Medal Symposia, the first of which, honouring Ngô Bao ả Châu, will take place in October 2012, as one of the highlights of the Institute’s scientific 20th Anniversary celebrations in 2012–13. Numerous further developments show the level of activity which make Fields such a vibrant place: over 50 Fields sponsored workshops and conferences took place within the last 12 months, the Fields-Mitacs Undergraduate Summer Research Program ran over two months with an increased number of participants, and the Fields Undergraduate Network has been growing steadily, bringing together once a month undergraduate students from our partner universities to learn from experts about varying topics in the mathematical sciences, through lectures and panel discussion. Also, in the fall of 2011, the online, peer-reviewed Fields Mathematics Education Journal was launched, providing a new exciting platform for the Institute’s numerous educational activities. The Institute’s international network of partner Universities has expanded as well, with the Université Lille 1 – Sciences et Technologies, in Northern France, joining as the first affiliate University in Europe in December 2011. Moreover, a memorandum of understanding between Fields and the French CNRS, which would establish the Institute as an Unité Mixte Internationale (UMI), is currently in preparation. Fields is a very special place for mathematicians in many regards, and in particular for those involved in steering its scientific direction. It is wonderful to be able to help making exciting mathematics possible, on the level of research administration, and, almost at the same time and certainly at the same place, to see it happening — with workshop participants giving talks, discussing mathematics in their offices and seminar rooms, on the blackboards in the hallways and around the fireplace, until late at night — and, of course, to switch oneself from working on reports and budgets to proving theorems. The unique atmosphere of the Institute, light and quietly lively, has a refreshing and energizing effect; as one of my research visitors put it: “I actually should be tired from jet lag, but here I just feel like doing math all the time!” It has been a very special privilege to meet and discuss — about mathematics, life or anything between the two which may come up after a talk or during dinner — with some of the world’s most distinguished scientists who visited the Institute, such as Yakov Sinai, Stephen Smale, Srinivasa Varadhan, Cédric Villani, and Shing-Tung Yau. What stays is the memory of joyful moments, a mixture of deep thoughts and beautiful anecdotes — from Yakov Sinai’s personal memories of Kolmogorov to Cédric Villani’s telling us, during dinner after his Fields Distinguished Lecture Series, about his lunch with President Nicolas Sarkozy. I am ending my term as Deputy Director of the Fields Institute on December 31, 2011 to take a professorship at the Université Lille 1, starting January 1, 2012. I look forward to maintaining very strong connections with the Fields family and the Canadian mathematical community, as well as Carleton University. My Canadian friends and colleagues make me hope, to say it with Sting’s words, that “you’ll remember me when the west wind moves”... Matthias Neufang THE FIELDS INSTITUTE for Research in Mathematical Sciences 222 COLLEGE STREET, TORONTO, ONTARIO, CANADA M5T 3J1 Tel 416 348.9710 Fax 416 348.9714 WWW.FIELDS.UTORONTO.CA</div> <div>that have been developed in the attempts to resolve this conjecture and connected problems. The workshop also included presentations on related algebraic topics, such as Maltsev Conditions and Tame Congruence Theory as well as presentations on CSPs over infinite templates, quantified CSPs, and connections with logic, finite model theory, and complexity. Approximate Solutions to Hard CSPs It is a fact of life that most interesting optimization problems are NP-hard. One way to cope with this intractability is to look for efficient (polynomial time) algorithms that produce solutions with guaranteed performance with respect to the optimum solution (e.g. off by at most 25%, or by a factor of 10, etc.) For example, given a 3-colourable graph G, it is known to be NP-hard to find a 3-colouring of G, but perhaps finding a 5-colouring of G is tractable. CSPs play an important role in the study of approximability of combinatorial optimization problems. For example, the celebrated PCP (Probabilistically Checkable Proofs) theorem can be formulated in terms of CSPs. Namely, it is equivalent to the statement that the maximum fraction of satisfiable constraints of an instance of a CSP is NP-hard to approximate within some constant factor. A prominent conjecture, the so-called unique games conjecture, also has a formulation in terms of approximability properties of certain instances of CSPs. This conjecture implies optimal inapproximability results for the max-cut problem and the vertex cover problem, amongst others. The third and final workshop of the Thematic Program was held from August 12 to 16 and was organized by Andrei Bulatov (Simon Fraser University), Johan Håstad (KTH, Stockholm), and Prasad Raghavendra (Microsoft Research and Georgia Tech). It focused on the recent advances in our understanding of the approximation threshold of various CSPs, based on progress in both algorithmic techniques and methods to show tight non-approximability results. The power of various convex programming relaxations for CSPs, the construction of gap instances highlighting limitations of such relaxations, and the connections of these to the complexity of approximating CSPs, were prominent themes of the workshop. One of the aims of the workshop was to foster a cross fertilization of ideas between the algebraic approach to characterize the tractability of CSPs and the analytic approach to characterize the approximability of CSPs, and draw parallels between the algebraic dichotomy conjecture and the Unique Games conjecture in order to shed some light on both of these prominent conjectures. Two of the highlights of the summer Thematic Program on the Mathematics of Constraint Satisfaction were the summer school on the CSP and the Coxeter Lectures given by Moshe Vardi of Rice University. The summer school was held during the last week of June and featured introductory lectures covering the main themes of the program. The lectures were given by Venkatesan Guruswami, Andrei Krokhin, Jaroslav Nešetril (Charles University), Ryan O’Donnell (Carnegie Mellon University), and Ross Willard. The summer school was attended by over 40 participants and set the stage for the rest of the summer program. During the week of July 11, Moshe Vardi presented a series of wonderful lectures on logic in computer science to a standing-room only crowd. The titles of Vardi’s talks were: And Logic Begat Computer Science: When Giants Roamed the Earth, From Philosophical to Industrial Logics, and Logic, Automata, Games, and Algorithms. Details of the lecture series can be found on page16. Matt Valeriote (McMaster) and Ross Willard (Waterloo) THE FIELDS INSTITUTE for Research in Mathematical Sciences | FIELDSNOTES 7</div> <div>‘Extreme Waves’ continued from page 13 detection, many other factors come into play in tsunami prediction, including analysis of the mode of ocean floor movement through real-time rapid analysis of seismic data, and detailed local considerations of impact on coastlines, as the destruction caused by a tsunami depends sensitively on the particular topography of the coastline and inshore bathymetry. The two major tsunami disasters of the last 10 years, one caused by the Sumatra earthquake of December 26, 2004 and the recent Tohoku earthquake in Japan, show that a better understanding of the phenomenon is vital to future coastal development in order to minimize loss of life and destruction of infrastructure. Mathematically, the dynamics of ocean waves include a number of other natural hazards that are relatively poorly understood, and which are topics of current research. In addition to tsunami waves, there are current research programs on rogue waves that represent an increasing danger to ship traffic as the importance of shipping is increasing in the modern economy, as is the increasing number of large vessels present in the Earth’s oceans at any one time. Rogue waves are presumably quite rare, and both their dynamics and the statistics of their occurrence are not well understood. They are difficult to measure accurately or to model in a fluid dynamics laboratory with any expectation of precision. Yet marine accidents due to rogue waves are believed to be the cause of on average ten major ship losses per year. This is therefore a challenge for mathematics and mathematical modelling to play a role in the research effort to understand and to predict the phenomenon. These giant waves were once considered a mariner’s myth, however they are now known to exist and to be a rare but significant marine hazard. A history of their modeling consists of a number of controversies, as do attempts at predictions of their statistics. There is in fact one precise scientific measurement of a rogue wave, taken on New Year’s Day in 1995 on the Draupner Platform in the North Sea; however for the most part there are either satellite traces picked from ocean wave fields, or anecdotal descriptions from ‘the bridge’ — one early such observation appears in the log of Ernest Shackleton, an encounter with an apparent rogue wave during his open boat voyage to seek help for his crew after his ship, the Endurance, foundered after being caught in polar ice. In any case there is an annual cost in economic terms as well as human life, and it seems a problem to which mathematics and mathematicians have the potential to make important contributions. The phenomenon of rogue wave formation is considered likely to be due to a focusing effect of two oblique wave fields; this is an effect that is enhanced in the presence of a current running counter to a prevailing wind, and thus it is a possibility that predictions can be made regarding sea conditions under which rogue waves are more likely. It is furthermore considered probable that the phenomenon of a rogue wave is transient, and therefore difficult to detect with certainty. One-dimensional models of Dyachenko and Zakharov show under certain conditions the generation of large amplitude waves from a wave field of substantially lower amplitude. Focusing effects have been studied in numerical simulations of Bateman, Swan and Taylor, using methods developed by Craig and Sulem. It is a field that will benefit from further attention and analysis, and one in which there is the potential benefit from collaborations between the mathematics community and ocean scientists. Our workshop took place over the days of June 13–16 2011, hosted by the Fields Institute. Fourteen plenary lecturers gave hour talks, roughly evenly divided between the topics (1) tsunamis and tsunami prediction and warnings, and (2) rogue wave phenomena. Eight of these speakers were practicing mathematicians, while otherwise we had university faculty from the departments of coastal and ocean science, geosciences, and representatives from the Canadian Institute of Ocean Sciences, the US National Oceanic and Atmospheric Administration (NOAA), the Russian Academy of Sciences, and the Australian Meteorological Office. The first lectures of the day were planned as overview talks, given by senior specialists in the topic, for the purpose of calibrating the background of the participants. These were followed by more focused and technical talks, on a wide variety of topics relevant to the research area. The first day was dedicated to tsunami research, with a general overview lecture by H. Segur (Applied Mathematics, University of Colorado), who explained the initial conditions of tsunamis generated by dip-slip seismic events (the most common major events in coastal subduction zones), describing the essentially linear but dispersive evolution of tsunami waves in the open ocean, and then showing film of the pile-up of such waves as they encounter the coast, and their resulting destructive power. This was followed by H. Yeh (Coastal & Ocean Engineering, Oregon State University) who discussed a theoretical limit of nonlinear amplification of a nonlinear wave at oblique incidence to a vertical wall (reminiscent of the tsunami barrier walls constructed at great expense in Japan). D. Greenslade (Bureau of Meteorology, Melbourne Australia) followed with a detailed description of the elements of a tsunami warning system, and in particular the Australian components of the Pacific Tsunami Warning Center. The day was capped by the fascinating and wide-ranging talk of E. Okal (Geophysics, Northwestern University) describing tsunamis in terms of the earth’s normal modes, and a variety of unusual and imaginative detection methods based (roughly) on this fact. The second day of the workshop was dedicated to rogue waves and their mathematical models. E. Pelinovsky (Department of Nonlinear Geophysical Processes, Russian Academy of Sciences, Nizhny Novgorod) gave a wide-ranging survey talk that gave a definition of a rogue wave, described their danger to maritime activities, and gave a context for mathematical models and their statistics. We then heard from G. Pedersen (Mathematics, University of Oslo) who described the phenomenon and the history of nonlinear waves in fjords that are generated by large landslides, and what the Norwegian geologists are doing to model these waves and to monitor potentially unstable geological formations lying over fjords. The final lecture was by J. Dudley (Institut d’Optiques, Université de Franche-Comté) describing mode equations for rogue waves in a modulational regime, and the special breather- 18 FIELDSNOTES | THE FIELDS INSTITUTE for Research in Mathematical Sciences</div> <div>CRM-Fields-PIMS Prize Lecture Mark Lewis on the Mathematics Behind Biological Invasion Processes “Biological invasions shape the environment as we see it. They have been estimated to be the largest influence on a species’ numbers, other than habitat destruction … Can an invading population establish itself in a new environment? Will it spread, and if so at what speed? What lasting impact will it leave?” The 2011 CRM-Fields-PIMS Prize Lecture began with questions, raised by prize winner Mark Lewis (Alberta). His lecture highlighted the mathematical theory and application of models for ecological population growth at both local and global levels. Historically, geneticist Sir R. A. Fisher derived equations to model the rates of change in population densities using a non-linear quadratic growth function with dispersal by diffusion. These tools were originally developed from the geneticist’s perspective, in a classical study of the spread of advantageous genes into a new environment. In his talk, Lewis noted that similar mathematical models were developed by both a team of Russian mathematicians during the time of Fisher and another team of American applied mathematicians. Lewis’ lecture focused on the speed of spread at a populations’ leading edge. Two equivalently effective techniques for modeling spread were emphasized: models utilizing diffusion processes and models utilizing traveling wave theory. Lewis gave a survey of technical tools and their applications. This covered the Kolmogorov-Petrovsky- Piskounov (KPP) equation, problem formulations in traveling wave theory, concavity as a criterion for linearization, level set methods for detection thresholds, non-local spread explained by stochastic equations, computation of kernels for dispersion distribution, integrodifference equations, and applications of the method of moments. A wide variety of specific empirical results were shown to have been accurately modeled using these tools. Examples of ecosystems to which his models can be applied include biological invasions of: muskrats, wolves, the bubonic plague, sea otters, spread of rabies, oak tree dispersion, cheatgrass, red maple, red squirrels, and the West Nile virus. The CRM-Fields-PIMS prize is awarded annually. Its main criterion for selection is outstanding contribution to the advancement of research. Mark Lewis is the Canada Research Chair in Mathematical Biology and works for the Centre for Mathematical Biology at the University of Alberta. Richard Cerezo THE FIELDS INSTITUTE for Research in Mathematical Sciences | FIELDSNOTES 17</div> <div>Workshop on the Mathematics of Extreme Sea Waves: Tsunamis, Rouge Waves, and Flooding WHEN NEWS CAME THROUGH OF A SERIOUS earthquake hitting Japan I was attending a conference at the American Institute of Mathematics (AIM) in Palo Alto, California. It was known within minutes that there was great potential for a serious tsunami, due to the operations of the earthquake and tsunami observations division of the Japan Meteorological Agency, and the Pacific Tsunami Warning Center run by the American NOAA. And indeed within ten minutes the Richter scale 9 earthquake drove waves of over 10 meters (and some reported at up to 50 meters) onto the Japanese coast, overwhelming the coastal defenses against such events. The result was the loss of 20,000 lives, enormous property damage, and a nuclear disaster whose effects continue today, months after the initial events. At the time I was tempted to drive to the California coast to witness the tsunami’s arrival some 12 hours after the earthquake (in competition for the Darwin award, as some would say). Several days after, Efim Pelinovsky emailed me with the urgent request to act; many facts are know about the dynamics of tsunamis, however many things are not known, including an understanding of their initial conditions and a detailed description of the effects of focusing and dispersion as these waves impact on the coast. As mathematicians interested in wave dynamics in the ocean, we felt a sense of urgency to act, to communicate with the ocean scientists and modelers of ocean waves, and to attempt to address some of the pressing outstanding problems in this field. The subtitle of the NOAA Center for Tsunami Research concisely states these activities as ‘developing methods and tools to reduce tsunami hazard and protect life’. This workshop on extreme ocean waves at the Fields Institute in June 2011 plays the role of a first step in this effort. Our event was of the character of a ‘hot topics’ workshop, intended to place mathematicians in touch with physical oceanographers, with the purpose of informing both communities of the existing challenges, up-to-date methods, and the most important outstanding problems of the topic. Our principal immediate goal was to articulate the most relevant problems in the discipline, and outlining the role that mathematics can play in addressing them. Subsequent to this, we hope to establish deeper collaborations and research links between the two research communities. The immediate reason for organizing this workshop is the recent destructive tsunami that followed the severe earthquake in Japan. In most known cases (but not all), tsunami waves are initiated seismically, they propagate at high speed across oceans, and their impact upon coastlines can be very destructive. Tsunamis are rare events, and because for the most part they are generated by large earthquakes, prediction is very difficult, a science grand challenge problem. In addition to earthquake ‘Extreme Waves’ continued on page 18 THE FIELDS INSTITUTE for Research in Mathematical Sciences | FIELDSNOTES 13</div> <div>LECTURES Coxeter Lecture Series Moshe Y. Vardi on Logic in Computer Science AS PART OF THIS PAST SUMMER’S THEMATIC Program on the Mathematics of Constraint Satisfaction, participants had the privilege of attending the Coxeter Lectures presented by Moshe Vardi (Rice). His lectures were on the topic of Logic in Computer Science and were held during the first three days of the program’s Workshop on Graph Homomorphisms, which ran from July 11 to July 15 at the Fields Institute. Vardi is the George Professor in Computational Engineering, as well as Director of the Ken Kennedy Institute for Information Technology, at Rice University. His research interests include database systems, computational-complexity theory, multi-agent systems, and design specification and verification. He is the recipient of numerous awards, including three IBM Outstanding Innovation Awards, the 2000 Gödel Prize, the 2008 ACM Presidential Award, the 2010 ACM Outstanding Contribution Award, and the 2011 IEEE Computer Society Harry H. Goode Award. Vardi is an editor of several international journals, and Editor-in-Chief of the Communication of ACM. He is a Guggenheim Fellow and was elected as a member of the US National Academy of Engineering, the American Academy of Arts and Science, the European Academy of Sciences, and the Academia Europea. Vardi’s first lecture, entitled And Logic Begat Computer Science: when Giants Roamed the Earth, consisted of a broadly accessible historical outline of the connection between formal logic and computer science. The audience was treated to a bird’s eye view of the last 2500 years—from Epimenides’ Liar’s Paradox to the Pentium chip— interwoven with amusing and insightful quotes from logicians as diverse as Aristotle and Lewis Carroll. Using Leibniz’s unfulfilled dream of mechanizing reasoning as a recurring theme, Vardi provided an entertaining and fascinating tour of the history of formal logic, visiting such characters as Ramon Lull, George Boole and Charles Peirce. He eventually returned to Euclid, whose great text has been in use for over 2000 years, in a discussion of what Wigner referred to as mathematics’ unreasonable effectiveness, the notion of mathematical proof and the consequences of the search by mathematicians of the late 19th and early 20th century to clarify this concept. This thread was followed, from Frege’s introduction of first order logic, via the discovery of Russell’s paradox and Russell and Whitehead’s Principia Mathematica, to the fall, at the hands of Gödel, Church and Turing, of Hilbert’s program to consolidate the foundations of mathematics. Vardi argued that it is precisely out of this quest that computer science was born; by the early 1950s, computers were built around the world, based on Von Neumann’s ideas, thus fulfilling Leibniz’s dream: “from reasoning, to patterns of reasoning, to logic, to computers, to computers that reason”. He closed the lecture with a moving quote from C. Papadimitriou on the sad fate of so many logicians such as Boole, Cantor, Frege, Gödel and Turing, and a remarkably prescient quote from Leibniz on the advent of the modern computer. Vardi’s second lecture focused on the last 100 years of this remarkable history, and consisted of a more technical presentation of several threads. First, a discussion of first order logic equipped only with unary predicates (Monadic FO), which was proved decidable by Löwenheim in 1915, via the bounded model property and quantifier elimination, and its extension to second-order logic (MSO) by Skolem in 1919. The second thread underlined the intimate connection between logic’s descriptive view and its operational view, as illustrated by Büchi, Elgot and Trakhtenbrot’s remarkable ‘Vardi’ continued on page 19 16 FIELDSNOTES | THE FIELDS INSTITUTE for Research in Mathematical Sciences</div> <div>NEWS Interview: Matheus Grasselli, New Deputy Director Matheus Grasselli (McMaster) steps in as Deputy Director of the Fields Institute on January 1, 2012. He recently spoke with Richard Cerezo about working at the Institute, the new job, and how it all ties in with his research and experiences. RICHARD CEREZO: You were a Fields Institute Thematic Program organizer at one point. How do you think that will help shape your term? MATHEUS GRASSELLI: It gave me a sense of how the Institute works. I know the impact that the Institute can have on a given area. Our 2010 Thematic Program [on Quantitative Finance: Foundations and Applications] was extremely influential, in the sense that I still go to conferences and meet people that make reference to the Program. I have a good sense of the reach of the Institute. Cerezo: How do you see yourself making an impact on research outside of your field? Grasselli: Identifying exciting mathematical areas that you don’t work in is part of the job and the challenge. You’re forced to be open-minded. You need to be able to recognize the excitement and potential contribution of the given area to mathematics. I’m coming in pretty open-minded and hope to learn a lot. Another thing that influenced me in accepting this position is that you experience a constant flux of ideas and need to keep up. Cerezo: Identifying different areas [in math] gives you a chance to explore outside your field. Are there things you’re particularly excited to see, and partnerships you’d like to build and promote? Grasselli: I am committed to strengthening the links [between] the Fields Institute and industry in general and in the financial industry in particular. My area is financial math, so this is not outside my area. But it is outside what I’ve been doing academically. There’s a good dialogue that can happen, because industrial partners have problems in which the institute can help and they realize the value in that. I would like to position the institute with other institutions that do similar things. For example, the Global Risk Institute in Financial Services. Their mandate is to promote the understanding of risk at a variety of levels. There is a lot that the Institute can and has contributed to Canada, Ontario, and Toronto in particular. It would be nice to showcase that at many different levels. For example, by way of the Fields Symposium. Also, there is the incubation program here [at the Institute], with companies that start at Fields independently—and they have all been successful. The fact that [Fields has] incubated a small number of companies, but they’ve all been successful is something that should also be promoted and advertised. That the institute hosts Forums for Mathematical Education and programs like Math Circles is something that the public should know about. Cerezo: At McMaster you helped develop both the Master in Financial Math (M-phimac) and the Integrated Science undergraduate program. Do you see your skill set as a program developer being something you bring forward? Grasselli: We are talking with different partner universities to see if there is demand, and whether logistics are in place, for a Master’s program to be hosted at Fields. For example, a Master’s in Industrial Mathematics. I would like to see the Fields-Mitacs [Undergraduate Summer Research] Program continue. I am a Mitacs investigator and have been a member of the network since I’ve arrived in Canada. I’m sure that Mprime will want to work together with Fields. Cerezo: What do you enjoy intellectually besides Financial Math? Grasselli: I’m a physicist by formation. My undergrad was in physics and my PhD was in physics, so I continue to be interested in some areas of physics. I’m more professionally interested in areas like quantum information, a course that I’m teaching now at McMaster and which I can see myself becoming more connected with in the future. Recently I’ve been occupying free time learning economics and branching out into areas of economics that mathematicians have traditionally not been very interested in. I’ve been reading about macroeconomics, which is considered the least mathematized area and—perhaps for exactly that reason—it is the most interesting. Also, the history of both economic thought and economic history itself, including crises and bubbles, have fascinated me a bit. In general, I like literature, history, history of science, biographies, and sometimes personal accounts of scientific discoveries. 8 FIELDSNOTES | THE FIELDS INSTITUTE for Research in Mathematical Sciences</div> <div>FIELDS NOTES FALL 2011/WINTER 2012 | VOLUME 12:1 THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES LECTURE SERIES Coxeter Lecturer Moshe Y. Vardi CRM-Fields-PIMS Prize winner Mark Lewis THEMATIC PROGRAMS Galois Representations and Automorphic Forms Inverse Problems and Imaging Mathematics of Constraint Satisfaction RECENT ACTIVITIES Connections in Geometry and Physics Mathematics of Extreme Sea Waves Fields-Mitacs Undergraduate Summer Research Program</div> <div>ations and Automorphic Forms of Galois representations and automorphic representations is thus a return to historical norms. One of the major successes of algebraic number theory in the first half of the 19th century was (arriving at) an understanding of the finite abelian extensions L of a number field K. Class field theory, as this subject is known, begins (roughly) with Kronecker and Weber’s identification of the abelian extensions of Q as subfields of cyclotomic fields Q ( ζ ) , and culminates in the general reciprocity law of Artin, which canonically identifies finite order characters of the absolute Galois group Gal( Q / K ) of K with the finite order characters of the idèle class group × × � K / K of K (the formulation of class field theory in terms of idèles is due to Chevelley and came after Artin’s work). Since that time, one of the central goals of number theory is to develop a theory of reciprocity for non-abelian extensions L / K. For example, given a general extension L / K, is there a way to describe the set of primes in K which split completely in L? If L / K is abelian, then class field theory shows that the primes that split completely are determined by congruence conditions, but the answer for general extensions seems to be much more mysterious. One of the first steps in the general study of non-abelian extensions was taken by Artin himself. Given a finite Galois extension L / K, for all primes p in the ring of integers O K outside some finite set S, one can define a canonical conjugacy class Frob p ∈ G : = Gal( L / K) (the Frobenius element). A theorem of Cebotarev implies that G — together with the labelled conjugacy classes Frob p — determines the field L. It thus makes sense to study L by studying Gal( L / K) together with its Frobenius classes. Given a representation ρ : Gal( L / K) → GL ( C), Artin defined a function L S ( ρ , s) in terms of the action of the image ρ( Frob p ) of the Frobenius conjugacy classes for primes outside S. The function L S ( ρ , s) is holomorphic for Re( s ) >1 , but Artin conjectured that an explicitly defined modified function L( ρ , s) (in which the ramified primes of K as well as the places at ∞ are taken into account) extends to a meromorphic function on all of C, and satisfies a functional * equation of the form L( ρ, 1− s) = W ( ρ) L( ρ , s) , where ρ * is n the dual representation and W ( ρ ) is a constant which depends somewhat subtly on ρ. Moreover, Artin conjectured that L( ρ , s) is an entire function, except when ρ contains a copy of the trivial representation, in which case L( ρ , s) is allowed to have a pole at s = 0 and 1 of order equal to the multiplicity of the trivial representation in ρ. Artin himself (modulo a key ingredient proved later by Brauer) was able to prove his conjecture except for the claim concerning holomorphicity. On the other hand, the special case of one dimensional representations ρ = χ followed from previous work of Hecke 1 . As an even more special case, one may consider what happens when K = Q and χ is the trivial character. In this case, the function L S ( χ , s) (where S is empty) is equal to the Riemann zeta function ζ ( s ). The functional equation of ζ was first proved by Riemann in two different ways. Riemann’s first proof was a clever application of the residue theorem, but Riemann’s second proof used a theorem of Jacobi (from 1829) that the (Jacobi) theta function ∞ n θ ( τ ) = ∑q , q = e −∞ 2 2πiτ satisfied the functional equation θ ( − 1/ 4τ ) = 2τθ ( τ ). Thus, right from the beginning, the theory of Galois representations and L-functions have been linked to modular forms. Somewhat later, Weil found a way to unify the class of Artin L-functions corresponding to characters with a different class of L-functions studied by Hecke (associated to Grossen characters). In particular, Weil reconciled both definitions as coming from algebraic representations of an enriched Galois group W K , now known as the Weil group. Skipping forward roughly twenty years, past Tate’s thesis, past the harmonic analysis of Harish-Chandra, and past the development of the theory of automorphic representations in an adelic framework, Langlands formulated a conjectural correspondence which on the one hand vastly generalizes the correspondence of Weil for � × K mentioned above, but also provides a conceptual understanding of Artin’s conjecture. (The theory of Weil is “Langlands for GL ( � 1 K )’’.) Associated to an irreducible Continued on next page 1 One immediate consequence of Artin’s conjecture is the theorem of Cebotarev that Frobenius classes are distrubuted in G in proportion to the order of the corresponding conjugacy classes. Cebotarev, amazingly enough, proved his theorem without using Artin’s conjecture (and indeed before the statement of Artin’s conjecture existed) by a clever reduction to the case of cyclic extensions. Perhaps in part due to this argument, Artin himself long harbored the suspicion that non-abelian class field theory itself could be reduced to the abelian case, although this does not seem to be the case. THE FIELDS INSTITUTE for Research in Mathematical Sciences | FIELDSNOTES 3</div> <div>Fields-Mitacs Undergraduate Summer Research Program THE FIELDS INSTITUTE IS RAPIDLY BECOMING an international hub of undergraduate mathematical activity. The 2011 Fields-Mitacs Undergraduate Summer Research Program raised the bar even higher for collaborative undergraduate research. Over the course of 10 weeks this summer, 23 student researchers gathered at the Institute to work with 15 supervisors on six different projects. The Program was a unique opportunity for students to gain experience at a world-classs institute and work in teams of highly motivated individuals. In addition to collaboration, students were also exposed to a diverse set of active research areas. These experiences influenced the students’ outlook on the research profession and the academic community. Though the topics chosen for the projects were unique to the Fields network of researchers, they were by no measure restrictive. Many of Fields’ established partnerships were featured during the Program. Students and supervisors were attracted from around the world to work at the Institute. On the first day of the Program, students met at the Institute to listen to their potential supervisors’ presentations, after which they ranked their preferred subjects. After team formation, an initiating period was used to bring students up to speed with research material. The Program participants then got to work on the following projects: Symmetries of Euclidean Tessellations and their Covers Participants: Kostiantyn Drach, Maximilian Klambauer, Fernando Lenarduzzi, Maksym Skoryk Supervisors: Isabel Hubard (UNAM), Mark Mixer (Northeastern), Daniel Pellicer (UNAM), Asia Weiss (York) Have you wondered what regular polyhedrons look like on different surfaces? In infinite spaces? On Möbius strips and Klein bottles? Tessellations are what give us tools to make these visualizations. The group studied various classifications of tessellations on surfaces, and in a variety of spaces. They were also investigating the existence and uniqueness of minimal regular covers, minimal quasi-regular covers, and minimal chiral-regular covers for various classes of regular polyhedrons and abstract spaces. Symmetries of the polyhedrons also formed topics of interest. The supervisors of this group also participated in the 2011 Thematic Program on Discrete Geometry and Applications. Constraint Satisfaction Participants: Zoltán Blázsik, Avinash Kulkarni, Martin Liu, Daniel Perkins, Anna Tossenberger, Young Wu Supervisors: Libor Barto (Charles University and McMaster), Matt Valeriote (McMaster), Ross Willard (Waterloo) Students of this group were fortunate that the 2011 S ummer Thematic Program on the Mathematics of Constraint Satisfaction was being held at Fields during the Undergraduate Research Program. As an emerging research niche, constraint satisfaction lies at the intersection of many different areas including combinatorics, complexity theory in theoretical computer science, and universal algebra. Due to the large pool of Thematic Program participants, this group had the largest number of supervisors. As a result, two undergraduate research groups were formed for this topic. Both groups began by working through a series of problem sets to get up to speed with the material. A concept central to their preliminary work was the majority polymorphism. A consequence of a constraint satisfaction problem (CSP) having a majority polymorphism is that established theory can be used to deduce the CSP’s tractability. One group explored an Ehrenfeucht-Fraïssé game using pebbles to determine whether certain CSPs fit into a particularly ‘easy’ category of polynomial time problems, namely the class of nondeterministic logarithmic space problems. The other group attempted to classify oriented trees having a majority polymorphism. In the process of their investigations, students were given the chance to learn high-powered theorems (which currently reside at the field’s frontier) directly from some of the field’s pioneers. Understanding Financial Crisis Participants: Lucas Bentivenha, Richard Cerezo, Eric Hyungvin Ihm, Euijun Kim, Nikita Reymer, Rafael Rocha, Garence Staraci Supervisors: Matheus Grasselli (McMaster), Oleksandr Romanko (McMaster) Group research is presented at the Fields-Mitacs Undergraduate Summer Research Program mini-conference. This project began prior to the Undergraduate Program by laying some of the initial groundwork for a bigger collaboration between McMaster, MITACS, and 14 FIELDSNOTES | THE FIELDS INSTITUTE for Research in Mathematical Sciences</div> <div>‘Vardi’ continued from page 16 like solutions that these equations admit. The third and fourth days of lectures featured talks on a wide range of topics involving the dynamics of ocean waves, their mathematical modeling and issues involved in their practical prediction and detection. D. Arcas (NOAA, San Diego) spoke on operational aspects of tsunami modeling and detection, complementing Greenslade’s description with the U.S. contributions to the global detection system. M. Onorato (Università di Torino) spoke on the role of currents in generating rogue waves. I. Fine described the Canadian contributions to the tsunami detection network, and the research potential that they have. O. Bokhove (Applied Mathematics, University of Twente) brought an experimental device to his lecture, which demonstrated beach and shoreline dynamics due to nonlinear wave erosion. D. Dutykh (Université de Savoie-CNRS) described a beautiful model equation for shallow water dynamics, including the run-up onto the beach, a potentially very useful modeling tool for large-scale tsunami impact studies. J. Gemmrich (Physics, University of Victoria) described a classical statistical approach to predicting the probability of rogue waves from extrapolation from known distributions of sea states. Ch. Kharif (IRPHE – Université de Marseille) gave a wide-ranging overview of models of extreme sea states, with applications to rogue wave modeling in both two and three dimensions. The workshop was a success, and this week spent with non-mathematician colleagues was very valuable both to them and to ourselves. It makes one reflect on the degree to which Canada could put more emphasis and interest in science whose aims are to protect its shores and its shipping. In the mean time we return to our own contributions, guided by what we learned. We are looking forward to a deeper and more broadly construed program on the Mathematics of the Earth’s Oceans to take place within the context of the Mathematics of Planet Earth initiative in the year 2013. Walter Craig (McMaster) result, proved in the late 1950s, identifying the languages describable in MSO with those describable by nondeterministic automata (NFAs). The decidability of MSO then follows from the emptiness problem for automata, but at the cost of non-elementary blowup. The third thread centered on sequential circuits, arguably the canonical model of circuit design; the problem of model-checking in this context reduces to the satisfiability problem on infinite words, solved by Büchi who introduced the correct notion of ω-regular languages. Then followed a discussion of the use of temporal logic for program behaviour specification, and key results by Kamp and Pnueli on LTL (linear temporal logic), giving decidability of model-checking via MSO, and Vardi and Wolper’s result giving a direct translation of LTL to Büchi automaton (with only exponential blowup) rather than via FO with nonelementary blowup. The talk concluded with Vardi’s hands-on experience at Intel, designing the hardware design language Forspec, built on Pnueli’s LTL, and incorporating useful engineering features based on regular expressions. Finally Vardi’s self-contained third lecture revisited in more technical detail most of the themes presented in his second lecture; once again two paradigms were underlined, namely that of logic as a declarative formalism versus logic as an imperative formalism, i.e. the connection between logic and automata. The Büchi, Elgot, Trakhtenbrot result was revisited, as well as Rabin-Scott’s proof that languages defined by NFA’s can be defined by a deterministic counterpart; the Büchi extension of this result to infinite words (ω-regular languages), and what Vardi argues is one of the most powerful decidability results in logic, together with the bounded model approach, Rabin’s 1969 extension of the above results to tree MSO and tree automata. The talk concluded with a thorough discussion of the powerful use of games in obtaining succinctness results, e.g. construction of a Büchi tree automaton from CTL formulae with only linear blowup. Benoit Larose (Champlain Regional College) ‘CUMC’ continued from page 10 to give a mathematical talk on a topic of their choice. Keynote speaker Yvan Saint-Aubin (Montreal) talked about randomness and conformal invariants—advanced subjects for most undergraduate students. An impressive aspect of this presentation was his interaction with the audience. Some students were asked to come to the front to answer questions. It kept the presentation dynamic, which is not always the case in classes or talks. With a touch a humour, Saint-Aubin helped participants understand the problem solved by Stanislav Smirnov (Geneva), the winner of a 2010 Fields Medal. It was a reminder of how popularization affects the understanding of new ideas, especially in mathematics. At the end of the event, Frederick Rickey (USMA) gave a talk on the history of the fundamental theorem of calculus. We learned how definitions of the integral have changed over the last two centuries. As mentioned during the talk, it is helpful to learn the history of a concept, as this helps in explaining it to others. This is an example of what the CUMC is all about: learning new and old aspects of mathematics. The quality of the presentations Ana Iorgulescu (Toronto) was one of many at the CUMC was undergraduate speakers at the 2011 CUMC. astonishing, and the subjects quite varied. There was something of interest for each participant, whether it was learning about a new area of mathematics or getting the hang of an advanced idea in their own domain. Dominique Maheux (Laval) THE FIELDS INSTITUTE for Research in Mathematical Sciences | FIELDSNOTES 19</div> <div>WHO WAS J.C. FIELDS? WHAT propelled him to create a medal for mathematics? And why did the International Mathematical Union (IMU) accept it? The Fields Institute and the American Mathematical Society announce the publication of a biography of J.C. Fields – Turbulent Times in Mathematics: The Life of J.C. Fields and the History of the Fields Medal. Authors Elaine McKinnon Riehm and Frances Hoffman draw upon scattered archives in North America and Europe to piece together J.C. Fields’ story. Fields created the medal at a time when international cooperation among scientists and mathematicians PUBLICATIONS Turbulent Times in Mathematics: The Life of J. C. Fields and the History of the Fields Medal was fractured by the aftermath of World War I and by the uncertainties of the 1930s. Included in the study are notes about the 52 mathematicians awarded the Fields Medal between 1946 and 2011 as well as 16 pages of illustrations that range from Fields childhood to photographs of the great mathematicians of the late-nineteenth and earlytwentieth centuries. Tom Archibald, (Simon Fraser University) writes: “highly readable and replete with period detail, the book sheds useful light on the mathematical and scientific world of Fields’ time, and is sure to remain the definitive biographical study.” Fields Institute Communications, Volume 61 Perspectives on Noncommutative Geometry Edited by Masoud Khalkhali (University of Western Ontario), and Guoliang Yu (Vanderbilt University) This volume consists of the proceedings of the Noncommutative Geometry Workshop that was held as part of the Thematic Program on Operator Algebras at the Fields Institute in May 2008. Pioneered by Alain Connes starting in the late 1970s, noncommutative geometry was originally inspired by global analysis, topology, operator algebras, and quantum physics. Its main applications were to settle some long-standing conjectures, such as the Novikov conjecture and the Baum- Connes conjecture. Next came the impact of spectral geometry and the way the spectrum of a geometric operator, like the Laplacian, holds information about the geometry and topology of a manifold, as in the celebrated Weyl law — this has now been vastly generalized through Connes’ notion of spectral triples. More recently, number theory, algebraic geometry and the theory of motives, and quantum field theory have influenced the development of noncommutative geometry. Almost all of these aspects are touched upon with new results in the papers of this volume. This book is intended for graduate students and researchers in both mathematics and theoretical physics who are interested in noncommutative geometry and its applications. Fields Institute Monographs, Volume 28 Introduction to Orthogonal, Symplectic and Unitary Representations of Finite Groups Carl R. Riehm (McMaster University and the Fields Institute) Orthogonal, symplectic and unitary representations of finite groups combine two more traditional subjects of mathematics — linear representations of finite groups, and the theory of quadratic, skew symmetric and Hermitian forms — and thus inherit some of the characteristics of both. This book is written is an introduction to the subject, with principal goal an exposition of the known results on the equivalence theory of these representations and related matters such as the Witt and Witt-Grothendieck groups, over the “classical” fields — algebraically closed, real closed, finite, local and global. It was A. Fröhlich who first gave a systematic organization of this subject in a series of papers beginning in 1969. His paper Orthogonal and Symplectic Representations of Groups represents the culmination of his published work on this subject. This book includes most of the work from that paper, extended to include unitary representations, and also provides new approaches, such as the use of the equivariant Brauer-Wall group in describing the principal invariants of orthogonal representations and their interplay with each other. THE FIELDS INSTITUTE for Research in Mathematical Sciences | FIELDSNOTES 11</div> <div>‘Inverse Problems’ continued from page 5 obtained by L1 minimization. On variational problems, there has been celebrated progress on faster techniques such as operator splitting, augmented Lagrangian, primal-dual methods needed to minimize non-smooth and non-quadratic terms. Another recent advance is the introduction of nonlocal methods for image restoration; these give impressive results in image denoising, and the search for faster computational implementation is ongoing. The program will focus on the following themes: 1. L1 minimization and applications (including Total Variation minimization). 2. Compressed Sensing by variational regularization methods. 3. Proximal point methods and iterative methods for solving ill-posed inverse problems (including iterative Bregman methods, hierarchical decompositions, surrogate functionals). 4. Geometric image analysis (denoising of surfaces, metrics in spaces of diffeomorphisms, computational quasiconformal geometry, spectral distance). 5. Optimal Transportation and Wasserstein Distance methods for registration and segmentation. 6. Nonlinear PDE methods for image processing. 7. Nonlocal methods (nonlocal means, nonlocal total variation, bilateral filtering). Emmanuel Candes of Stanford University will give the Distinguished Lecture Series in connection with this program. Summer Research School on the Mathematics of Medical Imaging This month-long program will have approximately 40 graduate students and postdoctoral fellows. During the first three weeks of the program, the participants will have an opportunity to attend an array of courses and research level talks. The students will be organized into small supervised teams of up to five people, based on the research project they choose. The fourth week will be devoted to collaborative work on this project. The following courses have been planned for the program: • Medical Image Registration Jan Modersitzki (University of Lübeck) • Research in Mathematical Image Processing Todd Wittman (Virginia Commonwealth University) • Variational Regularization Methods for Image Analysis and Inverse Problems Otmar Scherzer (University of Vienna) • Frontiers in Rapid MRI, from Parallel Imaging to Compressed Sensing and Back Michael Lustig (UC Berkeley) • Sparse and Redundant Representation Modeling of Images Michael Elad (Technion) • MRI for Mathematicians. Numerical Methods for Maxwell’s Equations Charles Epstein (University of Pennsylvania) • Numerical Methods for Distributed Parameter Identification Eldad Haber (UBC). • Microlocal Methods in Inverse Problems Gunther Uhlmann (University of Washington & UC Irvine) • Microlocal Analysis of Thermoacoustic Tomography Plamen Stefanov (Purdue University) By integrating a collaborative research workshop into a month-long program, we aim to provide an atmosphere where graduate students, postdoctoral fellows and senior researchers can learn from each other in an enjoyable and productive way. The Thematic Program will continue into August, with a Workshop on Microlocal Methods in Medical Imaging and conclude with a Fields-Mitacs Industrial Problem-Solving Workshop on medical imaging. Adrian Nachman (Toronto) ‘GAP’ continued from page 9 the Dirichlet problem for Einstein metrics, presenting various results on existence and uniqueness. Robin Graham (University of Washington) used conformal structures to construct new examples of pseudo-riemannian manifolds with split G2 holonomy. On the physics side, Pedro Vieira (Perimeter Institute) reported on certain exact computations of structure constants and scattering amplitudes in the AdS/ CFT correspondence. Aspects of a conjectured duality between Vasiliev’s higher spin gauge theory and vector models were presented by Xi Yin (Harvard). The shorter talks by postdoctoral fellows likewise covered many interesting topics. These included Floer theory of cleanly intersecting immersed Lagrangians (Ken Chan, Waterloo), new Einstein metrics on associated 3-sphere bundles over Fano Kähler-Einstein manifolds (Dezhong Chen, Toronto), cylindrical contact homology of universally tight sutured solid tori (Roman Golovko, Montreal), extremal Kähler metrics on projective bundles and stability (Hongnian Huang, CRM), a partial compactification of the moduli space of the Vafa- Witten equations (Ben Mares, McMaster), Legendrian knots and contact structures (Sinem Onaran, Waterloo), and CFT correlation functions as AdS scattering amplitudes (João Penedones, Perimeter). Throughout the conference, the excellent staff of the Fields Institute maintained a steady supply of refreshments and snacks, and saw to it that the visitors from far away were well-taken care of. They also organized a wonderful reception for the conference participants. The organizers hope that the Geometry and Physics (GAP) conference will become a regular event in the Canadian mathematical calendar, one in which the participants continue to enjoy high level but informal interactions between geometers and physicists from Canada and abroad. Especially with regard to graduate student participants, we hope that they come away energized, with new ideas and perspectives that enrich their research programs, and with new contacts and collaborations that will benefit their chosen careers. McKenzie Wang (McMaster) 20 FIELDSNOTES | THE FIELDS INSTITUTE for Research in Mathematical Sciences</div> <div>representation ρ, Langlands predicted that there exists a cuspidal automorphic representation π for GL n( � K ) such that the local representations of GL n( K p ) for primes p correspond (in a precise way) to the eigenvalues of the Frobenius conjugacy class Frob p under ρ (there is a more complicated recipe for the bad primes). Moreover, the L-function L( ρ , s) should be equal to the L-function L( π , s) (somewhat later, these automorphic L-functions were shown to be entire and satisfy the expected functional equations). Also around this time, in a manner whose precise historical development is somewhat murky, it came to be believed that attached to any motive M over K (say, an elliptic curve over Q) one could also attach an L-function L( M , s) that conjecturally could be identified with an automorphic form π. It is at this point that the subject started to break up into various different areas. On the one hand, there was detailed study of the arithmetic of modular curves (as generalized moduli spaces of elliptic curves with level structure) as well as the concomitant properties of classical modular forms. On the other hand, there was the general problem of functoriality, with a particular emphasis on understanding various special cases (like endoscopy) using the Fundamental lemma. It is only relatively recently with the proof of the Sato-Tate conjecture as well as the local Langlands conjecture that the focus of arithmetic interest has shifted away from classical modular forms and more towards the general case. This has happily coincided with the long awaited proof of the Fundamental lemma, which has already enabled progress in the construction of Galois representations associated to certain classes of automorphic forms for GL n . Throughout the duration of the semester, there will be two courses, two conferences, and a special workshop for graduate students and postdoctoral researchers. One course will concentrate on the arithmetic aspects of automorphic forms, whereas the other will concentrate on the representation theoretic aspects. The theme of the first workshop will be on the cohomology of Shimura varieties, which are the most natural generalization of classical modular curves. The second workshop will concentrate on recent developments in the p-adic Langlands program. The idea behind the workshop for graduate students and postdocs is not only to bring students up to speed with a significant amount of material, but also for slightly more advanced graduate students and postdoctoral students to relearn the theory of modular forms from a more automorphic perspective. One difficulty is that much of the current literature in automorphic representations is written for a substantially different audience — namely, those in representation theory. It is fundamentally important that graduate students and postdocs have the opportunity not only to become acquainted with the material, but also to learn a coherent story about the role automorphic representations play in answering arithmetic questions. Due to recent progress on a range of important problems in the Langlands program, the field seems poised for even more exciting developments. Ultimately, a key goal of the Thematic Program on Galois representations and Automorphic Forms is to help unify the subject in order to realize this possibility. Frank Calegari (Northwestern), Matthew Emerton (Northwestern), Florian Herzig (Toronto), Mark Kisin (Harvard), Stephen Kudla (Toronto) WINTER 2012: PROSPECTIVE Thematic Program on THE THEMATIC PROGRAM ON INVERSE PROBLEMS and Imaging will take place from January – August 2012. It has been organized in conjunction with the Mitacs International Focus Period on the Mathematics of Medical Imaging (June 2011 – August 2012). For the Thematic Program at the Fields Institute, we have chosen to focus in depth on a few selected active areas in inverse problems and image analysis, to establish connections between these fields and identify important new directions of investigation. The emphasis will be on longer events that foster research, learning, and collaboration in situ, rather than on workshops with many talks. Program on Geometry in Inverse Problems To non-invasively determine the properties of the interior of an inhomogeneous object, scientists analyze its effect on acoustic or electromagnetic waves. Mathematically, wave propagation is modeled by partial differential equations, and the problem consists in determining the coefficients (unknown inside the object) of such an equation from available information about its solutions. These problems have important applications in medical imaging, geophysical prospecting and non-destructive testing. When the object is anisotropic, these inverse problems lead to challenging questions which can be formulated in purely geometric terms. In electrical impedance tomography, the relevant differential operator is the Laplace-Beltrami operator; one seeks to recover the corresponding Riemannian metric modeling the anisotropic medium. On the other hand, for hyperbolic problems, singularities propagate along geodesics. A classical example is the kinematic inverse problem in seismology, where the information is travel time through geological layers. This corresponds to the boundary rigidity problem in geometry (studied by Michel, Gromov, Croke, Sharafutdinov, and others) where one seeks to determine the Riemannian metric of a manifold with boundary from knowledge of the distance between its boundary points. The problem has been solved in two dimensions in recent years by Pestov and Uhlmann, and important progress in higher dimensions has been achieved through very different methods by Burago and Ivanov. The interaction between differential geometry and inverse problems goes beyond the representative example mentioned above. For instance, when dealing with issues of stability under measurement errors in anisotropic inverse problems, one needs to consider a whole class of models which satisfy some a priori, coordinate invariant conditions, and consider what happens with these models under the small variations of the measured data. This framework places inverse problems into the area of geometric convergence, Gromov precompactness of Riemannian manifolds. Answers to some of the challenging questions that arise in the study of inverse problems are only beginning to emerge. The aim of the Program is to bring together top geometers 4 FIELDSNOTES | THE FIELDS INSTITUTE for Research in Mathematical Sciences</div> <div>Inverse Problems and Imaging who are interested in working on some of these challenging questions with theorists from inverse problems, attacking them with tools from partial differential equations. In addition, several important new directions for the field will be explored during the Program. Inverse problems are known to be ill-posed. One remedy, discussed above, is to consider them under some a priori conditions (often described geometrically in terms of the appropriate Gromov moduli space). Another approach is to restrict attention to finding some invariants of the object (such as volume, diameter, or various topological or geometrical characteristics) and investigate if they can be stably recovered. Important inverse problems for time-varying media lead to hyperbolic equations with timedependent coefficients. These require consideration of the space-time continuum as a whole, and thus need to be formulated on Lorentzian manifolds. The study of global geometry becomes crucial to solving such problems. Related problems concern inverse problems arising in models of field theory, leading to questions such as the collapse of Lorentzian manifolds, which will be considered during the Program. The deep recent results on Whitney’s extension problem and interpolation of functions may have natural geometric analogues very relevant to inverse problems. Indeed, since in practice one is working with discrete data, one of the goals of the future study of inverse problems will be to determine the existence and the properties of a continuous object (such as a Riemannian manifold or orbifold within a specified class) that is close, in an appropriate sense, to a finite metric space reconstructed from the available measurements. This will be a research-intensive program aimed at close collaboration between the participants. The focus will be on five main topics, briefly motivated above: • The boundary rigidity problem • Anisotropic inverse problems, geometric convergence and spectral geometry • Index theory and inverse problems; reconstruction of topological or geometric invariants • Inverse problems on Lorentzian manifolds • The geometric Whitney problem Program on Variational Methods and Compressive Sensing in Imaging In recent decades, the analysis and processing of images has become vital to many areas of science, medicine, engineering, manufacturing, and everyday use. Recent mathematical developments in the approach to imaging have brought about sophisticated methods for compression, reconstruction, restoration, registration, segmentation, and feature extraction. The advanced mathematical methods being studied for these problems include new developments in harmonic analysis, nonlinear optimization, numerical linear algebra, integral equations, partial differential equations, differential geometry, statistical estimation, and stochastic modeling. Among the most recent advances, the discovery and development of compressive sensing has already had a transformative impact in numerous disciplines. Due to the contributions of Candes, Tao, Donoho, and much subsequent work by many researchers, compressed sensing has become an area of explosive growth. It allows the solution of highly underdetermined problems when the unknowns are sparse (or approximately sparse). A prototypical example arising in important magnetic resonance imaging (MRI) applications is the recovery of a function (a sparse image) from highly incomplete knowledge of its Fourier transform. A deep result of Candes, Romberg and Tao (2004) showed that, with overwhelming probability, exact reconstruction can be ‘Inverse Problems’ continued on page 20 THE FIELDS INSTITUTE for Research in Mathematical Sciences | FIELDSNOTES 5</div> <div>Guelph Biomathematics and Biostatistics Symposium Climate Change and Ecology: A Mathematical & Statistical Perspective IT IS GENERALLY ACKNOWLEDGED THAT humanity’s transformation of the earth has increased the concentration of greenhouse gases, thereby causing climate change. Climate change is likely to alter the frequency of extreme weather events, cause global warming, and lead to extensive changes in many ecosystems. Global climate change is a central issue in ecology, and it is important to assess how anticipated changes might affect biodiversity, food webs, and natural resources. The biodiversity impacts of climate change can be seen in altered phenology, population density, and community structure. Changes in species’ geographical distributions have also been detected. Fluctuating ecosystems have the capacity to strongly affect human welfare. For example, the distribution and ecology of several species of pests and diseases of great significance to agriculture, fisheries, and forestry are strongly influenced by climatic factors. The complexity of these systems makes any predictions and potential mitigating strategies difficult, and there is room for uncertainty. In order to ensure that our ecosystems provide the services that society demands, we must be able to predict how ecological communities will respond to global climate change and, in turn, how changes in community composition will affect ecosystem services. The focus of the Guelph Biomathematics and Biostatistics Symposium on Climate Change and Ecology: a Mathematical and Statistical Perspective was the methodology that will play a crucial role in dealing with such problems. For example, climate envelope models have been used to predict the distribution of FIELDSNOTES DIRECTOR Edward Bierstone DEPUTY DIRECTOR Matthias Neufang MANAGING EDITOR Andrea Yeomans SCIENTIFIC EDITOR Carl Riehm DISTRIBUTION COORDINATOR Tanya Nebesna ON THE COVER: An early morning photograph of ice crystals, formed after a cold, foggy Ontario night. COVER PHOTO Mike MacLeod ADDITIONAL PHOTOS AND ILLUSTRATIONS Michael Bliefert (page 17), Elod Beregszaszi (page 9), Richard Cerezo (pages 8, 16, 19), Erin Murphy (book cover design, page 11), H. Radcanska (page 5), Keith Yeomans (page 2) The Fields Institute for Research in Mathematical Sciences publishes FIELDSNOTES three times a year. Questions? Comments? Email ayeomans@fields.utoronto.ca species under current, past, and future climatic conditions by inferring a species’ environmental requirements from localities where it is currently known to occur. The aim of the symposium was to provide exposure to the various mathematical and statistical techniques used to model and analyze the impact of climate change on the environment. There were some 55 confirmed participants, mainly from Canada and the United States, with additional participants registering for the day. Approximately half of the participants were graduate students and postdocs. The symposium was keynoted by two invited lectures. The first keynote speaker, James V. (University of British Columbia), gave the Gordon C. Ashton Memorial Biometrics Lecture. This was an exciting talk in the area of agroclimate risk management with applications to prediction of the bloom dates of perennial crops in the Okanagan region of British Columbia. The second keynote speaker was Marie-Josée Fortin (University of Toronto). She delivered an exciting talk on how statistical models and dynamic species distribution models can be used to study species range shifts. In addition to the keynote speeches, there were also seven excellent contributed talks. One of the exciting aspects of this symposium was the opportunity for researchers from various disciplines— Ecology, Mathematics, Statistics, and Geography—to share different perspectives on Climate Change. This unusual mix of researchers led to some very lively discussions between the various groups. Marcus Garvie and Julie Horrocks (Guelph) Canadian Undergraduate Mathematics Conference THIS YEAR, THE CANADIAN UNDERGRADUATE Mathematics Conference (CUMC) attracted over 160 students, of whom approximately 80 gave talks. The conference also featured eight keynote speakers, from Canada and abroad. The main talk subjects were pure and applied mathematics, statistics, computer science, physics and finance. At the beginning of the conference, André Fortin (Laval), Pamela Gorkin (Bucknell) and Frédéric Gourdeau (Laval) spoke about how to give a good talk. Many participants had never given a scientific talk, making the CUMC a perfect event at which to begin — one of the main objectives of the conference is to give undergraduate students the opportunity ‘CUMC’ continued on page 19 10 FIELDSNOTES | THE FIELDS INSTITUTE for Research in Mathematical Sciences</div> <div>CURRENT ACTIVITIES Connections in Geometry and Physics Conference (GAP2011) THE THIRD CONNECTIONS IN GEOMETRY AND Physics conference took place at the Fields Institute from May 13, 2011 to May 15, 2011. The conference received support this year from the Fields Institute, the Perimeter Institute, the Department of Mathematics of the University of Toronto, and the Faculty of Mathematics of the University of Waterloo. The organizers of the conference, Marco Gualtieri (Toronto), Spiro Karigiannis (Waterloo), Ruxandra Moraru (Waterloo), Rob Myers (Perimeter Institute), and McKenzie Wang (McMaster), welcomed over eighty participants from all over the world to the meeting. In addition to a good mix of senior and younger researchers, the participants included also a large number of graduate students from Canadian universities. The goals of this yearly conference are to foster interaction and cooperation between geometers and mathematical physicists, to highlight significant international developments in the two fields, and to support the research and training activities of Canadian researchers, their postdocs, and graduate students. Since the first conference, the organizers have selected for each year three topics which help to focus (but not constrain) the conference activities. This year the topics were: advances in Floer theory, geometric flows, and the AdS/CFT correspondence. For those who have not followed this yearly conference closely, the topics for 2010 were mathematical general relativity, gauge theory, and mirror symmetry, and those for 2009 were elliptic and parabolic equations in geometry, the geometry and topology of moduli spaces, and structures in symplectic geometry. The somewhat intensive schedule over three days of the 2011 conference included twelve hour-long lectures by leading experts working in the three topic areas and seven half hour talks by postdocs on their own research. The atmosphere of the talks was reasonably relaxed and included numerous questions and comments from the audience, some of which came from student participants. In the area of symplectic Floer theory, Octav Cornea (University of Montreal) reported on new developments in the study of Lagrangian cobordisms and their relation with decompositions in Fukaya categories. François Lalonde (University of Montreal) talked about properties and constructions of weakly exact Lagrangian submanifolds and their relationship with gluing theorems in symplectic geometry and quantum homologies. New filtrations in singular instanton knot homology and their applications were explained by Tom Mrowka (MIT). Lagrangian correspondences and their natural appearance in a functorial setting in extended Fukaya categories were discussed by Katrin Wehrheim (MIT). There were three talks on geometric flows. John Lott (Berkeley) presented new results regarding the long-time behaviour of the Ricci flow in 3-dimensions and indicated many open questions in this area. André Neves (Imperial College) reported on his work on the prevalence of singularities in using the mean curvature flow to produce special Lagrangian manifolds in Calabi-Yau manifolds. Natasa Sesum (Rutgers) gave a survey and a comparison between the properties of Ricci and Yamabe solitons. Robert McCann (Toronto) demonstrated the power of ideas from differential geometry, optimal transport and nonlinear PDEs by presenting new uniqueness and stability theorems for the multidimensional version of a famous problem in economic theory solved by Mirrlees and Spence regarding optimal pricing strategy by a monopolist who has only statistical information about the preferences of anonymous buyers. An important mathematical aspect of the AdS/CFT correspondence is the study of conformally compact Einstein spaces. Michael Anderson (Stony Brook) gave a survey on ‘GAP’ continued on page 20 THE FIELDS INSTITUTE for Research in Mathematical Sciences | FIELDSNOTES 9</div> <div>UPCOMING THEMATIC PROGRAMS GALOIS REPRESENTATIONS AND AUTOMORPHIC FORMS JANUARY TO JUNE 2012 Organizers: Frank Calegari (Northwestern), Matthew Emerton (Chicago), Florian Herzig (Toronto), Mark Kisin (Harvard), Stephen Kudla (Toronto) FEBRUARY 29–MARCH 2, 2012 Coxeter Lecture Series: Michael Harris (Paris VII) MARCH 12–16, 2012 Galois Representations and Automorphic Forms Workshop for Graduate Students and Postdoctoral Fellows MARCH 19–23, 2012 Workshop on Cohomology of Shimura varieties: arithmetic aspects and the construction of Galois representations APRIL 18–20, 2012 Distinguished Lecture Series: Christophe Breuil (Paris XI) APRIL 23–27, 2012 Workshop on the p-adic Langlands program: recent developments and applications For more information, please visit www.fields.utoronto.ca/programs/scientific/11-12/galoisrep INVERSE PROBLEMS AND IMAGING JANUARY TO AUGUST 2012 Organizers: Tony Chan (HK U. of Science and Technology), Charles Epstein (U. Pennsylvania), Allan Greenleaf (Rochester), Yaroslav Kurylev (University College London), Jan Modersitzki (Lübeck), Adrian Nachman (Toronto), Gunther Uhlmann (U. Washington), Luminita Vese (UCLA) MARCH 26–APRIL 27, 2012 Program on Geometry in Inverse Problems APRIL 30–MAY 31, 2012 Program on Variational Methods and Compressive Sensing in Imaging 2012 Distinguished Lecture Series: Emmanuel Candès (Stanford) JULY–AUGUST 2012 SUMMER THEMATIC PROGRAM ON THE MATHEMATICS OF MEDICAL IMAGING Organizers: Charles Epstein (U. Pennsylvania), Allan Greenleaf (Rochester), Jan Modersitzki (Lübeck), Adrian Nachman (Toronto), Gunther Uhlmann (U. Washington), Hongmei Zhu (York) JULY 3–31, 2012 Summer Research School on the Mathematics of Medical Imaging AUGUST 13-17, 2012 Workshop on Microlocal Methods in Medical Imaging AUGUST 20-24, 2012 Industrial Problem-Solving Workshop on Medical Imaging For more information, please visit www.fields.utoronto.ca/programs/scientific/11-12/inverseprob THE FIELDS INSTITUTE for Research in Mathematical Sciences | FIELDSNOTES 23</div> <div>GENERAL SCIENTIFIC ACTIVITIES UPCOMING EVENTS JANUARY TO JUNE 2012 For more information about our activities, please visit www.fields.utoronto.ca/programs FEBRUARY 3, 2012 Second Québec-Ontario Workshop on Insurance Mathematics at the Fields Institute FEBRUARY 22–24, 2012 Workshop on Surfactant Driven Thin Film Flows at the Fields Institute MARCH 8–9, 2012 Workshop on Coordinated Activity in Physiology: Measures, Concepts and Controversies at the Fields Institute Supported by the Centre for Mathematical Medicine APRIL 16–18, 2012 Workshop on Graphical Models: Mathematics, Statistics and Computer Science at the Fields Institute APRIL 19–22, 2012 Workshop on Exceptional Algebras and Groups at the University of Ottawa MAY 3, 2012 Nathan and Beatrice Keyfitz Lectures in Mathematics and the Social Sciences: Stephen Fienberg (Carnegie Mellon University) at the Fields Institute MAY 7–11, 2012 From Dynamics to Complexity: A Conference Celebrating the Work of Mike Shub at the Fields Institute MAY 29–31, 2012 Workshop on Rational Homotopy Theory and its Applications at the University of Ottawa JUNE 18–22, 2012 Workshop on the Corona Problem: Connections Between Operator Theory, Function Theory and Geometry at the Fields Institute JUNE 24–28, 2012 The 2012 Annual Meeting of the Canadian Applied and Industrial Mathematics Society (CAIMS) Hosted by the Fields Institute Held at the University of Toronto JUNE 24–JULY 6, 2012 2012 Séminaire de Mathématiques Supérieures at Centre de recherches mathématiques, Montreal CURRENT THEMATIC PROGRAM DISCRETE GEOMETRY AND APPLICATIONS JUNE TO DECEMBER 2011 Organizers: Karoly Bezdek (Calgary), Robert Connelly (Cornell), Antoine Deza (McMaster), Herbert Edelsbrunner (IST Austria), Asia I. Weiss (York), Yinyu Ye (Stanford) SEPTEMBER 13–16, 2011 Workshop on Discrete Geometry OCTOBER 17–21, 2011 Workshop on Rigidity and Symmetry SEPTEMBER 19–23, 2011 Conference on Discrete Geometry and Optimization SEPTEMBER 19, 21, & 23, 2011 Fejes Tóth Lecture Series: Thomas C. Hales (Pittsburgh) SEPTEMBER 26–29, 2011 Workshop on Optimization OCTOBER 11–14, 2011 Workshop on Rigidity OCTOBER 2011 Distinguished Lecture Series: Erik Demaine (MIT) OCTOBER 24–27, 2011 Workshop on Symmetry in Graphs, Maps and Polytopes NOVEMBER 2011 Coxeter Lecture Series: Stephen Smale (City University of Hong Kong) NOVEMBER 7–11, 2011 Workshop on Computational Topology NOVEMBER 14–18, 2011 Workshop on Sphere Arrangements 22 FIELDSNOTES | THE FIELDS INSTITUTE for Research in Mathematical Sciences</div>