## CRGs Starting Before 2005

### PIMS CRG on Dynamics and Related Topics: 2003-2005

**Overview**

The study of dynamical systems has had a long and distinguished history in mathematics. This study has ranged from applications involving differential equations and information theory, to more theoretical work focusing on systems with topological or algebraic structure. In the past few decades this field has grown dramatically, and completely new directions have opened up. One of the most significant recent developments has been the intense study of the joint action of several commuting transformations, or Zd-action. Although a comprehensive theory is currently out of reach, one class of such actions has been analyzed in great detail by Klaus Schmidt and his co-workers. These are the algebraic Zd-actions, which offer a rich and beautiful connection with commutative algebra and algebraic geometry, which are now being mined.

Another aspect of actions of higher-rank groups is the theory of aperiodic tilings and quasicrystals. The subject began with the work of Raphael Robinson and Roger Penrose, who showed that there were polygonal shapes that would tile Euclidean space, but only in aperiodic ways. The resulting tilings, however, are far from being completely random, and show a great deal of regularity. The dynamical point of view, where the transformations are shifts of tilings, has been particularly successful in analyzing these new structures. A singular physical discovery, namely crystal-like structures called quasicrystals which violated the traditional crystallographic rules, has led to very fruitful interactions with physical scientists and mathematicians.

During the last century, deep and fruitful connections were developed between dynamics and operator algebras. In recent years, interactions between topological dynamics and C*-algebras has grown with the development of tools in Connes's program for non-commutative geometry. This has been particularly evident in the theory of aperiodic order, largely through Bellissard's work connecting tilings, C*-algebras, K-theory, and solid state physics and the recent solution of the so-called gap-labeling problem.

There are quite a number of researchers dynamical systems and related areas at PIMS sites. Their interests are diverse, but they share expertise and create connections. The University of Washington has built up one of the strongest groups in dynamical systems in the world. It includes Christopher Hoffman, Douglas Lind, Steffen Rohde, Boris Solomyak, and Selim Tuncel. In addition, Manfred Einsiedler (a former Ph.D. student of Klaus Schmidt) will be starting a three-year appointment beginning the fall of 2002. Also Klaus Schmidt is planning to spend a sabbatical year in Seattle during 2002-03, and also possibly one or both of the adjacent summers. A number of other visitors in dynamics are expected that year as well, including Mike Boyle (Maryland), Christopher Denninger (Muenster), William Parry (Warwick), and Daniel Rudolph (Maryland).

At the University of Victoria, Ian Putnam works specifically in the area of interactions between operator algebras and dynamics. John Phillips and Marcelo Laca both work in areas of operator algebras that have strong ties with dynamics. Chris Bose works in ergodic theory. Rod Edwards works in the dynamics of biological systems and may have some interaction.

At the University of Alberta, Bob Moody works on aperiodic order and, to some extent, Al Weiss, also. Anthony Lau and Volker Runde work in operator algebras. At the University of Calgary, Berndt Brenken, Igor Nikolaev and Michael Lamoureux work in areas of operator algebras very closely linked with dynamics.

**Faculty**

U. Alberta: Robert Moody, Anthony Lau, Volker Runde, Al Weiss

U. Calgary: Michael Lamoureux, Berndt Brenken, Igor Nikolaev

U. Washington: Douglas Lind, Christopher Hoffman, Douglas Lind, Steffen Rohde, Boris Solomyak, Selim Tuncel, Manfred Einsiedler.

U. Victoria: Ian Putnam, John Phillips, Marcelo Laca, Chris Bose, Rod Edwards

Visitors and other contributors: Klaus Schmidt (Vienna), Mike Boyle (Maryland), Christopher Denninger (Muenster), William Parry (Warwick), Daniel Rudolph (Maryland).

### PIMS CRG on Mathematical Ecology and Evolution: 2003-2005

**Overview**

As the current revolution in biological information progresses, there is a well recognized need for new quantitative approaches and methods to solve problems in ecology. One challenge is to model complex ecological systems--systems which depend upon a myriad of inputs, but often with incomplete details regarding the inputs. Such systems range from spatial disease dynamics (eg. influenza, tuberculosis) to the response of biota to global environmental change (eg., vegetation shifts), to the impact of habitat fragmentation on species survival. Mathematical tools for addressing such biological problems include stochastic processes, nonlinear dynamical systems, adaptive dynamics, scaling methods from individuals to populations, statistical inference (particularly in the area of inverse problems), and parallel computational methods. Our primary goal in this period of concentration is to develop and strengthen the synergistic interactions between mathematics and ecology in PIMS universities. Research in any interdisciplinary field brings its own particular set of challenges: (i) The rigorous and unambiguous aspects of mathematics vs the overwhelming complexity of the biological world. (ii) The different languages of mathematics and biology can provide a major barrier to interdisciplinary research.

With this in mind, an interdisciplinary research program that significantly impacts both mathematics and biology has some basic requirements: (i) the opportunity for training outside the core discipline of mathematics or biology and (ii) a mechanism for effective collaborations and cross-communication between empiricists and theoreticians.

These requirements are, of course, in addition to mathematical and biological expertise. The above ingredients are central to the training of personnel in this research plan.

This period of concentration will incubate significant new original research, foster local interactions, provide leadership to the new researchers, and strengthen the international profile of mathematical ecology and evolution in our universities. Areas of mathematical ecology research at PIMS universities include: Nonlinear population dynamics; Spatially structured populations; Adaptive dynamics; Model selection and validation and inverse methods, Stochastic models for populations; Scaling laws--from individuals to populations.

**PIMS Distinguished Chair**

The period of concentration has funded two distinguished chairs at approximately 10K each. The first Bryan Grenfell, is described below. We have not yet determined the other PIMS distinguished chair and would welcome suggestions. We are also hoping to have additional visitors with other funding. If you have suggestions for these visitors or for other funding please inform us. Contact Thomas Hillen mathecol@math.ualberta.ca.

Bryan Grenfell is a preeminent mathematical epidemiologist at the University of Cambridge, UK. His specialty is analysis of patterns in epidemiological data, and understanding the spatio-temporal dynamics of diseases and populations using mathematical and statistical models. Recent awards include the Order of the British Empire for services to epidemiology and the control of infectious diseases. Recent research includes mathematical analysis of foot-and-mouth disease, measles modeling, and analysis of nonlinear population dynamics. In the year 2001 alone, five of his published papers appeared in "Science" or "Nature".

The goal of the PIMS Distinguished Chair is to have Grenfell visit for 3.5 weeks in September 2003 to interact with mathematicians and mathematical ecologists at PIMS universities. The two host of mathematics, statistics and epidemiology. Interactions arising from Grenfell's visit will help catalyze new research interactions between PIMS researchers.

Distinguished chairs are to spend time at multiple PIMS sites.

**Faculty**

Coordinator

Dr. Thomas Hillen

Associate Professor

Department of Mathematical and Statistical Sciences

University of Alberta, Edmonton, AB, T6G 2G1

Phone: (780) 492-3395 Fax: (780) 492-6826

Email: mathecol@math.ualberta.ca

Co-Organizers

Dr. Michael Doebeli

Associate Professor

Departments of Zoology and Mathematics

University of British Columbia Phone: (604) 822-3326

Fax: (604) 822-2416

Email: doebeli@zoology.ubc.ca

Dr. Mark Lewis

Professor

Dept.of Math.Stat. Sci. and Biological Sciences

University of Alberta,

Edmonton, T6G 2G1,

Phone: (780) 492-0197

Fax: (780) 492-6826

Email: mlewis@math.ualberta.ca

Dr. Edward McCauley

Professor

Department of Biological Sciences University of Calgary

Phone: (403) 220-5583

Email: mccauley@ucalgary.ca

Faculty Participants (tentative)

University of Alberta:

Mark Boyce boyce@ualberta.ca,

Herb Freedman hfreedma@math.ualberta.ca,

Thomas Hillen thillen@math.ualberta.ca,

Subhash Lele slele@ualberta.ca,

Mark Lewis mlewis@math.ualberta.ca,

Michael Li mli@math.ualberta.ca,

Jens Roland jroland@gpu.srv.ualberta.ca,

Joseph So Joseph.So@ualberta.ca

University of Calgary:

Ed McCauley mccauley@ucalgary.ca,

Shane Richards richas@ucalgary.ca

University of British Columbia:

Fred Brauer brauer@math.ubc.ca,

Michael Doebeli doebeli@zoology.ubc.ca,

Leah Keshet keshet@math.ubc.ca,

Dolph Schluter schluter@zoology.ubc.ca

University of Washington:

James Anderson jim@cbr.washington.edu,

Carl Bergstrom cbergst@u.washington.edu,

Daniel Grunbaum random@u.washington.edu,

Ray Hilborne rayh@u.washington.edu,

Mark Kot kot@amath.washington.edu

Simon Fraser University:

Eirikur Palsson epalsson@sfu.ca,

Bernard Roitberg bernard_roitberg@sfu.ca

University of Victoria:

Pauline van den Driessche pvdd@math.uvic.ca

**PIMS Postdoctoral Fellow **

For 2004-06, Mario Pineda-Krch, was a postdoc at UBC.

For 2003-04 our POC has been awarded two PIMS PDFs. Both of these PDFs may be renewed for 2004-05.

The value of the PIMS fellowship is $20,000 per year (salary including benefits). This must be matched this with a minimum of $20,000 from non-PIMS sources to provide a minimum annual salary of $40,000 (including benefits).

Approximately 20% of the postdocs time will be spent helping in activities associated with the POC (summer school, Banff ecology retreat, interacting and communicating with visitors and so forth). Some of this will require extended travel to PIMS sites or BIRS where the activities are being held.

The funding cycle of the Institute runs from April 1 to April 1. Therefore, PIMS requests that each fellowship commence prior to January 1 of the fiscal year for which it is allocated. For example, the 2003-04 fellowships should commence before January 1, 2004.

PIMS strongly urges the PIMS CRGs not to apply for additional PDFs through the general competition, so as to ensure access to the programme for other PIMS members.

**Notes**

There are two postdocs associated with the program. These are funded 50% from the program, and the other 50% is to come from private research grants and teaching.

On August, 1, 2003 Dr. Frithjof Lutscher started as PIMS postdoc. He works 50% at U of Calgary with Ed McCauly and 50% at U of Alberta with Mark Lewis.

On November, 1, 2003 Dr. Joanna Renclawowicz will start as PIMS postdoc. She will work 33 % at U of Victoria with Pauline van den Driesche and 66 % at U of Alberta with Thomas Hillen and Mark Lewis.

Note that, besides research and teaching, duties of the postdocs include 15% of the time devoted to helping with the POC (see below). As outlined in the proposal to PIMS, the postdocs are to work in interdisciplinary mathematical ecology and evolutionary biology.

Note that these postdocs are only available to Canadian Universities - sorry to UW.

### PIMS CRG on Number Theory: 2003-2005

**Overview**

Number theory is one of the oldest, deepest and most vibrant branches of modern mathematics. It centrally incorporates some of the most sophisticated and profound mathematical ideas that have been developed (witness the recent proof of Fermat's Last Theorem) and yet remains broadly useful in many areas of pure and applied mathematics. Indeed, it is remarkable how often number theory comes to bear both in other areas of mathematics and in applications. A notable recent example is cryptography and internet security whose protocols are based on number theoretic problems.

Number theory is particularly strong in Canada with the PIMS Number Theory Group featuring prominently. This group is large and well distributed across the PIMS Universities. It has a number of prominent senior world-class researchers leading a group of richly talented young mathematicians. The recent influx of new number theorists into several PIMS universities has created an exciting working group (S. Choi, I. Chen, and P. Lisoneck at SFU plus M. Bennett, G. Martin, and V. Vastal at UBC). Both UBC and SFU groups have been involved in modern computational number theory (D. Boyd, P. Borwein, W. Casselman). In addition to hiring two new number theorists in the past year, UBC has additionally supplemented its faculty in cognate areas such as harmonic analysis (I. Laba) and algebraic geometry (J. Bryan and Z. Reichstein). This CRG already organizes several ongoing joint activities including Pacific Northwest Number Theory seminars and mini-conferences, and supervises many successful PIMS PDFs. A multi-year period of concentration will allow coordination and integration between those activities that are already being organized and new opportunities (BIRS, PIMS Distinguished Visitors) in a single framework to maximize impact.

**PIMS Distinguished Chair**

Professor Jeffrey D. Vaaler (University of Texas at Austin) delivered a lecture series in the second and third weeks of June 2003 at SFU.

Professor Bjorn Poonen (University of California, Berkeley) delivered a lecture series in July 2004 at SFU.

Professor Sergei Konyagin (Moscow State University) delivered a lecture series in March 2004 at UBC. Winner of the Salem Prize in 1990, Prof. Konyagin has made numerous significant contributions in number theory, approximation theory and harmonic analysis.

**Faculty**

CRG Leaders:

Peter Borwein (SFU), David Boyd (UBC)

UBC:

Michael Bennett, David Boyd, Bill Casselman, Rajiv Gupta, Izabella Laba, Greg Martin, Nike Vatsal

SFU:

Peter Borwein, Imin Chen, Stephen Choi, Petr Lisonek

U. Alberta:

James D. Lewis

U. Calgary:

Richard Guy, James P. Jones, Richard Mollin, Renate Scheidler, Hugh Williams

U. Washington:

Ralph Greenberg, Adrian Iovita, Neal Koblitz, Boris Solomyak

Other institutions:

Amir Akbary (U. Lethbridge), Edward Dobrowolski (College of New

Caledonia), Matt Klassen (DigiPen Inst. of Tech.), Kristin Lauter (Microsoft)

**Postdoctoral Fellows**

Ben Green, PIMS PDF (2003-04) at UBC

Friedrich Littman, PIMS PDF (2003-05) at UBC

Christopher Rowe, PIMS PDF (2003-05) at UBC

Ron Ferguson, MITACS PDF (2002-03) at SFU

William Galway, PIMS PDF (2002-03) at SFU

Alexa van der Waall, MITACS PDF (2002-03) at SFU

### PIMS CRG on Scientific Computing: 2003-2005

**Overview**

Scientific computing, in connection with applied mathematics, has become essential in solving many real-world problems. Its methodologies are needed, for example, in modeling physical, chemical, and biomedical phenomena; in designing engineered parts, structures, and systems to optimize performance; in planning and managing financial and marketing strategies; and in understanding and optimizing manufacturing processes. Problems in these areas arise in companies that manufacture a host of industrial and consumer products such as aircraft, automobiles, engines, textiles, computers, communications systems, chemicals, drugs as well as in various service and consulting organizations. They also arise in many national and international initiatives such as those in global change, biotechnology, and advanced materials. Because today's research problems are complex and multi-faceted, they very often require highly trained scientists with strong mathematical and computational skills and an in-depth knowledge of the discipline from which the research problem comes.

The excitement within Applied Mathematics and Scientific Computing comes from their evolution at the speed of technology. The exponential increase in computing capabilities continually opens up new vistas accessible to numerical modeling. Every advancement in scientific measurement reveals new phenomena demanding explanation and produces yet larger datasets requiring analysis. The importance of mathematics has largely increased because for many of the industrial applications of computing, the central questions are mathematical ones.

Scientific computing plays a critical role in applied and industrial mathematics at the PIMS universities and across North America. Within PIMS universities, the scientific computing community has been a very active collaborative group with a tradition of multi-university activities particularly among researchers at U. of Washington, SFU, and UBC. Recently new activities have emerged, due in large part to the leading role played by PIMS. A special feature of this period of concentration is the promotion of a multidisciplinary approach to the subject and the inclusion of important research topics such as the earth and atmospheric sciences. The base of support for the proposed programs includes researchers from Mathematics, Applied Mathematics, and Computer Science.

**PIMS Distinguished Chair**

The CRG will have two Distinguished Chair in 2003 and two more in 2004. These chairs will visit the group for at least one month and give a minicourse of lectures.

Professor Otmar Scherzer will be at University of British Columbia as a PIMS Distinguished Chair in Scientific Computing for the period of July 1 to September 30, 2004

Professor Chris Budd will be at Simon Fraser University as a PIMS Distinguished Chair in Scientific Computing in the summer of 2004.

**Faculty**

CRG Leaders: Steve Ruuth (SFU), Manfred Trummer (SFU), Chen Grief (UBC), Randy Leveque (U. Washington), Yanping Lin (U. Alberta), Elena Braverman (U. Calgary).

SFU: R. Choksi, M.C. Kropinski, T. Möller, D. Muraki, K. Promislow, B. Russell, S. Ruuth, L. Trajkovic, M. Trummer, J. Verner, R. Zahar.

U. Alberta: Y. Lin, J. Macki, P. Minev, Y.S. Wong

UBC: U. Ascher, O. Dorn, S. Dunbar, I. Frigaard, A. Peirce, B. Seymour, B. Shizgal, J. Varah, M. Ward, B. Wetton, M. Yedlin

U. Calgary: T. Ware, R. Westbrook

U. Victoria: D. Olesky, P. van den Driessche

U. Washington: R. LeVeque, L. Adams, D. Durran, A. Greenbaum, G. Hakim, N. Kutz, R. O'Malley, P. Schmid, J. Burke, C. Bretherton

Ballard Corporation: R. Bradean, J. Kenna

Boeing Corporation: J. Lewis, S. Filipowski, M. Epton

Quadrus Financial Technologies: S. Reddy

**PIMS Postdoctoral Fellows**

Jian-Jun Xu, PIMS PDF at SFU

Jianying Zhang, PIMS/MITACS PDF at UBC

This CRG will include another PDF in 2003 and two more in 2004.

### PIMS CRG on String Theory: 2003-2005

**Overview**

Some of the most fundamental questions in science are at the cutting edge of modern mathematical physics. They address the very structure and origin of the universe, the nature of the constituents of all matter and their interactions and the mathematical structures necessary for a quantitative formulation of the fundamental laws of nature. Two concrete goals of current focus in fundamental physics are to find a quantum theory of gravity that avoids the inconsistencies that arise from trying to reconcile Einstein's general theory of relativity with quantum mechanics and to find a unified theory which encompasses all of the forces of nature and describes all of the particles which are subject to those forces.

Superstring theory is a promising candidate for a physical theory that could simultaneously achieve both of these goals. String theory is a physical model which is postulated to describe fundamental interactions at exceedingly small distances where quantum mechanical fluctuations of the geometry of spacetime would become important. As a physical dynamical system, it is still not completely understood. It is clear that the full power of superstring theory will only be realized once significant progress has been made in understanding its mathematical and dynamical structure. This will involve the development of new mathematics and will be an important frontier area for both mathematics and physics in the foreseeable future.

**PIMS Distinguished Chair**

Ashoke Sen (Harish-Chandra Research Institute), will give a series of lectures during July and August 2003 at UBC.

Ashoke Sen's significant contributions to string theory include his early work on string field theory, duality symmetries, and black hole solutions. This led him to the study of strong-weak coupling duality of supersymmetric gauge theories. His proof of the existence of the conjectured bound states required by those dualities is now one of the classic works in the field. This work has had enormous influence and is usually credited as the start of the "second superstring revolution" --- a shift of perspective which the field is still undergoing.

In 1998, Professor Sen initiated the study of non-supersymmetric states in string theory. Once again, this has proven to be a very fruitful study. Most recently, he has returned to the study of string field theory in the context of tachyon condensation on D-branes. This has led to his current work on the cosmological consequences of tachyon condensation.

The CRG will have another Distinguished Chair in 2003 and two more in 2004. These chairs will visit the group for at least one month and give a minicourse of lectures

**Faculty**

CRG Leaders: Gordon Semenoff (UBC), Eric Woolgar (U. Alberta)

U. Alberta: B. Campbell, V. Frolov, D. Page (Physics), T. Gannon, E. Woolgar (Math)

UBC: G. Semenoff, M. Rozali, M. Van Raamsdonk, K. Schleich, D. Witt, M. Choptuik, W. Unruh (Physics and Astronomy), J. Bryan, K. Behrend (Math).

U. Lethbridge: M. Walton (Physics)

Perimeter Institute: R. Myers, L. Smolin=

SFU: K. Viswanathan (Physics)

U. Toronto: A. Peet (Physics)

U. Washington: A. Karch (Physics)

Asia Pacific Center for Theoretical Physics, Korea: Taejin Lee

**PIMS Postdoctoral Fellows**

Jian-Jun Xu, PIMS PDF at SFU

Jianying Zhang, PIMS/MITACS PDF at UBC

This CRG will include another PDF in 2003 and two more in 2004.

**Graduate Students**

K. Chu, P. de Boer, M. Laidlaw, J. Gardezi, B. Sussman, B. Ramadanovic, D. Young, K. Furuuchi, R. Fazio.

### PIMS CRG on Topology: 2004-2006

**Overview**

The PIMS community has an active group of researchers in topology and related fields. Their research may be roughly divided into two major themes: geometric and algebraic. Among the geometric issues being studied by PIMS topologists are the classification of manifolds (particularly in dimension 3 and 4), group actions on Riemann surfaces, knot theory and its applications, and relating 3-manifold topology to relativity theory. A sample of the contributions in algebraic topology are: application of algebraic topology to robotics, developing equivariant minimal models in homotopy theory, applying subtle algebraic properties of projective spaces and bundles to solve classical problems in quadratic forms and combinatorics. Because of their geographic separation and diversity of interests, this community of scientists is particularly well-served by forming a collaborative research group.

Scientific Activities

**PIMS Distinguished Chair**

The CRG will have two Distinguished Chairs:

Roger Fenn (University of Sussex) at UBC in 2004

Fred Cohen (University of Rochester, New York) at UBC in 2005

**Faculty**

CRG Leaders: Laura Scull (Primary contact, UBC), George Peschke (U. Alberta), Dale Rolfsen (UBC), Peter Zvengrowski (U. Calgary)

U. Alberta: George Peschke, James Timourian

UBC: Jim Bryan, Kee Lam, Dale Rolfsen, Laura Scull, Denis Sjerve

U. Calgary: Kalathoor Varadarajan, Peter Zvengrowski

U. Washington: Eric Babson, Ethan Devinatz, Michael Freedman, Steve Mitchell, John Palmieri, Jack Segal

PIMS Post-doctoral Fellow

2005-2006: Antonio Ramírez (UBC)

**Visitors**

Dr. Elena Kudryavtseva will be arriving at the University of Calgary in September 2004 as a PIMS Visiting Research Fellow with the Collaborative Research Group in Topology and be staying for one year. Prior to her arrival she can be contacted through Dr Peter Zvengrowski at UC at zvengrow@math.ucalgary.ca. Once she arrives her office will be Math and Stats 570 and phone will be (403)210-8697.

### PIMS CRG on Probability and Statistical Mechanics: 2004-2006

**Overview**

Much of the original motivation for th study of spatially interactive stochastic systems came from stochastic models in statistical physics. An intensive area of recent research centres around the idea that complex local dynamics can lead to a small number of well-understood continuum models upon space-time rescaling. When the underlying system is at or near criticality the limit invariably seems to be closely related to super-Brownian motion. The list of such results obtained in recent years is remarkable and includes Fisher-Wright and Fleming-Viot Models in population genetics (Dawson Donnelly, Etheridge, Kurtz, March and Perkins), interacting particle systems including contact process and voter models (Bramson, Cox, Durrett, Le Gall, Perkins, and Sakai), lattice trees and animals above the critical dimension of 8 (Derbez and Slade), and percolation and oriented percolation at criticality above the critical dimensions of 6 and 4, respectively (Hara, van der Hofstad and Slade).

Other local interactions arising in models for competing species, predator-prey systems or symbiotic branching lead to more complex stochastic models which behave locally like superprocesses but with branching, migration and drift coefficients which depend on the current state of the system. Two challenging and related topics are therefore:

The development of a general theory of interactive superprocesses and in particular methods to characterize these processes and study their properties.

The use of such models in mathematical ecology and evolution.

The rescaling results of Slade and his co-authors have created some strong common interests between the statistical physics and spatial stochastic process communities. The scaling limits of low dimensional statistical physics, however, are not super-Brownian motion. It is a defining goal of statistical mechanics to identify them and to calculate their properties. At present there is excellent progress in two dimensions where the stochastic Loewner processes provide natural candidates for scaling limits (ongoing work of Lawler, Schramm and Werner). Another promising program is based on the renormalization group. The self-avoiding walk in 4 (and 4 - e ) dimensions can in principle be analyzed by these methods (on going work of Brydges, Imbrie and others).

A period of concentration in Probability and Statistical Mechanics at PIMS will start from April 2004 - August 2006.

There will be a number of short term and long term visitors, and several conferences (see `Scientific Activities '). Each summer there will be a summer school, lasting about 5 weeks, with two advanced courses on special topics in probability theory. Graduate students from Western Canadian Universities may attend these courses under the Western Dean's Agreement.

Administrative enquirers should be addressed to pims@pims.math.ca.

Scientific inquiries should be addressed to one of the coordinating committee at UBC:

Martin Barlow, David Brydges, Alexander Holroyd, Vlada Limic, Ed Perkins, Gordon Slade.

Participating departments: UBC, U Washington, Microsoft Research, U Alberta, U Saskatchewan, U Regina.

**PIMS Distinguished Chairs**

Prof. Frank Den Hollander (Leiden University and EURANDOM, The Netherlands) will be visiting UBC January 1-August 15, 2006.

The CRG had two Distinguished Chairs in 2004.

Prof. Richard Bass (U. Connecticut) spent an academic year at UBC, arriving in Aug. 04 and leaving the following summer.

Prof. Yaozhong Hu (University of Kansas) spent an academic year at U. of Alberta from August 1, 2004 to August 31, 2005.

**Faculty**

CRG Leaders: David Brydges (UBC), Chris Burdzy (U. Washington), Ed Perkins (UBC), Byron Schmuland (U. Alberta)

UBC: David Brydges, Joel Feldman, Alexander Holroyd, Vlada Limic, Gordon Slade, Martin Barlow, Ed Perkins, John Walsh.

U. Alberta: Byron Schmuland, Mike Kouritzin

U. Washington: Chris Burdzy, Zhen-Qing Chen, Bruce Erickson, Chris Hoffman, Lisa Korf, Steffen Rohde.

Microsoft Research: Jennifer Chayes, Christian Borgs, Oded Schramm, David Wilson.

U. Regina: Michael Kozdron

University of Saskatchewan: Chris Soteros, Raj Srinivasan

Other institutions: Remco van der Hofstad (Eindhoven), Don Dawson (McGill)

**PIMS Postdoctoral Fellows**

Dr. Omer Angel, PIMS PDF at UBC, 2004-06

Dr. Codina Cotar, PDF at UBC

Dr. Alexander Roitershtein, PDF at UBC

**Visitors**

2006

Jan. 2-8, 2006: Ted Cox (Syracuse University).

Jan. 1-Aug. 18, 2006: Frank den Hollander (Leiden University and EURANDOM).

Feb. 28-Mar. 9, 2006: Jean-Dominique Deuschel (Technische Universität Berlin).

May 21-Jun. 18, 2006: Akira Sakai (EURANDOM).

Jun. 26-Sep. 9, 2006: Malek Abdeselam (Universite Paris 13).

2005:

Jan. 4-Feb. 4, 2005: Nicolai Krylov (U. Minnesota) will visit UBC.

Feb. 13-Mar. 6 2005: Alain Sznitman (ETH) will visit UBC.

March, 2005: Bruno Remillard (HEC Montreal) will visit U. Alberta.

June 3-July 9, 2005: Lorenzo Zambotti, (Pisa) will visit UBC.

June and July, 2005: Dan Romik (Weizmann, Institute) will visit UBC.

June 6 - July 1, 2005: Yuval Peres (UC Berkeley) will visit UBC.

Two months summer 2005: Takashi Kumagai (Kyoto University) will visit UBC.

Summer 2005: Leonid Mytnik, (Technion) will visit UBC.

Summer 2005: Carl Mueller (U. of Rochester); Steve Evans, (Berkeley U.) will visit UBC.

2004:

May 1 - June 8, 2004: Doug Blount (Arizona State University) will visit U. Alberta.

May 19 - 31, 2004: Takashi Kumagai (Nagoya University) will visit UBC.

May 10-23, 2004: Don Dawson (Carleton U.) will visit UBC.

May 15-23, 2004: Yves Le Jan (U. Paris, Orsay) will visit UBC.

May 15-June 15, 2004: Pierre Tarres (U. Toulouse) will visit UBC.

May 17-23, 2004: Robert Dalang (U. Lausanne) will visit UBC.

May 19 - June 26, 2004: Greg Lawler (Cornell U.) will visit UBC.

June and Aug 15 - Sept 15: Erwin Bolthausen (U. Zurich) will visit UBC.

July 2004: John Imbrie, (U. Virginia), Jon Dimock, (SUNY Buffalo), Pronob Mitter, (U. Montpellier) will visit UBC

Aug. 2004 - Aug. 2005: Richard Bass (University of Connecticutt) will visit UBC.

August 23 - September 5, 2004: Remco van der Hofstad (Technical University Eindhoven) will visit UBC.

Sept. 2004: Ted Cox (Syracuse U.) will visit UBC.

Sept. 13-21, 2004: Andreas Greven (U. Erlangen) will visit UBC.

Oct. 11-14, 2004: Tony Guttmann (University of Melbourne) will visit UBC.

Fall Term, 2004: Zhenqing Chen (U. of Washington) will visit UBC.

Nov. 2004: Takashi Kumagai (Kyoto University) will visit UBC.

### PIMS CRG on AG-GC-RT: 2005-2007

*Note: AG, GC, RT stands for Algebraic Geometry, Group Cohomology, Representation Theory.*

**Overview**

Algebraic geometry is a mathematical discipline which uses the techniques and tools of algebra (e.g. rings, ideals and fields) to attack geometric problems. The fundamental objects which algebraic geometers study are algebraic varieties, the common zeros of a collection of polynomials. In the last four decades, beginning with the ground breaking work of Alexandre Grothendieck, the discipline has undergone phenomenal growth and has had a profound influence on the development of modern mathematics. Many of its celebrated works have led to Fields Medals: the proofs of the Weil Conjectures by Deligne, Mumford's work on geometric invariant theory, Hironaka's work on the resolution of singularities, Mori's work on the classification of algebraic varieties in dimension three and Wiles' proof of Fermat's Last Theorem which used arithmetic algebraic geometry. Furthermore, the work of Kazhdan, Lusztig, Kashiwara and others has made algebraic geometry an indispensable tool for representation theory. In the last fifteen years, exciting new connections between algebraic geometry and physics emerged, which led to unexpected new mathematical theories such as mirror symmetry and quantum cohomology and to many important developments in the field of mathematical string theory.

For the most part, these advances have been brought about by the fact that algebraic geometry poses intrinsically interesting and relevant problems, and has the property of developing the mathematical tools to solve them. It has therefore attracted many talented mathematicians, many of whom are not formally trained in the area, but have realized its value. This has further stimulated new connections between algebraic geometry and other disciplines: e.g. combinatorics, cryptography, statistics, and quantum computing.

Algebraic geometry has also given us new insight into the nature of algebraic groups and Galois cohomology. During the last two decades many exciting fundamental theorems have been established due to the introduction of new powerful techniques from algebraic topology and algebraic geometry. For instance, Voevodsky's use of homotopy and cobordism theory have resulted first in the solution of Milnor conjecture and, more recently, the Bloch-Kato conjecture. Further development of these ideas is crucial.

The PIMS CRG has many people working in the cutting edge in several of the above areas. Among the specialties represented by our varied group are algebraic stacks, geometric invariant theory, algebraic group actions, toric varieties and torus actions, algebraic cycles, Gromov-Witten theory, arithmetic algebraic geometry, classification theory, algebraic representation theory, Lie theory and Schubert varieties, group cohomology.

**PIMS Distinguished Chair**

Michael Thaddeus

(Columbia, New York), September 2005 - August 2006

**Faculty**

CRG Leader: Arturo Pianzola (Alberta), Jim Bryan (UBC)

SFU: Nils Bruin, Imin Chen

U. Alberta: Xi Chen, Gerald Cliff, Vladimir Chernousov, Terry Gannon, Jim Lewis, Arturo Pianzola

UBC: Alejandro Adem, Kai Behrend, Jim Bryan, Jim Carrell, Bill Casselman, Kalle Karu, Dale Peterson, Zinovy Reichstein

U. Calgary: Clifton Cunningham

U. Washington: Eric Babson, Sara Billey, Chuck Doran, Amer Iqbal, Sandor Kovacs, Paul Smith, Rekha Thomas, James Zhang

**Postdoctoral Fellows**

Postdoctoral fellows associated with the algebraic geometry group include Jacob Shapiro (PIMS-UBC), Anca Mustata (UBC), Andrei Mustata (UBC).

Postdoctoral fellows associated with the cohomology/representation theory group include Jochen Kuttler (PIMS-UBC) and Kevin Purbhoo (UBC).

Two CRG sponsored PIMS postdoctoral fellows joined the group in the fall of 2005: Hsian-Hua Tseng (UBC) and Iulia Pop (U. Alberta). Shuang Cai (U. Alberta) has joined the CRG in the fall of 2006.

**Visitors**

J. Carlson (U. Georgia)

P. Gille (CNRS, Universite Paris-Sud)

D. Harari (ENS Paris)

Canon Leung (University of Science and Technology, Hong Kong)

D. Maulik (Princeton)

Jan Minac (U.Western Ontario)

M. Roth (Queens)

S. Smith (U. Illinois at Chicago)

G. Soifer (Bar - Ilan University, Ramat Gan, Israel), March 2005: The Auslender Conjecture: history, results and open problems.

O. Mathieu (University of Lyon I, France), May 2005: On the homotopy of geometric quotients.

M. S. Raghunathan (Tata Institute of Fundamental Research, Mumbai, India), May 2005: Imbedding quasi-split groups of equal rank in isotropic groups.

I. Panin (Steklov Institute, S. Pitersburg, Russsia), September 2005: A purity theorem for linear algebraic groups.

Yongbin Ruan (Wisconsin-Madison), August 2-5 2005, UBC

Steven Mitchell, Feb 22, 2006

Ching-Li Chai, March 7-10, 2006

K. Zainoulline (Bielefeld University, Germany), March 2006: Motivic decompositions of projective homogeneous varieties.

A. Vishik (Independent University, Moskow, Russia), September 2006

Jesper Grodal, Nov. 27 to Dec. 16, 2006

### PIMS CRG on Inverse Problems: 2005-2007

**Overview**

Inverse Problems (IP) are problems where causes for a desired or observed effect are to be determined. An important example is to determine the density distribution inside a body from measuring the attenuation of X-rays sent through this body, the problem of "X-ray tomography". The mathematical problem was studied first by Radon in 1917. Much later, pioneering work by Hounsfield and Cormack led to the first working X-ray tomography machines and later to CAT scans and was honored with the Nobel Prize for Medicine in 1979. This development revolutionized the practice of medicine. Other more recent medical imaging techniques are MRI where the effect of a strong magnetic field on the body is measured, ultrasound where sound waves are sent through the body their reflections measures and Electrical Impedance Tomography where electrical measurements are made on the boundary of the body to name just a few. Earth sciences continue to be a generator of many compelling inverse problems. All of our knowledge of the Earth's interior is indirectly derived from surface measurements, as is a great deal of what we know about the surface and the atmosphere.

Reflection seismology in oil exploration is a well-known and economically important inverse problem. Here sound waves are generated at the surface of the Earth. By looking at the reflection of these waves one would like to determine the location and character of oil deposits. From an economic perspective, seismic imaging is by far the dominant geophysical inversion technique. Seismic imaging creates images of the Earth's upper crust using seismic waves generated by artificial sources and recorded into extensive arrays of sensors (geophones or hydrophones). The technology is based on a complex, and rapidly evolving, mathematical theory that employs advanced solutions to a wave equation as tools to solve approximately the general seismic inverse problem. In the year 2000, nearly $4-billion was spent worldwide on seismic imaging. The heterogeneity and anisotropy of the Earth's upper crust require advanced mathematics to generate wave-equation solutions suitable for seismic imaging.

Other inverse problems arise in non-destructive evaluation of materials. The structural changes due to cracks or flaws are used to identify the locations of those defects. Radar and sonar are based on inverse scattering methods. Mathematics plays a crucial role in the understanding and modeling of the inverse problem as well as in finding reconstruction algorithms. Bring the last twenty years or so there have been remarkable developments in the mathematical theory of inverse problems. These developments together with the enormous increase in computing power and new powerful numerical methods have made it possible to make significant progress on increasingly more realistic and difficult inverse problems.

Many of the physical situations indicated above are modeled by partial differential equations. The inverse problem is to determine the coefficients of the partial differential equation inside the medium from some knowledge of the solutions, usually on the boundary. Already the interaction between experts in partial differential equations and on inverse problems has produced significant advances.

**Postdoctoral Fellows**

UC: Hugh Geiger, 2005

UW: Kim Knudsen, August 2004-June 2005 Mikko Salo, February-July, 2005 Horst Heck, August 2005-February 2006 Xiaosheng Li (PIMS PDF), September 2005-present.

**Faculty**

CRG leaders: Gary Margrave (U. Calgary), Gunther Uhlmann (U. Washington)

UBC: Joel Feldman, Richard Froese, Nassif Ghoussoub.

U. Calgary: Paul Binding, Peter Gibson, Gary Margrave, Michael Lamoureux, Peter Lancaster, Larry Lines, Jedrzej Sniatycki and Tony Ware.

U. Washington: Ken Bube, Edward Curtis, James Morrow, John Sylvester, Gunther Uhlmann.

**PIMS Distinguished Chairs and Visitors**

Visitors at UBC

William Symes, Rice University, July 2005.

Visitors at UC

Lou Fishman, MDF International, September

2005.

Visitors at UW

PIMS Distinguished Lecturer at UW, David Isaacson (Rensselaer

Polytechnical Institute), visited UW and UBC in June 2006 and gave a series of lectures on "Recent Developments in Electrical Impedance Tomography."

April 1, 2005-December 31, 2005

Nurlan Dairbekov, Sobolev Institute of Mathematics, Russia, April 1, 2005-May 1, 2005.

Peter Markowich, University of Vienna and Wolfgang Pauli Institute, May 2-3, 2005.

Soenke Hansen, University of Paderborn, Germany, May 28-June 11, 2005.

Margaret Cheney, RPI, May 31-June 1, 2005.

Plamen Stefanov, Purdue University, July 24-August 6, 2005.

William Symes, Rice University, July 24-August 5, 2005.

Guillaume Bal, Columbia University, July 31-August 31, 2005.

Raluca Felea, Rochester Institute of Technology, July 31- August 27, 2005

Malabika Pramanik, Caltech, July 31-August 31, 2005.

Karthik Ramaseshan, University of Rochester, July 31-August 31, 2005.

Maarten de Hoop, August 8-11, 2005

Jenn-Nan Wang, National Taiwan University, Taiwan, August 31, 2005-December 31, 2005.

David Dos Santos Ferreira, Universite de Paris Nord, France, September 12-16, 2005.

Vladimir Sharafutdinov, Sobolev Institute of Mathematics, Russia, Vladimir Sharafutdinov, September 20, 2005-February 20, 2006.

Gen Nakamura, Hokkaido University, Japan, September 26-October 3, 2005

Christiaan Stolk, University of Twente, The Netherlands, October 23-30, 2005.

George Hagedorn, Virginia Tech, October 27-30, 2005.

Frederic Noo, University of Utah, November 17-18, 2005.

Gustavo Ponce, University of California, Santa Barbara, December 1-4, 2005.

Jenn-Nan Wang, National Taiwan University, Taiwan, January 1, 2006-August 24, 2006.

Valery Serov, University of Oulu, Finland, Jan 1, 2006-June 30, 2006.

Leonid Pestov, Ugra Institute of Information Technologies, Russia, January 7-February 7, 2006.

Ivar Ekeland, UBC, February 14, 2006.

Nassif Ghoussoub, UBC, April 11-12, 2006.

Yury Shestopalov, Karlstad University, Sweden, April 17-19, 2006.

Vladimir Sharafutdinov, Sobolev Institute of Mathematics, Russia, September 20, 2005 February 20, 2006.

Dmitri Burago, Pennsylvania State University, February 25-March 1, 2006.

David Finch, Oregon State University, March 5-7, 2006

Maarten de Hoop, Purdue University, March 12-15, 2006

Lassi Paivarinta, University of Helsinki, Finland, March 28-April 14, 2006

Axel Osses, CMM, Chile, April 17-May 5, 2006

Andras Vasy, Stanford University, May 7-May 10, 2006

Vladislav Kravchenko, National Polytechnical Institute, Mexico, May 7-19, 2006

Plamen Stefanov, Purdue University, June 5-15, 2006

Maarten de Hoop, Purdue University, August 13-15, 2006

Gen Nakamura, Hokkaido University, Japan, August 14-18, 2006

Leonid Pestov, Ugra Research Institute of Information Technologies, Russia, September 1-30, 2006.

Allan Greenleaf, University of Rochester, September 9-12, 2006

Guillaume Bal, Columbia University, October 22-23, 2006.

Arnold Kim, University of California, Merced, October 30, 2006.

Liliana Borcea, Rice University, November 4-7, 2006.

### PIMS CRG on Quantum Topology: 2005-2007

**Overview**

The problems of interest in this CRG are (i) the so-called "many-body problem" in non-relativistic physics, particularly on lattices in low spatial dimension; and (ii) the problem of finding a universal quantum computer which evades decoherence. Phrased this way, these problems seem almost parochial. However we now know that they are in many ways equivalent, and that moreover they are closely related to important problems in theoretical computation, graph theory, in topology, in black hole physics and string theory, and in non-commutative geometry. There is also a strong relation to problems in number theory.

The main purpose of this CRG is to bring together a group of mathematicians and physicists whose interests are united by the 2 problems stated above. Our aim is to resolve some critical issues, which are issues in both mathematics and physics. The work we plan will focus around the following projects:

1-dimensional Problems

Renormalisatoin Group

Topological Methods

Numerical Methods

New Field Theories

Quantum Environments and Decoherence

Spin Nets of Qubits

**PIMS Distinguished Chair**

2006/07

In October, 2006, Prof Alexei Kitaev will be visiting PITP and PIMS; he is also holding a 'Distinguished PITP/PIMS visiting Professorship'. During his time here he will be giving a short lecture course, which should be of interest to those in theoretical physics (particularly condensed matter physics and quantum information), in mathematics (particularly topology and related areas), and in computer science. Details of the course, and also some biographical details about Kitaev appear below:

Kitaeve was originally trained as a theoretical physicist (he did his PhD in the Landau institute with V Pokrovsky). However he branched out after that into both mathematics and computer science. Upon arrival in the USA he worked as a postdoc both in Microsoft and in Caltech, and then in a very unusual move, he moved from a postdoc to a full Professorship at Caltech in 2002, with positions both in Theoretical Physics and Computer Science. He now also has a position at Santa Barbara, in the new centre for quantum information there.

Kitaev is well known for fundamental work both in physics and mathematics. Perhaps his best known recent work, published in 2003, was the invention of the topological quantum computer, in a paper which was already cited over 500 times before even being published.

Alexei Kitaev has an office in the PITP visitor centre.

**Faculty**

CRG Leaders: Philip Stamp (UBC), Boris L. Spivak (U. Washington), and Joel Feldman (UBC)

UBC: Ian K. Affleck, Mona Berciu, Joel Feldman, George A. Sawatzky, Philip Stamp

U. Alberta: Frank Marsiglio

U. Calgary: Richard E. Cleve, John Watrous

SFU: Igor Herbut

U. Washington: Boris L. Spivak

Others:

M. Freedman (Microsoft Research), A. Kitaev (Caltech), C. Bourbonnais (Sherbrooke), D. Senechal (Sherbrooke), A. M. Tremblay (Sherbrooke), R. Gill (Utrecht), R.B. Laughlin (Stanford), A.J. Leggett (Urbana), S. Popescu (Bristol, UC Berkeley), P.B. Wiegmann (U Chicago), S.C. Zhang (Stanford), C. Nayak (UCLA)

**Postdoctoral Fellows**

Year 2005-07

Andrew Hine