SCIENTIFIC ACTIVITIES - Fields Institute - University of Toronto

OCTOBER 2010 | VOLUME 11:1
FIELDS
NOTES
SCIENTIFIC
ACTIVITIES
Workshop on Modelling,
Understanding, and Managing
River Ecosystems
Affine Schubert Calculus
Summer School and Workshop
First Montreal Spring School in
Graph Theory
Fields-Carleton Finite Fields
Workshop
Workshop on Random Matrix
Techniques in Quantum
Information Theory
Brain Neuromechanics
LECTURE SERIES
Jeremy Quastel on Directed
Polymers and Random Growth
Jianqing Fan on Vastdimensionality
and Sparsity
FIELDS-MITACS
SUMMER
UNDERGRADUATE
RESEARCH PROGRAM

In Memory of Jerrold E. Marsden
We are deeply saddened by the death of Jerry Marsden, September 21 at his home in Pasadena,
after a battle with cancer. Jerry was a founder and true friend of the Fields Institute. He served as
our first Director from 1992 to 1994, and organized several of our major programs over the years.
Our prestigious Marsden Postdoctoral Fellowship is named in his honour.
Jerry Marsden was a friend and mentor of many people in the Canadian mathematics community
and around the world. His ideas and inspiration will live on in his mathematical works and those of
his students and colleagues. A press release about Jerry and information about making donations in
his memory can be found at www.cds.caltech.edu/~marsden/remembrances
Modelling, Understanding, and
Managing River Ecosystems
RIVERS AND THEIR ASSOCIATED ECOSYSTEMS
are in danger from human use and alterations, despite
their essential contribution to human society—namely
freshwater and food supply, hydroelectric energy,
transportation, and recreation. Effective and efficient
management of these systems requires deep understanding
and sophisticated models. Complex hydrological models
that feature little or no biological mechanisms have long
dominated river management. Recently, the biological
dynamics of river ecosystems under simplified hydrological
and geomorphological assumptions have become a focus
of attention in spatial ecology modelling. The workshop
on Modelling, Understanding, and Managing River Ecosystems
brought together, for the first time, experts and young
researchers from different areas—hydrologists, engineers,
ecologists, mathematicians, and managers—and provided
an opportunity for participants to discuss recent advances
in all related fields as well as future integration of different
approaches.
The theme of the first morning session was Biology
and Data. Donald Baird (Canadian River Institute) spoke
about available and non-available Canadian river data in his
talk, Down by data: Data poverty in Canadian river ecosystem
research, and how and why we can enrich it. The second invited
speaker, Les Stanfield (Ministry of Natural Resources,
Canada), spoke about Challenges and opportunities for
quantifying the cumulative effects to stream conditions. Stanfield
focused particularly on fish habitat and the impact of new
2 FIELDSNOTES | FIELDS INSTITUTE Research in Mathematical Sciences
housing developments. The afternoon topic, Transport
and Flow, concentrated more on the geophysical aspects of
rivers. Peter Steffler (Alberta) presented elaborate software
for 2D computation and visualization of realistic flows and
biological mechanisms in his talk entitled Computational
modelling of stream hydrodynamics and ecological processes. Rob
Runkel (U.S. Geological Survey) presented his work in
Characterizing metal transport using OTEQ, an equilibriumbased
model for streams and rivers.
The second day morning session was devoted to the
topic of Population Dynamics. It began with a presentation
by Frank Hilker (Bath), Predator-prey systems in streams and
rivers. The afternoon focused on Synthesis and Policy.
Roger Nisbet (UCSB) gave a presentation on Population
response length: theory and applications; Shannon O’Connor
(Montreal) talked on NSERC’s new HydroNet, a national
research network to promote sustainable hydropower and
healthy aquatic ecosystems, and Ed McCauley (NCEAS)
closed the workshop with his outlook and future research
directions, entitled Big problems—Productive solutions: Insight
from a broad range of modelling approaches.
The workshop schedule allowed for many informal
discussions, over lunch and during the evening reception,
that participants enjoyed as much as the lectures. Students
and postdoctoral fellows were excited to meet some of the
leading researchers in the field, who in turn were inspired
by the younger generation and their enthusiasm.
Frithjof Lutscher (Ottawa)

This summer, 20 undergraduate students spent eight weeks at the
Fields Institute carrying out research on applied mathematics projects.
Topics ranged from placenta growth and dysfunction detection methods
to high dimensional combinatorial games.
FIELDS-MITACS
UNDERGRADUATE SUMMER
RESEARCH PROGRAM
Group supervisors came from the principle sponsoring universities and
affiliates of the Fields Institute. The students found the program
to be a unique experience, and many have developed new research
interests as a result.
Compiled by Richard Cerezo
FIELDS INSTITUTE Research in Mathematical Sciences | FIELDSNOTES 3

Thin Films Equations Group
4 FIELDSNOTES | FIELDS INSTITUTE Research in Mathematical Sciences
Daniel Badali (Toronto), Alexandra Kulyk (National Technical University at Kharkiv
Polytechnical Institute), Steven Pollock (McGill); Supervisors: Marina Chugunova
(Toronto), Dmitry Pelinovsky (McMaster)
We wanted to look at the stability of the thin-film PDE
More specifically, we wanted to investigate what was going on with the steady-state
solutions to the PDE. In other words, we were dealing with the equation
The first week was spent getting us up to speed on a boat-load of linear algebra, ODEs, and Numerical Methods. For the first
few weeks, a lot of the time was spent building the appropriate software to search for solutions to the above ODE, given varied
, q and the mass, M, of the system.
We had some challenges building plots for q vs. M for various , since the curve was neither a legitimate function in q or M
We needed to build software that could intelligently overcome the multiple turning points and loops in the q − M plane. The
main goal behind this software suite was to automate the generation of these curves as we let tend to 0. We quickly learned
−
that no matter how intelligent our turning-point algorithm was, ≤10
4 created a graph which was too difficult for our software.
−
Thus, we could build 3D plots of our q vs. M vs. for all with automation, but for ≤10
4 ht
− h h x h hx hxxx
x
, we ended up having to
guide the code by hand. Playing with the code and building these graphs dominated the first four weeks of our project.
At this point, the goal shifted from qualitatively investigating our ODE to performing bifurcation analysis. We realized
we had some serious bifurcation as became smaller, since our qualitative analysis would return more intricately knotted loops
the smaller we pressed . Since our ODE was non-linear, we spent a lot of time trying to gain some insight into the bifurcation
process through linearization, and projection into “truncated Fourier spaces.” That is, we assumed our ODE’s solutions had the
form h( x) = a0 + a1 cos( x) + a2 cos( 2x) + b1 sin ( x) + b2 sin ( 2x)
, and tried to see if we could recreate any form of bifurcation in this new,
smaller, and more easily understandable space.
This is where our project came to an end. We’re sitting on some qualitative information about h, when projected into this
Fourier space, but we’re not sure if we can see any bifurcation in this space, or in the space spanned by { 1,cos ( x) ,sin ( x)
}.
—Steven Pollock
∂ ⎛
⎞
⎜ − ( ) + ( + ) ⎟
∂ ⎝
⎠
=
1 3 1 3
cos
0 .
3 3
1 3 1 3
h − h cos( x) + h
( hx + hxxx ) = q .
3 3
.
−4 −2
∈ ⎡⎣ 10 , 10 ⎤⎦ Combinatorial Games Group #1
Qiu Hua Tian (Toronto), Hera Yu (Toronto); Supervisors: Peter Danziger
(Ryerson), Eric Mendelsohn (Toronto), Brett Stevens (Carleton)
During the program, I worked on problems concerning combinatorial game theory under
the supervision of Peter Danziger (Ryerson), Eric Mendelsohn (Toronto), and Brett
Stevens (Carleton). While Brett guided us through the first week to give us some insight
on the theory of combinatorics, Eric and Peter provided us with frequent and weekly help
on technical questions and gave us great emotional support during the entire program.
My partner Yu He, an exchange student from Nanjing University, and I worked on
proving that the tick-tack-toe game on an affine plane of order 5 always ends in second
player draw. This problem has been previously tackled by stepwise analysis, which is extremely long and therefore impractical.
We tried to find a simpler proof. We took two different approaches, one from proving the existence of a blocking set and one
from finding an algorithm that suited the game. However, we were unable to find a valid proof through these approaches.
By definition, an affine plane of order 5 has 30 winning lines from 5 parallel classes. X is the first player and O the second.
We first tried to prove that it is possible to form a blocking configuration that contains a line with 4 O’s on it, and the
minimal blocking set of each configuration always consists of 9 O’s. Once the blocking configuration is formed, X cannot win.
Nonetheless, we were not able to prove that the blocking configuration will always form before X wins. We spent the second
month trying to find an algorithm for the game. The basic idea that guided our research was the weight function that assigns a
value to each possible winning line. In the weight function defined by Erdős and Selfridge, the base of the weight function was
defined to be 2. A weight function with base 2 proves that tick-tack-toe on an affine plane of order 6 or higher is second player
draw, but it draws no conclusion on an affine plane of order 5. We first tried to modify the base, but it turns out that base 2 is
the only valid base. We then tried to prove that the best move for X would be to follow the weight function step by step. Even
so, he cannot win. We were unable to finish this proof because it is hard to predict the behaviour of the whole game.
—Qiu Hua Tian

Combinatorial Games Group #2
Matthew Patrick Conlen
(Michigan), Juraj Milcak
(Toronto); Supervisor: Brett
Stevens (Carleton)
During the Fields-MITACS
Undergraduate Research
program, I was engaged in
research in combinatorial game
theory led by Brett Stevens
(Carleton). In the first week of the program we were lectured daily in
this field. From then on we had meetings with our supervisors, who
guided us throughout our research. Brett gave us a list of about 20
unsolved problems, all of which were accessible to undergraduates.
From this long list, my colleague Matthew Conlen, from the
University of Michigan, and I chose a problem concerning a winning
strategy for the first player for a game of tick-tack-toe played on an
affine plane of order four. We found no previous results regarding
this problem. We only used properties of the affine plane and Latin
squares, which describe the winning lines of the game board. A 4x4
Latin square contains entries from / 4 ,
where equal numbers
represent a line in the affine plane. We also considered two other
non-Latin squares that represent the trivial horizontal and vertical
winning lines. Altogether, this gave us a representation of the 5
parallel classes of our geometry. We used mappings Ψi that map a
point of the game board to / 4 .
We can use Ψi ( A)
to determine
which line the point A belongs to in the i th parallel class. We then
considered sets of four points that have no three points on a line.
We proved that creating such sets always allows the first player to
win. We defined p to be a set as above also satisfying the following
property: there exist two parallel classes i and j such that
We found that p is the set the first player wishes to make the
most often, depending on some other particulars. After proving
that the first player can always create such a set with the first four
moves, we carefully examined all possibilities, classified them up to
isomorphisms, and proved that the first player can win by extending
the lines generated by the set. —Juraj Milcak
Pattern Avoiding Group
∪ Ψi ( P) = ∪ Ψ j ( P)
= ℤ / 4ℤ
P∈
p P∈
p
Placenta Modelling Group
Gaole Chen
(Rochester),
Liudmyla
Kadets (Kharkiv
National
V.N. Karazin
University),
Johnathan
Wagner (Hebrew
University of
Jerusalem), Zheng Wang (Toronto); Supervisor:
Michael Yampolsky (Toronto)
Established on the assumption that evolution seeks
to achieve energy minimization, Murray’s law relates
the radii of a parent vessel to that of the daughter
vessels in a biological piping system. Over the years,
new mathematical models for blood flow have been
created to explain Murray’s law. Large numbers of
experiments have been conducted to verify this law
in nature. Our objective was to determine if Murray’s
law is universal by reviewing current literature with
experimental data, summarizing their findings and
determining the validity of their results. In addition,
we presented and discussed some of the prominent
mathematical models that incorporate the physics of
pulsatile flow and non-Newtonian fluids.
We also studied the human placenta to verify
Murray’s law in this organ. We studied pictures of
the human placenta in which the blood vessels had
been traced professionally. We wrote a program in
C which processed these pictures and returned data
such as vessel dimensions and the type of branching
point. Finally, assuming Murray’s law is correct, we
constructed a “Murray’s Grade,” which is used to
rank different branching points in a vascular system
based on their deviation from optimality. This will
enable us to investigate how deviation from Murray’s
law affects the performance of organs such as the
placenta in future studies. —Liudmyla Kadets
Chris Berg (York), Samer Doughan (Toronto), Steven Karp (Waterloo); Supervisor:
Mike Zabrocki (York)
We started with the idea from combinatorics of enumerating permutations which avoid a
pattern. A permutation ‘contains’ a pattern if there is a subsequence of the permutation which
has the same relative order as the pattern and ‘avoids’ it otherwise. For example, 541632
contains the pattern 231 because, for instance, 462 is a subsequence of the permutation and
has the same relative order as 231. However, this same permutation 541632 avoids the pattern
123.
With this idea in mind, we considered a different notion of pattern avoidance when the permutation is thought of as a
minimal length word written in terms of the Coxeter group generators (elements s i which exchanges i and i +1). Every permutation
can be written (non-uniquely) as a minimal length word in the generators s i . We then considered the problem of enumerating
permutations whose reduced words did not contain subsequences that had a relative order that matched a pattern. Along the way
we found relationships with the Bruhat order, juggling sequences, and Hopf algebras. —Mike Zabrocki
FIELDS INSTITUTE Research in Mathematical Sciences | FIELDSNOTES 5

Infectious Disease Modelling Group
Cameron Davidson-Pilon
(Wilfrid Laurier), Preeyantee
Ghosh (Hyderabad), Yueh-
Ning Lee (National Taiwan);
Supervisor: Jianhong Wu
(York)
After a disease outbreak,
we can plot a curve for the
cumulative number of infected individuals and fit it to a curve.
Often, and with enough frequency to raise curiosity, the outbreak
will follow the Richard’s model, which is the solution to the
differential equation
a
⎛ ⎛ C
′(
( t)
⎞ ⎞
C t) = rC ( t)
⎜1−
⎜ ⎟ ⎟
⎜ ⎝ K ⎠ ⎟
⎝ ⎠
where C ( t)
is the cumulative number of infectives at time t.
The model is very similar to the logistic model but possesses the
parameter a, an index of the inhibitory effect of the deviation of
growth from the exponential relationship.
The problem presented to us was to mathematically derive
the Richard’s model from first principles. In particular, we were to
give epidemiological reasons for the characteristic growth of the
accumulated infected. We started by reading the current literature
on disease models. The most popular model is the deterministic,
compartmental SIR model or a derivative of it. The most
important feature of the standard SIR model is it gives relations
between the number of infected and susceptibles and their rate of
change in the form of differential equations. We started our task
by abandoning this assumption and left the rate as an unknown
and then assumed a priori that the cumulative total infected,
followed the Richard’s model. Using a very creative idea, Yueh-
Ning derived a closed form expression for the incidence rate. An
alternate novel approach to the problem, derived by Cameron, was
to look at an outbreak as a branching process. Using a dynamic
expected number of secondary infections per infection, Cameron
could derive the expected number of accumulated total infected in
terms of the mean duration of infection and a certain integration
of the expected number of infectives up to the time of interest.
Assuming the accumulated total follows the Richard’s model, we
could numerically examine how the disease spread.
Both methods show a defining characteristic of a disease
following the Richard’s model. Since the Richard’s model often fits
real life outbreaks, we decude with confidence that the outbreak
will have a constant number of secondary infections, but then
quite rapidly decrease to almost zero.
Preeyantee’s work focused on the appearance of multiple
strains or species of diseases in a single host, called a superinfection.
She noticed that during treatment, it is possible to
mistake resistance with super-infection, where the non-vanishing
symptoms are caused by the other diseases and not the treated
one. This idea led to a system of differential equations examining
how resistance may evolve under such circumstances. This idea
arose from discussions with several participants of the 2010
Summer Thematic Program on Mathematics for Drug Resistance
of Infectious Disease that took place during our summer research
program. We are trying to put everything together and submit it
for possible publication.
—Cameron Davidson-Pilon
6 FIELDSNOTES | FIELDS INSTITUTE Research in Mathematical Sciences
Cancer Stem Cell Modelling Group
Abhishek
Deshpande (IIIT
Hyerabad), Joy
Jing Liu (Ottawa),
Philip Marx
(Tulane), Tian An
Wong (Vassar);
Supervisors:
Matthew Scott
(Waterloo) Mohammad Kohandel (Waterloo),
Sivabal Sivaloganathan (Waterloo)
On the first day, we met with our advisers, who
introduced us to two potential paths for research—
modelling of hydrocephalus and stochastic
simulations of brain tumour growth. As both were
novel concepts to all of the group members, we
spent the first few days reading papers to improve
our understanding and decide which topic to pursue.
After choosing the brain tumour problem, we read
recent biology papers in the field, with the goal of
formulating simple yet descriptive mathematical
models, then testing them using MATLAB and C++.
Testing included setting parameters to fit the
given data and, in the case that conclusions in
the biology papers were inexplicable through the
mathematical models, the addition and testing of
additional assumptions about cell hierarchies and
dynamics.
We ended up focusing on a specific paper,
A Hierarchy of Self-Renewing Tumor-Initiating
Cell Types in Glioblastoma by Chen et. al, which
provided its own hypothesis on the still poorly
understood tumour cell hierarchy. We focused
on mathematically replicating the paper’s data
on neurosphere formation (through stochastic
simulations) and CD133 percentages (using the
system of deterministic equations). In order to
explain the high CD133 percentages reported in the
paper, we then inserted an element of dynamic dedifferentiation,
in which progenitor cells are able to
“de-differentiate” to cells with stem-like properties,
for example through the process of epithelialmesenchymal
transition (EMT). In a field where
the cell hierarchy is still not well understood, this
could yield interesting insight on current models and
provide ideas for future biological experiments.
The dynamic evolution of the population can
be described deterministically or stochastically.
Stochastic models are appropriate for small cell
populations during earlier stages of tumour growth.
In later periods, when the cell number increases,
stochastic fluctuations are minimized allowing a
deterministic description of the dynamics. The
great advantage of a deterministic framework is its
analytic tractability allowing a larger perspective of
the population growth and a facilitated estimate of
parameter-sensitivity in the model.
—Philip Marx and Tian An Wong

First Montreal Spring School
in Graph Theory
MUCH OF MATHEMATICS IS DRIVEN BY
conjectures, and this is particularly true of graph theory. Two
of the great conjectures, which drove, stretched, and teased a
generation of graph theorists, were Wagner’s Conjecture and
The Strong Perfect Graph Conjecture (due to Berge). In the past
decade both have been proved, the former by Robertson and
Seymour and the latter by Chudnovsky, Robertson, Seymour and
Thomas. Both proofs (and this is especially true of the former)
represent the culmination of a grand project of research that
has built up a whole structural theory surrounding the required
result.
For a new generation of graph
theorists, these results may be taken
as given. However, that does not mean
their proofs should be ignored. The
wealth of knowledge and techniques
built up in proving these results is a
bounty that the new generation is
lucky to inherit. The First Montreal
Spring School in Graph Theory was an
opportunity for young researchers
from Canada and around the world
to learn of this bounty from
three top academics in the
field. In all we had over 50
participants at the school,
coming from 15 countries.
In 1937, Wagner
proved that a graph G
is planar if and only if it
contains neither K5 nor
K as a minor. He then
3, 3
considered the more abstract
problem: does there exist,
for each surface Σ, a similar characterization (a finite list of
excluded minors) for graphs embeddable in Σ? Wagner noticed
that to ensure the existence of such characterizations, it suffices
to prove that in every infinite sequence of graphs there is one
that is a minor of another. It is this latter statement that became
known as Wagner’s Conjecture. Wagner’s Conjecture was
proved by Robertson and Seymour as part of their grand project
on graph minors. Results from this entire project were covered
in the lecture course Structural results obtained from excluding
graph minors given by Bruce Reed. Bruce is currently in the final
stages of completing a book on the graph minors project. The
key structural theorem of the course states (approximately) that
all graphs without a fixed graph H as a minor can be obtained
by gluing together (in an appropriate way) certain ‘topologically
simple’ graphs. At the end of the course we turned to Wagner’s
Conjecture. It is by no means trivial to deduce Wagner’s
Conjecture from the structure theorem; however, the structure
theorem does allow us to get a grip on the problem, which is
essential to its resolution.
One colours a graph by assigning a colour to each vertex in
such a way that no two adjacent vertices receive the same colour.
If there is a set of ω vertices which are all mutually adjacent (a
clique), then it is clear that at least ω colours will be needed to
colour the graph. It is a somewhat interesting property of a graph
if this number of colours suffices (i.e. if the number of colours
needed is equal to the size of the largest clique). It is much more
interesting still if this property
holds not only for G, but also for
all induced subgraphs (graphs that
can be obtained from G by deleting
vertices). If this is the case then
we say G is perfect. It is easy to
find graphs that are not perfect.
For example, odd cycles of length
at least five are imperfect, and so
are their complementary graphs
(i.e. those obtained by switching
edges and non-edges). Thus, for a
graph to be perfect it is certainly
necessary that it contains neither
an odd cycle of length at least five,
Clockwise from
top left: Maria
Chudnovsky; Paul
Seymour; workshop
participants.
(Photos by Juraj
Stacho)
nor the complement of such a
cycle, as an induced subgraph. The
Strong Perfect Graph Conjecture,
due to Berge (1960), states that this
trivial necessary condition is also
sufficient. The other course in the
school, Structural results obtained by
excluding induced subgraphs, included
a proof of the Strong Perfect Graph Conjecture. The course
was taught jointly by Maria Chudnovsky (Columbia) and Paul
Seymour (Princeton). In Paul’s lectures we were introduced to
a structure theorem for graphs (now known as Berge graphs)
that have neither odd cycles of length at least five nor their
complements as induced subgraphs. The structure theorem
shows that such graphs either belong to one of a few families of
basic graphs or admit one of a few types of decomposition. Since
the families of basic graphs are known to be perfect, and graphs
admitting such decompositions cannot be minimal imperfect
graphs, the proof of the Strong Perfect Graph Conjecture
follows. Maria’s lectures focused on newer results and open
conjectures, including recent advances related to the Erdős-
Hajnal Conjecture (a conjecture that is still open).
Simon Griffiths (McGill)
FIELDS INSTITUTE Research in Mathematical Sciences | FIELDSNOTES 7

Random Matrix Techniques in
Quantum Information Theory
THE FIELDS–PERIMETER
joint workshop on Random
Matrix Techniques in Quantum
Information Theory was held at
the Perimeter Institute from July
4-6, 2010. It was organized jointly
by Benoît Collins (Ottawa), Patrick
Hayden (McGill/Perimeter) and
Ion Nechita (Ottawa).
The aim of this workshop
was to bring together researchers
from the areas of probability
theory and random matrix theory
in mathematics with specialists from quantum information theory.
Over the last decade it was discovered that in order to tackle
important questions in quantum information theory, such as
additivity problems, and probabilistic methods, random matrix
methods could be of crucial help.
A deeper level of interaction between the quantum
information theory community and mathematics had already been
recognized as fundamental, and joint events were organized with
the operator algebra and operator space communities. However,
bringing together mathematicians working in probability theory
and quantum information theorists with an interest in statistical
methods had not yet been accomplished, and we believe that our
workshop quite efficiently filled that gap.
It was important to bring together people from probability
and random matrices with those from quantum information,
as huge breakthroughs have been achieved over the last few
years concerning the additivity of the minimum output entropy,
especially by Matthew B. Hastings (Microsoft), Hayden, and
Andreas Winter (Bristol).
The motivation for the workshop was the recent resolution
of quantum information theory’s best-known conjecture using
random matrix techniques. Updates on further developments
surrounding this additivity conjecture provided some of the
highlights of the workshop. The additivity conjecture was first
stated by Christopher King (Northeastern) and Mary Beth
Ruskai (Tufts), who were both present at the workshop. After
several classes of channels were shown to satisfy the conjecture,
Hayden and Winter showed that a stronger version of it, widely
believed to hold at the time, was false. They used a random
construction and their proof relied on concentration of measure
techniques, developed earlier in joint work with Debbie Leung
(Waterloo), also a workshop participant. The counterexample
for the original conjecture was constructed by Hastings in 2009,
his proof also relying on random matrix techniques. Talks on this
subject occupied a whole day of the schedule. King and Motohisa
Fukuda (UC Davis) gave an introduction to the conjecture and
8 FIELDSNOTES | FIELDS INSTITUTE Research in Mathematical Sciences
Hastings’ proof. Fernando Brandao
(UFMG) presented an alternative
approach to the problem, using
concentration of measure
techniques. Stanislaw Szarek (Paris
6) spoke about very recent joint
work of his, Guillaume Aubrun’s
(Camille Jordan) and Elisabeth
Werner’s (Case Western Reserve)
on another proof of the Hastings
result using Dvoretzky’s theorem.
Collins introduced free probability
techniques, helpful in studying
random quantum channels that can be used to give precise
results on the minimal output entropies. Finally, the Additivity
Problem Day was concluded by Aram Harrow’s (Bristol) talk on
the computational complexity of approximating entropies of
channels. This session was emblematic of the workshop, with
mathematicians and quantum theorists alternating on the podium,
presenting their research to a large audience.
The workshop was attended by over 40 participants, including
more than a dozen students. Twenty lectures were delivered.
As the audience members’ backgrounds were extremely diverse,
every speaker split their talk into two parts. The first half had to
be completely accessible to the other community, and the second
addressed research questions relevant to the conference. For the
mathematicians, this was a unique chance to learn firsthand about
the quantum information techniques and important problems. For
the quantum information community, it was a unique opportunity
to learn about recent and more classical techniques in random
matrix theory.
Time was set aside to allow for discussions between the
participants. In particular, there was a problem session that gave
rise to many new and interesting questions, providing material
for future research work. Audience members participated
enthusiastically in these sessions, offering problems, suggestions
and even making a start on some solutions.
This collaboration between the Fields Institute and the
Perimeter Institute enjoyed national media coverage when
Canadian Prime Minister Stephen Harper unexpectedly invited
himself to our conference. He took this occasion to greet Stephen
Hawking (who was visiting the PI at the time) and to make an
important announcement about the funding of postdoctoral
fellowships in Canada, as well as to share Ontario wine with the
participants.
The workshop was very timely, and the organizers hope it
will prove to be a first milestone on the road towards a fruitful and
intensive collaboration between the two communities.
Benoît Collins (Ottawa)

Extended Workshop on Groups and Group
Actions in Operator Algebra Theory
THE WORKSHOP TOOK PLACE AT THE UNIVERSITY
of Ottawa from July 12–16, 2010. It was the first in a series of joint
events between the mathematics departments of the Universidade
Federal de Santa Catarina, Florianópolis, Brazil, and the University
of Ottawa, and was the result of a recent agreement between the
two universities, initiated by the Mathematics departments. The
workshop was open to everybody, and the majority of speakers and
participants came from outside of the two signatory Universities.
Operator algebras originated in the work of John von
Neumann (in particular in his search for a natural mathematical
framework for quantum mechanics), Isreal Gelfand and Mark
Naǐmark. Von Neumann algebras incorporate the noncommutative
versions of measure theory, topology and differential geometry.
The theory of operator algebras is undoubtedly one of the domains
in mathematics most notable for the depth of its problems, the
richness of its ideas, its connections to many different fields, and its
great potential as a unifying language and source of illumination.
This area is recognized as being among a few major fields of
research strength of both the Canadian and Brazilian schools of
mathematics. There is an internationally recognized research group
in operator algebras in Canada and a strong subgroup in Ottawa.
The department of mathematics of the Universidade Federal de
Santa Catarina is building a large group of researchers in operator
algebras, led by Ruy Exel (UFSC).
The workshop consisted of two three-hour minicourses, as
well as a number of one-hour and 45-minute invited talks. The first
mini-course was given by Vadim Kaimanovich, who was recently
appointed as a Canada Research Chair Tier I at the University
of Ottawa. Kaimanovich’s course, entitled Markov Chains and
Groupoids, was devoted to various probabilistic models generalizing
random walks on groups (random walks in random environment
and with internal degrees of freedom, along classes of equivalence
NOTED
We note with great regret the death
of Richard Kane of the University of
Western Ontario on October 1.
Richard died of cancer at the age of
66. Richard is well-known for his
work in algebraic topology and is the
author two highly-regarded books,
on the homology of Hopf spaces and
on reflection groups and invariant
theory.
Richard has a remarkable history
of service to the mathematical
community in Canada, for which he
was honoured by the inaugural David
Borwein Distinguished Career Award
and the Distinguished Service Award
of the Canadian Mathematical
relations etc.).
Main attention was paid to the problem of triviality of
the Poisson boundary for invariant Markov operators and its
application to amenability of groupoids.
The second mini-course, entitled Cartan Subalgebras, Fell
Bundles and Twisted Actions of Inverse Semigroups, was given by Exel.
The course focused on the rich interplay between C*-algebras
and dynamical systems, beginning with the seminal work of Jean
Renault (Orléans) on groupoid C* -algebras and the description
of Cartan subalgebras in terms of twisted étale groupoids and the
non-commutative generalization obtained by the speaker using Fell
bundles over inverse semigroups.
Workshop participants.
Invited lectures were given by Alcides Buss (UFSC), Benoît
Collins (Ottawa), George Elliott (Toronto), Ilijas Farah (York),
Daniel Gonçalves (UFSC), David Kerr (Texas A&M), Ion Nechita
(Ottawa), Volodymyr Nekrashevych (Texas A&M), Matthias
Neufang (Carleton/Fields), Ping Wong Ng (Louisiana), Zhuang
Niu (Memorial), Catalin Rada (Ottawa), Renault, and Benjamin
Steinberg (Carleton).
A special afternoon session was dedicated to David
‘Groups’ continued on page 21
Society in 2006. He was a Fellow
of the Royal Society of Canada and
a Fellow of the Fields Institute.
Richard was a loyal and gracious
friend to his colleagues and students.
DONALD A. DAWSON, Professor
Emeritus and Distinguished
Research Professor at the School
of Mathematics and Statistics at
Carleton University and Adjunct
Professor at McGill, has been
inducted into the Royal Society.
MIROSLAV LOVRIC, Professor
of Mathematics and Statistics
at McMaster University, is the
recipient of the 2010 Adrien
Pouliot Award for his significant
contributions to Canadian
mathematics education.
JOHN MIGHTON, founder of JUMP
Math, has been awarded the Order
of Canada in recognition of his work
with JUMP.
CHRISTIANE ROUSSEAU, Professor
of Mathematics and Statistics at
the University of Montreal, has
been elected a Vice President of the
International Mathematical Union
for the 2011-2014 term. She is the
first Canadian to hold this position.
FIELDS INSTITUTE Research in Mathematical Sciences | FIELDSNOTES 9

GAP 2
Conn
CLASSICALLY, IT WAS IMPOSSIBLE
to distinguish between theoretical physics
and pure mathematics. For example,
consider Newton, Lagrange, and Hamilton.
Were they physicists or mathematicians?
In later years, however, as research in both
disciplines became more specialized, the
two sides began to drift apart. Since the
1950s and 1960s there has been a gradual
reconciliation, resulting in spectacular
success both in physics and in mathematics,
particularly in geometry and topology. More
recently, the unexpected phenomenon
of “mirror symmetry” of Calabi-Yau
manifolds was predicted by string theorists,
and in many cases verified rigorously by
mathematicians. Many aspects of the
current research in differential and algebraic
geometry are being fueled by predictions
from theoretical physics, opening new and
exciting rigourous mathematical roads that
may otherwise have been left undiscovered.
In order to take advantage of this
symbiotic relationship between geometry
and physics, on May 7–9, 2010, the
Perimeter Institute hosted the second
Connections in Geometry and Physics (GAP)
conference, with major financial support
from the Fields Institute and the Faculty
of Mathematics of the University of
Waterloo. The organizers of this year’s
conference were Jaume Gomis (Perimeter),
Marco Gualtieri (Toronto), Spiro
Karigiannis (Waterloo), Ruxandra Moraru
(Waterloo), Rob Myers (Perimeter), and
McKenzie Wang (McMaster.) As was
the case during the 2009 conference, this
meeting was an opportunity to increase
the interaction, both locally and globally,
between theoretical physicists and
mathematicians, particularly those working
in differential and algebraic geometry.
The list of principal speakers included
a healthy mixture from all three themes,
10 FIELDSNOTES | FIELDS INSTITUTE Research in Mathematical Sciences
both mathematicians and physicists, and
also consisted of both local area researchers
and international experts. In particular, we
had a significantly increased international
participation this time.
Every year, the organizers of GAP
choose three main themes for the
conference, emphasizing areas where
physics and geometry have enjoyed much
recent mutual benefit. Here is a short
overview of this year’s three themes:
MATHEMATICAL RELATIVITY
Although General Relativity was
formulated over 90 years ago, its impact on
mathematical research has only increased
steadily with time. This is due largely to
the tremendous progress made during the
last four decades in our understanding of
non-linear partial differential equations
and global differential geometry. The
Hawking and Penrose singularity
theorems result from formulating causal
relationships in topological terms and
exploiting the relationship between
curvature and focal points. The proof
of the positive mass theorem by Schoen
and Yau uses 3-manifold theory and the
theory of minimal surfaces, while the
proof by Witten relies on the existence of
asymptotically constant harmonic spinors.
More recent highlights include the proof
by Christodoulou and Klainerman of the
global stability of Minkowski space, the
proof of the Riemannian Penrose inequality
by Huisken-Illmanen and Bray, and the
on-going analysis of black-hole dynamics
and stability by Finster-Kamran-Smoller-
Yau. Here the mathematical tools came
from geometric flows (mean curvature
and inverse mean curvature flows), Fourier
analysis, and the theory of hyperbolic
equations. Intense efforts are currently
devoted to such topics as the unique
continuation of the Einstein equation
(Alexakis, Anderson, Herzlich), stability
of the Kerr solution (Alexakis-Ionescu-
Klainerman), the Einstein constraint
equation (Butscher, Chrusciel, Corvino,
Isenberg, Pacard, Pollack, Schoen),
concepts of quasi-local mass (M. Liu,
M.T. Wang, S. T. Yau), Poincaré-Einstein
manifolds (Anderson, Biquard, Chrusciel,
Chang-Gursky-Qing-Yang, Graham,
Fefferman, J. Lee, Mazzeo), and higherdimensional
black hole geometry (Gibbons-
Hartnoll-Page-Pope, Myers-Perry). The
last two topics are related to the AdS/CFT
(anti-de-Sitter space/conformal field theory)
correspondence in conformal field theory.
Many more exciting research directions
come into focus if one ventures beyond
the Lorentzian setting to the Riemannian
setting: for example, metrics with special
holonomy, Sasakian-Einstein geometry, and
quasi-Einstein metrics.
Mathematical Relativity is an area in
which historically there has been substantial
Canadian representation. Besides current
contributors such as Niky Kamran
(McGill), Richard Mann (Waterloo), Robert
Myers (Perimeter), Don Page (Alberta), one
should note the classical works of W. Israel
and J. L. Synge.
GAUGE THEORY
As the fundamental basis of the
theory of elementary particles, gauge
theory in the guise of Yang-Mills theory
is at the center of theoretical physics.
Since the end of the 1970s, however, it
has been a central topic in mathematics
as well, especially influencing differential
and algebraic geometry as well as lowdimensional
topology. The successes of
gauge theory in uncovering deep structure
in these fields are too numerous to list.
They include the understanding of flat

010
ections in Geometry and Physics
connections on surfaces by Atiyah and
Bott; the breakthroughs of Donaldson
concerning smooth 4-dimensional
manifolds using instantons; various
flavours of knot invariants starting
with Witten’s understanding of the
Jones polynomial; recent progress in
understanding 3-manifolds by Kronheimer,
Mrowka, and Taubes using monopoles;
and finally the recent revitalizing of the
geometric Langlands program by Gukov,
Kapustin, and Witten, using topological
supersymmetric 4-dimensional Yang-Mills
theory. The general pattern in all of these
developments (and many others in gauge
theory) is that gauge theory provides us
with natural moduli spaces which then
yield invariants, which we may associate to
the original objects of study. The resulting
invariants are, in many cases, extremely
deep.
Recently, there has also been a surge
of progress in our understanding of gauge
theory itself; in particular, the work of
Kontsevich and Soibelman, as well as
Nakajima, on stability conditions for gauge
theories, as well as the work of Costello
on the mathematical understanding of
renormalization in 4-dimensional Yang-
Mills theory. Finally there are spectacular
developments using twistor theory to
calculate amplitudes in supersymmetric
Yang-Mills theory.
MIRROR SYMMETRY
Discovered by physicists as a duality
between string theories with spacetimes
associated to different Calabi-Yau
manifolds, mirror symmetry has evolved
into a rich field within mathematics which
involves algebraic geometry, symplectic
geometry, and homological algebra. Mirror
symmetry is essentially a series of surprising
relationships between the complex and
Workshop participants
symplectic geometry of different Calabi-
Yau manifolds, surprising because they
seem to be quite indirect and lack an
obvious geometric explanation. The
relationships are even more remarkable
because they enable the calculation of deep
and difficult combinatorial and enumerative
data that were previously thought to be
inaccessible.
The first mathematical explanation
of mirror symmetry was proposed by
Strominger, Yau, and Zaslow, who outlined
a way of establishing the duality using
special Lagrangian fibrations of Calabi-Yau
manifolds, and involving both the Legendre
and Fourier transforms. This has led to a
very successful program, starting with the
results of Batyrev-Borisov for Calabi-Yau
hypersurfaces in Fano toric varieties, and
culminating with the work of Gross and
Siebert which involves the use of affine
geometry, tropical geometry and the
degeneration of Calabi-Yau manifolds to
establish a construction of mirror manifolds
with the required properties.
Another approach was suggested by
Kontsevich, and is known as homological
mirror symmetry. He proposed that a
large part of the mirror symmetry relations
could be explained as an equivalence of
categories between derived categories of
coherent sheaves (for a complex manifold)
and Fukaya categories (for symplectic
manifolds). The conjectured equivalence
of categories was then established in many
cases by Fukaya and Seidel, and has also led
to the use of tropical geometry in the study
of Floer theory in symplectic geometry.
The homological mirror symmetry
approach is notable for its introduction of
powerful algebraic techniques in symplectic
geometry, which have been used to great
effect in many other fields, including
categorification and differential topology.
We were very fortunate to attract
many excellent world-renowned researchers
to GAP 2010, including Shing-Tung Yau
(Fields Medal 1982, Wolf Prize 2010);
David Morrison (Clay Mathematics
Institute Senior Scholar 2005); and
Nikita Nekrasov (Hermann Weyl Prize
2004). We were also pleased once again
to have heavy participation by local area
graduate students in both mathematics
and physics. At least one third of the
registered participants were students. GAP
2010 featured six short talks by local area
postdoctoral fellows, intending to showcase
these promising young researchers to help
them succeed in the next stage of their
professional careers.
Spiro Karigiannis (Waterloo)
FIELDS INSTITUTE Research in Mathematical Sciences | FIELDSNOTES 11

Fields-Carleton Finite Fields Workshop
THE FIELDS-CARLETON FINITE FIELDS WORKSHOP
is held in order to partially supplement the larger series of
conferences, Finite Fields and their Applications, which occur in
odd-numbered years. It focused on three areas of finite fields
research: pseudo-random sequences, irreducible and primitive
polynomials, and special functions over finite fields. All three
topics have applications to digital communications, including
coding theory and cryptography.
The spirit of the workshop centres around promoting
collaborations between finite fields researchers and fostering
new and innovative ideas in each area of research. The
workshop attracted 35 participants, of which 17 were students
and post-docs from Canada, Iran, Ireland, and Singapore. In
addition to twelve talks by eight invited speakers, there were
eight contributed talks (mostly by graduate students) on current
research.
Invited lectures and mini-courses were given by Stephen
D. Cohen (Glasgow), Theo Garefalakis (Crete), Guang Gong
MATHEMATICIANS WORKING IN THE FIELDS OF
geometry, algebraic combinatorics and mathematical physics met at
the Fields Institute and the University of Toronto campus this July
for a summer school and workshop on the topic of Affine Schubert
Calculus.
The first four days of this event consisted of expository
lectures, and were followed by an additional four days of talks
highlighting recent research in this area. This meeting was the
closing event associated to an NSF Focused Research Group
(FRG) grant where the members’ research was concentrated on
mathematics related to this subject.
Schubert calculus refers to manipulations of subsets of
12 FIELDSNOTES | FIELDS INSTITUTE Research in Mathematical Sciences
(Waterloo), Gary McGuire (UCD Dublin), Gary Mullen (Penn
State), Arne Winterhof (Austrian Academy of Sciences), Qing
Xiang (Delaware) and Joe Yucas (South Illinois Carbondale).
Speakers presented talks and mini-courses which outlined the
current state-of-the art in each of these topics and provided
open problems and new avenues of research in each talk.
Winterhof presented a three-part mini-course on
Research methods for pseudo-random sequences. The methods
presented included the computation of linear complexities
and the evaluation of special exponential sums and measures
of randomness. In addition, Gong introduced some new
constructions of pseudo-random sequences.
Mullen opened the mini-course on Primitive and irreducible
polynomials with a survey of the state-of-the-art. Cohen followed
with an exposition of the main techniques in the area, which
centre around character sums and a new p-adic method.
In addition to the mini-course, Garefalakis presented new
results on self-reciprocal irreducible polynomials given some
prescribed coefficients over finite fields.
Highly nonlinear functions over finite fields are necessary
in the implementation and analysis of modern cryptosystems.
Mullen gave a survey of basic results on permutation
polynomials and value sets of polynomials over finite fields.
McGuire gave an in-depth analysis of various nonlinearity
properties of functions over finite fields, and their relations
with cryptography and coding theory. Xiang examined the
relationship between highly nonlinear functions and special
types of graphs.Yucas presented a generalization of the socalled
Dickson polynomials of the first and second kind to any
positive k kind, and outlined some research avenues for the
classical Dickson polynomials.
Daniel Panario (Carleton)
Affine Schubert Calculus Summer
School and Workshop
Grassmann varieties called Schubert cells. Schur functions
are an algebraic realization of the cohomology classes of the
Grassmannians. The title of the workshop and summer school,
Affine Schubert Calculus, refers to an extension of Schubert calculus
to affine Grassmannians. The algebraic realization corresponding
to Schur functions are the k-Schur functions of Lapointe-Lascoux
and Morse (where the ‘k’ here represents an affine grading).
k-Schur
functions were first discovered because of their relationship
to Macdonald symmetric functions. Later, Thomas Lam showed
that the k-Schur functions were connected to the geometry and
topology of the affine Grassmannian.
‘Schubert Calculus’ continued on page 21

Directed Polymers and Random Growth
JEREMY QUASTEL IS PROFESSOR AND ASSOCIATE CHAIR OF
mathematics at the University of Toronto. He was one of the delegates
representing Canada at the International Congress for Mathematicians in
Hyderabad this year (where he presented his work in Section 13: Probability
and Statistics) and is co-organizer of the Thematic Program on Dynamics and
Transport in Disordered Systems starting January 2011 at the Fields Institute.
His talk at the Fields Institute’s 2010 Annual General Meeting was
titled Directed polymers and random growth, a topic of probability theory and
stochastic differential equations. He spoke of the relationship between the
process of ballistic aggregation and discrete and continuum directed random
polymer models.
What does a random surface look like after a long period of time?
Ballistic aggregation is meant to answer this question, a discrete model in
one-dimension for blocks falling and building up on the 1D integer lattice.
The relevance of this theory to science is that understanding this process will
allow engineers to develop new tools to build materials by spraying atoms onto
a surface. Understanding of this and analogous models are one of the central
themes Quastel would like to see develop in his thematic program. He reports
that there have been a number of significant advances in the study of these
models and will certainly add excitement to the activity at Fields next year.
His lecture was split into two parts, background and recent results. The
directed random polymer models about which Quastel spoke were in the 2D
randomly weighted integer lattice. Each point in the lattice is represented by
W with an imposed random walk X that goes through the lattice collecting
i , j
i
the random values W and sums up the values. Taking the expectation
i , j
(summing all the possible paths and taking the logarithm of the sum) gives
us the free energy on the lattice. Physicists predict that the free energies of
the discrete random polymer model will give valuable information about the
object.
Quastel spoke about some reformulation of the GUE Tracy-Widom
models from random matrix theory in terms of the rescaled distribution of the
principle eigenvalue of a randomly chosen matrix from the GUE-TW. To his
surprise, these results are useful for the continuum directed polymer models.
In the context of the discrete model, Quastel mentioned his interest in
the behaviour emering from the strong coupling between the random walks
and the random lattice. The KPZ (Kardar-Parisi-Zhang) model governs
anything that experiences growth governed by randomness at different
sites, with added non-linearity. These models are extremely general. Quastel
mentioned his interest in having the theoretical tools to make robust
predictions over the KPZ universality class. In models studied in the past, as
the dimension increases, randomness behaviour undergoes a phase transition
1
3
where the n and Tracy-Widom distributions are replaced with normal
Gaussian behaviour and random walks.
The results obtained in work done with Gideon Amir and Ivan Corwin
are contained in a recent paper concerning the continuum directed random
polymer. The paper contains all the necessary formulas to construct a
distribution for the model. Quastel stated some results about solutions that
locally look like Brownian Motion, which start with smooth initial conditions.
Motivation for this problem comes from results in the field of liquid crystal
turbulence in which experiments were carried out in December 2009.
‘Jeremy Quastel’ continued on page 20
LECTURES
FIELDS INSTITUTE Research in Mathematical Sciences | FIELDSNOTES 13

Distinguished Lecture Series in Statistical Sciences
JIANQING FAN IS WELL-KNOWN FOR HIS WORK
in financial econometrics, computational biology, semiparametric
and nonparametric modeling, and other aspects of statistical
theory and methodologies. He is the Frederick I. Moore Class
of 1918 Professor in Finance, Director of the Committee of
Statistical Studies at Princeton, the Past President of the Institute
of Mathematical Statistics, and winner of the 2000 COPSS
Presidents’ Award.
In early May,
Jianqing gave the
Distinguished Lecture
Series in Statistical
Science at the Fields
Institute. His two
lectures, titled Vastdimensionality
and
sparsity and ISIS:
A vehicle for the
universe of sparsity,
focused primarily on
high-dimensional statistical modelling and feature selection.
These aspects became important with the advent of mass data
collection, advances in computation, and the discovery of new
interplay between various natural and social sciences. In his public
lecture, Fan outlined the problem of high dimensionality in fields
Coxeter Lecture Series
THE 2010 SUMMER THEMATIC PROGRAM ON THE
Mathematics of Drug Resistance in Infectious Diseases was held at
the Fields Institute during July and August. In association with
this thematic program, Professor Neil Ferguson was invited to
the Fields Institute to deliver the Coxeter Lecture Series on
Mathematical modelling of emerging infectious disease epidemics and their
control.
Ferguson, a Professor of Mathematical Biology in the
Division of Epidemiology, Public Health, and Primary Care of the
Medical School at Imperial College, is a world leader in the use of
mathematical models in infectious disease epidemiology. He is the
Director of the MRC Centre for Outbreak Analysis and Modelling.
In the first lecture, Ferguson reviewed the development of
outbreak modelling over the last two decades and discussed the
drivers which lead to more complex computational simulations
being increasingly used replacing simpler compartmental models of
disease transmission. The second lecture discussed ways in which
modelling can be optimally used to assist public health policymakers
in their planning for and reaction to emerging infectious disease
threats—an issue on which Ferguson is an expert, and which was
14 FIELDSNOTES | FIELDS INSTITUTE Research in Mathematical Sciences
Vast-dimensionality and Sparsity
as diverse as bioinformatics, genetics, physics, and economics,
discussing the twin challenges of noise accumulation and
spurious correlations. The notion of sparsity or, more generally,
homogeneity, was methodically described for feasible inference
to high-dimensional problems. After an overview of the penalized
likelihood approach for variable selection, Fan communicated the
idea of large-scale screening and moderate-scale selection together
with conditional inference as an effective solution to highdimensional
problems. Using this approach in an analysis of U.S.
housing price indices over a 30-year period, Fan demonstrated
impressive prediction improvements.
His second presentation was meant for a more specialized
audience. Exploiting sparsity, Fan outlined a unified framework
for solving high-dimensional variable selection problems. He
applied iterative vast-scale screening followed by moderate-scale
variable selection, resulting in a process called ISIS. Applying this
process to multiple regression, generalized linear models, survival
analysis, and machine learning, Fan demonstrated its overall
reach via marginal variable screening and penalized likelihood
methods. With tailored simulation studies and manipulation of
empirical data from disease classifications and survival analyses,
Fan demonstrated the advantages of a folded-concave over convex
penalty method on sure screening properties, false selection sizes
and model selection consistency.
Elif Fidan Acar (Toronto)
Mathematical Modelling of Emerging Infectious Diseases
of great interest to the thematic program participants. The third
lecture focused on the potential impact of antiviral resistance
during an influenza pandemic. He offered several explanations
for new findings that show the degree to which previous risk
assessments concerning antiviral resistance in influenza pandemics
have been over-pessimistic. In the lecture, Ferguson touched on
the critical issue of the dependence of the final impact of resistance
during a closed epidemic on the transmissibility of a sensitive
and resistant virus, the mutation rate from one type to the other,
and the level of seeding of both viral types at the beginning of
the epidemic. He argued that resistance is not likely to entail a
substantial reduction of effectiveness of antivirals during the start
of a pandemic, but that intensive drug use in this phase can lead
to a higher degree of resistance in later epidemics. His concluding
remark that “simple models suggest antiviral resistance could be a
major issue in the first wave of a new pandemic, but allowing for
spatial heterogeneity reduces speed of resistance” strongly echoed
the theme of transmission heterogeneity of the two-week block of
this entire thematic program on mathematics for drug resistance.
Jianhong Wu (York)

Optimization and Data Analysis
in Biomedical Informatics
CENTRE FOR
MATHEMATICAL
MEDICINE
DATA MINING AND BIOMEDICAL SCIENCE ARE TWO
of the fastest growing areas of engineering and scientific computing.
Modern data acquisition protocols have generated vast amounts
of data that require fast and rigorous analysis methods. New
challenges and problems are being posed for applied mathematicians,
statisticians, computer scientists, and engineers.
The workshop on Optimization and Data Analysis in Biomedical
Informatics explored recent progress in the field. Several challenges
and roadblocks in biomedical informatics with reference to the
application of data mining were discussed, and new computing
technologies and paradigms in optimization and control, as well as
systems engineering were presented.
During the two days of the workshop distinguished speakers
from universities in Canada, the U.S., and Europe gave presentations
about the state-of-the-art in data analysis methodologies.
Specific talks related to machine learning (e.g.
support vector machines, biclustering, generalized
eigenvalue classification), optimization, biomedical
information systems, biomedical imaging, and
signal processing were given. The applications
presented at the conference covered a large part
of biomedicine and included DNA microarray
analysis, Raman spectroscopy, biomarkers,
electroencephalogram analysis, and
radiation therapy treatment.
The conference attracted the interest
of many local faculty members and students
from both the University of Toronto and the
University of Waterloo. The organizers were
very satisfied with the high quality of the talks.
This workshop provided an excellent chance
to meet and interact with scientists from
very diverse backgrounds ranging from
mathematics, electrical engineering,
and industrial and systems
engineering, to biology,
medicine and information
technology.
Selected papers
based on the talks
given at the workshop
will be published as a Fields
Communications volume by the
American Mathematical Society.
Tom Coleman (Waterloo)
FIELDS
FIELDS
INSTITUTE
INSTITUTE
Research
Research
in
in
Mathematical
Mathematical
Sciences
Sciences |
FIELDSNOTES
FIELDSNOTES
15
15

Brain
Neuromechan
BRAIN TISSUE IS AN INHOMOGENEOUS, MULTIscaled,
multi-layered, and inter-connected set of neurons, glial
cells, and vascular networks. So far, the biophysics and dynamics
of the cell types within these networks have been studied
individually, as well as their network interactions.
On the other hand, the macroscopic mechanical response
of the brain to traumatic injuries has also been the object of
intense experimental and computational studies. However, a
full understanding of brain physics and dynamics can only be
achieved by linking biomechanical and biochemical processes
taking place in the brain at different length and time scales, and
by accounting for the interactions of and feedback among the
brain’s networks.
The aim of this first interdisciplinary workshop on
Brain Neuromechanics was to bring together experts from
different areas of brain research, such as applied mathematics,
neuroscience, engineering, neurosurgery, to present the latest
developments in their fields and discuss opportunities for longterm
research collaborations.
The first speaker, James Drake, Chief Neurosurgeon
at the Hospital for Sick Children in Toronto, set the tone
for the workshop by delivering a talk on the inseparability
of neurosurgery and neuromechanics. Improved medical
diagnoses, treatment strategies and clinical protocols can be
achieved only through discoveries in fundamental brain science.
In particular, hydrocephalus, a brain condition known from
the time of Hippocrates and characterized by an abnormal
accumulation of spinal fluid within the fluid-containing
spaces of the brain, remains a puzzle for neurosurgeons even
today; the two surgical treatments currently used display no
statistical difference with regard to the efficacy of treating
hydrocephalus. Some of the subsequent talks showed
promising new advances in the understanding of the underlying
mechanisms that give rise to hydrocephalus. Miles Johnston
(Sunnybrook) showed that, in the rat brain, interstitial fluid
pressures increased after antibody administration into a lateral
16 FIELDSNOTES | FIELDS INSTITUTE Research in Mathematical Sciences
ventricle, suggesting that capillary absorption might play a
pivotal role in the onset of hydrocephalus. This finding provides
the exciting possibility that some forms of hydrocephalus
may be treatable with pharmacological agents rather than
through surgical interventions. Richard Penn (Chicago)
presented a novel macroscopic biomechanical model of
fluid-structure interactions in the brain which could explain
the onset of hydrocephalus. So far, the model appears to be
in agreement with preliminary experiments on dog brains.
Kathleen Wilkie (Waterloo) introduced a new age-dependent
fractional viscoelastic model for the brain and showed that the
natural pulsations of the brain cannot be the primary cause
of either infant or adult hydrocephalus. Almut Eisentrager
(Oxford) presented a novel multi-fluid poro-elastic model
of hydrocephalus which incorporates blood pulsations on
the cardiac cycle time scale and thus can be used to simulate
spinal fluid pressure fluctuations in clinical infusion tests.
Finally, Corina Drapaca (Penn State) presented the first
neuro-mechanical models that couple the biomechanics and
biochemistry of the brain. One model, based on the triphasic
theory, shows that normal pressure hydrocephalus (NPH) can
be caused by an ionic imbalance in the absence of increased
intracranial pressure. This represents a significant finding in
NPH research, opening the door to the possibility of treating
hydrocephalus using pharmaceutical agents. The other model
can incorporate non-invasive neuro-imaging measurements
and can then be used to investigate the brain’s mechanics
under different clinical scenarios. Although Martin Ostoja-
Starzewski’s (Urbana-Champaign) talk did not focus on
hydrocephalus, his MRI-based finite element approach to
study traumatic brain injuries could also be used in computer
simulations of hydrocephalus. In addition, a novel extension of
continuum mechanics to fractal porous media was introduced to
address the random fractal geometry of the brain.
The talks given by Alan Wineman (UMichigan) and
Katerina Papoulia (Waterloo) served to remind the audience

ics
of the continuum mechanics framework and some important
mathematical concepts that any researcher should be aware of
when designing biomechanical models of the brain. Another
talk of pedagogical nature was given by K. Unnikrishnan
(UMichigan) on computational methods and associated
statistical significance tests used to detect patterns in multineuronal
spike trains which can uncover the functional
connectivity of the underlying neuronal networks.
Whilst the talks given by Joseph Francis (SUNY
Downstate) and Jürgen Germann (Toronto Phenogenomics)
presented novel experimental approaches to investigate
the plasticity of the brain, the lectures given by Paul
Janmey (UPenn) and Kristian Franze (Cambridge) focused
on the mechanosensitivity of healthy and diseased brain
cells and the experimental settings required to estimate
mechanical parameters at such small length and time scales.
The mechanics of individual cells and their networks are
essential in understanding the brain damage seen in deep
brain stimulation studies performed on animal brains. The
talks of Bruce Gluckman (Penn State) and Andrew Sharp
(Southern Illinois) served to raise awareness in the neuroscience
community of the damaging effects of electrode implants. In
particular, Gluckman’s experimental results showed not only
local damage near insertion locations, but also unexpected
non-local damage of brain tissue. Patrick Drew (Penn State)
presented a novel technological approach of imaging single
vessel vascular dynamics in the mouse cortex that provides
the first non-controversial link between functional signals and
their vascular origin. The talk of Leslie Loew (Connecticut
Health Center) presented yet another technological discovery
that uses voltage sensitive dyes to image neuronal physiology.
Loew also introduced briefly the Virtual Cell, a software system
used to simulate neuronal cell biology. All these technological
advancements can provide valuable data for mathematical
models of the brain.
Corina Drapaca (Penn State)
FIELDS INSTITUTE Research in Mathematical Sciences | FIELDSNOTES 17

IN CONVERSATION
IN LATE MAY, LEAH KESHET (UBC) PRESENTED
a lecture at the Centre for Mathematical Medicine entitled
Adventures in Mathematical Biology. She spoke mainly about
recent progress in cell motility modelling using a systems biology
approach. (Cells typically move by a process called chemotaxis,
and proteins regulate cell movement by enabling this process.)
Keshet described the progression of her model from inception,
using three simple partial differential equations to model the
protein dynamics, to a more robust model including proteins that
stay relatively dormant during movement. This expansion led to
the addition of three differential equations to the model.
Keshet’s lecture outlined the symmetry breaking that occurs
during initiation of cell polarization, as well as the related cell
signaling systems. She summarized her research, presented the
mathematical models she has developed, and hypothesized
underlying biological mechanisms produced from the insight she
has gained from her models.
The key points of her talk were related to the relevant
developments in biotechnology and their implications for
the modelling and understanding of cell motility. After her
introduction she showed movies made through microscopes
that illustrated cell motility. She then gave a general description
of the approach to understanding cell motility mechanisms.
She noted that using genetic circuit diagrams to identify the
relevant proteins involved in the process is the only way to study
it. After this, she explained the process of research in this field
18 FIELDSNOTES | FIELDS INSTITUTE Research in Mathematical Sciences
Leah Keshet
and the importance of interactions between mathematicians and
biologists.
From making initial biological observations and producing a
preliminary model, Keshet was able to validate it and make the
necessary modifications to produce more accurate explanations.
Her preliminary model consisted of three coupled partial
differential equations corresponding to the active proteins:
phosphoinositides, GTPases, and the actin cytoskeleton. To
capture the characteristic wave movement of the cells, she
improved the preliminary model by including inactive proteins.
Keshet noted that her understanding of the phenomenon
has been produced from the ground up and that it has taken
off to a great start. Keshet has given follow-up directions to
some of her collaborators, graduate students, and post-doctoral
fellows. They are creating a stochastic version of the model to
include inherent biological instabilities, a scheme of fluid based
computations to study the motion of shapes, and some biological
directions to study the cell’s ability to diffuse and re-orient itself.
Keshet was a student of applied mathematician Lee Segal.
In the 1960s, Segal pioneered many of the asymptotic and
quasi-steady state methods used to study complex mathematical
models in a more tractable format.
Following her lecture, Keshet sat down with Sivabal
Sivaloganathan, Co-Director of the Center for Mathematical
Medicine, to answer a few general questions about mathematical
biology.

Richard Cerezo: From my understanding
of the lecture, you characterized cell
motility by its activity with three proteins.
Leah Keshet: That was one aspect of a
bigger project in which the three proteins
play a major role, this is absolutely true.
When I give this lecture to biologists, I
tend not to do it quite in the same order…
Since I am giving the talk at the Fields
Institute I wanted to have a centerpiece
which was more mathematical.
RC: Are you a biologist or mathematician
by training, or both?
Siv Sivaloganathan: Her mother was a
pure mathematician and her father was a
biologist, so it was inevitable that she was…
LK: …stuck in between.
RC: I guess that comes across when you
have to tailor your talks to different
audiences.
LK: I try to, although you sometimes get
the opposite attitude and some people
say that to be in a field like this, you need
to be able to sit in two chairs, and you
need to have a very big bottom, because
biologists will say, ‘This is completely too
simplified’ and there is nothing biological
here. And mathematicians will come along
and say, ‘This is too soft, there is nothing
mathematically interesting here.’ So it’s
tricky.
SS: When you’re looking at a problem
that is essentially biological, how do
you go about thinking ‘what’s the crux
of the mathematical problem?’ How
do you go about formulating a problem
mathematically?
LK: Well it’s not trivial. It took many
years before we got to even asking the right
questions. We were stumped for a while
at the point where we saw, ‘these are the
three proteins, and these are the reactions.
We’ve simulated them but we don’t get
polarization. What’s going on here? Why
don’t we get what we want?’ To begin to
see what was happening took a long time.
To begin to formulate a simpler problem
that we could pursue analytically took even
more time. It’s been a total of seven or
eight years from when we began thinking
about these proteins.
SS: Would you say that apart from
experience, it’s a whole universe of things
outside the realm of mathematics that you
need to feel through to get a handle on the
problem?
LK: For the first five or six years, a lot
of the work is reading the literature and
figuring out what it means, rather than
having a biologist to work with directly.
This is because biologists rarely see the
value of models. The biologists who have
these values are rare and it takes a while for
us to build up enough of a background to
publish. I have collaborations with Condilis
in New York, which arose because I gave
a similar talk in Minnesota and he was one
of the people in the audience. He could
see that there was some value in those
directions.
RC: Was this approach to the problem out
of necessity or was this something that you
were trained during your PhD?
LK: I think it’s more a matter of luck.
That is, having the right people come
together at the right time. So when I began
thinking about this, I had a postdoc Stan
Moray [...] and he was a person who already
had these two dimensional simulations for
moving cell platforms—not so much as for
many cells interacting with each other, but
he could immediately see that this could be
done. While some students were working
out the biochemistry, we could then go to
them and say, ‘Here’s what we found,’ and
he could go then and make 2D simulations.
If we had to do everything from scratch, it
would have taken a long time.
RC: For the future generation of math
biologists, what attitude should we foster?
Since this is a highly non-traditional field
in mathematics.
LK: The good thing is that the field has
become more central and nowadays,
biologists typically have to show some
type of modelling component in their
grant proposals. They cannot simply
apply for NIH or NSF funding without
this balance. They have to show that they
have some way of taking that data from
their experiments and making sense of
it by working with theorists. Therefore,
they have much more motivation to be
connected to young people who’ve got
quantitative techniques. So I think it’s
important both to get the good math
background, which means PDEs, ODEs,
numerical simulation, a bit of computer
programming, knowing how to use
MATLAB, as well as taking the necessary
background courses in biology that you’re
interested in like immunology of cell
biology in my case, and then being very
open to talking to and finding people in
those fields to talk to.
SS: Because in many ways, biology up
until now has been just observation and
acquiring of lots of data. But trying to do
mathematics with objects that are not in
your chemical equations is something that
is just slowly sinking in.
RC: In what way, if any, did Professor
Lee Segal influence your work, your
approach, and your philosophy?
LK: Good question. I think first of all
that he was a great applied mathematician.
And I have to say that, despite all his
good intentions, he did not get me all
excited about asymptotic analysis. But
I did appreciate its usefulness, so other
people working with me deal with those
things. He had a way of taking problems
and saying, ‘How can we simplify this as a
first cut? How can we take something very
complicated and try and write down what
are the key things that we’ll go after as a
first version? Once we understand that,
we’ll add extra details.’ He was very good
at that.
[...] there are many scientists in every
area who are extremely possessive and have
the attitude of saying ‘this is my idea, this
is how it works’, ‘everybody else is a fool’,
‘this is the only way’. Segal was very much
willing to see all kinds of points of view and
debate, ‘well this kind of model can explain
this, the other kind of model is not so good
at explaining that’ and vice versa. He was
known in the field as being a great man,
almost a father figure, and I think that is
really important.
RC: So a human aspect was very
important, as well as his ability to relate
to other people, not only his ability to be
solely a scientist?
LK: To bridge between different
perspectives and not be too put off if
someone thought that ‘no actually, things
work a different way’, to be tolerant of
different points of view and be open.
SS: This is interesting because we
had a workshop a couple of weeks ago
on Mathematical Oncology and one
of the students of Weinberg at MIT
was speaking and I was talking to him
afterwards and he was saying how he got
to work with Weinberg. Weinberg said to
him ‘anybody could be a great scientist, I
want you to be a great human being’ and I
thought that was a wonderful thing to say.
FIELDS INSTITUTE Research in Mathematical Sciences | FIELDSNOTES 19

New Fields Institute Publications
Fields Institute Communications, Volume 57
GANITA Seminars on Algebraic Curves and Cryptography
Edited by V. Kumar Murty, University of Toronto
In 2001, the GANITA Lab was founded at the University of
Toronto to study the applications of mathematics to problems
in Information Technology. The Sanskrit word ganita means
computation or calculation which is one of the themes of the
research activities in the lab. GANITA is also an acronym for
Geometry, Algebra, Number Theory and their Information
Technology Applications. Over the past nine years, the lab has
mostly concentrated on applications related to information
security. Part of the mandate of the lab was to contribute to
the training of students and postdoctoral fellows interested in
entering the area. For this purpose, a weekly seminar was held
to discuss background material as well as to learn about recent
research. This volume is a small selection of some of those
seminar talks. They are arranged around the theme of point
counting on various classes of abelian varieties over finite
fields. The presentations are mostly suitable for independent
study by graduate students who wish to enter the field, both in
terms of introducing basic material as well as providing a guide
to the literature.
Fields Institute Communications, Volume 58
New Perspectives in Mathematical Biology
Edited by Siv Sivaloganathan, University of Waterloo
This volume provides a glimpse of the vibrancy and
excitement felt in mathematical biology and medicine as
it emerges from its period of infancy in the last century.
Recently, the field has taken centre stage as a major theme
of modern applied mathematics with strong links to the
empirical biomedical sciences, and has become one of the
most rapidly growing areas of modern science.
The lectures on which the book is based were delivered at
the Society for Mathematical Biology (SMB) Conference held
in Toronto in 2008, and reflect the broad spectrum of current
research interests and activity in the field. The conference
The Fields Institute for Research in Mathematical Sciences
publishes FIELDSNOTES three times a year.
Director: Edward Bierstone
Deputy Director: Matthias Neufang
Managing Editor: Andrea Yeomans
Scientific Editor: Carl Riehm
Distribution Coordinator: Tanya Nebesna
Cover Photo: Mike MacLeod
Additional Photos: Richard Cerezo (pages 4, 5, 6, 13); Ruy
Exel (page 9); Spiro Karigiannis (page 11); Mike MacLeod
(page 2); Juraj Stacho (page 6).
20 FIELDSNOTES | FIELDS INSTITUTE Research in Mathematical Sciences
brought together a group of world-renowned scientists,
researchers, postdoctoral fellows and graduate students for
four days of cutting edge talks and lectures.
Fields Institute Monographs, Volume 27
Polyhedral and Semidefinite Programming Methods in
Combinatorial Optimization
By Levent Tunçel, University of Waterloo
Since the early 1960s, polyhedral methods have had a central
role to play in both the theory and practice of combinatorial
optimization. Since the early 1990s, a new technique,
semidefinite programming, has been increasingly applied to
some combinatorial optimization problems. The semidefinite
programming problem refers to optimizing a linear function
of matrix variables, subject to finitely many linear inequalities
and the positive semidefiniteness condition on some of the
matrix variables. On certain problems, such as maximum cut,
maximum satisfiability, maximum stable set and geometric
representations of graphs, semidefinite programming
techniques yield new and important results. This monograph
provides the necessary background to work with semidefinite
optimization techniques, usually by drawing parallels to the
development of polyhedral techniques and with a special focus
on combinatorial optimization, graph theory and lift-andproject
methods.
The core of this book is based on ten lectures given at the
Fields Institute during the academic term Fall 1999, as part
of the Fields Institute Thematic Program on Graph Theory and
Combinatorial Optimization. These lectures were expanded and
developed by the author in several courses at the University
of Waterloo, evolving over the past 10 years into the present
monograph.
As prerequisites for this monograph, a solid background
in mathematics at the undergraduate level and some exposure
to linear optimization are required. Some familiarity with
computational complexity theory and the analysis of
algorithms would also be helpful.
‘Jeremy Quastel’ continued from page 13
With a special set of initial conditions, Quastel presented
a visual corner growth model and the exact formula for ASEP
(Asymmetric Simple Exclusion Process) by Tracy and Widom
(published in 2008). Results were found independently by
Sasawoto-Spohn. Some cases were resolved for specific initial
conditions. If p = q,
we can estimate the KPZ solution.
Techniques for other exclusion processes can be extrapolated
from these special cases.
The formula obtained proved a long standing scaling
conjecture. The lecture was a glimpse into the world of polymer
models and the upcoming thematic program.
Richard Cerezo (Toronto)

‘Schubert Calculus’ continued from page 12
One of the motivating open problems
of this subject is to understand the algebra
of k-Schur functions well enough to
develop a combinatorial model of the
structure constants. It is known that the
Gromov-Witten invariants appear as special
cases of the structure constants of the
k-Schur functions; answering this particular
aspect of the affine Schubert calculus would
help answer long standing open problems
in the area of mathematical physics and
geometry.
The summer school opened with a talk
by Jennifer Morse (Drexel) who, in the first
of three presentations, gave an explanation
of Schur symmetric functions and the Pieri
rule, which she generalized in later talks.
She showed in her second and third talks
how changing one element of the definition
of Schur functions gives a definition of
k-Schur functions, and changing it in a
different way gives a definition of dual
k-Schur functions.
Her three lectures gave background
that was used in the presentations by Luc
Lapointe (Talca). His first leture told the
story about how the k-Schur functions were
originally discovered as the ‘largest’ basis
of a subspace of Macdonald’s symmetric
functions for which the Macdonald
symmetric functions were positive. In
the second lecture he outlined a list of
properties, conjectures, and open problems.
Thomas Lam (UMichigan) gave a series
of lectures for which he had produced
lecture notes in advance of the summer
school, covering a very useful array of
mathematics for algebraic combinatorics.
He showed the definition of Stanley
symmetric functions that are a generating
function for the reduced words of a
permutation and then generalized them
to affine Stanley symmetric functions and
showed how they were related to k-Schur
functions.
A portion of the FRG grant was
dedicated to computational aspects of
affine Schubert calculus. Jason Bandlow
(UPenn) and Nicolas Thiéry (Paris Sud
11) gave a number of tutorials on the open
source mathematics software Sage which
has programs to compute with
k-Schur functions and an extensive
algebraic combinatorics toolbox. Mark
Shimozono (Virginia Tech) gave a
series of lectures laying out an explicit
method for computing affine Stanley
symmetric functions for all types and made
connections between the geometry and
the algebra in detail. Lenny Tevlin (NYU)
gave an introductory lecture on the last day
of the summer school on quasi-symmetric
functions.
Participants had a day off and the
workshop portion of the event was held
in the following four days. It was a really
pleasant experience to hold this meeting
during the summer. We had a beautiful day
between the summer school and workshop;
a few of the participants took the ferry to
see the Toronto islands.
The topics of the workshop were more
focused on recent research results and
opened with a presentation by Sami Assaf
(MIT). She spoke about joint work with
Sara Billey (MIT) that showed the k-Schur
functions were Schur positive. Bandlow
spoke about joint work with Anne Schilling
(UC Davis) and myself on the Murnaghan-
Nakayama rule for k-Schur functions which
gives an expansion of k-Schur functions in
the power sum generators of the algebra.
Jonah Blasiak (Chicago) spoke about
the representation theory of graded S n
modules which are conjectured to have
a decomposition into S n representations
given by k-Schur functions.
Some of the talks presented
constructions concerning problems related
to Affine Schubert Calculus, and others
discussed closely related topics such as
crystals and quantum Schubert Calculus.
Hugh Thomas (New Brunswick) gave
an interesting talk about how to derive
Littlewood-Richardson rules using Pieri
rules and jeu-de-taquin. Luis Serrano
(UMichigan) spoke on work to generalize
non-commutative Schur functions of the
methods by Fomin and Greene to other
types. Thomas Lam had used similar
techniques to define non-commutative
k-Schur functions.
Mike Zabrocki (York)
‘Groups’ continued from page 9
Handelman (Ottawa) on the occasion of his
forthcoming 60th birthday.
Many animated mathematical
discussions took place. New research
collaborations are certainly expected to
result from this event, and a conference
sequel (Brazilian Operator Algebra
Symposium) will be held in Florianópolis
from January 31 to February 4, 2011.
Thierry Giordano and Vladimir Pestov (Ottawa)
THANKS
to our
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MAJOR SPONSORS
Government of Ontario—
Ministry of Training, Colleges,
and Universities; Goverment of
Canada—Natural Sciences and
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(NSERC)
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corporations, or foundations, and is
a registered charity.
The Fields Institute is grateful to all
its sponsors for their support.
FIELDS INSTITUTE Research in Mathematical Sciences | FIELDSNOTES 21

Call for Proposals,
Nominations, and Applications
For detailed information on making proposals or nominations, please see the website: www.fields.utoronto.ca/proposals
CRM-Fields-PIMS Prize — Submission deadline November 1, 2010
The CRM-Fields-PIMS Prize is the premier Canadian award in recognition of exceptional research achievement in the
mathematical sciences. Nominations for this joint prize are solicited. The candidate’s research should have been conducted
primarily in Canada or in affiliation with a Canadian university. Please send nominations to
crm-fields-pims-prize@fields.utoronto.ca. Nominations for the CRM-Fields-PIMS prize should reach the Fields Institute no
later than November 1, 2010. For further details, please visit www.fields.utoronto.ca/proposals/crm-fields-pims_prize.html.
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. If you are considering a proposal, we recommend that you contact the Director or Deputy
Director (proposals@fields.utoronto.ca). For further details, please visit www.fields.utoronto.ca/proposals/other_activity.html.
Thematic Programs
Letters of intent and proposals for semester-long programs at the Fields Institute are considered in the spring and fall each
year and should be submitted preferably by March 15 or September 30. Organizers are advised that a lead time of several
years is required, and are encouraged to submit a letter of intent prior to preparing a complete proposal. The Fields Institute
has started a new series of two-month long summer thematic programs focusing on interdisciplinary themes. Organizers
should consult the directorate (proposals@fields.utoronto.ca) about their projects in advance to help structure their proposal.
Fields Research Immersion Fellowships
This program supports individuals with high potential to re-enter an active research career after an interruption for special
personal reasons. To qualify, candidates must have been in a postdoctoral or faculty position at the time their active
research career was interrupted. The duration of career interruption should be at least one year and no more than eight
years. Examples of qualifying interruptions include: a complete or partial hiatus from research activities for child rearing;
an incapacitating illness or injury of the candidate, spouse, partner, or a member of the immediate family; or relocation to
accommodate a spouse, partner, and/or other close family member. The Research Immersin Fellow will participate fully in
the thematic program, with the expectation that this will allow candidates to enhance their research capabilities and establish
or re-establish a career as a productive, competitive researcher. The award is to be held at the Fields Institute, but there are
no restrictions on the nationality or country of employment of the candidate.
For programs in a given program year (which runs July to June) the closing date will be the preceding March 31.
Applications should be sent by email to the Director. Late applications will be considered if the position has not yet been
filled. For further details, please visit www.fields.utoronto.ca/proposals/research_immersion.html.
Outreach Proposals
The Fields Institute occasionally 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 who consider organizing such an activity
and seek Fields Institute support, are invited to submit a proposal to the Fields Outreach Competition. There are two submission
deadlines each year, June 1 and December 1, with the second competition scheduled for December 1, 2010. Proposals should
include a detailed description of the proposed activity as well as of the target audience. A budget indicating other sources of support
is also required. Submissions should be sent to proposals@fields.utoronto.ca. Questions about this program may be directed to the
Director or Deputy Director.

FIELDS ACTIVITIES
Bridging Research, Education, and Industry
Current and Upcoming Thematic Programs
ASYMPTOTIC GEOMETRIC ANALYSIS,
JULY TO DECEMBER 2010
Organizers: V. Milman (Tel Aviv), V. Pestov (Ottawa),
N. Tomczak-Jaegermann (Alberta)
SEPTEMBER 13–17, 2010
Workshop on Asymptotic Geometric Analysis and
Convexity
SEPTEMBER 14–16, 2010
Distinguished Lecture Series: Avi Wigderson (IAS)
SEPTEMBER 17, 20, 21, 2010
Coxeter Lecture Series: Shiri Artstein-Avidan (Tel Aviv)
OCTOBER 12–16, 2010
Workshop on the Concentration Phenomenon,
Transformation Groups and Ramsey Theory
NOVEMBER 1–5, 2010
Workshop on Geometric Probability and Optimal
Transport
DYNAMICS AND TRANSPORT IN DISORDERED
SYSTEMS , JANUARY TO JUNE 2011
Organizers: D. Dolgopyat (Maryland), K. Khanin (UTM),
R. de la Llave, (UT Austin), A. Neishtadt (IKI), J. Quastel
(Toronto), B. Tóth (BME)
FEBRUARY 14–19, 2011
Workshop on Disordered Polymer Models
APRIL 4–8, 2011
Workshop on the Fourier Law and Related Topics
JUNE 13–17, 2011
Workshop on Instabilities in Hamiltonian Systems
FEBRUARY 22–24, 2011
Distinguished Lecture Series: Yakov Sinai
(Princeton)
APRIL 13–15, 2011
Coxeter Lecture Series:
Srinivasa Varadhan (Courant)
General Scientific Activities — October 2010 to January 2011
All activities take place at Fields unless otherwise stated. Detailed informaiton: www.fields.utoronto.ca/programs
OCTOBER 8, 2010
4th Symposium on Health Technology
at the University of Waterloo
OCTOBER 9–10, 2010
RECOMB Satellite Workshop on
Comparative Genomics
at the University of Ottawa
OCTOBER 20, 2010
Computational Neuroscientists in Upper
Canada (CNUCs) Workshop
OCTOBER 22, 2010
CRM-Fields-PIMS Prize Lecture:
Gordon Slade (UBC)
OCTOBER 25, 2010
Big Ideas in Mathematics Symposium
OCTOBER 28, 2010
2010 IFID Conference on Models
for Lifecycle Finance, Insurance and
Economics
OCTOBER 29, 2010
Workshop on Technology Integration in
Teaching Undergraduate Mathematics
Students
NOVEMBER 6-7, 2010
Workshop on Algebraic Varieties
DECEMBER 6-10, 2010
Workshop on Discrete and
Computational Geometry
at Carleton University
FIELDS
DECEMBER 8-10, 2010
First Joint North American Meeting
on Industrial and Applied Mathematics
SMM-SIAM-CAIMS
at the Universidad del Mar, Huatulco,
Mexico
DECEMBER 9, 2010
Computational Neuroscientists in Upper
Canada (CNUCs) Workshop
JANUARY 21-23, 2011
Combinatorial Algebra meets Algebraic
Combinatorics Conference
at Lakehead University
FIELDS INSTITUTE Research in Mathematical Sciences | FIELDSNOTES 23

Message from the Director
Thoughts on the IMU General Assembly
Many of our readers will know that the Fields Institute
has been in a competition with two other research
institutes to host a Stable Office of the International
Mathematical Union (IMU). The competing institutions
were the Instituto Nacional de Matemática Pura e Aplicada
(IMPA) in Rio de Janeiro, and the Weierstrass Institute for
Applied Analysis and Stochastics (WIAS) in Berlin. A decision
in favour of WIAS was made by a majority vote of the IMU
General Assembly meeting in Bangalore, India, August 16–17.
I participated in the General Assembly as a member of the
Canadian delegation, and also to present the Fields Institute
bid.
Much of my time and energy during the past year, as
well as dedicated efforts by staff and friends of the Fields
Institute went into the development of our bid. I would like
to express my heartfelt appreciation to all who contributed.
Particular thanks are due to City of Toronto Mayor David
Miller, University of Toronto President David Naylor and
Canadian Mathematical Society President Tony Lau for their
support. Invest Toronto provided expert help in preparing our
presentations.
My opinions on the result are of course partisan! But
it is certain that the host institution and community will
create a perception of the Stable Office that will mark the
future development of IMU activity. I believe that the
Fields Institute and the City of Toronto offered the IMU
a unique opportunity, in terms of openness to people and
ideas, diversity of the Institute’s scientific activities, and its
global outreach (both to traditional centres of mathematics
excellence and to the developing world). Neither we nor
IMPA, however, were able to offer the same amount of
government financial support as WIAS.
Our participation in the competition for the IMU Stable
Office has helped raise the international profile of the Fields
Institute and the Canadian mathematical community. It
has opened opportunities for continuing collaboration with
the IMU that I hope to report further on in future issues of
the Fields Notes. As a part of our bid for the Stable Office,
the Fields and Perimeter Institutes offered the IMU to
jointly fund a Fields–IMU–Perimeter Fellowship to bring young
researchers from Africa to participate in Fields or Perimeter
programs. We plan to implement this initiative as the Fields–
Perimeter Fellowship.
The amount of government support that would have
been needed to produce a winning bid was extremely modest
in relation to higher-profile government initiatives and the
potential benefits to Canada of hosting the headquarters of
international scientific organizations. A prominent German
senator enthusiastically described WIAS’s winning bid as
“an accolade for Berlin as an excellent place for science. [...]
Mathematics in the region has established itself as a major
factor that strengthens the link between science and the
economy.”
It is impossible to avoid a comparison also with South
Korea’s winning bid for the next International Congress of
Mathematicians (ICM 2014). South Korea’s presentation
to the General Assembly was an eloquent testimony to a
successful twenty-year initiative to become a world power in
mathematics. The effort was sustained by Korea’s traditional
reverence for mathematics, and confidence that an investment
in the development of mathematical talent represents perhaps
both the most effective and the least expensive way to build
an economy based on innovation and technological advances,
with payoff within a generation. At the ICM, I heard that
more than half of South Korea’s Olympic athletes go on to get
degrees in Medicine or PhDs in Mathematics!
On July 6 at the Perimeter Institute, Prime Minister
Stephen Harper announced 20 million dollars in government
funding to speed the growth of science in Africa by
establishing a network of five centres of the African Institute
for Mathematical Sciences across Africa within the next few
years. The Canadian government is to be congratulated on
this investment in a vast pool of potential mathematical and
scientific talent, and for heeding the vision of the nations of
Africa who recognize the value to their societies of nurturing
the scientific culture of their young people.
What about Canadian society and young people in our
country? Canada is slipping in its competitiveness ranking
by the World Economic Forum, particularly when it comes
to measurement of innovation factors. NSERC seems
to be losing its way — under-funded, micro-managed by
the government, less and less responsive to the scientific
community. Funding of the Canadian mathematics institutes
seems under threat. A worrying number of Canada’s best
young mathematicians are leaving, after taking university
positions here just a few years ago because of Canada
Research Chairs or other opportunities. Our provinces badly
need plans to improve the quality of mathematics education at
all levels.
It is a reasonable working assumption that human
scientific potential is evenly distributed in the world’s
population. I believe that we in Canada have a lot to learn
from policies of rising economic powers like China, India and
South Korea on the development of this potential. Are we
leaving our own children behind?
Edward Bierstone
FIELDS INSTITUTE Research in Mathematical Science
222 COLLEGE STREET, TORONTO, ONTARIO, CANADA M5T 3J1
Tel 416 348.9710 Fax 416 348.9714 WWW.FIELDS.UTORONTO.CA