A 0-dimensional scheme is said to be "smoothable" if it deforms to a disjoint union of points. Determining if a given 0-dimensional scheme is smoothable seems to be quite a difficult problem in general, and I will survey some of the main results in this area of research, including some recent progress that is joint work with David Eisenbud and Mauricio Velasco. In particular, I will explain how Gale duality provides a geometric obstruction to smoothability.

This is part of our ongoing Lunch Series for Teaching & Learning.

Description: This session will feature a panel of six UBC Math instructors of various stripes (three faculty and three grad students, one of whom is teaching for the first time) who are using clickers this term or did last year. There will be a brief introduction on instructional use, driven in part by clickers in the audience, so you will be able to see a sample of the technology in action. This will not be a technical how-to, however; the bulk of the time will be devoted to hearing from the panel members about their classes:
Did the technology help to achieve instructional goals?
What were the challenges in terms of lesson design and technology use?
What was surprising?

After experiencing the pain of the tech bubble crash Sunny Wong decided to break all the rules and throw in the towel to lose the rest of his money. Instead of losing any money he makes money.
He combines all 3 markets into one. He is somewhat of a “Jack of all Markets” and a “Master of None”.But he still manages to consistently make profits every day.He also calls his trading style a business and not investing.Sunny is not a fundamental or technical trader. Sunny will now discuss his Active Neutral Trading as a Business in a nutshell for all of us to hear.

Dr. Calvin Winter CFA will introduce Sunny Wong and frame his trading in the context of the Behavioral Finance Cognitive Biases which Sunny is exploiting. Ample time will be available for questions and answers.

(Joint work with C. Procesi and M.Vergne) Let G be a compact
Lie group with Lie algebra g. Given a G-manifold M with a G equivari-
ant one form w we consider the zeroes M^{0} of the corresponding moment
map and define a map, called innitesimal index, of S[g*]^{G}-modules
from the equivariant cohomology of M^{0} with compact support to the
space of invariant distributions on g*.
In the case in which G is a torus, N is a linear complex representation
of G, M = T*N with tautological one form we are going to explain
how this is used to compute the equivariant cohomology of M^{0} with
compact support using certain spaces of polynomial which appear in
approximation theory.

This colloquium will examine the broad themes of nonlinear dispersive PDE, with the nonlinear Schrodinger and wave equations as examples. We'll discuss the underlying linear theory (Strichartz estimates + conservation laws), weakly nonlinear phenomena (scattering), full nonlinear behaviour (standing waves), and instability (blowup). The emphasis will be on motivation and not proof. This talk is intended for mathematically mature non-experts.

We study sumsets A+B:={a+b: a in A, b in B} where A and B are sets of integers, and B has positive density. We show that under an equidistribution condition, A+B must be highly structured, and in particular must contain all the finite configurations from some Bohr set. This generalizes work of Renling Jin, and Bergelson, Furstenberg, and Weiss. We will see how a description of sumsets follows from a description of ergodic averages, and how embeddings of a set of integers A into compact abelian groups influence the structure of summands A+B where B is a dense set of integers. This raises questions, some new and some old.

We consider the situation in which a weak solution of the Navier-Stokes equations fails to be continuous in the strong L^2 topology at some singular time t=T. We identify a closed set S_T in space on which the L^2 norm concentrates at this time T. The famous Caffarelli, Kohn Nirenberg theorem on partial regularity gives an upper bound on the Hausdorff dimension of this set. We study microlocal properties of the Fourier transform of the solution in the cotangent bundle T*(R^3) above this set. Our main result is a lower bound on the L^2 concentration set. Namely, that L^2 concentration can only occur on subsets of T*(R^3) which are sufficiently large. An element of the proof is a new global estimate on weak solutions of the Navier-Stokes equations which
have sufficiently smooth initial data.

The chordal Schramm-Loewner evolution (SLE) is a measure on curves
connected boundary points of a domain. Schramm defined the
process for simply connected domains, but it is not immediate
how to extend the definition to multiply conneceted domains.
I will discuss an approach using the Brownian loop measure
and some work in progress showing that the measure is well
defined. The recent work uses ideas from a recent paper of
Dapeng Zhan on reversibility of whole plane SLE.

I present an inverse function theorem for differentiable maps between Banach spaces, the proof of which relies on Ekeland's variational principle. As a consequence, the assumptions are substantially weakened: the map F to be inverted is not required to be C^{1}, or even Fr\'{e}chet-differentiable. I then state a inverse function theorem for differentiable maps between Fr\'{e}chet spaces which contains the classical theorem of Nash and Moser as a particular case. In contrast to the latter, the proof does not rely on the Newton iteration procedure, but on Lebesgue's dominated convergence theorem. As a consequence, the assumptions are substantially weakened: the map F to be inverted is not required to be C^{2}, or even C^{1}, or even Fr\'{e}chet-differentiable.

Note for Attendees

Please plan to attend a Special Tea to be held at 2:15 pm prior to Ivar Ekeland's colloquium. This will take place in MATH 125 and special cakes will be served.

One of the central theorems of classical Lie theory is that all split Cartan subalgebras of a finite dimensional simple Lie algebra over an algebraically closed field are conjugate.This result, due to Chevalley, yields the most elegant proof that the type of the root system of a simple Lie algebra is its invariant. In infinite dimensional Lie theory maximal abelian diagonalizable subalgebras (MADs) play the role of Cartan subalgebras in the classical theory. In the talk we address the problem of conjugacy of MADs in a big class of Lie algebras which are known in the literature as extended affine Lie algebras (EALA). To attack this problem we develop a bridge which connects the world of MADs in infinite dimensional Lie algebras and the world of torsors over Laurent polynomial rings.

When a thin rectangular sheet is clamped along two opposing edges and stretched, its inability to accommodate the Poisson contraction near the clamps may lead to the formation of wrinkles with crests and troughs parallel to the axis of stretch. A variational model for this phenomenon is proposed. The underlying energy functional includes bending and membranal contributions. Motivated by work of Cerda, Ravi-Chandar, and Mahadevan, the functional is minimized subject to a global constraint on the area of the mid-surface of the sheet. Analysis of a boundary-value problem for the ensuing Euler–Lagrange equation shows that wrinkled solutions exist only above a threshold of the applied stretch. A sequence of critical values of the applied stretch, each element of which corresponds to a discrete number of wrinkles, is determined. Whenever the applied stretch is sufficiently large to induce more than one wrinkle, previously proposed scaling relations for the wrinkle wavelength and root-mean-square amplitude are confirmed. Comparisons with experimental measurements and numerical results indicate that the analytical results are remarkably robust.

I state and prove an inverse function theorem between Fr\'{e}chet spaces, which does not require that the function to be inverted is C^{2}, or even C^{1}, or even Fr\'{e}chet-differentiable.

The problem of finding a combinatorial rule for computing Kronecker
coefficients i.e. the multiplicities appearing when the tensor product
of two irreducible representations of the symmetric group is
decomposed into irreducibles, is a long standing open problem in
combinatorial
representation theory.

I'll start with a crash course in Symmetric functions and give an
account of the work done on the aforementioned problem and the
motivation behind it. I'll point out the cases in which answers were
obtained and end by mentioning the special cases that I've tackled.

The Navier–Stokes-alpha-beta equation regularizes the Navier–Stokes equation by the addition of dispersive and dissipative terms. The dispersive term is proportional to the divergence of the corotational rate of the symmetric part of the velocity gradient. The dissipative term is proportional to the bi-Laplacian of the velocity. The coefficients of these terms involve factors alpha and beta, respectively, both having dimensions of length. Calculating the energy spectrum for an assembly of stretched spiral vortices reveals an inertial range where Kolmogorov's -5/3 law holds and shows that choosing beta less than alpha yields a better approximation of the inertial range of the Navier–Stokes equation. Direct numerical simulations of three-dimensional periodic turbulent flow confirm this and also show that vorticity structures behave more realistically when beta is less than alpha. However, the simulations indicate that optimal choices of alpha and beta are resolution dependent. This suggests the possibility of developing multigrid methods that capitalize on resolution dependence by using the Navier–Stokes-alpha-beta equation at coarse grid levels, with different choices of alpha and beta at each level, to accelerate convergence to solutions of the Navier–Stokes equation at the finest grid level. Results obtained from a two-dimensional spectral multigrid algorithm of this type show promise.

Note for Attendees

Refreshments and cookies will be served at 1:15pm in the PIMS Lounge downstairs near WMAX 110.

Littlewood-Paley functions are used to characterize some function spaces such as Hardy space, Sobolev space and Lipschitz space. In the last 30 years, Brownian motion has been widely used to give such characterizations in terms of Lp-norm. Recently, there has been an increasing interest in discontinuous processes, particularly in stable processes. An interesting question is if there is an analogue of the Lp characterization in the case of discontinuous processes. In this talk, first I will discuss the Dirichlet problem which forms the core of this problem and then I will talk about its solution, corresponding L-P functions, and my recent results in this topic.

To each finite group G one associates an algebraic invariant rk(G), called
the rank of G, and a topological invariant hrk(G), called the homotopy rank
of G. The number hrk(G) is defined in terms of free G-actions on products
of spheres. In this talk we define the two invariants and discuss the rank
conjecture, which states that rk(G)=hrk(G).

The speaker for UBC/UMC this week is Josh Zukewich.

Title: Evolution of cooperation in structured populations

A quick introduction to evolutionary game theory as a powerful tool to model frequency dependent selection. We'll develop a spatially structured model where individuals are represented by nodes in a graph (or network). Analysis of the model reveals simple and surprising implications for the evolution of cooperation.

I will define and discuss some basic properties of algebraic groups over a field, including the classification of the split simple (connected) algebraic groups (i.e. Dynkin diagrams). Then I will discuss G-torsors, give examples of their related algebraic structures, and define the essential dimension and essential p-dimension of algebraic groups. Finally, for some (all, if there is time) split simple groups I will survey what is known about these invariants.

Since Riemann's 1859 monograph, the study of the distribution of prime numbers has been dominated by the study of the zeros of the Riemann zeta function and Dirichlet L-functions. Although there have been ad hoc elementary proofs of some of the key results, there has been no coherent alternative approach to that of Riemann. In this talk we will introduce a new way to develop analytic number theory, without zeros, stemming from the concept of "pretentiousness". This is joint work with Soundararajan.

Note for Attendees

---A note from the speaker about preliminaries

There are not really any notes since this is an introduction to a new way of thinking about an old topic. If anything, a student should look at Davenport's book "Multiplicative Number Theory" for the best version of the classical approach, but I doubt someone will do that within the next week! I will explain things without assuming much background (from profs as well as students!)

To any polynomial over a perfect field of positive characteristic, one may associate an invariant called the F-pure threshold. This invariant, defined using the Frobenius morphism on the ambient space, can be thought of as a positive characteristic analog of the well-known log canonical threshold in characteristic zero. In this talk, we will present some examples of F-pure thresholds, and discuss the relationship between F-pure thresholds and log canonical thresholds. We also point out how these results are related to the longstanding open problem regarding the equivalence of (dense) F-pure type and log canonical singularities for hypersurfaces in complex affine space.

The modulation spaces are spaces of distributions that behave like
the Besov spaces, by essentially replacing the dilation in their
definition with a time-frequency shift (modulation).
The main goal of the talk is to convey the idea that modulation space
estimates arise as natural alternatives when estimates on other
classical function spaces fail. We will discuss this idea through two
examples: wave or Schr"odinger multipliers, and a class of bilinear
pseudodifferential operators.

Abstract: When identical information has to be delivered from a source to multiple destinations, we would like to use a transmission scheme known as multicast. Multicast delivers the information simultaneously to all receivers in a single transmission over the network, and thus, in general, uses less transmission energy and time than the traditional scheme consisting of multiple independent deliveries to different destinations. Mathematically, the network is modeled as a directed graph, and information as streams of numbers in some finite field. Often, when network resources are limited, it is beneficial to implement multicast with network coding, that is, to allow network nodes to linearly combine the incoming information streams and forward the resulting merged streams to their neighbors. We show to which extent, how, and at what cost, the information can be hidden from a wiretapper who can observe all (possibly merged) information streams on a fixed number of network edges of his choice, and knows how the information streams are linearly combined throughout the network. We are interested in perfect (unconditional) secrecy rather than encryption which relies on the limited computational power of the wiretapper.

A linear space is an incidence structure of points and lines such that
every line is on at least two points and any two distinct points are
on exactly one line. This is also known as a pairwise balanced
design. The dimension of such a structure is the maximum positive
integer D such that any D of its points generate a proper subspace. In
more detail, let's consider linear spaces on n points such that any d
points generate subspaces of size at most s.

The main result I will present is the construction of arbitrarily
large linear spaces having an upper bound on s (the largest generated
subspace size) depending only on d (the number of generating points).
A somewhat surprising consequence in extremal graph theory will be
discussed as well.

Delay is one of the most important performance measures of a communication system. In scenarios as different as mobile, ad hoc, wireless networks and content distribution networks, delay is caused by seemingly very different random phenomena, which are often statistically similar. Thus, in a wide range of applications, the delay can be reduced by following the same idea. To help understand this idea, we consider the following scenario: Imagine an agent having a piece of information that he wants to communicate to his partner. The agent knows that his partner resides in an occupied city but does not want to be seen talking with him, or with anyone else on the street for a long time. The agent and his partner have a number of friends walking in the same city, who are willing to relay small pieces of information between the secret couple. Because of that, the agent decides to split his data in small chunks which he can then inconspicuously pass to his friends. To increase the data lifetime among his friends who are moving in an adverse environment, the agent also decides to make these data chunks redundant by erasure coding, that is, by generating a larger number of chunks s.t. the original data can be recovered based on any sufficiently large subset of the redundant chunks. Assuming that all participants in this multi-agent information transfer perform simple random walks over a finite, random, regular graph, and can exchange information only when they are on the same node of the graph, we describe how coding, at the expense of introducing redundancy and processing complexity, not only increases the lifetime of the data, but also reduces the average time necessary for the transfer of information between the agent and his partner.

Multiplicative differential forms are relevant whenever considering an object with a smooth groupoid of symmetries. One can ask what is the corresponding infinitesimal object, and in fact some of the most important examples arise from this direction. The geometric structures of Hamiltonian mechanics - Poisson manifolds, Dirac structures, etc. - can be viewed as infinitesimal data which, when integrated, yield multiplicative 2-forms on Lie groupoids.

I will explore the relationship between multiplicative structures on Lie groupoids and their infinitesimal counterparts on Lie algebroids.

Let T:X \to X be a measure preserving transformation of some infinite-measure space (X,\mathcal{B},\mu) with \mu(X)=\infty.

Associated with T is a natural probability-preserving map T_* which acts on discrete countable subsets of X, with respect to the probability measure defined by Poisson processes on X. This map is called the Poisson suspension of T.

I will review some basic properties of Poisson suspensions.

Under the assumption that the transformation T is recurrent and ergodic, I will prove ergodicity of the map T \times T_*, which acts on Poisson processes with one ``marked particle''.

Given a nonzero integer d and a positive integer n we know, by Hermite's Theorem, that there exist only finitely many degree n number fields of discriminant d. It is thus natural to ask whether there are refinements of the discriminant which completely determine the isomorphism class of a number field. In this talk we will consider the integral trace form as such refinement. By using one of Bhargava's composition of cubes, we show that the integral trace form is a complete invariant for cubic fields with positive fundamental discriminant. If time allows we will discuss some further results for higher degree number fields.

Note for Attendees

Refreshments will be served between the two talks.

The boundary rigidity problem consists in determining the Riemannian metric of a compact Riemannian manifold with boundary by measuring the lengths of geodesics joining points of the boundary. The lens rigidity problem consists in determining the Riemannian metric of a compact Riemannian manifold with boundary by measuring the scattering relation or lens relation: We know the point of exit and direction of exit of a geodesic if we know its point of entrance and direction of entrance.

These two problems arise in travel time tomography in which one attempts to determine the index of refraction of a medium by measuring the travel times of waves going through the medium.

We will survey what is known about this problem and some recent results.

I will discuss the duality between algebraic tori and their character modules. Then I will introduce the notion of symmetric rank of an integral representation, and use this to explain a technique for computing the essential p-dimension of algebraic tori. This technique can be extended to finding the essential p-dimension, ed(N;p), of a normalizer of a split maximal torus in a simple group G, which in turn gives an upper bound for ed(G;p). We will work out at least one specific example, which will show the essential 2-dimension of the exceptional group E_8 is at most 120 - a new upper bound.

In 1980 A. P. Calder\'on wrote a short paper entitled "On an inverse boundary value problem". In this seminal contribution he initiated the mathematical study of the following inverse problem: Can one determine the electrical conductivity of a medium by making current and voltage measurements at the boundary of the medium? There has been substantial progress in understanding this inverse problem in the last 30 years or so.

In this lecture we will survey some of the most important developments.

The goal of this talk is to describe the sheaf model for the infinite loop space of the Thom spectrum. Since the main theorem of Galatius, Madsen, Tillmann, Weiss is that this sheaf model is homotopy equivalent to the classifying space of the cobordism category understanding this sheaf model is an important step in proving the Mumford conjecture.

We construct new smooth CY 3-folds with 1-dimensional Kaehler moduli by pfaffian method and determine their fundamental topological invariants. The existence of CY 3-folds with the computed invariants was previously conjectured by C. van Enckevort and D. van Straten. We then report mirror symmetry for these non-complete intersection CY 3-folds. We explicitly build their mirror candidates, some of which have 2 LCSLs, and check the mirror phenomenon.

Cooperation has been often studied in the framework of evolutionary game theory. Usually each player adopts a single strategy against everyone: cooperation or defection. But humans can discriminate and adopt different strategies against different opponents. In this talk I am going to present some analytical and simulational results for the case where the players can distinguish the opponents and, in the second part, I am going to talk about the extension of these ideas that has been developed jointly with prof. Christoph Hauert.

We consider the problem of existence and uniqueness of multi-soliton solutions for the LÂ²-supercritical generalized Korteweg-de Vries equation. We recall that a multi-soliton is a solution which behaves as a sum of N solitons in large time. After a survey of existing results in the subcritical and critical cases, and also in the 1-soliton case, we will state the theorem of existence and uniqueness of an N-parameter family of N-solitons in the supercritical case. Finally, we will sketch a proof of the classification part of this theorem.

The Schramm-Loewner evolution is a one-parameter family of random growth
processes in the complex plane introduced by Oded Schramm in 1999. In the
past decade, SLE has been successfully used to describe the scaling limits
of various two-dimensional lattice models. One of the first proofs of
convergence was due to Greg Lawler, Oded Schramm, and Wendelin Werner who
gave a precise statement that the scaling limit of loop-erased random walk
is SLE with parameter 2. However, their result was only for curves up to
reparameterization. There is reason to believe that the scaling
limit of loop-erased random walk is SLE(2) with the very specific natural
time parameterization that was recently introduced by Greg Lawler and
Scott Sheffield, and further studied by Greg Lawler and Wang Zhou. I will
describe several possible choices for the parameterization of the discrete
curve that should all give the natural time parameterization in the limit,
but with the key difference being that some of these discrete time
parameterizations are easier to analyze than the others. This talk is
based on joint work in progress with Tom Alberts and Robert Masson.

Toric schemes admit a combinatorial description which, in turn, allows
one to describe sheaves of modules by certain diagram categories. I
will explain these basic constructions in some detail, and then give a
non-standard approach to constructing the derived category of a
regular toric scheme; in technical language, the derived category will
appear as the homotpy category of a colocalisation of a simple diagram
category.

The speaker for UBC/UMC this week is Professor Leah Keshet.

Title: Getting Together: Flocks and the single bird

In this talk, I will present a few examples of the
phenomena that accompany the formation and dynamics
of swarms, schools, bird flocks and other collective social groups. I
then describe some of the common questions that scientist are
interested in addressing. How do such flocks stay together?
how do individuals keep from colliding or getting too crowded?
what exactly are the individuals doing inside that flock?
I'll describe some recent work that combines mathematics and
the real world. Thanks to the work of Ryan Lukeman, my former
PhD student, we were able to answer some of these questions
for a local flock of ducks (a.k.a. surf scooters).

The quadratic reciprocity law (dating back to Gauss) is a cute result of algebra letting you decide almost instantly whether a given integer is a square modulo a prime. This theorem is famous for having about 100 different proofs, and counting. I'll explain what it really says and give one proof which is particularly elementary.

I shall present an energetic variational phase field model for multiphase incompressible flows which leads to a set of coupled nonlinear system consisting a phase equation and the Navier-Stokes equations. We shall pay particular attention to situations with large density ratios as they lead to formidable challenges in both analysis and simulation.

I shall present efficient and accurate numerical schemes for solving this coupled nonlinear system, in many case prove that they are energy stable, and show ample numerical results which not only demonstrate the effectiveness of the numerical schemes, but also validate the flexibility and robustness of the phase-field model.

I'll explain a new method for computing the structure constants according to a method implicit in an old paper of Jacques Tits. This will be an expansion of the talk I gave last Spring. Then I'll recall how this allows one to compute in reductive groups, following Chevalley, Carter, and Cohen-Murray-Taylor.

Special Seminar Abstract: Many scientific, engineering and financial applications require solving high-dimensional PDEs. However, traditional tensor product based algorithms suffer from the so called "curse of dimensionality".

We shall construct a new sparse spectral method for high-dimensional problems, and present, in particular, rigorous error estimates as well as efficient numerical algorithms for elliptic equations.

We shall also propose a new weighted weak formulation for the Fokker-Planck equation of FENE dumbbell model, and prove its well-posedness in weighted Sobolev spaces. Based on the new
formulation, we are able to design simple, efficient, and unconditionally stable semi-implicit Fourier-Jacobi schemes for the Fokker-Planck equation of the FENE dumbbell model.

It is hoped that the combination of the two new approaches would make it possible to directly simulate the five or six dimensional Navier-Stokes Fokker-Planck system.

MATX 1100. Special PIMS Distinguished Lecture --CANCELED

Fri 26 Nov 2010, 3:00pm-4:00pm

Abstract

At the International Congress of Mathematicians held in Hyderabad, India this past August, Louis Nirenberg, from the Courant Institute for the Mathematical Sciences at New York University, was awarded the inaugural Chern Medal for his role in the formulation of the modern theory of non-linear elliptic partial differential equations and formentoring numerous students and post-docs in this area.

In addition to this prestigious prize, Professor Nirenberg has received many
other awards and honours, including: the American Mathematical Society’s Bôcher Prize in 1959, the Jeffrey-Williams Prize of the Canadian Mathematical Society in 1987, the Steele Prize of the AMS in 1994 for Lifetime Achievement, the Crafoord Prize in 1982and the U.S. National Medal of Science in 1995. Please seewww.icm2010.in/prize-winners-2010/chern-medal-louis-nirenberg for a full citation.

Professor Nirenberg was born in Hamilton, Ontario and obtained his undergraduate degree from McGill University before emigrating to the United States. He has shown a longstanding interest in the Canadian mathematical community, mentoring and supporting many of our colleagues. This fact and his enormous contributions tothe mathematical sciences will be recognized by the University of British Columbia by awarding him an honorary degree during its 2010 Fall Congregation (November 24-26, 2010). On November 26 he will deliver a special PIMS Distinguished Lecture at 3pm at UBC.

Everyone is invited to attend and celebrate Louis Nirenberg's wonderful career in mathematics.

After talking about how a sheaf of categories gives rise to a topological category, I will finish the proof of the main theorem of Galatius, Madsen, Weiss and Tillmann which states that the classifying space of the cobordism category is weakly homotopy equivalent to the infinite loop space of the Thom spectrum.

Log canonical thresholds are invariants of singularities that play an important role in birational geometry. After an introduction to these invariants, I will describe recent progress on a conjecture of Shokurov predicting the Ascending Chain Condition for such invariants in any fixed dimension. This is based on joint work with Tommaso de Fernex and Lawrence Ein.

Abstract: Mark has been experimenting with the Livescribe Echo, a pen that records what you write along with audio commentary you may make while writing. By producing "pencasts" using this pen, he has been able to post responses to students' questions and solutions to problems that include discussion in the natural way that we do when working with students during our office hours or presenting ideas in class. The technology is simple to use: you just write and speak as you normally would. There are many possibilities for using this technology in mathematics classes.

Invariants of singularities are defined in birational geometry via divisorial valuations, and are computed by resolution of singularities. In positive characteristic, one defines similar invariants via the action of the Frobenius morphism. The talk will give an overview of the known results and conjectures relating the two sets of invariants.

This presentation starts with a quick epidemiological overview that puts emphasis on neglected diseases and health disparities in the context of developing and/or poor nations. The primary emphasis is however on Tuberculosis (TB). A review of mathematical models and results on issues related to the transmission dynamics and control of TB, under various degrees of complexity is provided. The presentation continues with a discussion on the relationship between urban growth and TB decline in the USA. The observations are supported using demographic and TB epidemiological time series that capture the observed patterns of disease prevalence in growing urban centers in the States of Massachusetts and a large aggregate of cities in the USA, over a long window in time.

## Note for Attendees

Tea and cookies afterwards!