We will begin with some basic background on shifts of finite type (SFT) and then talk about the recent paper by K. McGoff with the title above. Given an SFT X, fixed n, and alphat in [0,1], one defines a random sub-SFT of X by selecting words of length n, independently with probability alpha. McGoff shows that the probability that the resulting SFT has various properties as n -----> infty can be described explicitly in terms of alpha. (This will be the first talk in an informal weekly seminar on various topics within Symbolic Dynamics and Ergodic Theory).

Cooperation represents a key organizing principle in genetic and cultural evolution. Yet cooperation is a conundrum because cooperators make a sacrifice to benefit others at some cost to themselves. Exploiters or defectors reap the benefits and forgo costs. Despite the fact that groups of cooperators outperform groups of defectors, Darwinian selection or utilitarian principles based on rational choice should favor defectors. Nevertheless, cooperation is ubiquitous in biological and social systems. Public goods games have established as the leading mathematical and game theoretical metaphor to study such social dilemmas, which are characterized by the conflict of interest between the group and the individual. An analysis based on evolutionary game theory shows that cooperation can be stabilized by punishing defectors. Punishment is also ubiquitous in animal and human societies - ranging from toxin producing microorganisms to law enforcement institutions. However, it remains unresolved how initially rare, costly punishment behavior can gain a foothold and spread throughout a population. In nature, animals and humans often carefully select their interaction partners or adjust their behavioral patterns. In the simplest case individuals simply refuse to participate in risky enterprises. Such voluntary participation in social endeavors describes a simple yet efficient mechanism to prevent deadlocks in states of mutual defection and thus represents a potent promoter of cooperation - but fails to stabilize it. However, the combined efforts of punishment and volunteering are capable of changing the odds in favor of cooperation - but only in finite populations. Under the stochastic dynamics of finite populations the freedom to withdraw leads to prosocial coercion. This implements Garret Hardin’s principle to overcome the 'Tragedy of the Commune': 'mutual coercion mutually [and voluntarily] agreed upon'. To date, theory and experiments emphasize the role of such peer-punishment, which is of crucial importance in various animals, but at least in most human societies peer-punishment has been largely superseded by sanctioning institutions and vigilantism deemed illegal. This can be modeled by introducing pool-punishment, which represents a precursor of executive power and echoes Elinor Ostroms self-governing principles in her work on 'Governing the Commons'. Pool-punishment always incurs costs to those committed to it even if no one requires reprimanding. Interestingly, our model predicts that individuals still trade the higher efficiency of peer-punishment for the increased stability of pool-punishment to maintain cooperation.

Note for Attendees

Tea and cookies will be served in the Math Lounge (MATH 125) at approximately 2:45pm.

I will present a joint work with Behrend.
For any smooth projective variety, we construct a differential graded scheme (stack) structure on the moduli space of
complexes of coherent sheaves. The construction uses the Hochschild cochain complex of A infinty bi-modules.
As an application, we show that the DT/PT wall crossing can be intepreted as change of stability conditions on
dg schemes.

We give a criterion under which a solution g(t) of the Kahler-Ricci flow contracts exceptional divisors on a compact manifold and can be uniquely continued on a new manifold. This is a joint work with Jian Song.

Implicit time stepping applied to discretized incompressible fluid flow problems results in discrete Stokes-like (saddle-point, KKT) problems. There is a class of so-called Projection Methods in which these Stokes-like problems are efficiently approximated by the solutions of standard, second order elliptic problems. Over the last 20 years, these methods have been increasingly well understood. In this talk, a review of the methods and the analytic results are given. Leading order errors (both character and order) from the methods can be precisely described using asymptotic techniques. This asymptotic error analysis is applied to a number of commonly used projection methods. The results are confirmed by careful numerical computations using a spectral method on a representative model problem.

Let F be a field of characteristic different from 2, let WF denote the Witt ring of quadratic forms over F and IF its fundamental ideal. The maximal number of additive generators of I^{n}F/I^{n+1}F (or infinity, if no such maximum exists), is called the n-symbol length of F and denoted by lambda_{n}(F). While lambda_{0}(F)=lambda_{1}(F)=1, the 2-symbol length lambda_{2}(F) is of particular interest and was already studied by Bruno Kahn. On the other hand, if F is a Pythagorean field, its stability index is an invariant that measures the complexity of the space of orderings of the field F. We show that a Pythagorean field (more generally, a reduced abstract Witt ring) has a finite stability index if and only if it has finite 2-symbol length. We give explicit bounds for the two invariants in terms of one another. To approach the question whether those bounds are optimal we consider some examples of Pythagorean fields. This is joint work with Karim Becher.

In these talks we describe our proofs of the Hanna Neumann Conjecture. This
conjecture of the late 1950's can be described both as a problem in group
theory or as one in graph theory. Our first proof is longer and interprets the
problem using homology of sheaves on graphs; our second proof is very short and
uses only simple graph theory, but was inspired from the type of induction used
in our first proof. Both proofs demonstrate the strengthened form of the
conjecture formulated by Walter Neumann, and both proofs use earlier resolved
cases of the conjecture.

A crucial idea of the proof is to express the seemingly awkward notion of
"reduced cyclicity" of a graph in simpler terms, involving limits over covering
maps. This "limit homology theory" may be of independent interest, and is
related to the Atiyah Conjecture; our theory requires some curious linear
algebra that also may be of independent interest.

Our talks will not assume any previous knowledge of sheaf theory.

We consider a system of bosons in a box, under the influence of a mutually
repellent pair potential. The particle density is positive and is kept
fixed while the volume of the box is increased. We discuss the following
result: the identification of the limiting free energy, at positive and sufficiently
small temperature, in terms of an explicit variational formula. We
use a large deviations approach combined with the representation of the
system of bosons through a system of interacting Brownian bridges.

This is a joint work with S. Adams (Warwick) andW. Konig (T.U. Berlin
and Weierstrass Institute).

This is the first talk this term for UBC/UMC, the undergraduate mathematics colloquium. The speaker is Professor Brian Wetton.

Title: Approximating the arctan function

Calculators can perform the same basic operations that humans can do by hand: add, subtract, multiply and divide. How these basic operations can be used to approximate more complicated functions (square roots and the arctan function are used as examples) to arbitrary precision will be discussed. Some history is given as to how these functions were computed routinely before calculators were available.

Graphs naturally pop up everywhere e.g. the Internet, networks of social contacts etc. Random graphs (and thus probability theory) are used to study these networks which are too large to study analytically. I will talk a bit about the questions which random graph theory hopefully can resolve. Also I will show introduce random graph models and discuss some results.

----

Simon Rose on "Why are generating functions cool?"

A generating function is a formal power series (that is, an infinitely long polynomial) where the coefficients are chosen to be some numbers we find interesting. In this talk I will show a few surprising results that can be proven quite easily within this theory, and encourage you to no longer care about whether or not something converges.

We will continue with background on symbolic dynamics, in particular the dynamical zeta function. Then we will state and prove some of McGoff's results on random shifts of finite type.

We consider the complex Monge-Amp\`ere equation on a compact K\"ahler manifold $(M, g)$ when the right hand side $F$ has rather weak regularity. In particular we prove that estimate of second order and the gradient estimate hold when $F$ is in $W^{1, p_0}$ for any $p_0>2n$. As an application, we show that there exists a classical solution in $W^{3, p_0}$ for the complex Monge-Amp\`ere equation when $F$ is in $W^{1, p_0}$.

Polarization occurs when cells segregate specific proteins and other factors to opposite ends of the cell in response to some signal. A cell with a symmetric distribution of proteins must have a symmetry breaking event in order to become polarized, resulting in a stable asymmetric protein distribution. In this informal talk, I will discuss possible mechanisms used by embryos of the nematode worm C. elegans to initiate the process of polarization, including new experimental evidence produced this summer.

In these talks we describe our proofs of the Hanna Neumann Conjecture. This
conjecture of the late 1950's can be described both as a problem in group
theory or as one in graph theory. Our first proof is longer and interprets the
problem using homology of sheaves on graphs; our second proof is very short and
uses only simple graph theory, but was inspired from the type of induction used
in our first proof. Both proofs demonstrate the strengthened form of the
conjecture formulated by Walter Neumann, and both proofs use earlier resolved
cases of the conjecture.

A crucial idea of the proof is to express the seemingly awkward notion of
"reduced cyclicity" of a graph in simpler terms, involving limits over covering
maps. This "limit homology theory" may be of independent interest, and is
related to the Atiyah Conjecture; our theory requires some curious linear
algebra that also may be of independent interest.

Our talks will not assume any previous knowledge of sheaf theory.

Gersgorin theorem says that the eigenvalues of finite matrices lie on
the so called Gersgorin region in the complex plane. This theorem implies
in particular that if a matrix is strictly diagonally dominant, then it
is invertible. Taussky improves: if a matrix is irreducibly diagonally dominant,
then it is invertible (quite long time ago, in 1949!).
In this talk, we discuss this problem in probabilistic way. We introduce
a Markov process from the given diagonally dominant matrix. Then by
using Feynman-Kac formula and large deviation principle, we can bound
the components of the inverse.

In this first talk we will introduce the notion of a symmetry of a PDE system. A symmetry of a PDE system is any transformation that maps all solutions into other solutions of the same PDE system. Lie’s algorithm provides an effective way to find directly the local symmetries of a given PDE system. Since the continuous symmetries of a PDE system are defined topologically, they are not restricted to local symmetries. It turns out that a natural way to find such nonlocal symmetries is through the construction of nonlocally related systems arising from the conservation laws of the given PDE system. A local symmetry of such a nonlocally related system can yield a nonlocal symmetry of the given PDE system. In this series of talks, it will be shown how to construct further nonlocally related systems (and consequently further nonlocal symmetries) through various means.

This is the first talk this term for the TAAP Seminar Series. Our speaker is Isabeau Iqbal, from the Department of Educational Studies and the Centre for Teaching, Learning and Technology.

Title: What is the value of the peer review of teaching?
Abstract: The literature suggests that the peer review of teaching process can foster a culture of teaching and learning and can enhance the level of collegiality in departments. In this session, I will present a snapshot of the peer review of teaching at UBC, highlighting distinctions between "formative" and "summative" peer review, and making links to the improvement of teaching. I will draw from the higher education literature and my own research to feature how departmental culture can support or hinder the peer review of teaching process.

I will begin by discussing some recent theoretical and numerical joint work with Liang and Sujatha on the Tate-Shafarevich group of elliptic curves over Q with complex multiplication and rank at least 2. In the latter part of the lecture, I will explain some rather wild general speculation which grew out of these computations.

Note for Attendees

Refreshments will be served between the two talks.

The Yoshida lift is a theta lift that takes a pair of automorphic forms on a definite quaternion algebra to a holomorphic Siegel modular form. We show that a natural refinement of the classical Yoshida lift in fact preserves p-integrality in a suitable sense. Since the result is rather technical, we will motivate it by casting it as the first step in a program sketched out by Harris-Skinner-Li aimed at relating the p-divisibility of the special values of certain automorphic L-functions to the existence of non-trivial elements of a certain Selmer group. Time permitting, we will also discuss a non-vanishing modulo p result of the integral Yoshida lift.

Mordell's conjecture is a fundamental question regarding rational points on curves of genus greater than one. Starting from this question, we will discuss a generalization proposed by Lang, which was solved by Faltings.Then we will formulate a generalization of Lang's conjecture, which we study by ultimately reducing it to a question on linear algebraic groups.

MATH ANNEX 1100 (PIMS/UBC distinguished colloquium)

Fri 24 Sep 2010, 3:00pm-4:00pm

Abstract

I will try to explain the general background to the subject, and discuss the formulation of the main conjectures of non-commutative Iwasawa theory, on which there has been important recent progress.

The Ricci flow is by now well-known as the technique used to prove the Poincare conjecture and several other remarkable mathematical results. It is natural to search for applications in physics. I will describe two, both related to mass-energy.

The first is that the Ricci flow is an approximation to a renormalization group flow in bosonic string theory. In the RG flow, a natural question to ask is how does the spacetime mass behave at different scales. In the Ricci flow approximation, we can sometimes answer this question.

The second is that, while mass-energy is a global property of spacetime, nonetheless various proposals for quasilocal mass exist and have sometimes proved useful. Bartnik has suggested a quasilocal mass which has many desirable properties, including both nonnegativity and manifest monotonicity as the quasilocal region is expanded. Nonetheless, his definition presents practical difficulties. This led Bartnik to make certain geometrical conjectures which, if true, would improve the situation. One of these is the Static Minimization Conjecture, much of which is now a theorem due to recent work of Anderson and Khuri. The minimization conjecture leads to the study of a version of Ricci flow. I will explain Bartnik's quasilocal mass and give the results of early attempts to study the minimization conjecture via this method.

In this talk we will describe the underlying construction of the main
theorem of Galatius' paper. This main theorem is a generalization of the
Pontryagin-Thom construction and establishes a weak homotopy equivalence
between the classifying space of a cobordism category and the infinite loop
space of the spectrum MTSO(n). We then compute the rational cohomology of
this infinite loop space and describe how the generalized MMM-classes come
from it.

Let G be a simple Lie group or Kac-Moody group and P a parabolic
subgroup.
One of the goals Schubert calculus is to understand the product
structure
of the cohomology ring H^*(G/P) with respect to its basis of Schubert
classes. If G/P is the Grassmannian, then the structure constants
corresponding to the Schubert basis are the classical
Littlewood-Richardson
coefficients which appear in various topics such as enumerative
geometry,
algebraic combinatorics and representation theory.

In this talk, I will discuss joint work with A. Berenstein in which
we give
a combinatorial formula for these coefficients in terms of the Cartan
matrix corresponding to G. In particular, our formula implies
positivity
of the “generalized” Littlewood-Richardson coefficients in the
case
where the off diagonal Cartan matrix entries are not equal to -1. Moreover, this positivity result does not rely on the geometry
of
the flag variety G/P.

Vortex solitons are standing wave solutions with complex phase that is an (integer) multiple of the angular polar coordinate. This multiple we call the 'spin', and indexes a family of solutions with increasing L2 norm. In the case of no spin, Merle and Raphael have shown that there exists a range of data that blowup with the Townes profile (the regular soliton) and whose H1 norm grows at a precise 'log-log' rate. We prove that in the case of spin 1, there is comparable data that blows up with the vortex profile and the log-log rate. The case of spin 2 and 3 will be discussed. This is joint work with Gideon Simpson (Toronto)

Title: A mathematical model of antigen bonds on immune cells

The bond between an antigen and a T-cell is one of the most important elements of our adaptive immune system. How the T-cell has such high sensitivity to slight differences in antigen remains a mystery. It is also a mystery why the antigen bond's lifetime is different depending on what the antigen is attached to. I will introduce a model of a single antigen bond. The model uses diffusion-advection equations, elasticity mechanics, and stochastic processes. As with many physical processes, the entire model can be understood in terms of energy minimization. We developed this model to explain experimental data on bond lifetimes, but it ended up generating a new hypothesis about T-cell sensitivity.

## Note for Attendees

Tea and cookies will be served in the Math Lounge (MATH 125) at approximately 2:45pm.