
Fri 3 Apr 2009, 3:30pm
Department Colloquium
WMAX 110 (PIMS)

The Ant in the Labyrinth: Random Walks and Percolation

WMAX 110 (PIMS)
Fri 3 Apr 2009, 3:30pm4:00pm
Abstract
Percolation was introduced by Broadbent and Hammersley in 1957. The simplest version to describe is on the Euclidean lattice Z^{d}. Let p be a fixed probability between 0 and 1. Each bond in Z^{d} is retained with probability p, and removed with probability 1p, independently of all the others. The percolation cluster containing a point x, denoted C(x), consists of those points which can be reached from x by a path of retained bonds. There is a critical value p_{c}∈(0,1) such that if p < p_{c} then all clusters are finite, while for p > p_{c} there is an infinite cluster.
Random walks on percolation clusters were introduced by De Gennes in 1976: he called this the problem of 'the ant in the labyrinth'. If p = p(n,x,y) is the probability that a random walker ('the ant'), starting at x, is at y at time n, then p describes diffusion of heat on the cluster.
For the supercritical phase (p > p_{c}) this problem is now quite well understood, and p(n,x,y) converges to a Gaussian distribution as n→∞. PDE techniques introduced by Nash in the 1950s, play an important role in some of the arguments.
The critical case p = p_{c} is much harder, since the clusters have fractal properties. One expects that p(n,x,x) ∼ x^{ ds/2}, where d_{s} is called the spectral dimension of the cluster. Alexander and Orbach conjectured in 1982 that d_{s} = 4/3 in all dimensions: this has recently been proved in some high dimensional cases.
hide

Leah Keshet, Brian Marcus and Malabika Pramanik
Department of Mathematics, UBC

Tue 14 Apr 2009, 2:05pm
SPECIAL
One Time Event
WMAX 110 (PIMS)

PIMS/UBC Info Session on Grant Opportunities: "Funding Possibilities"

WMAX 110 (PIMS)
Tue 14 Apr 2009, 2:05pm3:00pm
Details
We will discuss funding opportunities in pure and applied mathematics, and strategies for obtaining grants, fellowships, and so on. We will focus on NSERC grants in Canada and National Science Foundation grants in the U.S.
hide

The College of William and Mary

Wed 15 Apr 2009, 2:00pm
Mathematical Biology Seminar
WMAX 216

Emergent network properties in the preBotzinger Complex: the cellular and synaptic mechanisms of respiratory rhythm generation

WMAX 216
Wed 15 Apr 2009, 2:00pm3:00pm
Abstract
Breathing is an interesting and essential lifesustaining behavior for humans and all mammals. Like many rhythmic motor behaviors, breathing movements originate due to neural rhythms that emanate from a central pattern generator (CPG) network. CPGs produce neuralmotor rhythms that often depend on specialized pacemaker neurons or alternating synaptic inhibition. But conventional models cannot explain rhythmogenesis in the respiratory preBotzinger Complex (preBotzC), the principal central pattern generator for inspiratory breathing movements, in which rhythms persist under experimental blockade of synaptic inhibition and of intrinsic pacemaker currents. Using mathematical models and experimental tests, here we demonstrate an unconventional mechanism in which metabotropic synapses and synaptic disfacilitation play key rhythmogenic roles: recurrent excitation triggers Ca2+activated nonspecific cation current (ICAN), which initiates the inspiratory burst. Robust depolarization due to ICAN also causes voltagedependent spike inactivation, which diminishes recurrent excitation, allowing outward currents such as Na/K ATPase pumps and K+ channels to terminate the burst and cause a transient quiescent state in the network. After a recovery period, sporadic spiking activity rekindles excitatory interactions and thus starts a new cycle. Because synaptic inputs gate postsynaptic burstgenerating conductances, this rhythmgenerating mechanism represents a new paradigm in which the basic rhythmogenic unit encompasses a fully interdependent ensemble of synaptic and intrinsic components.
hide

Department of Mathematics, UBC

Wed 15 Apr 2009, 3:00pm
Probability Seminar
WMAX 216

Branching processes and real world networks

WMAX 216
Wed 15 Apr 2009, 3:00pm4:00pm
Abstract
The aim of this talk is to highlight the usefulness of continuous time branching process theory in understanding refined asymptotics about various random network models. We shall exhibit their usefulness in two different contexts:
(1) First passage percolation: Consider a connected network and suppose each edge in the network has a random positive edge weight. Understanding the structure and weight of the shortest path between nodes in the network is one of the most fundamental problems studied in modern probability theory. In the modern context these problems take an additional significance with the minimal weight measuring the cost of sending information while the number of edges on the optimal path (hopcount) representing the actual time for messages to get between vertices in the network. In the context of the configuration model of random networks we shall show how branching processes allow us to find the limiting distribution of the minimal weight path as well as establishing a general central limit theorem for the hopcount with matching means and variances.
(2) Spectral distribution of random trees: Many models of random trees (including general models embedded in continuous time branching processes) satisfy a general form of convergence locally to limiting infinite objects. In this context we find via soft arguments, the convergence of the spectral distribution of the adjacency matrix to a limiting (model dependent) non random distribution. For any \gamma we also find a sufficient condition for there to be a positive mass at \gamma in the limit.
Joint work with Remco van der Hoftsad, Gerard Hooghiemstra, Steve Evans and Arnab Sen.
hide

Department of Mathematics, UBC

Wed 15 Apr 2009, 3:00pm
Topology and related seminars
WMAX 110 (PIMS)

Stable decompositions and almost commuting elements in Lie groups

WMAX 110 (PIMS)
Wed 15 Apr 2009, 3:00pm4:00pm
Abstract
In this talk I show that the space of almost commuting elements in a compact Lie group G splits after one suspension.
hide

Meiji Institute for Advanced Study of Mathematical Sciences

Thu 16 Apr 2009, 2:00pm
Mathematical Biology Seminar
WMAX 216

Origin of culture: an evolutionary model of social learning

WMAX 216
Thu 16 Apr 2009, 2:00pm3:00pm
Abstract
Social learning is an important ability seen in a wide range of animals. Especially, humans developed the advanced social learning ability such as language, which triggered rapid cultural evolution. On the other hand, many species, such as viruses, rely on genetic evolution to adapt to environmental fluctuations. Here we propose an evolutionary game model of competition among three strategies; social learning, individual learning, and genetic determination of behavior. We identify the condition for learning strategies to evolve.
hide

University of Georgia

Wed 22 Apr 2009, 3:00pm
Department Colloquium / Topology and related seminars
WMAX 1110 (PIMS)

Modules of constant Jordan type

WMAX 1110 (PIMS)
Wed 22 Apr 2009, 3:00pm4:00pm
Abstract
This is joint work with Eric Friedlander, Julia Pevtsova and Andrei Suslin. We consider modules over an elementary abelian group on which every element in the radical, but not the square of the radical, has the same Jordan canonical form. Such modules can be used to define bundles on projective spaces and Grassmanians. They have many interesting properties. We can get them as submodules of any module of the group algebra. In this talk I will discuss some of the constructions and their generalizations.
hide

BC Centre for Disease Control

Thu 23 Apr 2009, 2:00pm
Mathematical Biology Seminar
WMAX 216

Early Realtime Estimation of the Basic Reproductive Number

WMAX 216
Thu 23 Apr 2009, 2:00pm3:00pm
Abstract
The basic reproductive number, R_0, which is generally defined as the expected number of secondary infections per primary case in a totally susceptible population, is an important epidemiological quantity. It helps us to understand the possible outcome of an initial infection seeding in a social setting: whether it leads to a small outbreak, or it evolves into a largescale epidemic. The basic reproductive number encapsulates the information about the biology of disease transmission as well as the structure of human social contacts. We use concepts from network theory to present a novel method for estimating the value of the basic reproductive number during the early stage of an outbreak. This approach will greatly enhance our ability to reliably estimate the level of threat caused by an emerging infectious disease.
hide

University of Utah

Thu 23 Apr 2009, 3:30pm
Probability Seminar
WMAX 216 (PIMS)

The nonlinear wave equation in high dimensions: existence, Holdercontinuity and ItoTaylor expansion

WMAX 216 (PIMS)
Thu 23 Apr 2009, 3:30pm4:30pm
Abstract
hide

UC Irvine

Fri 24 Apr 2009, 3:00pm
Department Colloquium
WMAX 110 (PIMS)

Getting to the heart of smooth 4manifolds

WMAX 110 (PIMS)
Fri 24 Apr 2009, 3:00pm4:00pm
Abstract
In this talk, designed for a broad mathematical audience, we will describe what is known (and unknown) about the classification of smooth 4manifolds. In particular we will uncover a new mechanism that makes progress towards the conjecture that every simplyconnected closed 4manifold has either zero or infinitely many distinct smooth structures.
hide

Vancouver Island University

Mon 27 Apr 2009, 2:30pm
SPECIAL
One Time Event
MATH 225

Skylight Candidate Interview  Subject: TBA

MATH 225
Mon 27 Apr 2009, 2:30pm4:00pm
Details
TBA
hide

Trent University

Wed 29 Apr 2009, 3:00pm
Probability Seminar
WMAX 216 (PIMS)

Myopic strategies and impact of forecast errors

WMAX 216 (PIMS)
Wed 29 Apr 2009, 3:00pm4:00pm
Abstract
We introduce and discuss some new stochastic models of optimal portfolio selection with reduced impact of forecast errors. In particular, we found some new examples of optimal myopic strategies, including some discrete time models with serial correlations. In addition, we found some new cases when the strategies that don't leave the efficient frontier even if there is an error in the forecast. It may happen for nonmyopic strategies if the required information about the future is limited.
hide

Note for Attendees
There will be tea and cookies in the math lounge at approximately 2:45pm.