Mathematics

Random Maps 1

Speaker: 
Gregory Miermont
Date: 
Mon, Jun 4, 2012 - Tue, Jun 5, 2012
Location: 
PIMS, University of British Columbia
Conference: 
PIMS-MPrime Summer School in Probability
Abstract: 
The study of maps, that is of graphs embedded in surfaces, is a popular subject that has implications in many branches of mathematics, the most famous aspects being purely graph-theoretical, such as the four-color theorem. The study of random maps has met an increasing interest in the recent years. This is motivated in particular by problems in theoretical physics, in which random maps serve as discrete models of random continuum surfaces. The probabilistic interpretation of bijective counting methods for maps happen to be particularly fruitful, and relates random maps to other important combinatorial random structures like the continuum random tree and the Brownian snake. This course will survey these aspects and present recent developments in this area.

Interacting Particle Systems 1

Speaker: 
Omer Angel
Date: 
Mon, Jun 4, 2012
Location: 
PIMS, University of British Columbia
Conference: 
PIMS-MPrime Summer School in Probability
Abstract: 
Particles attempt to follow a simple dynamic (random walk, constant flow, etc) in some space (interval, line, cycle, arbitrary graph). Add a simple interaction between particles, and the behaviour can change completely. The resulting dynamical systems are far more complex than the ingredients suggest. These processes (interchange process, TASEP, sorting networks, etc) have diverse to many topics: growth processes, queuing theory, representation theory, algebraic combinatorics. I will discuss recent progress on and open problems arising from several models of interacting particle systems.

Balanced self-interacting random walks

Speaker: 
Yuval Peres
Date: 
Mon, Jun 4, 2012
Location: 
PIMS, University of British Columbia
Conference: 
PIMS-MPrime Summer School in Probability
Abstract: 

N.B. Due to microphone problems, the audio at the beginning of this recording is poor.

It is well known that a random walk in d>2 dimensions where the steps are i.i.d. mean zero and fully supported (not restricted to a hyperplane), is transient. Benjamini, Kozma and Schapira asked if we still must have transience when each step is chosen from either μ1 or μ2 based on the past, where μ1 and μ2 are fully supported mean zero distributions. (e.g. we could use μ1 if the current state has been visited before, and μ2 otherwise). We answer their question, and show the answer can change when we have three measures instead of two. To prove this, we will adapt the classical techniques of Lyapunov functions and excessive measures to this setting. No prior familiarity with these methods will be assumed, and they will be introduced in the talk. Many open problems remain in this area, even in two dimensions. Lecture based on joint work with Serguei Popov (Campinas) and Perla Sousi (Cambridge).

Asymptotic dimension

Speaker: 
Mladen Bestvina
Date: 
Mon, May 28, 2012
Location: 
PIMS, University of British Columbia
Conference: 
PIMS Distinguished Lecturer
Abstract: 
Most of the talk will be an introduction to (second) bounded cohomology of a discrete group. I will explain classical constructions of bounded cocycles and recent results (joint with Bromberg and Fujiwara) regarding mapping class groups and a construction of bounded cocycles with coefficients in an arbitrary unitary representation.

Second Bounded Cohomology

Speaker: 
Mladen Bestvina
Date: 
Mon, May 28, 2012
Location: 
PIMS, University of British Columbia
Conference: 
PIMS Distinguished Lecturer
Abstract: 
Most of the talk will be an introduction to (second) bounded cohomology of a discrete group. I will explain classical constructions of bounded cocycles and recent results (joint with Bromberg and Fujiwara) regarding mapping class groups and a construction of bounded cocycles with coefficients in an arbitrary unitary representation.

On growth and form: geometry, physics and biology

Speaker: 
Lakshminarayanan Mahadevan
Date: 
Thu, May 24, 2012
Location: 
PIMS, University of British Columbia
Conference: 
2012 Niven Lecture
Abstract: 
The diversity of form in living beings led Darwin to state that it is "enough to drive the sanest man mad". How can we describe this variety? How can we predict it? Motivated by biological observations on different scales from molecules to tissues, I will show how a combination of biological and physical experiments, mathematical models and simple computations allow us to begin to unravel the physical basis for morphogenesis.

Mathematical Cell Biology Summer Course Lecture 36

Speaker: 
Dimitrios Vavylonis
Date: 
Thu, May 24, 2012
Location: 
PIMS, University of British Columbia
Conference: 
Mathematical Cell Biology Summer Course
Abstract: 
TBA

Mathematical Cell Biology Summer Course Lecture 35

Speaker: 
Dimitrios Vavylonis
Date: 
Thu, May 24, 2012
Location: 
PIMS, University of British Columbia
Conference: 
Mathematical Cell Biology Summer Course
Abstract: 
  • Regulatory circuits in Bacterial Chemotaxis and motility
  • Introduction to molecular motors, porters vs rowers and cooperativity of myosin in muscle
  • Microtubule dynamics
  • Cytokinesis
  • FRAP studies of microtubule dynamics in the mitotic spindle

Cell Polarity Models & Simulating Cell Motility Using the Cellular Potts Model (CPM)

Speaker: 
Leah Edelstein-Keshet
Date: 
Wed, May 23, 2012
Location: 
PIMS, University of British Columbia
Conference: 
Mathematical Cell Biology Summer Course
Abstract: 
First, the universal features of polarizing cells are listed, and details of the Mori-Jilkine wave-pinning model are assembled and discussed biologically and mathematically. A short review of thelocal pulse analysis is provided to indicate the usefulness of this method of analysis. Then, I discuss the survey of polarizing cells from a paper by Jilkine and LEK (2011) that appeared in PLoS Comput Biol 7(4): e1001121. Here, common and distinct attributes of different cell types and of several models for cell polarization are compared. The responses of models to a set of computational perturbations mimicking stimuli protocols are described. This lecture introduces the topic of 2D cell motility simulations, but focuses on one specific method, the CPM (as implemented by Maree et al in Bull Math Biol (2006), 68(5):1169-1211 and PLoS Comput Biol (2012) 8(3): e1002402). I explain the details of the method, the biological facts that were included (signaling from GTPases and phosphoinositides to actin assembly and myosin contraction). I illustrate typical results, and then discuss some of the technical aspects of the method, emphasizing its link to the (previously discussed) Metropolis-Hastings algorithm. I also show how Stan Maree was able to chose CPM parameters to phenomenologically mimic the known relationship between actin filament ends and cell protrusion speed.

Mathematical Cell Biology Summer Course Lecture 33

Speaker: 
Dimitrios Vavylonis
Date: 
Wed, May 23, 2012
Location: 
PIMS, University of British Columbia
Conference: 
Mathematical Cell Biology Summer Course
Abstract: 
Introduction to molecular motors, porters vs rowers and cooperativity of myosin in muscle
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