Department of Mathematics

 Spring 2015

We will meet on Thursdays at 12:00pm.
Here is a tentative list of speakers (to be extended): Matthew Junge (University of Washington), Peter T. Otto (Willamette University), Ilya Zaliapin (University of Nevada, Reno), Benjamin Dalziel (Princeton University)
Registration information: Mth 607, Sec 003 - CRN 54851



Title and abstract

Thursday, April 16, 12:00pm
GILK 115
Matthew Junge
University of Washington

"The frog model on trees."

Abstract. On a d-ary tree place some number (random or otherwise) of sleeping frogs at each site, as well as one awake frog at the root. Awake frogs perform simple random walk and wake any "sleepers" they encounter. A longstanding open problem: Does every frog wake up? It turns out this depends on d and the amount of frogs. The proof uses two different recursions and two different versions of stochastic domination. Joint with Christopher Hoffman and Tobias Johnson.
Thursday, April 30, 12:00pm
Furman 202
Joint with Math Bio Seminar
(job talk)
Benjamin Dalziel
Department of Ecology and Evolutionary Biology
Princeton University

"Deterministic chaos in US measles epidemics."

Abstract. Regular fluctuations in the incidence of immunizing infections exemplify the emergence of stable patterns in complex populations. Stable annual or biennial cycles in disease incidence occur in a variety of host-pathogen systems because of a common demographic clockwork, consisting of depletion of the susceptible population by infection or vaccination followed by recruitment through birth or waning immunity, modulated by seasonal fluctuations in transmission rates. High amplitude fluctuations in transmission rates can cause nonlinear dissonances in the demographic clock, leading to unpredictable variation in epidemic sizes that show sensitive dependence on initial conditions/deterministic chaos. However, this is hypothesized to be rare, because sufficiently large oscillations in transmission rates are uncommon, and would result in deep epidemic troughs that predispose the system to stochastic extinction. I will discuss recent work with collaborators analyzing epidemic data that describe a ubiquitous path to locally persistent deterministic chaos through small shifts in the seasonal pattern of transmission, rather than through high amplitude fluctuations in transmission rates. We base our analysis on a comparison of measles incidence in 80 major cities in the prevaccination era US and UK. Unlike the regular limit cycles of the UK series, the US data exhibit spontaneous shifts in epidemic periodicity, due to a slight lengthening of the seasonal period of low transmission associated with school summer holidays. These local dynamics resulted in spatially decorrelated epidemics across the US during the early 20th century. This shows that subtle systematic changes in host behavior can fundamentally alter the spatiotemporal coherence of epidemics, without significantly impacting pathogen persistence, globally or locally. Routes to deterministic chaos in population dynamics may therefore be prevalent.
Thursday, May 14, 12:00pm
GILK 115
Peter T. Otto
Willamette University

"Expected length of random minimum spanning trees."

Abstract. Consider a graph where each edge is given an independent uniform [0,1] length. In 1985, Frieze proved that the expected length of the minimum spanning tree with these random edge lengths of the complete graph converge as the number of vertices go to infinity. Since then there have been numerous refinements and generalizations of this result. In this talk, I will give a survey of some of these results including the work we completed during the Willamette Valley REU Consortium for Mathematics Research in 2008 where we derived a polynomial representation of the expected length of the minimum spanning tree.
Thursday, June 4, 12:00pm
GILK 115
Ilya Zaliapin
University of Nevada, Reno

"Horton and Tokunaga self-similarity for random trees: Empirical evidence and rigorous results"

Abstract. Nature exhibits many branching tree-like structures beyond the botanical trees. River networks, Martian drainage basins, veins of botanical leaves, lung and blood systems, and lightning can all be represented as tree graphs. In addition, time-oriented trees describe a number of dynamic processes like spread of disease or transfer of gene characteristics. This would sound like a trivial observation if not for the following fact. Despite their apparent diversity, a majority of rigorously studied branching structures exhibit simple two-parametric Tokunaga self-similarity and Horton scaling. The Horton scaling is a weaker property that addresses the principal branching in a tree; it is a counterpart of the power-law size distribution for system’s elements. The stronger Tokunaga self-similarity addresses so-called side-branching; it ensures that different levels of a hierarchy have the same probabilistic structure (in a sense that can be rigorously defined). The solid empirical evidence suggests an existence of a universal self-similarity mechanism and prompts the question: What probability models can generate Horton/Tokunaga self-similar trees with a range of parameters? This talk reviews the existing results and recent findings on self-similarity for tree representation of branching, coalescent processes and time series. We show that the essential models, including white noises, random walks, critical Galton-Watson branching and Kingman’s coalescent produce trees with Tokunaga and/or Horton self-similarity. Our results explain, at least partially, the omnipresence of Tokunaga and Horton structures and suggest a framework for their statistical analysis. The results are illustrated using geophysical applications.
This is a joint work with Yevgeniy Kovchegov (Oregon State U).

Past probability seminars: Fall 2005, Winter 2006, Spring 2006, Fall 2006, Winter 2007, Spring 2007, Fall 2007, Winter 2008, Spring 2008, Fall 2008, Winter 2009, Spring 2009, Fall 2009, Winter 2010, Spring 2010, Fall 2010, Winter 2011, Spring 2011, Fall 2011, Winter 2012, Spring 2012, Fall 2012, Winter 2013, Spring 2013, Fall 2013, Winter 2014, Spring 2014, Fall 2014, Winter 2015,