Anja Sturm
Assistant Professor of Mathematics

Department of Mathematical Sciences University of Delaware 501 Ewing Hall (Office 313) Newark, DE 19716-2553 USA Phone: 01 - 302 831 2776 Fax: 01 -302 831 4511 Email: sturm*at*math.udel.edu Seminar on stochastic processes University of Delaware April 3-5, 2008

I am an Assistant Professor at the Department of Mathematical Sciences at the University of Delaware. I received my DPhil from Oxford University in 2002, where I was supported by a Rhodes Scholarship and an EPSRC award. I started my undergraduate work at the University of Tübingen, Germany, in Mathematics and Physics. I then received a Master of Science in Applied Mathematics from the University of Washington. Before joining the University of Delaware, I was a postdoctoral fellow at the Weierstrass Institute of Applied Mathematics and Stochastics in Berlin and at the University of British Columbia, Vancouver, as well as a Junior Professor of Applied Probability at the Technical University Berlin.

Research

My research interests lie in Probability theory and its applications. The main focus of my work has been on probability models and stochastic processes that describe the evolution of interacting particle systems and population models forward and backward in time. Many of these models have implications for the applied sciences, especially for Biology and Physics. My particular interest has been on models in population genetics and ecology. Here, realistic mathematical models of the underlying processes that govern which genes are passed on from one generation to the next are of ever increasing importance as they are prerequisite to the correct quantitative analysis and interpretation of the unprecedented quantities of genetic data available today.

My own recent research in this area has centered on suitably chosen models of populations (of individuals or genes) that realistically incorporate the influence of two factors which have a significant impact on genetic data: firstly, spatial substructure of the population manifesting itself through local reproduction and interaction as well as locally varying environments; secondly, variations in reproductive success as they may arise due to inherent randomness or a selective advantage or disadvantage stemming from the genetic type of an individual, its location, or its surrounding environment (of other competing individuals). Recently, I have received funding from the National Science Foundation to continue to develop the mathematical theory on various aspects of such spatial models with varying offspring laws.

Backward in time the focus is on analyzing the genealogical tree of a small sample from a population and their diffusion limits (as the overall population size is large). These models of genealogies are also known as coalescent processes because the ancestral lines forming the genealogical tree of the sample coalesce as we follow them into the past. Such coalescent processes connected to populations models with singular offspring distributions -corresponding to potentially large individual families- have attracted much recent attention as several deep connections to different classes of stochastic processes have been discovered and exploited - albeit in a non-spatial setting. On the other hand, spatial settings arising from models with population subdivision and migration (and small offspring variances) have been studied for some time. In my work, one aim will be to bring these two directions of research together by studying the genealogies of spatial population models with larger offspring variances. For applications, the goal is to describe the distribution of genetic variability one should expect to see in the sample given these characteristic influences. Comparison with data then allows for inference of the population's history. On the level of individuals this answers to the question "Where did we come from?", on the level of genes it also sheds light onto the importance and function of particular parts of our genetic code. Forward in time one of the central questions being addressed is to determine parameter regimes for long-term survival without explosion of the population size and -if multiple types of particles are present initially- of long-term coexistence of various or all types. Stochastic particle models that may exhibit such phenomena are generally spatial models with an interaction mechanism between particles that gives small populations (of a particular type) an advantage. Unlike these particle systems with self-regulation those with independent reproduction of all individuals, which are mathematically simpler and have been well studied, do not have this property and are known to either go extinct or explode. The ultimate goal is to determine whether there are phase transitions in the parameter space that separate survival of the populations from extinction and coexistence from dominance of single types. This allows to conclude which mechanisms and forces lead to a stable population with lasting diversity.

Back to top

Publications

• "Tightness of voter model interfaces" . With J. M. Swart
  (to appear in the Electronic Communications in Probability )

• "Voter models with heterozygosity selection" . With J. M. Swart
  Annals of Applied Probability Volume 18, Number 1 (2008), 59-99.

• "On pathwise uniqueness for stochastic heat equations with non-Lipschitz coefficients" .
  With L. Mytnik, and E. Perkins
  Annals of Probability Vol. 34 (2006), no. 5.

• "The spatial Lambda-coalescent" . With V. Limic
  Electronic Journal of Probability Vol. 11 (2006) , no. 15, 363-393.

• "A super-stable motion with infinite mean branching" . With K. Fleischmann,
  Annales de l'Institut Henri Poincare Probabilites et statistiques Vol. 40 (2004), no. 5, 513-537.

• "Coalescence in a random background" . With N.H. Barton , A.M. Etheridge
  Annals of Applied Probability Vol. 14 (2004), no. 2, 754-785.

• "On convergence of population processes in random environments to the stochastic heat equation with colored noise" ,
  Electronic Journal of Probability Vol. 8 (2003) Paper no. 6, pages 1-39.

• DPhil thesis: On spatially structured population processes and relations to stochastic partial differential equations, Oxford University, 2002.

• ``On mechanisms to synchronize neuronal activity'', With P. König.
  Biological Cybernetics, Vol. 84 (2001) no. 3, 153-172.

• Master thesis: On mechanisms to synchronize neuronal activity, University of Washington, Seattle, 1998.

Back to top

Teaching

This spring semester I am teaching two courses:


MATH350 Probability Theory and Simulation

MATH826 Martingales, Brownian motion, and stochastic calculus


Courses taught at the University of Delaware in past semesters include:

MATH850 Theory of Probability

MATH631 Introduction to stochastic processes

MATH630 Probability Theory and Applications

MATH450 Mathematical Statistics

MATH302 Ordinary differential equations


Before coming to the University of Delaware, I have taught Introductory Probability at the University of British Columbia and a graduate level course on population processes- referrring to branching processes, superprocesses and coalescent processes- at the Technical University Berlin. I have also been a teaching assistant and tutor at Oxford University for the following courses: Basic Probability, Applied Probability, Stochastic Processes, Statistics, Differential Equations, Discrete Mathematics and Linear Programming.

UD Probability seminar

Jointly with Wenbo Li I am organizing the probability seminar at the University of Delaware. This semester seminars are generally on Thursdays at 3:30. Please contact me if you would like more information.

Seminar schedule

Web links

Visualizing probability

Virtual laboratories in Probability and Statistics
This link includes notes on probability and statistics as well as many simulated experiments and problems (applets).

For more applet simulations also check out
The birthday problem
The matching problem
Coupon collector problem
Say red card game
Poisson point process
Joint and marginal density
The central limit theorem
More central limit theorem
Galton ball drop - central limit theorem

Interesting quotes


"The most important questions of life are, for the most part, really only problems of probability."
                     (Pierre Simon de Laplace, Theorie Analytique des Probabilites)

"How dare we speak of the laws of chance? Is not chance the antithesis of all law?"
                     (Joseph Bertrand, Calcul des probabilites)

"The theory of probability as a mathematical discipline can and should be developed from axioms in exactly the same way as geometry and algebra."
                     (Andrei Kolmogorov, Foundations of the Theory of Probability)

"Mathematicians are like a certain type of Frenchman: when you talk to them they translate it into their own language and then it soon turns into something entirely different. " [Die Mathematiker sind eine Art Franzosen: Redet man zu ihnen, so uebersetzen sie es in ihre Sprache, und dann ist es alsobald ganz etwas anderes.]
                     (Johann Wolfgang von Goethe, Maxim 1279)

More probability quotes