Evolutionary dynamics in finite populations reflects a balance between Darwinian selection and random drift. For a long time population structures were assumed to leave this balance unaffected provided that the individuals' fitness is frequency independent, i.e., both mutants and residents have fixed fitness values. This result indeed holds for a certain (large) class of population structures or graphs. However, other structures can tilt the balance to the extend that either selection is eliminated and drift rules or drift is eliminated and only selection matters.

In nature, fitness is generally frequency depended, i.e., is affected by interactions with other members of the population. The most important case refers to the evolution of cooperation under Darwinian selection and represents a major challenge in evolutionary biology and behavioral ecology. The essence of the evolutionary conundrum is captured by social dilemmas where cooperators provide a benefit to the group at some cost, while defectors attempt to exploit the group by reaping the benefits without bearing the costs of cooperation. The most prominent game theoretical models to study this problem are the Prisoner's Dilemma (PD) and the Snowdrift game (SD). In the PD cooperators are doomed if interactions occur randomly but in structured populations, they may form clusters and thereby reduce exploitation by defectors. This results in stable co-existence of cooperators and defectors and has lead to the general belief that space is beneficial for cooperation. Interestingly, however, when relaxing the social dilemma and considering the SD, this no longer holds. Due to the less stringent conditions, cooperators persist in random interactions but spatial structure often tends to be deleterious and may even eliminate cooperation altogether.

In many biological situations it is more appropriate to consider a continuous range of cooperative investment levels instead of two a priori fixed strategic types. This situation can be analyzed using the framework of adaptive dynamics. The continuous SD exhibits rich dynamics but most importantly provides an intriguing natural explanation for phenotypic diversification and the evolutionary origin of cooperators and defectors. It turns out that selection may not always favor equal contributions but rather promotes states where two distinct types co-exist--those that fully cooperate and those that exploit. In the context of human societies and cultural evolution this could be termed the 'Tragedy of the Commune' because differences in contributions to a communal enterprise have significant potential for conflicts on the basis of accepted notions of fairness.

(Refreshments precede the talk at 3:30 in Kerchof 314)

November 17, 2005:
Minimizing the Probability of Lifetime Ruin under Borrowing Constraints Virginia Young (University of Michigan)
4:00pm, KER 317
(Refreshments precede the talk at 3:30 in Kerchof 314)
We determine the optimal investment strategy of an individual who targets a given rate of consumption and who seeks to minimize the probability of going bankrupt before she dies, also known as lifetime ruin. We impose two types of borrowing constraints. First, we do not allow the individual to borrow money to invest in the risky asset nor to sell the risky asset short. However, the latter is not a real restriction because in the unconstrained case, the individual does not sell the risky asset short. Second, we allow the individual to borrow money but only at a rate that is higher than the rate earned on the riskless asset. We consider two forms of the consumption function: (1) The individual consumes at a constant (real) dollar rate, and (2) the individual consumes a constant proportion of her wealth. The first is arguably more realistic, but the second is closely connected with Merton's model of optimal consumption and investment under power utility. We demonstrate that connection in this paper, as well as include a numerical example to illustrate our results. [Paper in pdf]

September 27, 2005:
Topological Quantum Computation Paul Fendley (UVa Physics Department)
4:00pm, CAB 424
One consequence of quantum mechanics is entanglement, where the properties of the system as a whole are very different from those of the individual parts. This can remain true even if the parts are very far from each other. Exploiting this behavior makes a quantum computer possible, but making it fault-tolerant is a major difficulty. A radical proposal for avoiding these difficulties is to find a topological quantum computer. Here the computations are done using systems with non-trivial topological properties, so that local disturbances do not affect the computation. I'll explain what all this means, and describe some specific models which exhibit the necessary behavior.

March 24, 2005:
Mathematical Challenges in Chemistry and Biology Mario Geysen, Alfred Burger Professor (Department of Chemistry, UVa)
4:00pm, KER 317
(Refreshments precede the talk at 3:30 in Kerchof 314)
Protein engineering is a numerically challenging problem for which mathematics has provided at least a first pass solution to a practical method of preparing an optimized phage library for the identification of binding peptides. A more difficult case involves the design of a candidate enzyme library aimed at identifying non-natural catalytic proteins. A systematic approach to rediscovering existing chemistry and also identifying possible novel chemistries using highly automated robotic instrumentation benefits greatly from the application of mathematics.

February 24, 2005:
Stable Distributions: Models for Heavy Tailed Data
John Nolan (American University)
4:00pm, KER 317
(Refreshments precede the talk at 3:30 in Kerchof 314)
Stable random variables are the r.v.s that retain their shape when added together. These distributions generalize the Gaussian distribution and allow skewness and heavy tails--features found in many large data sets. We give an overview of univariate and multivariate stable laws. These distributions are now computationally accessible--statistical and computational examples will be given.

February 3, 2005:
Alex Szimayer (University of Western Australia)
4:00pm, KER 317
(Refreshments precede the talk at 3:30 in Kerchof 314)
This paper gives a tree based method for pricing American options in models where the stock price follows a general exponential Lèvy process. We develop a multinomial model for approximating the stock price process which can be viewed as generalizing the binomial model of Cox, Ross and Rubinstein (1979) for geometric Brownian motion. Under mild conditions, it is proved that the stock price process and the prices of American options on the stock, calculated from the multinomial model, converge to the corresponding prices under the continuous time Lèvy process model. Explicit illustrations are given for the variance gamma model and the normal inverse Gaussian process. Our approach overcomes some practical difficulties that have previously been encountered when the Lèvy process has infinite activity.

January 21, 2005:
Multivariate heavy tails, asymptotic independence and beyond
Sidney Resnick (Cornell University)
3:00pm, KER 317
A random vector having a distribution which is multivariate regularly varying at infinity can have a dependence structure which is hard to specify in practice. One extreme situation is "asymptotic independence" which roughly describes the situation where the random vector's components are not simultaneously large. Asymptotic independence makes it difficult to specify probabilities of extreme events unless one makes further assumptions. We discuss one subclass of heavy tailed asymptotically independent distributions where the joint distribution possesses hidden regular variation. We discuss detection of this phenomena along with other extensions into conditional models. Some applications to network and exchange rate data is provided.

December 2, 2004:
No Matter Where You Go, There You Are: Secure Localization Techniques for Mobile Wireless Networks
David Evans (Department of Computer Science, UVa)
4:00pm, location Kerchof 317
(Refreshments precede the talk at 3:30 in Kerchof 314)
http://www.cs.virginia.edu/~evans/talks/sam/
Many wireless networking applications require nodes to know their location, but it is often infeasible to manually configure each node's location or incorporate a GPS receiver in each node. Hence, we consider a localization scheme in which a few seed nodes determine their locations directly, and other nodes estimate their location from the messages they receive from seeds and other non-seed nodes. Although mobility would appear to make localization more difficult, we describe a localization method based on Monte Carlo techniques that exploits mobility to improve the accuracy and precision of localization. We report on results from simulated experiments and consider security issues involved in localization. Seed-based localization techniques can be disrupted by attacks including seed impersonation and replay attacks. A particularly damaging attack is a wormhole attack in which an attacker controlling two transceivers in the network connected by a high quality link can replay packets heard at one location at a different location. We propose a defense that uses properties of the physical world in which computing devices are deployed to mitigate wormhole attacks.

November 19, 2004:
Minimal Description Risky Asset Modeling
Chris Heyde (Columbia University)
3:15pm, Cabell 345
The geometric Brownian motion (Black-Scholes) model for the price of a risky asset which provides the theoretical underpinning of the huge financial derivatives industry stipulates that the log returns are independent and identically distributed and Gaussian. However, the empirical evidence shows the distribution to be leptokurtic (higher peak and heavier tail than the Gaussian) and it also reveals strong dependence. The talk will be concerned with (1) the stylized features which one finds in real data and how these can be incorporated into models which are still simple enough to be practically useful, (2) the controversy over choice of the distribution of returns and (3) some of the challenging mathematical questions which arise, including issues of scaling, fractal dimension and self-similarity.

October 14, 2004:
A Survey of Recent Monte Carlo Techniques
Todd Williams (Sagamore Hill Capital Management)
KER 317 (Refreshments precede the talk at 3:30 in Ker 314)
In this talk, we will give a short introduction to Monte Carlo simulation and discuss common variance reduction techniques, some of which may surprise our audience. We will also discuss some financial applications, including path-dependent derivative securities and computing hedge ratios (the "Greeks"). Finally, we will present recent advances in pricing American-style derivative securities, in which holders have the right to exercise early. These recent techniques involve formulating dual problems.

Monday, April 19, 2004:
Microstructural Biases in Empirical Tests of Option Pricing Models
Patrick Dennis (UVa, McIntire School of Commerce)
Cabell 215, 4:00pm (refreshments precede the talk at 3:30pm in KER 314)
This paper examines how noise in observed option prices arising from discrete prices and other microstructural frictions affects empirical tests of option models, and techniques for backing out implied risk-neutral parameters from option prices. Using a conservative estimate for the amount of noise, we demonstrate that it can be very difficult to distinguish alternative option models using a cross-section of noisy option prices. We also find that the accuracy of traditional implied volatility calculations, implied volatility regression forecast tests, the implied risk-neutral moment estimators of Bakshi, Kapadia, and Madan (2003), and the univariate diffusion test proposed by Bakshi, Cao, and Chen (2000) are all quite sensitive to the amount of noise in option prices. Our results suggest that even in active, liquid markets such as the S&P500 index option market, observation error significantly reduces the power of tests, and in some cases represents an important source of bias. The problem tends to be much more severe for less active options on lower priced stocks.

Thursday, February 5, 2004:
Protein structure determination via x-ray
Jeffrey Roach (University of North Carolina, School of Medicine)
4:00pm, Rouss Hall 202
Structural studies are responsible for much of our current understanding of protein science and enzymology. In spite of significant advances in other techniques, X-ray crystallography remains one of the most effective and commonly used methods of structure determination. The method consists of placing a protein crystal in an x-ray beam and measuring the intensity of the diffracted rays.

Assuming that the density of electrons in a crystal can be reasonably approximated by a periodic function in three dimensions, the square of the magnitudes of the Fourier coefficients of the electron-density function are proportional to the intensity of diffracted x-rays. Unfortunately, the argument or "phase" of the Fourier coefficient is not easily captured.

Mathematically speaking the crystallographic "phase problem" consists of determining suitable arguments to accompany the measured magnitudes of the Fourier coefficients of the electron-density In this limited context, without any further information regarding the nature of the function, any set of phases for the Fourier coefficients is as valid as any other set. Physically, of course, the reconstructed function corresponds to the electron density of some real molecular system; therefore, certain chemical constraints must be satisfied. Associating a phase set with a particular molecular model provides a means to evaluate different potential phase sets. Phase sets derived from molecular models that are chemically valid and produce Fourier coefficient magnitudes coincident with the measured data are superior to those than either lack chemical validity or would produce significantly different diffraction data.

In this talk we will discuss the mathematics of diffraction and structure determination paying special attention to applications of the representation theory of locally compact abelian groups and compact nonabelian groups.

Thursday, December 4, 2003:
An Information Theory of Biological Networks with Feedback
Toby Berger (Cornell University; currently UVa, Department of Electrical Engineering)
4:00pm, Physics 203 (Refreshments in Math Common Room, 3:30pm)
Abstract (pdf file)

Thursday, November 6, 2003:
Use of Simulation Models in Early Phases of Studying an Environmental Problem
George Hornberger (Dept. of Environmental Studies and Associate Dean for the Sciences)
4:00pm, Physics 203 (Refreshments in Common Room, 3:30pm)
Models used to describe the dynamics of environmental systems are typically simulations based on concepts of transfer of matter and energy between "compartments" of the system. These simulation models often are a set of differential equations with at least a modest number of unknown parameters. Thus, the utilization of a simulation model at an early stage of research on an environmental problem, the causes of which are poorly understood, is usually infeasible because of a lack of data. That is, the behavior of the system that defines the observed problem is known at least qualitatively, but data for calibrating a model are not available. If the problem of interest is generically similar to others that have been reported in the literature, however, a considerable amount of applicable information may be available and, if so, the parameters of a simulation model may be specified via a priori statistical distributions. Once parameter distributions have been set, Monte Carlo experiments can be utilized to examine whether the model is able to simulate the salient qualitative aspects of the problem-defining behavior. One objective of such an approach might be to identify areas of critical uncertainty in knowledge of the system and thereby to derive information that might be useful in focusing data collection. The problem of cultural eutrophication in Peel Inlet, Western Australia, where the system behavior of interest is the excessive growth of the alga Cladophora serves to illustrate the application of the method.

Thursday, October 16, 2003:
Understanding Biological Computation Via Constraint-based Optimizations
William B. Levy (Professor of Neurosurgery & Psychology, UVa School of Medicine)
4:00pm, location TBA
The brain is an information processing system whose fundamental elements are cells called neurons. Signals between neurons are usually binary, and the signals within a neuron are usually analog. One way to understand neuron-based information processing is to analyze the transmitted, transformed, and recirculated signals via the methods of Shannon's information theory and via methods that are essentially at the intersection between information theory and Bayesian statistical theory. But there is more to consider.

In general, physics constrains computation. The notable constraints include the available time, energy, and space (including number of computational elements), as well as the rates of information transmission attainable by individual elements. Although living organisms are not optimal, they are the result of an optimizing process. Because (1) the random design process of natural selection has had 800 million years to respond to computational constraints at the microscopic level and 150 million years to work on some not-so-microscopic aspects peculiar to mammalian computation and because (2) the microscopic environment of neurons has remained essentially constant over the eons, we assume that, microscopically, neuronal computation achieves an optimal compromise among the computational constraints. These ideas lead to the following research strategy to understand neurobiological computation:

Optimized computational constraint formulations that predict observed biology are evidence favoring these formulations as a valid theoretical perspective while formulations that produce incorrect predictions are presumed invalid.

The talk will explain these statements with examples.

Monday, April 28, 2003:
Mathematical Modeling of Mechano- and Chemo-Sensors in Arthropods
Joseph A. C. Humphrey (Dept. of Mechanical and Aerospace Engineering)
4:00pm, Clark 108

Thursday, March 27, 2003:
Dynamic Pricing and Learning in Electricity Markets
Alfredo Garcia (Dept. of Systems and Information, UVa); joint work with Enrique Campos and Jim Reitzes
4:00pm, Ruffner G004C
We analyze the price-formation process in an infinite-horizon oligopoly model where hydroelectric generators engage in dynamic price-based competition. We provide a simple characterization of a Markov Perfect Equilibrium (MPE) in terms of "indifference" prices--i.e., prices that equate the gains from releasing or withholding water. Although the MPE solution represents an equilibrium consistent with dynamic strategic behavior, it requires computational sophistication by market participants. However, under certain assumptions we prove that a simple "learning" procedure converges to the Markov Perfect Equilibrium (MPE).

Monday, February 10, 2003:
Applications of Bayesian Tracking Theory: From Submarines to Terrorist Operations
Christian Hellings and Christopher Boner (Metron, Inc.)
3:30pm, Maury 209
In this talk we will give an overview of some of the technical projects at Metron, Inc. We will discuss two projects in detail and illuminate the mathematical theories and algorithms that each involves. We will begin by introducing the basic theory of Bayesian tracking and describing an implementation that tracks submarine targets using multiple non-linear sensors with high false alarm rates. An extension to "track-before-detect" systems, which involve the monitoring of a region that is not known to contain any targets, will also be discussed. We will then describe work done on a second project to extend Bayesian tracking to track the state of an operation (such as a terrorist plan to carry out a chemical attack). We will illustrate how Bayesian nets may be used to probabilistically incorporate structured, linked data that contains conditional dependencies. If time permits, we will discuss ideas for the application of random graph theory to pattern analysis: identifying small subpatterns that, based on structural analyses, constitute a signature that may be exploited to enhance pattern matching/data mining efficiency.