The Undergraduate Summer School provides opportunities for talented undergraduate students to enhance their interest in mathematics. This program is open to undergraduates at all levels, from first-year students to those who have just completed their undergraduate education. Based on the backgrounds of the accepted students, they will be divided into two groups - introductory and advanced. There will be several activities organized for these groups, with some specifically intended for either the introductory or advanced group. There will be time for study groups and individual projects guided by advisors, as well as other activities.

2005 Course Descriptions:

1. Introductory Course: Dynamics, diseases and diversity
Fred Adler, Professor, University of Utah

Mathematical biologists study a wide range of biological processes, including population growth, physiology and genetics.  The underlying dynamics are mathematically interesting due to the strong positive and negative feedback among the many inter-connected components of complex living systems.  This course will cover three broad areas.  Firstly, we will use differential equations to study how diseases spread, do damage, and eventually die out.  Secondly, we will use discrete-time dynamical systems to study how consumers, such as ourselves, interact with biological resources with potentially distressing consequences.  And finally, we will use some elegant, but elementary, methods of probability theory to study how genetic systems maintain diversity, and show how modern genetic methods cast light on problems throughout biology.

2. Advanced Course: An Introduction to Cancer Modeling with Optimal Control
Lisette de Pillis, Professor, Harvey Mudd College

The modeling of cancer growth and treatment is one that does not admit only narrow knowledge, but requires skills from multiple disciplines.  This field of study lies at the intersection of biology and medicine, with mathematics at the core.  Cancer development and the dynamics of the immune system have been a significant focus of mathematical modeling in recent decades.  Immunotherapy, a treatment approach that enhances the body's natural ability to fight cancers, is becoming increasingly prevalent in many multi-stage treatment programs that also include chemotherapy, radiation, and surgery.  The critical importance of the immune system in combating cancer has been verified both clinically and through mathematical models.

In this course, we will begin with an overview of the growing field of cancer modeling, surveying the broad number of mathematical techniques that have been taken to attacking this large and complex problem.  We will then focus on specific models of cancer at the cellular level that include immune system responses, chemotherapy and immunotherapy.  Model dynamics will be explored through bifurcation analysis techniques and numberical experiments.  We will then introduce the calculus of variations and the mathematical theory of optimal control, which will be applied to these cancer models to determine theoretically improved treatment protocols.

3.  Special Supplementary Course:  The Mathematics of Phylogenetic Trees

Elizabeth S. Allman, University of Southern Maine, and John A. Rhodes, Bates College

Until recently, the inference of the evolutionary history of currently living species was based primarily on painstaking studies of their morphological similarities, together with comparison to the fossil record.  Now a vast new source of evolutionary data is available through genetic sequencing.  While similarities in DNA sequences among species suggest close ancestoral relationships and differences suggest greater evolutionary divergence, how to infer an entire evolutionary tree from biological sequences is a rich mathematical question.

This course begins with an overview of the sorts of biological questions of interest, and a look at thenature of biological sequence data.  We then develop several of the modern approaches to sequence-based phylogenetics, focusing on the modeling of the process of molecular evolution along a tree.  Shortcomings of the various methods and models, both theoretical and practical, will be used to motivate new ones.

Necessary mathematical and biological background will be kept minimal: basic probability and linear algebra are helpful but can be picked up along the way.  The course will also include elements of combinatorics, algorithmics, Markov models and statistics, as well as hands-on computer work with real and simulated data.

The Undergraduate Summer School is supported in part by the National Security Agency and in part by the National Science Foundation grant no. 0437137.