New Tools for Old Problems
When mathematicians notice connections between two distinct areas of their discipline, you can be sure something interesting will develop. That is exactly what happened at a recent workshop at the American Institute of Mathematics (AIM). The scientific advisory board of AIM noticed that new methods in ergodic theory, a subject arising from mathematical physics, may have applications in other...
Honeybee Olfaction
A honeybee may forage on 1,000s of flowers for nectar and pollen in its lifetime. Scent is one of the primary means that it uses for identifying rewarding flowers. How honeybees and other animals learn to associate complex and variable scents with important events is still not well understood. Honeybees are an excellent model system for studying olfaction because their physiology and behavior...
Low X-ray Exposure Imaging
The ability to “see” inside a human body has been of tremendous value to medical diagnostic. For example, it is almost impossible to properly diagnose brain tumor without being able to image the patient’s head, and very difficult to plan the complex surgery of removing the tumor without accurate three-dimensional visualization of the brain itself. A prevalent method to see our way inside a...
Stratification Learning
Data in high dimensions is becoming ubiquitous, from image analysis and finances to computational biology and neuroscience. This data is often given or represented as samples embedded in a high dimensional Euclidean space, point cloud data, though it is assumed to belong to lower dimensional manifolds. Thus, in recent years, there have been significant efforts in the development of methods to...
Homological Mirror Symmetry
During the academic year of 2006-2007 the School of Mathematics conducted a special program in algebraic geometry. This subject, with deep classical roots, is one of the most active areas in contemporary mathematics. Especially notable are its interconnections with number theory, mathematical physics and topology. The scientific activities during the year reflected the depth and breadth of...
Traffic Gridlock
In studying traffic flow in a large city grid, the following questions arise. Is there a largest traffic density that permits free flow? Is there a density above which gridlock is inevitable? In 1992 Biham, Middleton and Levine introduced a simplified model for the study of these questions, called the BML model. This simplified model proved to be surprisingly complex – in a series of papers...
Improved Electron Gun Design
Electron beams are employed in vacuum devices to generate very high power microwaves for many critical applications, including amplifiers in communications, television, radar guns and computer monitors, and high energy accelerators. The quality of these electron beams directly impacts the output power and efficiency that can be achieved. In a SAMSI workshop, a group of graduate students was...
Shedding Light on Blackouts
The control of communication and power networks through regulation and deregulated market mechanisms presents tremendous challenges and affects almost every citizen of the United States. Theoretical understanding in this area is built on the mathematical field called stochastic networks, a field whose growth has been closely tied to the Institute for Mathematics and its Applications (IMA). In...
An Eye for Aphids
Soybeans are a key livestock feed in the US, an important human food source in many parts of the world, and–in the form of biodiesel fuel–a promising source of renewable energy. Informed pest management increases yield and reduces pesticide application: farmers use population estimates obtained by counting the aphids on sample soybean leaves in planning their crop dusting schedules. However,...
Expander Graphs
A central computational theme is “parallel processing”: allowing a large number to processors to work simultaneously and thus potentially speed up computation. A major issue for this development is the architecture of the communication network among these processors. The problem of how to design such a network is translated into a mathematical problem of finding graphs called “expanders” which...