*Is it harder for a mathematician to learn biology or for a biologist to learn mathematics?*

**A**s director of the Institute for Pure and Applied Mathematics (IPAM), a national research institute based at the University of California, Los Angeles (UCLA), and funded by the National Science Foundation (NSF) that hooks up mathematical scientists with researchers from other disciplines, I get asked this question all the time. The first time someone asked me, I wondered: Which would be easier, explaining a difficult mathematical concept to a biologist, or trying to get a mathematician to grasp the rudiments of the polymerase chain reaction? The answer I give now is this: What's easier than either of these alternatives is for both the biologist and the mathematician to learn enough of each other's language to be able to talk to one another.

If I had one piece of advice to give to young people interested in interdisciplinary work, it would be this: You don't have to know everything. The more areas you develop deep insights into, the better, but just a feel can be sufficient. The crucial skill is dialogue. This involves humility--not a quality researchers in either mathematics *or* biology are known for--and a willingness to meet your opposite number on their own turf.

An interdisciplinary, dialogue-based research center like IPAM cannot exist in isolation; fortunately, we are located in the middle of the UCLA campus, which also hosts the medical school. UCLA has developed a climate on campus that is unusually well suited to interdisciplinary research. To back this up, there is an infrastructure of interdisciplinary training grants (such as multiple IGERTS) and interdisciplinary research institutes.

The other question I get asked a lot by young mathematicians is this: *What is the right kind of mathematics to learn if I'm interested in applications to biology?*

For some areas of biology there is a specific answer. We know at least some things that work. For others areas, however, the only answer is, "I wish we knew." And, almost by definition, those areas where we don't know the right math yet--where the right math probably hasn't even been developed yet--are the most exciting to work in.

**Various IPAM programs and partners**

IPAM works in various modalities. We have 3-month-long programs that bring together a mix of senior and junior researchers from several disciplines. Starting with tutorials where representatives of each community give an overview of its perspectives, problems, and tools, we move on to a series of workshops covering important areas of potential interaction. Funding is available for postdocs and grad students to spend 3 months at IPAM, and those who have attended come away with an incredible array of knowledge and contacts. Upcoming programs include "Proteomics: Sequence, Structure, Function" this 8 March to 11 June, and "Cells and Materials: at the Bioengineering Interface," which will happen in Spring of 2006. The details of these and all of the programs mentioned here can be found at www.ipam.ucla.edu.

IPAM also holds a graduate summer school. This summer, the topic will be "Mathematics in Brain Imaging." This 2-week program will have lectures by world leaders on computational anatomy (12 to 16 July) and functional brain mapping (19 to 23 July).

For undergraduates, IPAM holds a Research in Industrial Projects for Students program for 9 weeks every summer; this year it's 27 June to 27 August. Here, students with a mathematical background work on real-world projects that companies and national labs want solutions to. Recent sponsors range from Pixar to Protein Pathways.

IPAM's biological partners on campus are numerous. Two marvelous groups do brain mapping: the Laboratory of Neuro-Imaging and the Crump Institute. What kind of math is involved? One issue--which comes up in all of the sciences--is how to get the clearest image out of an imaging device such as positron emission tomography (PET) or magnetic resonance imaging. Perhaps the most powerful approach in recent years was pioneered by mathematicians Stanley Osher at UCLA and James Sethian at UC Berkeley: the level set method. UCLA mathematicians Tony Chan and Luminita Vese detect anatomical brain structures by lassoing them with a loop, which then shrinks tightly around the structure using a level-set algorithm to capture its exact shape.

What is surprising in this method is that the mathematics of shape, in the form of what differential-geometers call the curvature flow, is coupled with techniques that originated in the numerical solution of partial differential equations to deblur and denoise an image. It doesn't hurt that Michael Phelps at UCLA is coinventor of PET imaging and was a pioneer in realizing that what had been an abstruse mathematical tool--the Radon transform--allows three-dimensional reconstruction of a living brain. Phelps combined this with the insight that positron emission allows detection of metabolic processes in the brain as they are happening.

Equally fascinating is the problem of how to compare two brains. This could, for example, be a normal brain and a diseased brain, or the same brain at two different stages of development. A successful approach is to stretch one brain image to fit the other, trying to minimize the sum of two terms-a "penalty term" for how much stretching goes on, and a "mismatch term" for how different the two brains are after the stretching. What is the optimal penalty term? Where should it come from? Some of the best work has been done by Arthur Toga and Paul Thompson here at UCLA, two neuroscientists of exceptional mathematically savvy. What makes this a great area to work in is that there is a lot still to be done, in this area as in so many others.

Microarrays allow researchers to acquire data about expression levels of thousands of genes at one fell swoop. But what do you do with the data once you have it? Can you classify tumors and even recommend treatment using microarray data? Is being expressed simultaneously a sign that two proteins are metabolically linked? So far, the greatest successes in this area have come from the application of statistical methods, but new techniques are giving them a run for their money. In Stanley Nelson's lab at UCLA, mathematician Barry Merriman has developed promising new methods of analysis.

Another tack, taken by Christopher Lee, has been to develop powerful algorithms to search databases of genomic data for new single nucleotide polymorphisms, many of which have subsequently been confirmed using microarrays in Nelson's lab. Statistician Ker-Chau Li has found a cubical generalization of the correlation function, which he has used to detect, automatically, candidates for three-way interactions of proteins using microarray data. These are interactions that are more complicated than "A suppresses B" or "A and B are both expressed at the same time and suppressed at the same time."

In David Eisenberg's lab, statistical techniques have been used to find clues for which proteins are part of the same metabolic pathway, creating an enormous graph of potential interactions. What is especially interesting to a mathematician in this work is the challenge of bringing together data of very different types--a challenge that forms a crosscutting theme in many applications of mathematics and statistics. Undergraduate students in IPAM's summer program used combinatorics to analyze some of Eisenberg's data.

**Genetics, statistics, and Monte Carlo**

UCLA has strong biomathematics and biostatistics departments, both of which are active partners of IPAM. It also helps that the chair of UCLA's human genetics department, Kenneth Lange, was trained as a mathematician. Lange has worked on an amazing range of projects, for example creating a highly effective way to do genetic epidemiology by unraveling how a genetic defect was passed along through a complicated community genealogy. Tom Chou in the biomathematics department brings math to bear on problems ranging from the statistical mechanics of protein deformation to DNA recombination. Elliot Landaw uses differential equations to model transport across the blood-brain barrier and applies methods of adaptive control to chemotherapy for cancer.

James Lake managed to tease out information about the original separation of eukaryotes from prokaryotes using genomic data. Janet Sinsheimer from the biostatistics department used phylogenetic methods to determine the rate of evolution of HIV. Chiara Sabatti uses Bayesian statistics and Markov Chain Monte Carlo methods--known affectionately as MCMC to devotees--to analyze datasets from across the genome. Mathematician James White made the fortunate discovery that the same abstract pure mathematics that he used in his thesis unlocks some of the secrets of the geometry of circular DNA. Even UCLA's School of Dentistry counts as a serious player, with a collaboration between Benjamin Wu, DDS-PhD, and mathematician Stan Osher to determine, using partial differential equations, the optimal shape for a scaffold on which to grow tissue for transplants.

It would be really intriguing, since I am a mathematician, to try to draw a graph of all of the collaborations that have gone on at UCLA between mathematicians, statisticians, and computer scientists, on one hand, and biologists and medical researchers on the other. Many of those links pass through IPAM, as will even more as time goes on. Because nobody really knows where the next big biological breakthrough will come from, or what next area of math biologists will want to use, a young person who wants to try a life at the interface between these two exploding fields should go to a place with many interactions between the two groups, and where there is a focal point where the two groups can come together. This is an exciting time to be a young researcher, and the prospects for synergy between math and biology are staggering. Please consider this an invitation to become a part of this. See you at IPAM.