Victor Camacho

University of Utah
Department of Mathematics
Office: LCB 317
camacho@math.utah.edu



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Welcome to my World!

Introduction

I am a sixth year graduate student at the University of Utah studying mathematical biology. I completed my masters degree in May of 2009 and anticipate completing my PhD in 2013. I received my Bachelors of Science degree at Harvey Mudd College in Claremont, CA where I majored in mathematics and participated in varsity cross country and track. While I plan to return to California some day I anticipate living in Salt Lake City for at least two more years or until I complete my PhD.

Mathematical Biology

In a nutshell, mathematical biology is the field in which mathematicians with some background in biology attempt to use mathematics to answer certain questions about biology. We are mainly concerned with modeling how systems change over time, such as population growth, the spread of disease, blood flow, ion exchange dynamics in cells, immune response, neural networking, muscle movement and MUCH more!! Scientists and mathematicians have spent the last 350 years perfecting chemistry and physics to the point that we have very accurate mathematical models to describe a great majority of the phenomena studied in these fields. We have such a rigorous understanding of mechanical physics that we can calculate how stable a bridge or building will be before building it. A computer can calculate the course of a plane flight or a space shuttle mission before take-off. We can chemically engineer materials materials that can withstand high amounts of force, friction, temperature changes etc. Yet in biology, we are not yet at the point where we have the power to make such predictions. For example, our mathematical models in biology are not yet sophisticated enough to accurately calculate how a series of chemo-therapy treatments will affect a patient's body. These are the types of problems we work on in mathematical biology, a subject that is still in its infancy but will have great potential in the future.

Society and Mathematics

I find it difficult to discuss my academic interests with most people because the average U.S. citizen gives up on math too easily. When I tell people I am a math major the most common responses I hear are: "I hate math!", "I am terrible at math", "Why would you want to major in math?", "What can you possibly do with a degree in math?". These types of responses are understandable comming from people who have not studied math as extensively as I, but whether they believe it or not, everybody has the capacity to understand mathematics and use it to their advantage in the real world. These days it is more important than ever to be able to use and understand basic mathematical principles.

After high school we are normally left with the false impression that math nothing more than a mind excercise and has no practical value in everyday life. While it is true that studying and understanding complicated mathematical concepts can improve a person's intellegence, it is also true that mathematics has played a significant roll in almost every major breakthrough in human history. Mathematics has not always been at our disposal, it has been developed slowly throughout the centuries by extremely creative minds who sought new ways of understanding and sovling real world problems. When we learn of these creative methods in school, they are taught as boring, mechanical approaches for arriving at the solutions of very particular types of problems. If instead we were given a real world problem initially, before learning a systematic method for solving it, we would develop a much greater appreciation for the usefulness of the method as well as understand the great deal of creativity that was required to develop it. Mathematical creativity does not end when a particular branch of mathematics is developed, rather it becomes a tool to inspire more creativity among those who are trained to use it. Pure mathematics is the art of developing new and more powerful mathematic tools while applied mathematics is the art of using these tools in creative ways to solve real world problems.

In order to get ahead in today's world most we require a much better understanding of mathematics than our ancestors. Imagine in ancient times how useful it would have been for a merchant to know how to add, subtract, and multiply. It would have benefited even a common person to understand how to count, measure, and have a system for recording quantities. Around the time of the industrial revolution society began to rely much more heavily on mathematics. Not only were scientists and mathematicians developing new mathematical tools in order to understand the new challenges that faced them, but the average person would be greatly benefited by understanding how interest is compounded, taxes, currency exchange, inflation, productivity, scheduling and so on. Today we live in the information age, a time when mathematics and science is very highly developed, calculators can perform basic mathematical operations, and computers allow us to access many information from around the world. We are bombarded by mathematics far more than most people realize. Anytime we purchase or refinance a home, or purchase a new car we typically have a choice between different loan types. Anytime we invest money we must consider both the risk and potential return of our investment. When making purchases with certain companies on your credit card you are offered reward points for each dollar spent but are also charged a high interest rate on each dollar owed. When on a diet you must keep careful track of such things as calorie, fat, sodium, cholesterol, carbohydrate and protein intake. One could certainly get by in today's world with only a modest understanding of mathematics, but imagine how much more you can achieve during your lifetime and how much money you could save with a deeper understanding of mathematics.


A World Without Mathematics

Many inventions and human acheivements were origianally conceived of and constructed without the use of mathematics. However, most of these inventions or ideas were not perfected to the form we are accoustomed to today until their interworkings could be explained or modeled mathematically. Imagine for a moment how the world would be different in the absense of mathematics. Let's work backwards through time and think of all the major human acheivements that we would be without.

- Digital Systems: Anything that is digital relies on a binary number system, birnary operations, and binary representations of characters, programming code, sound bytes, graphics and so on. If we were not sophisticated enough to operate in binary then we could not have invented anything digital. Thus computers, software, the internet, cell phones, digital cameras, digital guidance systems, and many other things invented as a result of computer technology would not be here.

- Combustion Engine: While somebody may have realized, without the use of mathematics, that natural gases are combustable, it was a very deep mathematical and scientific endeavor to invent the combustion engine. All four stages of the combustion cycle require very careful calculations in order for the whole process to work properly. The engine could not have been designed or built without the use of mathematics and the application of well established physical principles. The combustion engine is the necessary mechanism for producing the majority of the world's energy, whether it is mechanical or electric. Without such efficient production of energy the economies of the world and our ability to travel and communicate would be greatly diminished.

- Radio Waves: Marconi's invention of the radio in the late 19th century was an amazing breakthrough. He demonstrated how one could send an electromagnetic pulse through the air and receive it in a remote location. Morse code was the first step toward using the radio to tansfer useful informaiton from one source to another. However, in order to transmit sound we needed a standardized way of translating information into radio signals and then transcribing them at the other end. The notion of amplitude moduluation (AM) and frequency modulation (FM) are two ways in which this is accomplished, both of which would have been impossible to develop without our modern knowledge of sines, cosines, and Fourier transformations. The ability to send information through the air and receive it somewhere else instantaneously was completely revolutionary and tremendously enhanced our ability to communicate across the globe. Without our highly mathematical system for transmitting and receiving information through radio waves, we would be without televisions, radios, two way radios, radar systems and many other inventions including sonar devices that operate using similar mathematical formulations.

- Electricity: The effects of static electricity were known to the ancient Greeks and Phonecians long before the birth of Christ. However, electricity was not truely harnessed until the brilliant scientist and applied mathematician, Benjamin Franklin, did extensive research on electricity. Through his experiments he developed mathematical models to describe the governing dynamics of electricity. In order to use electricity, physicists needed to be able to quantify things like charge, voltage, current, capacitance, and inductance, as well as understand the mathematical relationships among them. Later scientists and mathematicians such as Alessandro Volta, Adre-Marie Ampere, Georg Simon Ohm, and Michael Faraday made tremendous contributions to our mathematical understanding of electricity. Without such a thorough understanding of electrodynamics, ancient civilizations were only able to observe the effects of electricity, they could not use it to their advantage. Thus all electrically powered devices could not exist in a world without mathematics. We would not have electric lighting, refrigerators, radios, telephones, television, telegrams, and many other important devises that we rely on today.

- Steam Engine: As with the combustion engine, the design of the steam engine required careful calculations and a deep understanding of the physical behavior of water vapor. Important physical principles, such as the ideal gas law, were necessary to develop working steam engines of different sizes. As many know, the steam engine is responsible for initiating the industrial revolution. For the first time in history we had a non-living and portable source of power which could generate the mechanical energy necessary to run very large machines that could do useful work. This work could be done with great speed, precision and consistency. Before the industrial revolution there was no industry, instead people would produce goods in their own homes and trade with other people. Also, without a steam engine we could not have invented the train which was responsible for transporting goods and raw materials throughout many nations thus improving the rate at which they could develop their infrastructure.

- Siege Weapons: It is believed that catapults and ballistas where invented around 500 BC, but at that time they were crude and inaccurate. It was not until the time of Archimedes (200's BC) that very effective siege weapons were built. Archimedes is revered as one of the three greatest mathematicians to have ever lived. He derived mathematical models for lever and pulley systems which he could then use to build very large but precise weapons. Among these was a greatly improved and accurate catapult system, as well as a mechanical crane which could lift entire ships out of the water with the slightest effort. While siege weapons were originally intended as devices of war, it was not long before civilizations found ways to use the crain and catapult to aid in the construction of magnificent works of architecture. Imagine how our ability to design structures would diminish without the use of cranes and pulley systems. Even many modern day inventions such as the mountain bike, the jack, and the transmission of a car were designed using principles discovered by Archimedes over 2000 years ago!

- Architecture: One can certainly argue that basic structures such as huts or tents do not require any knowlege of mathematics. However, imagine the great precision of measurements and geometric genius that was needed to construct ancient structures such as the Pyramids, the Great Wall of China, the Mayan temples, the Aqueducts, the Parthenon, and the numerous other ancient structures that are scattered around the world. Such wonders were conceived by ancient architects who employed the mathematical tools available to them at the time. Basic geometric knowledge such as the pythagorean theorem, similar triangles, ratios, trigonometry and an approximation of pi were extremely useful in ancient architecture. Though not all of these ancient civilizations could provide logical proofs for the mathematical concepts, they used them to their advantage all the same. People did not always know about geometry, it was developed by brilliant minds who had revolutionary ways of thinking about the world. Ancient architecture serves as the basis of our modern architecture which employs much more complicated but powerful mathematical tools.

- Economics: Economics is the corner stone of civilization. It allows for a large group of people to live together peacefully as they can have a fair means to trade with one another. Primitive economies relied heavily on the use of basic mathematical operations and the ability to represent quantities with numbers. The decimal system took thousands of years to perfect. We take for granted the fact that we can represent the number 123 by writing down the digits 1, 2 and 3 in sequence rather than writing 123 dots on a page. Ancient people needed a universal representation of quantities that everybody could understand, thus began the developement of the language of mathematics. It was inefficient to write down 123 dots, so more intellegent and sophisticated methods for representing numbers were invented. With the developement of a standardized number system, entire civilizations could not only count and discuss qnatities of objects, they could also discuss measurements of basic things such as length, weight, time etc. Mathematics greatly enhanced the amount of information people could communicate and allowed governments to set economic rules that were fair and understood among all citizens. Without very basic mathematical principles an economy would not be fair, nor would citizens understand how to make it fair except on extremely small scales. Large civilizations could not remain stable without a fair economic system that is communicated in the language of mathematics.

If you remove all of the above human accomplishments from existence, we are essentially reduced to small groups of hunter/gatherers. The important lesson to be learned here is that math is extremely useful and it is constantly being improved. Like any other field of study, there is a reason to do mathematical research because there is more yet to be discovered. There will always be a need for more sophisticated mathematical tools in every area of science. My particular field of study is mathematical biology. If you have ever taken a course in biology you will have noticed that a background in math was not necessary, unlike physics and chemistry. The reason is that we have developed the necessary mathematical tools to explain almost all physical and chemical phenonina. It is much less obvious how to explain most biological phenomena in terms of mathematics. However, many believe that we should be able to mathematically model biological systems. There is a great movement among biologists and mathematicians to perfect the science of biology by introducing more mathematics into known biological models. Perfecting biology in this way will greatly enhance our understanding of modern medicine, DNA, evolution, protein structures, cell functions, diseases, environmental effects of global warming and much more. Mathematical biology will change our perspective on the world, just like the many great breakthroughs of the past.