The Hardest Math Problem and My Graduate Work in Biophysics
As a graduate student pursuing a PhD in Biophysics - Neurophysiology half a century ago, my research involved solving complex mathematical problems to understand the conductance trajectories in nerve cells. Specifically, I developed several programs to support my research and solve intricate mathematical challenges. This article delves into one of the most significant and multifaceted problems I encountered during my doctoral studies and the innovative solutions I devised.
Setting Up for Success: Access to Advanced Computational Tools
During my graduate studies, I was fortunate to gain access to the PLATO system at the University of Illinois, which utilized a full graphics display terminal powered by a Control Data Corporation supercomputer capable of 2 Mflops. This powerful system allowed me to perform complex calculations and develop interactive real-time environments for my work.
A Program for Differential Equations and Curve Fitting
One of the critical aspects of my research was to develop a program that could solve up to six differential equations and display the results on a customized axes. Simultaneously, I created another program for non-linear curve fitting, enabling me to work with up to six parameters and visualize both the data and the resulting fitted curve. These tools allowed me to gain deep insights into the conductance dynamics of nerve cells, which was essential for my research.
Experimental Protocol and Data Management
In addition to these analytical tools, I developed programs and hardware that would specify and automate the experimental protocol for 'voltage clamping' a nerve membrane. My experimental protocol included 300 steps, each to be executed six seconds apart, and involved complex interactions with physiological parameters. To manage the resulting data, I created a database system that treated sampled signals as vectors, allowing for efficient storage and manipulation of the large amounts of data collected.
Predictive Modeling and Data Analysis
To predict system outputs from specified inputs, I implemented a program to perform convolution integrals. This program was an interactive real-time environment that enabled me to simulate and analyze the conductance trajectories and other key parameters. These tools were instrumental in advancing my understanding of the underlying biological mechanisms and contributing to my research goals.
An Unsolved Problem in Applied Mathematics
While my research primarily dealt with applied mathematics, I also encountered unsolved problems that sparked my curiosity and inspired further investigation. One such example is the problem of finding a formula for the electrical capacitance of a conducting sphere placed close to a large conducting plane. This is a classic problem in applied mathematics, requiring a particular solution to equations describing electro-magnetic fields. My friend attempted to solve this problem by employing the method of 'images,' which involved defining a collection of point charges that would create an identical electric field.
The Solution and Its Limitations
Although my friend managed to find a solution using the method of 'images,' he realized that an infinite number of point charges were required to achieve perfect accuracy. He resorted to using a computer to approximate the solution by summing the effects of the smallest few thousand point charges, resulting in a good enough approximation. Unfortunately, he never found a general formula to express the solution in a closed form. This problem remains 'solved' in the sense that a workaround exists, but it remains interesting due to its underlying complexity and open-ended nature.
The Role of Pride and Curiosity in Mathematical Discovery
As a mathematician, I don’t use words like 'hard' or 'easy' to describe problems. An unsolved problem may have a simple but undiscovered solution. Sometimes, encountering a proof for a previously unsolved problem can suggest a new approach to an old challenge. Some mathematicians consider any solved problem as 'trivial' and any unsolved one as 'interesting."