Need to Tackle a Complex Problem? Applied Mathematics Can Provide the Solution

You’ve likely used math to solve common tasks, like figuring out a restaurant tip or calculating a room’s square footage. But how does math contribute to tackling more complex challenges, such as finding a cure for a disease?

As an applied mathematician, I use mathematical tools to analyze and solve complex biological problems. My work has involved studying gene and neural networks, such as cell interactions and decision-making processes. To tackle these challenges, I translate real-world scenarios into mathematical terms—a process known as modeling.

Mathematical Modeling in the Real World

If you’ve ever solved an arithmetic problem about train speeds or grocery costs, you’ve engaged in mathematical modeling. However, with more complex issues, simply framing the real-world scenario as a math problem can be challenging. This process demands creativity and a deep understanding of the problem, often involving collaboration between applied mathematicians and scientists from various fields.

For instance, a game of Sudoku can be represented as a mathematical model. In Sudoku, the player fills empty boxes with numbers from 1 to 9, following rules such as no repeated numbers in any row or column.

Once a mathematical problem is created to represent a real-world situation, the next step in modeling is solving that problem.

Diverse Mathematical Tools for Complex Problems

For Sudoku, this involves solving a system of equations with 81 variables. In the aspirin example, it requires solving an equation that describes how concentrations change over time. This is where various areas of math, such as algebra, analysis, combinatorics, and others, come into play. These fields, often used in combination, help tackle the complex math problems that arise from real-world applications.

The third step of modeling is translating the mathematical solution back into the context of the original problem. For Sudoku, the solution provides the correct number for each box in the puzzle. In the aspirin example, the solution would be a set of curves showing the concentration of aspirin in the digestive system and bloodstream over time. This is the essence of applied mathematics.

The Challenge of Unsolvable Problems in Applied Math

Or is it? While this three-step process represents the ideal workflow in applied math, reality is often more complex. When I reach the second step and need a solution for the math problem, it’s common—if not typical—to discover that no one knows how to solve it. In some cases, the necessary mathematical tools to analyze the problem don’t even exist yet.

For instance, analyzing cancer models is challenging because the interactions between genes, proteins, and chemicals are far more complex than the relationships between boxes in Sudoku. The main challenge is that these interactions are “nonlinear,” meaning the combined effect of two inputs isn’t just the sum of their individual effects. To tackle this, I’ve been developing new methods to study nonlinear systems, such as using Boolean network theory and polynomial algebra. With these and more traditional approaches, my colleagues and I have explored areas like decision-making, gene networks, cellular differentiation, and limb regeneration.

When dealing with unsolved problems in applied math, the line between applied and pure mathematics often blurs. Concepts once seen as highly abstract have proven essential for addressing modern challenges. This underscores the value of math for everyone—today’s pure mathematics could become tomorrow’s applied math, offering the tools needed to solve complex, real-world problems.

Read the original article on: Phys Org

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