1. The Primes
The properties and patterns of prime numbers — whole numbers that are divisible only by themselves and one — have been a source of wonder across cultures for thousands of years, and the study of prime numbers is fundamental to mathematics. This unit explores our fascination with primes, culminating in the million-dollar puzzle of the Riemann Hypothesis, a possible description of the pattern behind the primes, and the use of the primes as the foundation of modern cryptography.
2. Combinatorics Counts
Counting is an act of organization, a listing of a collection of things in an orderly fashion. Sometimes it's easy; for instance counting people in a room. But listing all the possible seating arrangements of those people around a circular table is more challenging. This unit looks at combinatorics, the mathematics of counting complicated configurations. In an age in which the organization of bits and bytes of data is of paramount importance — as with the human genome — combinatorics is essential.
3. How Big Is Infinity?
Throughout the ages, the notion of infinity has been a source of mystery and paradox, a philosophical question to ponder. As a mathematical concept, infinity is at the heart of calculus, the notion of irrational numbers — and even measurement. This unit explores how mathematics attempts to understand infinity, including the creative and intriguing work of Georg Cantor, who initiated the study of infinity as a number, and the role of infinity in standardized measurement.
4. Topology's Twists and Turns
Topology, known as "rubber sheet math," is a field of mathematics that concerns those properties of an object that remain the same even when the object is stretched and squashed. In this unit we investigate topology's seminal relationship to network theory, the study of connectedness, and its critical function in understanding the shape of the universe in which we live.
5. Other Dimensions
The conventional notion of dimension consists of three degrees of freedom: length, width, and height, each of which is a quantity that can be measured independently of the others. Many mathematical objects, however, require more — potentially many more — than just three numbers to describe them. This unit explores different aspects of the concept of dimension, what it means to have higher dimensions, and how fractional or "fractal" dimensions may be better for measuring real-world objects such as ferns, mountains, and coastlines.
6. The Beauty of Symmetry
In mathematics, symmetry has more than just a visual or geometric quality. Mathematicians comprehend symmetries as motions — motions whose interactions and overall structure give rise to an important mathematical concept called a "group." This unit explores Group Theory, the mathematical quantification of symmetry, which is key to understanding how to remove structure from (i.e., shuffle) a deck of cards or to fathom structure in a crystal.
7. Making Sense of Randomness
Probability is the mathematical study of randomness, or events in which the outcome is uncertain. This unit examines probability, tracing its evolution from a way to improve chances at the gaming table to modern applications of understanding traffic flow and financial markets.
8. Geometries Beyond Euclid
Our first exposure to geometry is that of Euclid, in which all triangles have 180 degrees. As it turns out, triangles can have more or less than 180 degrees. This unit explores these curved spaces that are at once otherwordly and firmly of this world — and present the key to understanding the human brain.
9. Game Theory
Competition and cooperation can be studied mathematically, an idea that first arose in the analysis of games like chess and checkers, but soon showed its relevance to economics and geopolitical strategy. This unit shows how conflict and strategies can be thought about mathematically, and how doing so can reveal important insights about human and even animal behaviors.
10. Harmonious Math
All sound is the product of airwaves crashing against our eardrums. The mathematical technique for understanding this and other wave phenomena is called Fourier analysis, which allows the disentangling of a complex wave into basic waves called sinusoids, or sine waves. In this unit we discover how Fourier analysis is used in creating electronic music and underpins all digital technology.
11. Connecting with Networks
Connections can be physical, as with bridges, or immaterial, as with friendships. Both types of connections can be understood using the same mathematical framework called network theory, or graph theory, which is a way to abstract and quantify the notion of connectivity. This unit looks at how this branch of mathematics provides insights into extremely complicated networks such as ecosystems.
12. In Sync
Systems of synchronization occur throughout the animate and inanimate world. The regular beating of the human heart, the swaying and near collapse of the Millennium Bridge, the simultaneous flashing of gangs of fireflies in Southeast Asia: these varied phenomena all share the property of spontaneous synchronization. This unit shows how synchronization can be analyzed, studied, and modeled via the mathematics of differential equations, an outgrowth of calculus, and the application of these ideas toward understanding the workings of the heart.
13. The Concepts of Chaos
The flapping of a butterfly's wings over Bermuda causes a rainstorm in Texas. Two sticks start side by side on the surface of a brook, only to follow divergent paths downstream. Both are examples of the phenomenon of chaos, characterized by a widely sensitive dependence of the future on slight changes in a system's initial conditions. This unit explores the mathematics of chaos, which involves the discovery of structure in what initially appears to be randomness, and which imposes limits on predictability.
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