IAP 2017 Classes

The sequence of prime numbers has fascinated mathematicians for thousands of years. Euclid proved that there are infinitely many prime numbers, but how they are distributed is mysterious. About 1800, Gauss and Legendre noticed that the number of primes less than a large number N seems to be approximately N/ln N. This observation is the Prime Number Theorem, and it was proven by Hadamard and de la Vallee Poussin in 1896.

Even today, nobody understands the error term--the difference between N/log N and the number of primes less than N. There is a very famous conjecture that the size of the error term is something like square root of N. This conjecture is one version of the "Riemann hypothesis," which is certainly the biggest unsolved problem in mathematics.

When faced with an impossibly hard problem, mathematicians like to change the subject; and that's my plan. I'll talk not about prime numbers (used to factor integers) but about irreducible polynomials (used to factor polynomials). I'll explain how to formulate a problem about counting irreducible polynomials that's analogous to the prime number theorem. The difference is that this counting problem has a precise answer. The main term of the answer looks just like the main term in the prime number theorem, and the error term behaves just like the Riemann hypothesis says it should.

"Big Data" describes a new era in the digital age where the volume, velocity, and variety of data created across a wide range of fields (e.g., internet search, healthcare, finance, social media, defense,...) is increasing at a rate well beyond our ability to analyze the data. Many technologies (e.g., spreadsheets, databases, graphs, linear algebra, ...) have been developed to address these challenges. The common theme amongst these technologies is the need to store and operate on data as whole collections instead of as individual data elements. This class describes the common mathematical foundation of these data collections (associative arrays) that apply across a wide range of applications and technologies. Associative arrays unify and simplify Big Data leading to rapid solutions to Big Data volume, velocity, and variety problems. Understanding these mathematical foundations allows the student to see past the differences that lie on the surface of Big Data applications and technologies and leverage their core mathematical similarities to solve the hardest Big Data challenges.

Wavelets are the smartest choice of basis for functions since Fourier analysis. They are at the same time localized in space and in frequency. It was a major discovery in the 1980s that one could build orthonormal bases of wavelets with favorable properties (small support, smoothness). They are now in extensive use in image processing and data analysis.

How many ways are there to stack a pile of cubes in the corner of the room? The answer is given by an elegant Macmahon's triple product formula. We will discuss how to address this enumeration question and its relatives by using symmetric polynomials.

At the exhortation of Gauss, number theorists pursued a variety of methods for determining whether or not a given integer is prime throughout the 19th and 20th centuries; some methods were fast but not always correct, while others were correct but not always fast. It was not until the 21st century that an algorithm was found that was provably both fast and correct, when Agrawal, Kayal, and Saxena announced that "PRIMES is in P" (Annals of Mathematics 160 (2004), 781-793). We will briefly review the history of this problem and then prove their remarkable result.

It's far easier to solve linear systems of equations than non-linear ones. In the former case, you can find the entire solution set all at once. In contrast, the state of the art in solving a nonlinear system is still Isaac Newton's method---or a sophisticated variant thereof---in which you start with an initial guess, approximate the derivative there, and hope it leads you iteratively toward a solution.

In this talk I'll present a totally different way to solve systems of equations: plot each equation, think of this plot as an array of pixels---either on or off (1 or 0)--- and multiply these arrays together in a prescribed way. It's surprising at first that multiplying arrays (or matrices) returns the approximate solution set for the whole system. But it does and in fact is surprisingly fast.

As a final selling point, the pixel array method seems to be one of those rare examples of an idea which was born out of category theory but which A. requires no category theory to learn, and which B. solves a real-world problem numerically.