IAP 2018 Classes

A piece of paper can be twisted into a cone, or a cylinder. Euler, in 1772, showed that there is a third kind of surface that can be made in this way: the "tangent developables." I will explain what tangent developables are, and show how to find the pieces of paper that twist into them. Bring scissors!

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. Machine Learning has emerged as a powerful tool for transforming this data into usable information. Many technologies (e.g., spreadsheets, databases, graphs, linear algebra, deep neural networks, ...) 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 lecture 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 and Machine Learning. Understanding these mathematical foundations allows the student to see past the differences that lie on the surface of Big Data and Machine Learning applications and technologies and leverage their core mathematical similarities to solve the hardest Big Data and Machine Learning challenges.

The traditional algorithm for computing square roots is cumbersome. In this lecture, I will discuss an alternative approach, one that involves some elementary number theory.

Discrete optimization can model a wide range of problems in business, science and engineering. Some discrete optimization problems can be very computationally challenging in theory. However, solutions with guaranteed high-quality can sometimes be quickly found in practice using state-of-the-art solvers. In this lecture we will see how the power of these solvers can be easily accessed through the Julia-based modeling language JuMP. We will also give an overview of the mathematical concepts and tools that are the basis for the effectiveness of these solvers.

We will define eigenvalues and eigenfunctions of the Laplacian on intervals and two dimensional domains, and see in particular why they play a role in understanding the vibrations of an elastic string or membrane. In certain cases these eigenfunctions can be computed explicitly, and we will see to what extent the eigenvalues (vibrating modes) determine the membrane itself.

What is the probability that two students in this class are born on the same day (independently of the year)? This is the birthday problem. We are going to answer this question based on the assumption that the probability of being born on a given day is the same for all 365 days of the year. What is this assumption is not true? How many students does it take to test if it holds? We'll also give an algorithm that does that. The class will use elementary probability.

The dynamics of small particles in fluids affects a wide range of physical and biological phenomena, from sedimentation processes in the oceans to transport of chemical messenger substances between and within microorganisms. After discussing these and other relevant examples, we will introduce the mathematical equations that describe such particle motions and study their solutions for basic test cases.