# Courses

Go with the Flow!

Instructor: Sarah Patterson

TA: Karina Cho (Harvey Mudd), Sophia Schein (Bryn Marwr)

Are you ready to make some waves? Navier-Stokes equation is used to model waves, aquatic animal locomotion, aerodynamics, blood flow, thin film lubrication, and much more! These models help researchers predict the weather, develop new medications, and create realistic animation in films. In this course, we will look at how to mathematically model fluid motion and many other natural phenomena. You will learn the meaning behind several differential equations including Navier-Stokes equation. You will also numerically solve various differential equations using basic programming techniques. No calculus or programming experience required.

Randomness through Markov Chains

Instructor: Erin Beckman

TA: Natalie Meacham (Bryn Marwr), Stephanie Kolden (Gonzaga)

Randomness is all around us! Understanding randomness is a big challenge, and Markov chains are here to help. Markov chains can be used to break simple codes and to make predictions about financial markets. In fact, many of us have interacted with Markov chains without even knowing it; they are used when your phone tries to predict what word you want to type next. Markov chains are one of the most fundamental objects in the study of randomness. They can help us predict what is going to happen in a complicated system by incorporating information from the current state of the system, “forgetting” any past information.In this course, we will look at probability and randomness through the lens of Markov chains. We will start with a short introduction to probability and then take a look at some properties and applications of Markov chains. We will also look at how to use them to develop models of real world systems. No calculus or probability experience is required, only a willingness to learn something new!

# Lectures

• Professor Jonathan Mattingly (Duke), Wednesday June 19

Quantifying Gerrymandering : Probing the Geopolitical Geometry of a state.

In October 2017, I found myself testify for hours in a Federal court. I had not been arrested. Rather I was attempting to quantifying gerrymandering using analysis which grew from asking if a surprising 2012 election was in fact surprising. It hinged on probing the geopolitical structure of  North Carolina using a Markov Chain Monte Carlo algorithm. I will start at the beginning and describe the mathematical ideas involved in our analysis. This project began as a sequence of undergraduate research projects and undergraduates continue to be involved to this day.

I will also explain the current state of the court cases and rulings and their implications for the mathematics needed moving forward. As time permits I will also discuss some open mathematical problems and issues which come from these investigations.

• Professor  Laura Miller (UNC), Thursday June 20

The dispersal of aerial plankton: Spider ballooning and flapping with bristled wings

A vast body of research has described the complexity of flight in insects ranging from the fruit fly to the hawk moth. Over this range of scales, flight aerodynamics are surprisingly similar. The smallest flying insects, including thrips and parasitoid wasps, may only have a wingspan of a few hundred microns and use bristled wings to fly. The aerodynamics of these unusual flyers has received far less attention. At a similar scale, spiders use a type of aerial dispersal called “ballooning” to move from one location to another. In order to balloon, a spider releases a silk dragline from its spinnerets and when the movement of air relative to the dragline generates enough force, the spider takes flight. In this presentation, I will give an overview of the research in my group to measure flight kinematics, quantify wing shape and dragline properties, measure flow velocities and forces using physical models, and perform numerical simulations to obtain aerodynamic forces.

• Professor Samit Dasgupta (Duke), Friday June 21

From local to global in number theory

In this talk, I will discuss the principle of going from “local” to “global” in the field of number theory.  In number theory, considering an equation locally means reducing it modulo a prime p.  Many questions in number theory boil down to the question: if we understand an equation completely modulo p, for every prime p, can we understand the equation over the ring of integers Z?  We will explore this question with specific examples and take a tour of what is known and what is unknown. Along the way we will describe the Riemann Hypothesis and the Birch—Swinnerton-Dyer Conjecture, two of the biggest unsolved problems in number theory.

• Professor Patricia Hersh (NCSU), Monday June 24

Counting by inclusion-exclusion topologically

The chromatic polynomial P_G(q) of a graph G counts how many ways its vertices may be colored with q colors so that no two neighbors have the same color.  I will explain how this is an example of a counting problem that can be explained/solved by the method of inclusion-exclusion, using objects from combinatorics known as partially ordered sets along with a function called the Möbius function of the partially ordered set.  Then I will discuss how the Möbius function in fact may be calculated using a different area of math, namely topology, by interpreting it as an “Euler characteristic”, namely as a function from topology that starts with a triangulation and outputs the number of vertices minus the number of edges plus the number of two dimensional faces and so on.

• Professor Ezra Miller (Duke), Tuesday June 25

Topology for statistical analysis of brain artery images

Statistics looks for trends in data. Topology quantifies geometric features that don’t change when shapes are squished, stretched, or bent continuously. What does one have to do with the other? When data objects are already geometric, such as magnetic resonance images of branching arteries, topology can isolate information of statistical relevance. This talk explains what we have learned about the geometry of blood vessels in aging human brains using topological methods in statistics. The main results are joint with Paul Bendich, Steve Marron, Alex Pieloch, and Sean Skwerer (at the time, a Math postdoc, Stat faculty, Math undergrad, and Operations Research grad student).

• Professor Colleen Robles (Duke), Wednesday June 26

The Gauss-Bonnet Theorem

The Gauss-Bonnet Theorem is the “crown jewel” of surface geometry. I will explain this beautiful result, what makes it so remarkable, and how one can prove/verify it using calculus.

• Professor Ingrid Daubechies (Duke), Thursday June 27

Teeth, bones and manifolds

• # Projects

• Project 1: Cleaning Up the Great Lakes

The great lakes in North America are the largest surface freshwater system in the world. They hold 21% of the world’s freshwater supply and supply water to 10% of Americans and 30% of Canadians. However, chemicals, toxic pollutants, pesticides, and heavy metals entered the Great Lakes from heavy industry, manufacturing, and agriculture. Students working on this project will investigate what it would take to clean up the lakes.

• Project 2: Code Breaking

A substitution code is a type of encryption where letters of the alphabet are permuted in a fixed way. If we are told how to decode the encrypted text, it is not hard to get back the original message. But what if we aren’t told how to decode the message? In this project, you will learn how to decode substitution codes by finding the most likely decoding instructions through Markov chains and Markov chain Monte Carlo methods.

• Project 3: Period Doubling and Chaos

The logistic equation is often used to model the growth of a population with a carrying capacity. In studying this dynamical system, the phenomena of period doubling and chaos are observed when computing numerical solutions. In this project, students will investigate this interesting behavior.

• Project 4: Modeling Epidemics

One area of health research that mathematicians have gotten involved in is researching the spread of infectious diseases. Using both random and non-random methods, mathematicians try to predict factors which affect the spread of disease through a population and work to prevent epidemics from occurring. In this project, you will learn about some of the models which are used to study diseases and what math can contribute to the world health discussion.

• Project 5: Stopping Fake News

In modern society, creating and disseminating information is easier than ever. While this allows access to information on unprecedented scales, the validity of the information can be difficult to determine. In this project, you will model the spread of fake news and investigate ways to deter distributing misinformation.

• Project 6: Random Walks and Random Motion

One of the most famous Markov chains is called a simple random walk. Imagine you are walking around a city and at each street corner, you pick a new direction to go randomly: left, right, forward, or back the way you came. How far would you walk? Would you ever come back to where you started? How does the way you make your random choice effect the answers to these questions? In this project, you will explore random walks as a way to answer these questions. We will also briefly look at real life examples of random walks and their continuous counterpart – Brownian motion.

• Project 7: Thanos Population Models

At the end of the “Avengers Infinity War”, the villain Thanos snaps his fingers and turns half of all living thing to dust with the hope of restoring balance to the natural world.   In this project, you will investigate the validity of his claim using mathematical modeling.

• Project 8: Markov Chains in Biology

In this project, you will learn about a Markov chain model for evolution of the proportion of a gene in a species. While evolution is a very complex process, you can use this simplified model to get an idea of how diploid or haploid evolution progresses and see the influence of natural selection and mutation on this process. You will use Markov chain tools to answer questions like about long it takes until a gene disappears from the population entirely and which genes will disappear.