# Crash Courses

## Crash Courses

### Quick overviews of fields of mathematics which are common to undergraduate research projects, especially the active undergraduate research areas, to help you decide what you might be interested in applying for.

### Abstracts of talks will be available under the collapsible headings below as we receive them.

## Session I: Saturday, December 4, 11:00 am - 12:00 pm US Central Time

### An Invitation to Combinatorics

### Dr. Alexander Díaz -Lopez (Villanova University) and Dr. Erik Insko (Florida Gulf Coast University)

Combinatorics is the study of the number and ways in which sets of objects can be combined in accordance with certain constraints. As a mathematical discipline, combinatorics, is one where the problems are usually easy to state, but the answers are sometimes hard to see unless you look at the problems from the right perspective. Hence combinatorics is one area of mathematics where fresh perspectives prove just as useful as heavy theoretical machinery, and in my experience the more diverse a group of collaborators is, the better its chances are of finding the right perspective to solve the problem they are studying. In this crash course, I will introduce some of the essential combinatorial techniques and share some important life lessons I have learned from my friends and collaborators in combinatorics. Time permitting, I will also try to share some interesting opening problems that undergraduate researchers can tackle.

### Numerical Methods for Differential Equations (A Topic in Numerical Analysis)

### Dr. Hristo Kojouharov (University of Texas at Arlington)

Numerical analysis is a branch of mathematics concerned with the study of algorithms for solving various mathematical problems numerically. In this presentation, we will give a quick review of one of the primary topics of numerical analysis: the numerical solution of ordinary differential equations. We will begin by going through some of the most common numerical methods, such as the Euler and Runge-Kutta methods. Following that, we will introduce the nonstandard finite difference (NSFD) methods and give several numerical examples motivated by specific biological systems. Our presentation will conclude with a look at some open problems in the field of NSFD methods.

### A Survey of Diophantine Equations (A Topic in Number Theory)

### Dr. Edray Herber Goins (Pomona College)

There are many beautiful identities involving positive integers. For example, Pythagoras knew 3^2 + 4^2 = 5^2 while Plato knew 3^3 + 4^3 + 5^3 = 6^3. Euler discovered 59^4 + 158^4 = 133^4 + 134^4, and even a famous story involving G. H. Hardy and Srinivasa Ramanujan involves 1^3 + 12^3 = 9^3 + 10^3. But how does one find such identities?

Around the third century, the Greek mathematician Diophantus of Alexandria introduced a systematic study of integer solutions to polynomial equations. In this talk, we'll focus on various types of so-called Diophantine Equations, discussing such topics as Pythagorean Triples, Pell's Equations, Elliptic Curves, and Fermat's Last Theorem.

## Session II: Saturday, December 4, 2:00 pm - 3:00 pm US Central Time

### Mathematics Education

### Prof. Márta Barbarics (Eötvös Loránd University, Budapest, Hungary)

This crash course will be a short introduction to the basics of research in mathematics education. We will look at the different stages of research projects, the types of research, the literature review, and based on these how to form research questions. You can learn about quantitative and qualitative methods, their instruments, and data analysis. I will show some examples of mathematics education research of varying scales, hoping that you find some inspiring ideas that are feasible in your context. As I have carried out research in connection with discovery learning, gamification, and alternative assessment in Hungarian secondary math classes, I am happy to answer any questions you may have in these topics as well.

### Mathematical Modeling of Complex Adaptive Systems (A Topic in Biomathematics)

### Dr. Yun Kang (Arizona State University)

A complex adaptive system (CAS) is a system that is complex in that it is a dynamic network of interactions, but the behavior of the ensemble may not be predictable according to the behavior of the components. It is adaptive in that the individual and collective behavior mutate and self-organize corresponding to the change-initiating micro-event or collection of events. Typical examples of complex adaptive systems include: climate; markets; governments; industries; ecosystems; social insect (e.g. ant) colonies; the brain and the immune system. Human social group-based endeavors, such as political parties, communities, war, and terrorist networks are also considered CAS. Mathematical models are powerful tools that can provide us quantitative approaches to elucidate complicated ecological and evolutionary processes on the numerous spatial, temporal and hierarchical scales at which CAS such as social insect colonies operate. Social insects such as ants, bees, wasps and termites, among the most diverse and ecologically important organisms on earth, live in intricately governed societies that rival our own in complexity and internal cohesion. They are excellent examples of CAS. In this talk, I will present some of our recent modeling work with closed collaborations with biologists that have been addressing important and interesting biological questions of social insect colonies such as (1) How are foraging behaviors of social ants dynamically regulated in response to environmental changes? (2) How do colony size, social communication, and age polyethism affect the task organization in the adaptive dynamical environment? (3) How information spreads in the social insect colonies.

### Symmetries (A Topic in Abstract Algebra)

### Dr. Julia Plavnik (Indiana University, Bloomington)

We encounter mathematics all around us. One beautiful and important example is symmetry. The intuitive perception of symmetries relates to balance, beauty, proportions, and harmony. Mathematically, we make the notion of symmetry precise via something called group theory. A symmetry of a cube is a transformation that leaves it unchanged, and the set of all such symmetries forms a structure we call a group. In this crash course, I will introduce the notion of a group and how they relate to symmetries. I will also present some examples that can be studied using diagrammatic. If time permits, I will also connect this with an important area in algebra, called representation theory by relating the elements in the group with some matrices.

## Session III: Sunday, December 5, 11:00 am - 12:00 pm US Central Time

### Statistics

### Dr. Trent Gaugler (Lafayette College)

What do sonic booms, eating strawberries, autism, mortality and marriage quality, and college football have in common? These are all the subjects of statistical analyses on which I have worked! In this talk, I'll give some very brief background on these projects, in the hope that I will give you some idea of the breadth of opportunities out there for people with a background in doing statistical analysis. In addition to describing some of my own research, I'll also outline the topics of some of the thesis, independent study, and REU projects with which I've been involved. I'll also touch on some of the skills I think are the most important for you to have if you want to pursue these kinds of opportunities.

### Low-Dimensional Topology: Knots and Surfaces

### Dr. Kenji Kozai (Southern Connecticut State University)

In topology, we study the “shapes” of objects up to continuous deformations. Low-dimensional topology focuses on objects that are 1, 2, 3, and 4-dimensional and their properties that are invariant under continuous deformations. In this talk, we will introduce the ideas of knots and links in 3-dimensional space, surfaces, and some examples of tools used to study properties of both of these types of objects that come up in undergraduate research.

### Playing Mathematical Billiards (A Topic in Dynamical Systems)

### Dr. Renato Feres (Washington University in St. Louis)

Mathematical billiards are a widely studied subject in the theory of dynamical systems. They display a range of behaviors, from very regular to fully chaotic, and provide mathematical models for a variety of systems of physical interest. This talk will introduce some ideas from billiard theory using a simple but surprising example.

## Session IV: Sunday, December 5, 4:30 pm - 5:30 pm US Central Time

### Geometric Modeling: Getting your Computer into Shape

### Dr. Kathryn Leonard (Occidental College)

Shape understanding—looking at a shape and intuitively understanding which parts are, e.g., body, arms, legs, toes, and ears—is almost effortless for humans. Training a computer to understand shapes in a similar way presents substantial challenges. Thoughtful choice of a geometric model for shapes can help address those challenges. This talk will discuss shape perception and the challenges of automation and then describe a promising mathematical shape model, the Blum medial axis. Using the Blum medial axis, we will propose a method for automatically decomposing a shape into a hierarchy of parts and determining the similarity between those parts. We will also present several undergraduate research projects emerging from this work.

### Graph Theory

### Dr. Feryal Alayont (Grand Valley State University)

Graph theory is the study of relationships. Visually, we represent objects as dots and pairwise relationships between them as lines connecting the dots. These graphs have many applications. As examples of applications, graphs can be used to represent social networks, in optimization and logistics problems, and to model molecules. In this talk, after learning relevant basics of graph theory, we will describe some general themes of research areas in this field. No previous knowledge of graph theory is required.

### Mathematics Communication

### Dr. Yen Duong (North Carolina Health News)

Math doesn't exist without communication: you'll get a zero if you turn in a blank problem set, even if you solved all the problems in your head. Math communication is a key skill for all mathematicians, but doing it as a career takes it to the next level: we'll talk about what math communication is, how I got to it as a career as a freelance math and science writer, and practice communication for different audiences.