# 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.

## Tentative Topics and Speakers:

- Graph Theory

- Dr. Caitlin Owens, DeSales University

- Combinatorics
- Dr. Pamela E. Harris, Williams College

- Algebra and Number Theory
- Dr. Joshua Holden, Rose-Hulman Institute of Technology

- Biomathematics
- Dr. Necibe Tuncer, Florida Atlantic University

- Mathematical Modelling with Graph Theory
- Dr. Darren Narayan, Rochester Institute of Technology

- Analysis and Dynamical Systems
- Dr. Jeffrey Schenker, Michigan State University

- Topology and Knot Theory
- Dr. Rolland Trapp,
**California State University San Bernardino**

- Discrete Geometry
- Dr. Pablo Soberón, City University of New York (CUNY) Baruch College

- Statistics
- Dr. Maria Tackett, Duke University

## Abstracts of Talks - Click Here

If you are planning to attend these talks, please use this information to decide which parallel talks you will attend. **Each talk will last 30 minutes.**

### Set 1 (1:25pm Pacific/4:25 Eastern):

**Knot/Braid Theory and Topology with Dr. Trapp**: My talk begins with a very brief background into geometric structures on topological spaces. These are big words, but think of creating a torus by gluing opposite sides of a rectangle. Some animations (and who doesn't like animations?) will follow that illustrate how to put a hyperbolic structure on certain link complements. Topological, geometric and combinatorial tools are introduced that allow one to address questions in this area, and some good starting references will be suggested. It'll be an enjoyable half hour if you can make it.**Graph Theory with Dr. Owens:**This talk will be an introduction to graph theory. We will not be looking at graphs in the way that many of you may think of graphs. I promise we will not be graphing parabolas or other conics. We will be looking at graphs composed of dots, called vertices, and lines, called edges (a more rigorous definition to be given in the talk). After formally defining what a graph is, we will talk about special types of graphs, and then about problems in graph theory. This talk will focus on the popular and fun problems involving graph coloring. So get out your colored pens (digital or real), and get ready to have some fun! We will also number our colors for those of you who may have difficulty distinguishing between colors.**Statistics with Dr. Tackett:**In this talk, we will provide an overview of opportunities for undergraduate students in statistics. We will begin by presenting examples of undergraduate research in statistics and strategies for getting started. We’ll then discuss ways students can share their work and get more connected with the statistics and data science communities. Lastly, we’ll conclude with a few upcoming events where students learn more about statistics research, get a glimpse of statistics in practice, and present their own projects.

### Set 2 (2:20pm Pacific/5:20 Eastern):

**Discrete Geometry with Dr. Soberón:**In this talk we will see a glimpse of discrete geometry. Since this is a broad topic, we will focus on the role of convexity in this area, and the study of families of convex sets. In particular we will discuss: Why do we care about convex sets? What kind of interesting combinatorial properties do convex sets have? What tools are useful to study these properties? And what are good examples of problems for undergraduate research?**Graph Modeling with Dr. Narayan:**Graph theory has powerful applications that can be used to analyze social, biological, and transportation networks. In this course we will give an overview of related "small world network" properties that can be used to analyze the efficiency of social networks, the Atlanta Metro system, and functional connectivity of the human brain. This talk will be accessible to students at any level.**Analysis and Dynamical Systems with Dr. Schenker:**Analysis and Dynamical Systems are the fields of mathematics concerned with limits and evolution equations. Many tools in these fields were developed to understand the solutions of equations modeling the physical world. I will describe some current research areas in harmonic analysis, partial differential equations, probability theory, and mathematical physics.

### Set 3 (3:15pm Pacific/6:15pm Eastern):

**Combinatorics with Dr. Harris:**My research is in the area of algebraic combinatorics. I like to use combinatorial arguments and techniques to enumerate, examine, and investigate the existence of discrete mathematical structures with certain properties. In this talk, I will introduce parking functions, some of their generalizations, results related to their enumerations, and I will provide ample evidence that combinatorics is a "low floor and high ceiling” mathematical area where everyone can contribute to the creation and discovery of new knowledge.**Biomathematics with Dr. Tuncer:**For this short course, we will develop a mathematical model which simulates a spread of a new infection in a population. We will then determine the biologically important threshold value called basic reproduction number, R0. R0 determines whether a disease will die out or persist in a population. This epidemiologically important value gives the number of secondary infections generated by one infect individual in a totally susceptible population during her/his infectious time. We will then see how vaccination will change the basic reproduction number.**Algebra and Number Theory with Dr. Holden:**Numbers, vectors, matrices, functions, permutations, paths in space, cryptographic systems, and lots of other things have something in common --- you can put them together according to certain rules. Abstract algebra studies how these systems can be classified according to what the rules have in common and how they are different. Understanding these similarities and differences gives a powerful tool for solving problems in one area by translating them into a different area and solving them there. Number theory is often grouped (no pun intended) with abstract algebra because they have many techniques in common. However, number theory also uses techniques from combinatorics, analysis, geometry, topology, logic, and pretty much all areas of mathematics in order to solve problems rooted in properties of whole numbers.