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, November 19, 11:00-11:50 AM US Central Time

Math Education

Dr. Fabiana Cardetti (University of Connecticut)

After several decades of educational reforms aimed at improving mathematics teaching and learning, we continue to observe inadequate understanding of mathematics at all grade levels. The disruption in academic learning caused by the COVID-19 pandemic has highlighted and exacerbated this issue for many of our students. The story is not all grim, however, as teams of researchers around the world are conducting studies not only to increase student achievement and engagement in mathematics but also to understand and test what teachers need in order to support and sustain such increase. In this presentation, I will describe some of the research projects I have engaged in over the years to offer an introduction to some of the types of studies in educational research. Along the way, I hope to engage with you in a discussion about important components of educational research such as: topic selection, research questions formulation, methodological approaches, and more.

Graph Theory

Dr. Jessica De Silva (California State University, Stanislaus)

The more we learn about mathematics, the more we realize how often mathematics appears in our daily lives. Percentages pop up anytime we make a purchase, and checking our speed while driving reminds us of the derivatives at play. Representing real-life situations in terms of mathematical concepts and language is one of the most valuable skills we have as mathematicians. In this talk, we will introduce a mathematical structure, called a graph (or network), which can be used to model many situations in our modern world. With winter just around the corner, we will focus on a graph model of a ski resort and discover the types of questions, both applied and theoretical, that this model motivates.

An Invitation to Linear Dynamics (A Crash Course in Analysis)

Dr. David Walmsley (St. Olaf College)

It is an astonishing fact that there exists a special function with the following property: any continuous function on the unit interval looks like one of the derivatives of this special function! In the research area of linear dynamics, we focus not so much on this spectacular function but rather on the operation that produces it, which in this case is differentiation. In this talk, we will give an introduction to the research area of linear dynamics with the ultimate goal of describing several projects undergraduate students have worked on in this area.

Session II: Saturday, November 19, 5:10-6:00 PM US Central Time

Operations Research

Dr. Allen Holder (Rose-Hulman Institute of Technology)

Operations Research (OR) is an area of applied mathematics that combines theory with computational skill to solve problems. OR's origins go back to World War II, with the original research groups helping the allied powers win the war. The field subsequently flourished due to its wide-scale applicability in business, engineering, and science. The principal mathematical domains are optimization, probability, and statistics, but the OR community regularly adopts/adapts/advances other areas as it addresses contemporary problems. We will sample the mathematics of OR by skipping through some of the problems that students and I have approached , including problems in medicine, economics, manufacturing, and computational biology.

Introduction to Combinatorial Number Theory (A topic in Combinatorics)

Dr. Zhanar Berikkyzy (Fairfield University)

Combinatorics is an exciting field of mathematics that studies discrete objects. It has applications to diverse areas of mathematics and other sciences. In combinatorics problems are often easy to state, but difficult to solve. In this talk, we will mainly focus on combinatorial number theory, a study of structured sets of integers. We will introduce several important theorems in this area. Then we will consider analogous theorems in a rainbow colored setting. We will share recent research results and interesting open problems.

Arithmetic statistics and number field asymptotics (A topic in Number Theory)

Dr. Allechar Serrano López (Harvard University)

In this talk, I will introduce the area of arithmetic statistics with classical examples. Then we will discuss number field asymptotics by discussing the case of quadratic fields as an example.

Session III: Sunday, November 20, 11:00-11:50 AM US Central Time

How Do We Communicate Mathematics? (A Crash Course in Mathematics Communication)

Dr. Kelsey Houston-Edwards (University of Tübingen, Freelance)

We’ll explore the mathematics communication landscape — from teaching and proof writing to podcasting and journalism. I’ll describe some of my own experiences and profile the work of others. Along the way, I’ll highlight some ways that you can get involved in communicating mathematics, either as a career or integrated into other jobs. We’ll also compile a list of general advice: how do you tailor your mathematical message to various audiences? This talk will have an (optionally) interactive format.

An Introduction to Fractals (A topic in geometry and topology)

Dr. Ethan Berkove (Lafayette College)

Roughly speaking, fractals are sets that exhibit self-similarity at many scales, that is, they contain smaller versions of themselves as subsets. Sets with fractal properties show up in many natural systems and have broad applications. The term “fractal” is actually relatively recent—it was coined by Benoit Mandelbrot in the mid 1970’s. However, examples of fractal sets, including some familiar mathematical objects, go back well over a century. In this talk we’ll take a look at some members of the fractal family. I will also describe some recent problems involving fractals that undergraduates have worked on.

Machine Learning and Mathematical Approaches for Pattern Recognition in Mathematical Biology

Dr. Mario Bañuelos (California State University, Fresno)

This talk will focus on two areas of mathematical biology: genomics and endemic disease modeling. An organism's phenotype, the physical expression of one or more genes, is greatly influenced by genomic variation in that organism or population. We consider applying machine learning frameworks to both detect longer variants as well as cluster mutations that may positively or negatively affect phenotype. In particular, epistasis, the phenomenon of one mutation affecting the resulting quantitative or qualitative phenotype, is used to assess gene variation in an attempt to find a combination of single nucleotide variants (SNVs) that contribute to a certain phenotype. Since one SNV rarely completely describes an organism’s phenotype, detecting these groups, or coalitions, of mutations without relying on an exponential number of numbers is one of the main challenges in this field. To alleviate these computational bottlenecks, we propose a neighborhood-based collaborative filtering approach by viewing this data with a recommender system formulation. As such, we are able to detect statistically significant higher order SNV interaction phenotypes related to muscle mice genomic variants. Next, we focus on the math modeling process for endemic disease modeling, the importance of data cleaning and visualization, and discuss open areas of research.

Session IV: Sunday, November 20, 3:30-4:20 PM US Central Time

“Statistics” What is it and why does it matter?

Dr. Marco V. Martinez (North Central College)

Predictive modeling, business analytics, data analytics, big data, machine learning and artificial intelligence are some of hottest topics in the corporate world, the government and therefore in higher education. We will talk of the role of statistics in some of these areas and the differences between them. Finally, we will discuss how we can use statistics to make you more marketable regardless of your career path.

Topology? Isn’t that the study of mapmaking? (A Crash Course in Topology)

Dr. Candice Price (Smith College & Spelman College)

In this crash course, we will talk about the informal mathematical definition of topology as well as see some examples of areas of topology and some interesting research questions in these areas.

Mathematics Related to Meaning and Inference in Language Processing (A Crash Course in Logic)

Dr. Larry Moss (Indiana University)

These days, everyone with a computer interacts with it in ordinary language. This interaction involves a lot of mathematics. For example, there is a whole field of math related to grammar, and this is related to how a computer would 'parse' your sentences -- and perhaps also how you are reading this very text. My talk will highlight a math topic that is related to a different topic: "inference", the study of "what follows from what?". The topic relates to logic (my main field) but also to increasing and decreasing functions from calculus. So we'll see one of the most beautiful aspects of mathematics, the fact that one math subject (increasing/decreasing functions) can illuminate a very different subject: language.