Crash Courses

Topology, and particularly algebraic topology, seeks to develop computable invariants to quantify the shape of abstract spaces. This talk will be about how such invariants can be used to analyze scientific data sets, in tasks like time series analysis, semi-supervised learning and dimensionality reduction. I will use several examples to illustrate real applications of these ideas.

 Join us to talk about pathways into math education and the importance of teachers with marginalized identities!  Many of us have experienced "traditional" K-12 math classrooms where "drill and kill" are king. This often plants and perpetuates the idea that mathematics is a subject focused on speed and compliance, and often isolated. For many people there is a comfort in the supposed structure of mathematics, but this also creates a barrier for students who want to work collaboratively, tinker, and experiment with mathematics. Unlearning the "traditional" math classroom for pre-service teachers can be tough to do in isolation, in this talk I will be discussing how challenging this schema in community can support students with marginalized identities, who have often been excluded from success in mathematics classrooms.

In essence, the field of graph theory is the mathematical study of connections and interactions. In this exciting field we use graphs-- but not *those* graphs you might be thinking of! Instead we study an entirely different type of "graph" consisting of dots and lines connecting those dots. The first part of this talk serves as an overview of the history, basic tools, and many of the real-world applications of graph theory. Then we dive into an example of research recently done in the field, and a discussion about research programs where such research is being done.

In this crash course, we will talk about problems at the crossroads of combinatorics and geometry.  In particular, how do geometric conditions on sets affect their combinatorial properties?  Some instances we will talk about are intersection patterns of families of convex sets, fair division problems, and combinatorial properties of finite families of points in R^d.  A surprising feature of many of these problems is that their solutions often borrow tools from topology.  The course will be accessible even if you don't have prior experience with these topics.

Poets often use metaphors and similes. Statisticians use models: abstractions that capture the essential features common to seemingly disparate situations. Common models include coin flips and drawing balls from urns. Statisticians don’t just care about coins and urns, though; we care about using these simple models as metaphors to help solve other problems. We’ll explore how models can help us estimate the size of crowds, the strength of enemy forces in World War II, and the age of the extinction of the dinosaurs.

Success in Algebra is one of the strongest predictors for high-school graduation and future success. But for many students, algebra is a gatekeeper, not a gateway. What makes algebra so hard? For one, it’s symbol, language, and representation systems are very abstract, and second, it requires focusing both on many specific rules and a larger structure and bigger picture at the same time. In this talk, I will present work that has shown that math and algebraic reasoning is grounded in perceptual processes, and that embedding perceptual action routines into mathematics learning by using innovative technology can be useful for improving conceptual understanding. I will demo two math technologies that I have designed, developed, and tested over the past several years, Graspable Math and From Here to There! Next, I will share findings from two large scale randomized controlled classroom studies that demonstrate their impact on mathematics performance. Third, I will discuss ways that I am using data analytic approaches on the log data collected in these technologies to advance basic and applied research on perceptual learning and explore possible mechanisms by which these tools impact student mathematics engagement and learning. 

Animal skin patterns, disease transmission, opinion dynamics, wound healing, tumor growth, schooling fish, leaf venation,… These are all examples of things that mathematical biologists work on, drawing on many sub-areas of mathematics, including differential equations, network science, and topology and geometry, among others. Here I’ll introduce mathematical biology and chat about some projects for undergraduate students interested in connecting math and biology.

Core Partitions are a subfamily of integer partitions that arise from the representation theory of the symmetric groups. They have, however, been found to be connected to deep concepts in number theory and rational combinatorics. We introduce this field and find several interpretations, using Young diagrams, the abacus, and lattice paths.

The field of Quantitative Finance, or Mathematical Finance (sometimes referred to as Financial Engineering), provides mathematicians with opportunities to work on interesting research problems and pursue careers that are exciting, intellectually challenging, and lucrative. I plan to start off by giving a brief introduction to the types of mathematical problems that arise in quantitative finance. Then I will discuss the mathematical concepts that are needed to solve these problems. If time permits, I will show some results from undergraduate research projects. I will also say a bit about research opportunities, graduate programs, and job opportunities. I hope to make the presentation interactive, so please come prepared to ask me lots of questions. NO previous knowledge of finance is required to follow this presentation/discussion.

Fluid motion is described by classical mechanics, that is, by the Newton’s laws of motion. This means that the basic tools of Calculus gives us access to studying the incredibly rich and complex motion encompassed by fluids. The study of fluids has not only fascinated physicists and engineers for centuries, but also mathematicians. In fact, the first ever partial differential equation was written down by Euler, who had conceived of what he referred to as a “perfect fluid,” one that moves along its own velocity field unobstructed by frictional forces and may experience deformations only through its internal pressure. These equations have since been referred to as the Euler equations for incompressible fluids. Although these equations were written down nearly 300 years ago, there nevertheless remain fundamental problems regarding solutions to these equations that remain unresolved to this day. This talk will introduce the famous Euler equations, using only calculus, and discuss issues that continue to confound mathematicians to this day.

Number theory is cool! You want to learn about it! It's connected to lots of other cool things, like Algebra, Geometry, Logic, Computer Science, and Art. Even Analysis, believe it or not. Because it's about integers, and integers are everywhere. I'll talk about some different types of number theory and give some examples of student projects. Also, we'll race some primes and see who wins.

Topology studies the most essential properties of shapes (or "spaces"), those that are the same no matter how you stretch or squeeze them.  Those properties can be especially interesting if the points in the space themselves have meaning: they can represent data, or ways of tiling the plane, or even other spaces.  I will give some examples of places in mathematics where this idea shows up.