When one type of convergence isn’t enough
In calculus, when we say a sequence converges to a certain limit, we mean that the terms of the sequence are getting closer to the limit in some sense. For a numerical sequence, we can define “closeness” in a natural way—the distance between two numbers is the absolute value of their difference—and it turns out that all other notions of convergence are equivalent to that. But what if the "points" of a sequence are something other than individual numbers? For example, we often approximate complicated functions using a sequence of nicer functions. As it turns out, there is more than one possible way to say that one function is "close" to another, leading to different notions of convergence that are not equivalent. This motivates the idea of normed spaces, one of the foundational concepts of the mathematical field of functional analysis. In this talk, we will explore these ideas with plenty of examples and pictures.
Total positivity of matrix and cluster algebras
An n x m matrix is said to be totally positive if every minors are positive. In the case of 2 x m matrix, there are (m(m-1))/2 minors to check before determining the total positivity of the matrix. As m gets larger, this number grows fast. However we only really need to check 2m-3 minors. We can prove this using a relation between the minors. We can extend this relation to define a very useful math object called the cluster algebra of a quiver. Cluster algebras were introduced by Fomin and Zelevinsky in 2002 and it has been widely studied since then.
Complex numbers: Why to like them, and how to avoid them
The complex numbers, also called imaginary numbers, must have hired the worst marketing firm in the world. After all, if they're complex and imaginary, then why bother? What's the benefit of talking about something that admits "I'm complicated and also fictional" in its own name? In this talk, I'll try to pull back the curtain of the truly unfortunately nomenclature so we can see the power and beauty of the complex numbers. We'll see how the complex numbers are precisely the tool needed to answer some natural and important questions in both geometry and algebra. But once I've convinced you that the complex numbers are great, I'll show you how to conspicuously avoid them with Sturm's Theorem, which fills in some nice details that are left out of the fundamental theorem of algebra.
Think like a Pac-Man ghost
The decision process that a Pac-Man ghost makes can be modeled by a mathematical model of a computer called a finite state automaton (FSA). We'll discuss another real-life example of an FSA and the details of this mathematical view of computation.
Integral Projection Models in Mathematical Biology
In this talk, I will introduce the concept of integral projection models (or IPMs), which are a type of discrete-time model often used by biologists. They are like matrix models, which divide a population into stages like egg, larva, pupae, adult, etc. However, IPMs are most useful when the stages are a continuous variable, and are thus used when biologists want to track the size of individuals in the population. I will review matrix models, introduce some IPMs which model plants, and then finally I'll discuss my work using IPMs to model fish populations. These are all linear models, but if time permits I'll show plots of a non-linear IPM which incorporates decreased growth when there's more competition for food. It will be helpful to know some linear algebra and the idea of an integral, but I will include lots of pictures!
Identity Crisis! Loosening the Definition of a Group
A group is an important mathematical object that captures the idea of symmetry. But what happens if we take the definition of a group and remove the requirement for an identity element? Does it still make sense? What kind of thing are we left with? In this talk, I will define what a group is and give some examples of groups. Then we will look at what can happen when we change the definition by removing the identity requirement.
Euclid's Fifth Postulate and non-Euclidean geometry
For around 2000 years, mathematicians have been perplexed by Euclid's Fifth Postulate. While Euclid's first four postulates are simply stated in one sentence each, the fifth postulate is often thought to be cumbersome and inelegant, taking some time to understand what is being said. Annoyingly, the fifth postulate is also required to prove seemingly obvious statements, such as the fact that two parallel lines never intersect, and that the sum of angles in any triangle is 180 degrees. Being so, many mathematicians have striven to show that the fifth postulate can be proved from the first four postulates. In this talk, we will discuss the history and techniques of a few of these attempts; namely the Khayyam-Saccheri Quadrilateral and Lambert quadrilateral. No knowledge beyond that of high-school geometry will be assumed.
Graph theory to the extreme
The study of graphs dates back to Euler in the 18th century. Two centuries later, thanks to mathematicians like Paul Erdös, extremal problems became a major focus in the study of graphs. These problems deal with maximizing (or minimizing) certain parameters in graphs with a given structure. Perhaps the most famous of these is Turán’s Theorem for the maximum number of edges in a graph that does not contain a large complete subgraph. In this talk, I will provide a basic introduction to extremal graph theory and graph theory as a whole. I will mention some initial results in the area, highlight some of the key components to a proof of Turán’s theorem, and give a glimpse into how a single problem has motivated so many more.