2018-19

Local moves on knots

Mathematical knots are smooth embeddings of a circle into 3-space.  Knots are sometimes examined by considering diagrams in the plane, which leads to the question:  How do properties of a knot change when we alter a region of the diagram?  In this talk we discuss some local moves on knots and their applications to knots in proteins.

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.

Bounded operators and when size matters

In a plane, there's a natural way to measure length: usually, straight-line distance.  This isn't the only way to measure length, though.  The distance from your house to the grocery store might instead be measured in driving distance, for example.  It turns out there are many ways to think about length in the plane.  We can also think about what length should mean in less intuitive spaces; for example, what should the "length" of a function be?  We will explore these different ways of thinking about length and use them to motivate the idea of bounded linear operators.

Structured population models in mathematical ecology

Ecologists and wildlife managers want to know how they can keep biological populations healthy; that is, they don't want a population to die out or to explode.  However, they can't run experiments on wild populations because that usually costs too much money, or they risk irreparably damaging a population.  Thus, biologists use mathematical models and computers to study populations.  In this talk, I will introduce a number of "structured" population models.  These models make use of physiological differences, like the size of a fish or the developmental stage in grasshoppers.  Biologists use matrices, differential equations, and more recently integral operators, to stimulate structured populations in the wild.  I will give examples of each of these models, with lots of pictures.  The mathematical details should be accessible to anyone who has taken calculus.

Fractional calculus: an analog of traditional calculus as we know it

The derivative is at the heart of mathematics and specifically calculus, so why not extend this idea?  What does it mean to take a half derivative?  A "pi-th" derivative?  These are the questions that sparked the area of fractional calculus.  In this talk, we will investigate these intriguing questions and draw comparisons between traditional calculus and fractional calculus.  Time permitting, we will explore some of the various applications for which fractional calculus is well-suited.

The mathematics of gerrymandering

Gerrymandering, or the act of drawing political boundaries to favor a certain candidate or party in elections, has been a reality in the U.S. for most of its history.  In fact, the term "gerrymander" itself refers to a particularly amphibious-looking state senate district approved in 1812 by the then-Massachusetts governor Elbridge Gerry.  Since then, hundreds of congressional districts in the U.S. have gained infamy because of their peculiar shapes--examples include the late North Carolina 12th district and the still-active Illinois 4th.  However, not all gerrymanders contain offensively-shaped districts, and indeed some of the most effective gerrymanders appear geometrically sound.  How can we determine that a plan is a gerrymander in the absence of poorly-shaped districts?  In recent years, mathematicians have developed tools which are able to do just that.  I'll discuss some of these tools and the ways in which they're being applied to real cases of gerrymandering today.  Mathematical topics will include graphs, Markov chains, and some geometry, but no prior knowledge is needed.

Using optimal control theory to model resource allocation in annual plants

The fitness of an annual plant can be thought of as how much fruit is produced by the end of its growing season.  Under the assumption that annual plants grow to maximize fitness, we can use techniques from optimal control theory to understand this process.  In this talk we will discuss some introductory optimal control theory as well as two different models for resource allocation to roots, shoots, and fruits in annual plants.  In each model we will examine how optimal control theory can be applied to determine the optimal resource allocation strategy for the plant throughout its growing season.

Accurately modeling fluid flow:  Data assimilation and parameter recovery

Scientists and mathematicians apply the continuum hypothesis to model fluid flow, i.e. the flow of substances like air or water.  One of the challenges of the accurate simulation of turbulent flows is that initial data is often incomplete.  Data assimilation circumvents this issue by continually incorporating the observed data into the model.  In this talk, I will discuss my work using a new approach to data assimilation in order to both accurately model fluid flow and to identify certain physical properties of the fluid being modeled.

Computational commutative algebra

What does the word "algebra" make you think of?  The quadratic formula?  Solving for x?  In this talk, I'll show you a few of the wild and esoteric things that algebra means to me, via the story of a recent research project that used the University of Nebraska's supercomputing resources to investigate a certain conjecture.  We'll see how technology can find uses even in the highly abstract, non-applied world of symbolic powers and commutative algebra.  Using audience suggestions, we'll even be able to do some of our own calculations live.

Low dimensional topology:  Knots and manifolds

Topology is the field of mathematics concerned with the shape of space.  An important question is this one: up to stretching these spaces without poking holes in them, how many different spaces are there?  Low dimensional topology aims to answer this question, among others, in dimensions 2, 3, and 4.  Moreover, a case can be made these are the only dimensions in which these questions matter.  It turns out that knots play a crucial role in describing 3 and 4 dimensional spaces.  In this talk we will explore some 3 and 4 dimensional spaces, discuss their descriptions and representations, and understand how we can sometimes tell them apart.

Multiplication -- more than meets the eye

It's easy to think that at this point in your life, you know all there is to know about elementary mathematics.  After all, how many different ways are there to multiply?  In this talk, we will first explore the type of knowledge that exists at the intersection of purely mathematical content and pedagogical knowledge.  In particular, we will investigate the idea that there is more to know about multiplication and division than pure math experts know.  In the second part of the talk, we will look at examples of this type of knowledge by discussing scratch work from students taking a standardized assessment.  This work comes from an internship with Educational Testing Service (ETS) in 2019.  The goal is that we all leave the talk appreciating the nuance in what mathematics students know and can do.

Kruskal's Algorithm

Consider the following situation and question:  A power company would like to lay down some wires in a local neighborhood, but wish to minimize costs by not laying down too many long and redundant wires, while making sure everyone is connected to power.  Is there a way to find such a minimum solution?

In this talk, we'll discover that the answer to this question is yes -- viewed in the right way, the above situation is an application of a well-known graph theory result called Kruskal's Algorithm.  We will go through all the mathematics we need to know to understand Kruskal's Algorithm, state and prove the algorithm does what it says it does, and show how it applies to the power company situation above.