Math Models of Population Dynamics
Math models are a useful tool in many different applications. We will discuss what math models look like and some of these applications before focusing on population models. We will build up from exponential growth models to Lotka-Volterra models for competing species. From there, we will introduce equilibrium points and a few tools to evaluate their stability. Finally, we will discuss an application of this model to cancer treatment.
Zombies? RUN (maybe)!
The zombie apocalypse has been the focus of a number of movies, TV shows, and video games for several decades now. A few examples include World War Z, The Walking Dead, and Resident Evil. A common thematic question lies within many of these: will humanity survive? We can use mathematics to answer this question and figure out the best strategy if we are going to survive. Did The Walking Dead get it right, or is there a better way? How well can humans actually outrun zombies? In this talk, I will explain how mathematics can be used in this context and explore what math says about the best survival tactics for a zombie apocalypse.
A crash course in neural nets, object detection, and semi-supervised learning
This summer, I worked as an intern at Ocuvera, a company based in Lincoln that prevents falls in hospitals using artificial intelligence. As a math graduate student entering the internship, I had zero experience developing AI and knew barely anything about deep learning. After 9 weeks, I had implemented a state-of-the-art semi-supervised learning algorithm for object detection. In this talk, I want to share with you a bit of what I learned from this experience, both about artificial intelligence, and about how surprisingly accessible the field is for anyone with a bit of programming experience.
Classroom social support: A phenomenological case study of mathematics graduate teaching assistants’ decision-making in the classroom
Every day educators make numerous subconscious that occur "in the moment" and therefore do not receive much thought. Many of these decisions are related to social support, actions that are intended to prevent or alleviate negative emotions such as stress or anxiety. Research suggests that there are different types of social support and that thesocial support offered by an instructor can have a significant impact on students' performance and affective well-being. Although social support is important to the student experience, there does not appear to be research about instructors' decisions about the support they offer or the extent to which social support is priority for them. A phenomenological case study design was used to explore the goals and beliefs of mathematics Graduate Teaching Assistants (GTAs) related to social support with the goal of understanding how GTAs support their students and how they decide how to do so. This talk will present an overview of the underlying theories of this study related to decision making and social support. We'll illustrate the utility of these theories by examining two GTAs in this study, and then we will discuss some interesting results of the study and the implications that have for GTA professional development.
Two dimensional dynamical systems with the matrix exponential
In physics and chemistry, we learn about processes like nuclear decay and spring oscillation. To motivate these concepts, we introduce some basic models that capture the dynamics, the models being given by ordinary differential equations. We then lean heavily on our knowledge of derivatives from calculus 1 to solve these equations. In this talk, we will introduce an alternative way to solve these equations, the matrix exponential. The matrix exponential allows you to compute $e^{matrix}$, and can be used to solve dynamical systems modeled by systems of ordinary differential equations of arbitrary dimension N (though we will just stick to N=1,2). No prior knowledge beyond calculus 1 is assumed or necessary.
Building brick walls in a billion dimensions
If you were building a brick wall, you would probably lay down bricks one layer at a time and shift each layer of bricks over by half of a brick length from the layer below. Doing this makes the wall quite strong and rigid, and it is why almost all brick walls look the same. A brick wall is basically a 2-dimensional structure, and the typical brick wall is the way to achieve the maximum rigidity in 2 dimensions. But what if we wanted to build brick walls in higher dimensions which also have the maximum rigidity? In this talk, we will (1) discuss what a high-dimensional brick wall even is, (2) see one or two famous theorems in mathematics and how they apply to this problem, and (3) construct a brick wall in every dimension and prove that each has nearly the maximum possible rigidity. No math background beyond high school will be needed to attend this talk.
Introduction to neural codes, rings, and ideals
In mathematical neuroscience, we wish to answer the question; What can be inferred about the environment from neural activity alone?
Neural codes are the brain’s way of representing, transmitting, and storing information about the world. Combinatorial neural codes describe a pattern of neural activity in terms of which neurons fire together, as opposed to the precise timing or rate of neural activity. Combinatorial codes can be analyzed using an algebraic object called the neural ring. In this talk, I will give biological motivation and combinatorial background in order to introduce neural rings and ideals. We will explore basic mathematical neuroscience through examples and correspondences.
Research in mathematics education: Unexpected crossroads between mathematics, sociology, and the arts
In this talk I'll give a friendly introduction to the field of Mathematics Education, including its intersections with sociology and the arts. I'll also share my experience working on four different NSF- and internally-funded projects focused on: (a) teachers' conceptions of functions, (b) mathematics departments' efforts to transform teaching practices, (c) partnerships between mathematics instructors and undergraduate learning assistants, and (d) poetic transcriptions of instructor experiences. There will be opportunities during and after the seminar presentation for Q&A and discussion about mathematics education, and graduate school, more broadly.
Intro to game theory: How to win games and impress your friends
When involved in adversarial contest (like playing a game), a natural question to ask yourself is “Is there a strategy I could take that would give me the best chances of winning?” Luckily, there is a branch of mathematics dedicated to answering questions exactly like that. In this talk, we will give an overview of what game theory is, how it can be used, and practical examples of game theory in math, economics, biology, and games of leisure. By the end of the talk, it is the goal that we all see what strategic elements are shared between companies, chess players, and lizards.
Harnessing Data with Continuous Data Assimilation
Data assimilation plays a vital role in numerous scientific disciplines, such as meteorology, oceanography, geophysics, and more. By incorporating observational data into physical models, data assimilation enables more accurate and robust predictions of complex systems like weather forecasting, directly impacting our daily lives. In this talk, we will explore a recently developed algorithm for data assimilation, known as the Azouani-Olson-Titi (AOT) algorithm.
An Introduction to Nonlocal Modeling
In this talk we will introduce and discuss a relatively new framework introduced by Stewart Silling in 2000 called nonlocal modeling. The original motivation was to study fracture in elastic materials, but it has grown to be a framework that has been used to model things like: population swarming, image processing, general fracture and damage (called Peridynamics in the literature), sand piling and corrosion. The main idea is that in some modeling contexts, the solution function may inherently be discontinuous or not differentiable, so classical PDE may not be ideal. To deal with this, we replace the PDE (local equations) with integral operators that ”add up” interactions between nearby points (nonlocal equations) and can handle non-differentiable and discontinuous functions. We’ll also discuss the subtleties that need to be addressed if these models are used and the type of questions mathematicians are interested in related to these models.
Plant compensatory growth in plant-herbivore interactions
Herbivory is often perceived as having a negative effect towards the growth of plants, but there are some plant species that actually benefit from getting eaten. These plants have a phenotypic trait known as “compensatory growth” which can be described as an accelerated growth in plant biomass as a response to damage caused by herbivory. Very few mathematical models that describe plant-herbivore interactions have accounted for plant compensatory growth. In this talk, we will present and analyze a system of nonlinear ordinary differential equations used to model the interactions between a plant with compensatory growth and an herbivore. The analysis consists of nondimensionalization, as well as determining the existence, uniqueness and stability of steady-state equilibria. We also present numerical estimates obtained from MATLAB to determine coexistence conditions between a plant with compensatory growth and an herbivore.
Geometry of the braid group
Groups can be used to understand many real-world objects, including braids. We will explore the definition and geometry of the braid group and see that the word problem in the braid group is solvable. Prior knowledge of groups and the word problem will not be assumed!