Properties for Solutions to Integral and Differential Equations
Partial differential equations (PDEs) and integral equations (IEs) are incredibly useful tools that can be used to model a wide variety of physical phenomena. Due to their complexity, it is usually not feasible to obtain exact solutions to PDEs or IEs, so we instead investigate the behavior of properties of solutions, or through estimates on the magnitude of different quantities (generally called energies). Hadamard's definition of well-posedness includes uniqueness, existence, and continuous dependence of solutions. We will discuss these aspects in the context of PDEs and IEs, by offering ideas into classical tools used in investigations, such as energy minimization, which is useful for showing uniqueness and existence. Finally, we present some ideas behind continuous dependence or stability of solutions.
Mathematical Control Theory and Applications
In classical differential equations theory (both ordinary and partial), we look for solutions to differential equations with a certain starting position and then study how they behave. Control theory is a different flavor, where we try not only ask for a certain starting position, but a certain ending position as well. In essence, we want to get from point A to point B and dictate how the path changes along the way. Doing this requires that we change the differential equation depending on the ending position, via a control function. This talk will introduce some of the basic ideas in control theory and discuss applications to cruise control and (time permitting) cake baking. No knowledge of differential equations is necessary.
A Gentle Introduction to Geometric Group Theory
Geometric group theory is a relatively new area of math that allows us to connect abstract algebraic concepts with geometry and topology. From the definition of a group, we can introduce two fundamental building blocks of geometric group theory: the Cayley graph of a group and its word metric. This forms a metric space that the group can act upon, and we may classify groups based on the resulting spaces they create. We will discuss several examples, and also delve deeper into the special properties of hyperbolic groups.
Multiplicative factorization in numerical semigroups
In elementary school, one learns the Fundamental Theorem of Arithmetic - that any positive integer greater than 1 can be represented uniquely as a product of positive prime integers. What we do not learn is that this uniqueness property can fail when examining some multiplicative subsemigroups of the positive integers. Not only is factorization not guaranteed to be unique, but neither is the amount of elements used to express a positive integer's factorization. A spectacular amount of nonunique factorization occurs in a class of semigroups known as numerical semigroups. In this talk we will discuss how to construct numerical semigroups and see some results on their factorization properties.
Classifying polytopes in dimensions 4 and beyond
It is well-known that there exist infinitely many regular polygons and that there exists exactly five regular polyhedra (more commonly known as the Platonic solids). In this talk, we will explore the geometric properties of these 2- and 3-dimensional shapes to understand the restrictions that lead to there being exactly five Platonic solids. We will extend these ideas into the fourth dimension to understand what the 4-dimensional analogue of a regular polygon “looks” like, and how many of these shapes exist and why. Building on the patterns that arise from dimensions 3 and 4, we will be able to make claims about such shapes in dimensions 5 and beyond.
Exploring the Navier Stokes Equations and their consequences
As an undergraduate, I wondered how people went from being calculus students to doing research in mathematics. I've learned that sometimes, all it takes is a problem to inspire you. In this talk, we will explore the (problematic) set of equations that inspired me--the Navier Stokes Equations. Additionally, I will discuss some of my current interests and projects involving non-local (peridynamic) operators and computations in Matlab.
Error-free communication
How often do we trust that the text or email we just sent or received came through correctly? Do you ever wonder why it works every time? In this talk, we will look at error-correcting codes, which keep our data accurate, whether on physical devices like CDs or over 3G, 4G, 5G, or any-G wireless communication. We will discuss how it is possible for a message to correct itself or to indicate whether there have been some errors in the transmission. Afterwards, we will talk about graduate school, so please come with questions!
Are we there yet? A beginner’s guide to the infinite