Our MAT392 Math Seminar class recently finished giving their presentations of semester long research on a topic of their choosing. Listed below are our presenters along with a description of their talk.

**Allie Ternet, “Modeling Cancer: The Role of Mathematics and Statistics within Oncology”**

Cancer research has greatly progressed over several years, and new discoveries are being made every day. Mathematical modeling has been used help model the formation of cancer and subsequently treat the disease. Over the past fifty years, two models have been pivotal in our treatment of cancer today: the Armitage Doll model and the de Pillis model. Allie explored how these models have changed oncology today as well as the clinical trials and statistical experiments that supported these studies.

**Hannah King, “Semigroups, C* Algebras, and ZM(C)”**

Hannah examined the connection between semigroups and C* algebras, looking at general definitions of these two areas in mathematics. Motivated by her research from the summer in inverse semigroup theory, she looked at the applications of the theorem we derived and applying that theorem within this context.

**Amish Mishra, “An Exploration of the Numerical Range of Toeplitz Matrices over Finite Fields”**

In this presentation, we characterize the *k*th numerical range of all Toeplitz matrices with a constant main diagonal and another single, nonzero diagonal where the matrices are all over the field with *p,* a prime congruent to 3 mod 4. For *k* in , the *k*th numerical range is always equal to with the exception of the scaled identity. Similar techniques are used to discover a general connection between the 0^{th} numerical range and the *k*th numerical range. Amish also investigated new directions to continue this research.

**Drew Anderson, “Matrices of Permutations”**

The study of matrices has been a staple in mathematics for years, especially the permutation matrix from Linear Algebra. However, very little research has been done on similar matrices, which Drew will call matrices of permutations. These are n by n square matrices that have each row containing some permutation of n numbers. We can find applications that have to do with systems of equations, combinatorics, math modeling, sudokus, and Latin squares. In this presentation, then, we examined all these matrices in detail and the patterns to be discovered within.

**Savannah Porter “Defensive Strategies in Baseball”**

Baseball’s lifelong fascination with statistics has been an ongoing study since the game began. The game involves more mathematics than what meets the eye. There are vast amounts of research done on the offense of baseball, but not as much is said about the defensive side of baseball. Savannah aims to answer why some of the key features of baseball exist, such as why there are nine players and why they are positioned on the field where they are.

**Luke Wilson “Tilt the Dice: A Combinatorial Game You Can Win”**

Combinatorial Games are all around us. From tic tac toe to chess, some of best games out there can be fit into the category of study called combinatorial game theory. This presentation took a look at an unstudied game called “Tilt the Dice” – a variant of Nim. We explored the strategy of Nim as well as looking at how we can assure victory nearly every time with little luck involved for Tilt the Dice.

**Stevanni McCray, “The Golden Ratio and its Applications to Sunflowers and Pineapples”**

The golden ratio is a numerical value equal to (1+√5)/2. Even though this astonishing number comes out to be a numerical number, this number is most known as the “most irrational number” for reasons we explained. The golden ratio appears in many places we would least expect—places such as art, architecture, nature, and more. In this research, we explored how the golden ratio appears in spirals in nature—more specifically sunflowers and pineapples.

**Ellie Grace Moore, “Fractals: History, Formulation, and their Artistic Application”**

This presentation gave a brief history of fractals as well as how they can be formed. Ellie Grace touched base on topics such as iterated function systems, Julia and Mandelbrot sets, as well as Newton’s method and complex dynamical system. Then she explained how fractals can be taken in an artistic way and how we see them in our world.

**Deborah Settles “The Symmetries of Dance”**

Have you ever considered the mathematical concepts in dance? What about representing movement with mathematics? Contrary to what you might be thinking right now, the art of dance and field of mathematics are indeed intertwined. Symmetry can be used to describe body movement of dancers. Both frieze patterns and the Klein Four group work well to represent the repetitive foot movement and whole group movement in dances.

**Caleb Holleman “Mathematical Logic and Mind: Use or Abuse?”**

Perhaps the greatest accomplishment in the field of mathematical logic, Kurt Godel’s incompleteness theorems have substantially shaped our understanding of fundamental mathematics. Godel’s two, paired results––which together permanently undermine any claims of mathematics as consistent totality––have since captured the public consciousness. Inevitably, the coupling of the theorems’ profundity and popularization has led to widespread misrepresentation, not the least of which has manifested as unjustified extension of the theorems to multifarious non-mathematical questions; yet some serious scholars have advanced arguments claiming Godel’s theorems bear critically on certain outstanding questions, such as questions related to mind. Following a review of Godel’s incompleteness results, Caleb investigates the function of incompleteness theorems in argument schemes surrounding Turing machines, mechanistic theories of mind, and artificial intelligence.

**Jamie Netzley, “Maximizing Your Probability of Scoring a Penalty Kick”**

While they seem so simple to a bystander watching a soccer game, penalty kicks can decide the winner of an important game. Therefore, when rewarded with a penalty kick, the player taking the kick must give himself the best chance possible of making the shot. However, to do this, there are many variables that the kicker must consider. Where is the goalkeeper standing? Where is the goalkeeper going to dive? Where do I want to aim the ball? How hard should I kick the ball? There are several factors that go into the “unstoppable” penalty kick, that is a kick that is perfectly placed so that the goalkeeper has no chance of saving it. This presentation answered the above questions and give more insight into how a player can improve his penalty kick.

**Hannah Peters, “Circles: Relationships between Angles, Arcs, Secants, and Tangents”**

Circles are everywhere and can be used in many different ways as a tool to find measurements or distances. There are connections between circles, angles, and arcs that create ways for one to find the length of an arc or an angle. These connections are made with the help of geometry and the introduction of lines such as radii, tangents, and secants. There are angles inside of the circle and outside of the circle and equations that were created to discover the corresponding lengths desired. What if you are given an angle and need to find the length of the opposite side? Or you are given the arc length and are wanting to find the angle? All of these instances are touched on in this presentation along with some conjectures on inverse of a point and tangent lines.

Students spent the entire semester researching these topics to write their paper and then present on their research in a twenty minute mathematical presentation.