Mathematics Capstone

Title: Introduction to Module Theory: A Conceptual Expansion from Vector Spaces to Rings and Fields

Abstract: Many equations or systems of equations are simple to solve because multiplicative inverses exist. Once this property is removed, solving can become much more complicated. This reading-based capstone project serves as an introduction to Module Theory with applications of solving equations over rings where such inverses may not exist. Ultimately, the focus of the capstone project is on understanding R-modules (i.e. modules over a ring R) and related structures.

Advisor: Dr. Hongde Hu

Title: Introduction to Young tableux and RSK-Algorithm.Introduction to Young tableux and RSK-Algorithm.

Abstract: In mathematics, a Young tableau is a combinatorial object useful in the study of representation theory of finite groups. It provides a convenient way to describe the group representations of the symmetric and general linear groups and to study their properties. Young tableaux were introduced by Alfred Young, a mathematician at Cambridge University, in 1900 and then were applied to the study of symmetric group by Georg Frobenius in 1903. This talk will give an introduction to Young tableaux. We will talk about one of its beautiful applications, a proof to a very important combinatorial identity using the famous Robinson-Shensted-Knuth (RSK) algorithm.

Advisor: Dr. Lipika Deka

Title: Symmetric Group and its connection to Young Tableaux.

Abstract: Symmetric group is the most well understood finite group and this serves as the building blocks for all finite groups. This talk will introduce the main definitions and theorems about Symmetric group with n elements including cycle types, conjugacy classes, and its group algebra. We will discuss a bijection between the conjugacy classes of symmetric group and partitions of n and show how this connects Symmetric group to Standard Young Tableaux. We will also show how Symmetric group acts on Standard tableaux.

Advisor: Dr. Lipika Deka

Title: Representation Theory of Symmetric Group using Young Tableaux

Abstract: For more than a decade, the Representation theory of Symmetric group has been the inspiration for research in the study of the Representation theory of both finite and infinite groups. In this talk we will give an introduction to the representations of Symmetric group. We will show the construction of all finite dimensional irreducible representations of Symmetric group using Standard Young tableaux. We will show how this serves as a building block for all finite dimensional representations of Symmetric group.

Advisor: Dr. Lipika Deka

Title: Shock-Tubes and the Riemann Problem in Gas Dynamics

Abstract: We solve a special case of the Riemann problem in gas dynamics. The Riemann problem can be characterized by a shock-tube. A shock-tube is a long thin, cylindrical tube containing a gas separated by a thin membrane. The gas is at rest on both sides of the membrane, but has different constant pressures and densities on each side. We assume the special case that the pressure of the gas varies inversely as a power of the volume. This problem is modeled by a system of partial differential equations referred to as the p-system. This system is the simplest set of hyperbolic conservation laws that model gas dynamics. Solutions of the p-system consist of rarefaction and shock waves. In a rarefaction solution, the density and pressure change continuously from one side of the shock-tube to the other. However, a shock-wave solution contains discontinuities in the pressure and density across the shock-tube. The existence of discontinuous solutions requires special techniques known as weak solutions. In this talk, we introduce the Riemann problem and then describe a full set of solutions for a specific gas.

Advisor: Dr. Michael B. Scott

Title: Effects of Manipulatives and Visual Models in the Algebra One Classroom

Abstract: The purpose of this capstone talk is to discuss the results of an experiment done on the implementation of manipulatives and visual models in the Algebra One curriculum. To test the effects of manipulative implementation, four periods of Algebra One at a local high school were tested. Two periods acted as a control group, in the other two periods activities were done approximately once a week. Observation was done on student behavior and attitude, student reaction to the activity, and teacher attitude during both the manipulative activity and the alternative, lecture based lesson. Example activities will be demonstrated and overall outcome and ideas for further research will be examined as well.

Advisor: Dr. Michael B. Scott

Title: A Mathematical Model of Fluid Flow Across an Aerofoil

Abstract: We introduce a model of fluid flow past an aerofoil, for example an airplane wing. The model consists of a system of partial differential equations derived from conservation laws. We first develop the simpler model of flow past a circular cylinder. Then we generalize the cylinder flow to model flow past an aerofoil. Lastly, we describe the characteristics of this flow.

Advisor: Dr. Michael B. Scott

Title: Winning the "SOS" game through Combinatorial Game Theory

Abstract: A combinatorial game is a two-players game, where both players are given complete information about the game. There are no chance moves, and in case of a draw, an ending condition is applied to decide a winner. Combinatory game theory guarantees a winning strategy for one of the players. This capstone explores the game of SOS, an impartial combinatorial game, to determine the winning strategy. We will show that any SOS game can be split into smaller SOS games with a known nimber value. Thus, our findings suggest that the sum of the smaller games will be the nimber value of the original game.

Advisor: Dr. Rachel Esselstein