An Alternative Graph Theoretic Proof of the Amitsur-Levitzki Identity

*Sam Rudy, Department of Mathematics, Washington University in St. Louis*

March 24, 2014 - 3:00 pm to 4:00 pm*Location: Cupples I, Room 199 | Host: Prof. John Shareshian*

*A Senior Honors Thesis Presentation.*

Abstract: The Amitsur-Levitzki identity is a polynomial identity of degree 2n on the on the ring *Mn(R)* over any commutative ring *R*. Since its original proof by Amitsur and Levitzki in 1950 the identity has been shown using various distinct methods. A 1963 paper by R. Swan showed the equivalence of the identity to a certain property of Eulerian paths on directed graphs that he was able to prove using induction on the number of vertices. I use an alternative method to prove the graph theoretic result, creating an adjacency matrix for the digraph over a non-commutative polynomial ring and proving the result via the Caley-Hamilton Theorem.

### Nash-Moser Inverse Function Theorem and its application to Ricci Flow

*Ziming Shi, Department of Mathematics, Washington University in St. Louis*

March 26, 2014 - 3:00 pm to 4:00 pm*Location: Cupples I, Room 199 | Host: Prof. Quo-Shin Chi*

*A Senior Honors Thesis Presentation.*

Abstract: Nash-Moser Inverse Function Theorem was originally discovered by John Nash, who used it to prove his famous embedding theorem of Riemannian manifold. It has now become an indispensable tool for many areas of mathematics, in particular PDE and differential geometry. In this presentation I am going to give an outline to the proof of the Nash-Moser theorem and highlight the most important ideas involved, and then I will demonstrate an application of Nash-Moser theorem in geometric analysis: proving the short-time existence of Ricci Flow.

### The Relativistic Heat Equation;

*Evan Miller, Department of Mathematics, Washington University in St. Louis*

March 4, 2013 - 4:30 pm to 4:30 pm*Location: Cupples I, Room 199 | Host: Prof. John Shareshian*

*A Senior Honors Thesis Presentation.*

Abstract: In this presentation, I will discuss some theoretical problems with the standard heat equation related to special relativity and the infinitely fast propagation of signals. I will explore an alternate heat equation developed by Brenier and McCann and Puel. The bulk of this talk will then focus on the extension of the the classical theory of the heat and Laplace equation to this new equations to this new equation. In particular, we will focus on the cases in which the strong and weak maximum principles from the standard heat equation or Laplace's equation do or do not extend to the relativistic heat equation and its stationary states. I will prove the strong maximum principle for the equilibrium states of the relativistic heat equation and a weak maximum principle for the relativistic heat equation. Finally, I will discuss a counterexample that shows we do not have a strong maximum principle for the relativistic heat equation.

### Modifications to Nowak et al.'s Mathematical Population Genetic Model of The Evolution of Eusociality

*Stephen Rong, Department of Mathematics, Washington University in St. Louis*

March 31, 2013 - 1:00 pm to 2:00 pm*Location: Cupples I, Room 6 | Hosts: Profs. Renato Feres (Mathematics) & David Queller (Biology)*

*A Senior Honors Thesis Presentation.*

Abstract: Abstract: Nowak et al.'s (2010) paper critiquing the standard use of kin selection theory for understanding the evolution of eusociality has been heavily criticized by the evolutionary biology community for misunderstandings and misrepresentations of the kin selection and eusociality literature. However, less attention has been paid to the mathematical population genetic model of the evolution of eusociality that they consider in their supplement. We extend and critique their paper: (1) that eusociality is hard to evolve, (2) that relatedness is not important to the evolution of eusociality, and (3) that there is no conflict of interest between queen and offspring. We find that (1) is due primarily to assumptions about the functional response of fitness in their model; (2) cannot be made from their model, and thus we modify their haploid model to consider the effects of relatedness, which we find to be important for evolving eusociality; and (3) ignores the potential for queen-offspring conflict and queen-control of offspring reproductive fate, which we explore through modifying heir haploid model to consider both queen-control and worker-control scenarios. We frame their model using the formalisms of Petri Net theory to easily derive ODEs which agree with those derived by Nowak et al.'s, stochastically model the system using Gillespie's algorithm, and derive stochastic ODE's describing the stochastic dynamics of the system.

### Categories, Groupoids, and Groupoidifications

*Ben Pollack, Department of Mathematics, Washington University in St. Louis*

March 31, 2014 - 3:00 pm to 4:00 pm*Location: Cupples I, Room 199 | Host: Prof. Xiang Tang*

*A Senior Honors Thesis Presentation.*

Abstract: In this talk I will provide an introduction to basic category theory and categorification, with a particular emphasis on groupoids. I will first discuss some of the essentials of category theory and then summarize some of the work of Baez and his co-authors in which they apply category theoretic results to obtain a categorification of L^{2} functions. Finally, I will present some original research that lays the foundation for a theory of categorifying continuous functions between topological spaces.

### Automatic Classification of Formal and Informal Text;

*Yinghui Xia, Department of Mathematics, Washington University in St. Louis*

March 27, 2013 - 1:00 pm to 2:00 pm*Location: Cupples I, Room 199 | Host: Prof. Ed Spitznagel*

*A Senior Honors Thesis Presentation.*

Abstract: This thesis explores the factors that indicate the documents formality level and constructs a formality classier based on iteratively reweighted least squares(IRLS) method. A random sample of 374 emails drawn from the Enron Email database are used for feature selection and evaluation. The sample is manually divided into formal section and informal section. The study filters all tokens in the set, including words, phrases and punctuations and aims at selecting a limited number of features to reduce the dimensions in automatic classification. In Chapter 4, I show the performance of these markers in some common classification methodologies, which considerably reduces the load of analyzing a large dataset. I believe the method proposed in the paper can be applied to categorization of genres other than emails because the study focuses on lexical attributes of an email and eliminates the bias caused by the correlation between category and formality. The number of features in text classification can reach tens of thousands. The IRLS classifier introduced in this paper is constructed on top of a small training corpus and reduces the number of features to five hundred.