## Teaching Mathematics in Our Times

Mathematics is a very powerful tool in multiple activities from solving an equation in one variable or to decode a complex code. The teaching in general and of the mathematics particularly are matters of greater importance for the contemporary society.

**Ivan Chavez Lopez**

## The Lorenz Atractor

In 1963, Edward Lorenz developed a mathematical model for atmospheric convection. The model is a system of three ordinary differential equations, now known as the Lorenz Equations. This system has chaotic solutions for certain parameter values and initial conditions, culminating into what Lorenz called the Butterfly Effect.

**Advisors**: Janet Best and Joaquin Rivera Cruz**Rafael Cizeski Nitchai**

## Connection Between PDEs and Biology

A partial differential equation (PDE) is a differential equation that involves its function and the partial derivatives. PDEs are important because they help apply mathematics into different sciences. One science in particular is biology, as PDEs are used to describe the spread of genes or diseases through diffusion.

**Advisor**: Joaquin Rivera**Reference**: Partial Differential Equation Textbook**Joel Coppadge**

## A Mathematical Model of Ischemic Cutaneous Wounds

Ischemic wounds happen when the blood flow is blocked from an area of the body, limiting the oxygen and nutrients that are essential to wound healing. Partial Differential Equations make the relationship between the Extracellular Matrix and some important components in wound healing. The solution of this partial differential equation is numerical.

**Advisor**: Chuan Xue**Reference**: A mathematical model of ischemic cutaneous wounds**Renan Lazarini Gil**

## Poincaré Conjecture

A poster and PowerPoint were prepared on Poincaré Conjecture including history, important people involved in the conjecture. The conjecture was explained as well as concepts needed to understand, and some important applications of the conjecture. The relationship of Perelman and conjecture was included as well as the relationship with the topology.

**Advisors**: Thomas Kerler, Juan Ariel Ortiz Navarro**Aura Carina Marquez Martinez/Mariana Paola Ramos Gordillo**

## Borel Summability

We try to solve the equation z^2g'+g=z using power series. We use a rewrite of the factorial and the change of variables z=1/x to modify a power series in order to arrive to a convergent function that turns out to be the solution of the DE. Finally we formalize this process defining the concepts of Borel Summability and justify it calling some theorems about it.

**Advisor**: Ovidiu Costin**Francisco Martinez**

## Connection between Mathematics, Computational Science and Biology

Throughout these years we have seen some great advancements in the field of biology that couldn’t have happened if no not for the help of mathematics and biology. Now we ask ourselves what is the connection between these sciences and how do they work together to produce such fantastic results. This is the exploration that is taken here using a mathematicians look into the way of how has his subject of study has mixed with others.

**Advisors**: Ching-Shan Chou and Ana C. Gonzalez**References**:

- Committee on Mathematical Sciences Research for DOE s Computational Biology, National Research Council. Mathematics and 21st Century Biology. (2005). The National Academy Press.
- Dutch Society for Theoretical Biology (NVTB). MATHEMATICS and BIOLOGY: THE INTERFACE CHALLENGES and OPPORTUNITIES. Website used: august 5. Mathematics and Biology: The Interface Challenges and Opportunities
- Murray, J.D. Mathematical Biology: I. An Introduction. Third Edition (2002). Springer.
- Ranganath, H. A. Nothing in Biology makes Sense without the Flavour of Mathematics. (2003) Nothing in Biology makes Sense without the Flavour of Mathematics [pdf]

**Juan Martinez Rivera**

## Fourier Series

Suppose that a function f defined in the closed interval [-pi, pi] can be expanded in a series whose terms consist of the trigonometric functions cos(nx) and sin(nx). The expansion is called the Fourier series of the function and is denoted by f-hat. The function f(x)=x was defined in the closed interval [-pi, pi] and its Fourier series was used to prove that the p-series with p=2 converges to pi^2 / 6.

**Advisor**: Aurel Stan**Reference**: Differential Equations Textbook by Dennis G. Zill and Michael R. Cullen, 2005**Joeseph J. Martinez**

## An Introduction to Differential Topology

This project includes definitions of Differential Topology and differential manaifold (variety). Explains and illustrate how to build a Differential manifold. Extends the concepts of Calculus to Topology for defining Tangent spsace. Includes definitions and examples of what is a Quotient manifold considering the Implicit Function Theorem and space of orbits. Includes a theorem for determine which is a manifold and examples. Briefly defines and exemplifies Riemannian varieties and connections.

**Advisor**: Juan Ortiz Navarro**References**:

- WIkipedia article on Manifold (accessed: 08/06/13)
- Interview the researcher in Differential Topology of The Ohio State University: Jean-Francois Lafont

**Jesus Oyola**

## Complexity Classes

A brief presentation about complexity classes and Cook's Theorem and relating these theoretical topics to real life; showing their importance on the development of efficient algorithms for solving decision problems or at least help us know if there exists, or not, an efficient algorithm for a particular problem.

**Advisor**: Ana C. Gonzalez**Harold Padilla**

## Chemotaxis

Chemotaxis is the phenomenon by which unicellular organisms move in response to chemicals in their environment. Currently there are two main steps in its study, the first belonging to the work of biologists and the second to mathematicians. Parting from basic models of movement done with partial differential equations, with chemotaxis one must also consider the concentration gradient to describe the behavior of agents affected by the presence of overriding external cues. Knowledge of this phenomenon is applied widely in various areas of medicine.

**Advisors**: Ching Shan-Chou and Joaquin Rivera Cruz**References**:

- Edelstein-Keshet, L. (2005). Partial Differential Equation Models in Biology. In: Mathematical Models in Biology. United States: SIAM. 437-442.
- Hall J. (2013) Discussion of Reading 15 Stability of Equilibrium Solutions

**Harold Padilla /Natalia Hernandez Santiago**

## Connes’ proof of Morley’s Theorem

In a 20 minute talk we will go briefly over a new proof of Morley’s theorem. The theorem states that pairwise angle trisectors of a triangle intersect forming an equilateral triangle. This proof links Euclidean geometry to group theory. Though we don’t give the actual proof, which involves some long factoring, we will be able to observe how Conne’s theorem is applicable to solve the given problem.

**Advisor**: Henri Moscovici**Reference**: Connes, Alain. A new proof of Morley’s theorem, 1998. Web. 1 July 2013. A new Proof of Morley's Theorem [pdf], Alain Connes**Juan Fernando Valdés**

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