## SAMMS Courses and inquiry activity topics

### Topology

The course will cover the following topics:

- Point set topology: open, closed sets, quotient topology, continuity, homeomorphisms.
- Surfaces: Orientability, examples, Boundaries, classification of surfaces, Euler characteristic.
- Knots: Reidemeister moves, tricolorability invariants, types of knots, basic knot polynomials.

### Real Analysis

The course will cover the following topics:

- Inequalities: Arithmetic and Geometric Means, Bernoulli, Cauchy-Schwarz, Jensen.
- Sequences: ϵ–δ definition of convergence, Cauchy, monotone and bounded, recursively defined.
- Series: conditionally and absolutely convergent, criteria of convergence.
- Continuous Functions: ϵ–δ definition, uniform continuity, some functional equations.

### Abstract Algebra

This course will be a gentle introduction to the concept of ring in Abstract Algebra, with interesting and accessible applications. Starting essentially from scratch (axiomatics of groups, rings and modules), this course will guide you towards a celebrated structure theorem, Wedeburn-Artin Theorem.

- Rings in algebra: Axiomatics, definitions, and examples.
- Properties of rings: Arithmetic properties of the elements in rings and the lattice of ideals.
- Division rings, fields, simple and semisimple rings: Wederburn-Artin theorem and the structure of semisimple Artinian rings.
- Applications of rings: Research problems and Algebraic Theory of Error-Correcting Codes.

### Computational Science. Problem solving using MATLAB

Most problems which arise in calculus and differential equations cannot be solved analytically. For example, most integrals cannot be evaluated analytically, most max/min problems cannot be solved analytically, and most differential equations cannot be solved analytically. However, you will not see such problems in these classes, because the problems are specifically constructed so they can be. In a similar vein, most problems which arise in linear algebra either cannot be solved analytically (such as eigenvalue problems if the matrix is 5x5 or larger), or are too computationally intensive (such as solving a linear system of equations with more than 2 unknowns). Maple and Mathematica are specifically designed to analytically solve "nasty" problems, while MATLAB is specifically designed to numerically approximate solutions to problems that arise in mathematics, the hard sciences, and engineering.

The purpose of this course is to give you a brief introduction to MATLAB, and then have you "explore" some problems. What problems?

- There are many problems in probability theory which are easy to describe but hard, or impossible, to solve analytically. It is easy to code Monte Carlo simulations to have the computer repeat an experiment tens of thousands of times and use the statistics generated to approximate the analytical solution. And, frequently, these computer simulations will provide valuable insight into what is actually happening. In fact, this course is designed to mesh with the Combinatorics & Probability course so that you can analytically and numerically explore problems as a unified whole.
- MATLAB makes it easy to generate graphical images to explore amusing curves in two dimensions and surfaces in three dimensions.
- There are many problems which have chaotic solutions and are easily viewed graphically by using computer simulations
- Also, fractal images such as the Cantor set, the Koch snowflake, Julia sets, and Mandelbrot sets can be easily explored.
- Another topic is how random the world around us really is. In sports it is often asserted that a player is "on a roll" or "in the groove". But is this necessarily true, or is it possible it is just chance? After all, someone somewhere is going to win the lottery twice, it's just that we only hear of it because it's a good human interest story.
- *) Feel free to come up with some of your own.

### Combinatorics & Probability

Combinatorics is one of the oldest branches of mathematics. For instance, assume, that we have three bacteria, two algae cells, and, also assume, that bacteria eat algae. A bit of thought shows that at least one of our bacteria will not be able to eat. This particular bit of reasoning is a special case of a “combinatorial principle” known as the pigeonhole principle and this particular consequence is instinctually understood by all life on earth. Probability is another old branch of mathematics. To illustrate this subject, assume that we have a petri dish divided into 1000 squares. Assume further that we uniformly distribute 768 bacteria randomly among the squares and that each square can contain more than one bacteria. What is the probability that one square will contain more than one bacteria? This is a harder problem that illustrates a probabilistic version of the pigeonhole principle, commonly known as the birthday problem.

- Basic principles: Induction, Pigeonhole principle, induction, “Structural” flexibility.
- Applications: Dirichlet’s approximation theorem, Szemerédi’s affine cube lemma, the Birthday Paradox, and related generalizations.
- Introduction to Ramsey Theory: Ramsey’s theorem, Schur’s theorem, Fermat’s last theorem modulo a prime, van der Waerden’s and geometric applications.
- Use of computational methods, coordination with Computational Science course.

### Differential Equations and Modeling

The course will cover the following topics:

- Models of single-species populations: Basic concepts and techniques of ODEs, bifurcations, logistic and Gompertz models, Allee effect, harvesting in populations models.
- Population models of two interacting species: Systems of linear DEs, qualitative analysis of non- linear DEs, Predator-Prey models, models in epidemiology and Basic Reproductive Ratio.
- Models for populations with spatial structure: Metapopulation models, elementary techniques of linear PDEs, reaction-diffusion equation, applications to genetics, ecology and chemistry.

### Applied Statistics

The course will cover the following topics:

- Graphical and numerical summary of data.
- Statistical inference: estimation and hypothesis tests.
- Tests for comparing two means.
- Linear regression analysis.
- Group projects: analyzing and obtaining meaningful conclusions from real-life data.