1. Rate of growth of populations (exponential model using linear regression and a differential equation).
2. Flow of water from a cylinder (square root model).
3. Growth of sunflower plants (logistic model).
4. Harvesting of trees (equilibrium solutions of an autonomous differential equation).
5. Analyzing temperature (sinusoidal model).
6. Determination of amount of solute in two or more containers (systems of first order linear differential equations using eigenvalues).
7. Predator-prey model (system of first order non-linear differential equations using PC-Matlab or Maple software).

1. Cantor and cardinality.
2. Ordinal numbers.
3. Some of Godel's work on undecidability.
4. Russel and Whitehead's work and Principia Mathematica.
5. Frege's work on the foundations of arithmetic.
6. Peano's axioms of arithmetic.
7. Dedekind's ideas of "cuts" of the real line and completeness.
8. Hilbert and the foundations of geometry.
9. Cauchy's introduction of rigor in analysis.
10. The four color theorem and what "proof" means.
11. The classification of finite simple groups and how the proof of that came about.

1. Choose a modern application of number theory (such as cryptography) and prepare a 2-3 page paper discussing the application.
2. Investigate a topic not covered in the class and prepare a lecture (15-20 minutes) which is to be presented in class.
3. Write a 2-3 page historical discussion of a central topic in number theory.

1. Solve problems at a higher degree of difficulty than other non-honors students.
2. Become familiar with probability distributions not normally covered in class. Also become familiar with families and systems of distributions. The SU and SB Johnson distributions, Tukey's lambda, the four parameter Lambda distribution, the Weibull, Gamma, and Extreme Value distributions are some examples. The study can include theoretical apsects, careful analysis of the shape in relation to the parameters, simulations, applications, etc. Write a 2-3 page paper detailing the distributions studied.
3. Reading (and eventually working on) applications of probability in the real world.
4. Get acquainted with classical probability topics or problems not covered in the course. Give a presentation in class on the topic.

1. Investigate the applications of dihedral groups in molecular modeling. Write a 2-3 page paper summeraizing your findings.
2. Explore the role of transformation groups in the study of geometry. Describe the group structures of both Euclidean motions and similarity transformations in detail and include a derivation of the equations for these transformations.
3. Write a descriptive analysis of the differences between the concepts of group, ring, field, and module. View this as an explanation for fellow students who may not understand the concepts. Include examples and illustrations of each of your main points. Your analysis should consist of at least 2-3 pages.
4. Read a book on the history of algebra (group theory, symmetry, simple groups, algebraic solvability), prepare a written report and a PowerPoint presentation to be given in class or in the math department seminar.

1. Student must select a current research paper related to real analysis and give a report on that paper either at a student seminar, at the math department colloquium, or in class.
2. Read a book on the history of analysis (a biography of a key figure, Newton and Leibniz's calculus war, the rigorous revolution of the 19th century), prepare a written report and a PowerPoint presentation to be given in class or in the math department seminar.

1. Investigate applications of Riemann-Stieltjes integration.
2. Investigate the mathematics behind generalizations of Riemann integration (such as Lebesgue integration).
3. Explore the historical setting which necessitated the development of generalizations to the Riemann integral (such as the Lebesgue integral).

1. Use a software package (such as F(z), Mathematica, Maple, or Matlab) to geometrically explore some topics from the class. Present the results of these explorations in class.
2. Study solutions to various applied problems which involve complex variables (such as partial differential equations or conformal mapping as applied to aerodynamics). Present the results to the class.

1. Chemical graph theory: DNA is a chain consisting of bases, where each base is one of four possible chemicals. The sequence of bases encodes certain genetic information. In particular, it determines long chains of amino acids which are known as proteins. How long does a part of a DNA chain have to be for there to be enough possible sequences to encode 20 different amino acids? Numerous projects from this topic can be developed. In fact, there are research journals devoted entirely to combinatorial chemistry and this field is currently one of the "hottest" fields of research.
2. Scheduling: A basketball team plays 30 games in 20 days playing at least one game every day. Show that there must be a period of consecutive days during which the team plays exactly 9 games. Is there necessarily a period of consecutive days when exactly 10 games are played? From problems like this, enough expertise can be developed to design more complex schedules.
3. Graph models: In a city without one-way streets, a math professor drives to school, which is 8 blocks west and 5 blocks north of his house. In how many ways can he drive to school without ever repeating his entire route if he always avoids a dangerous intersection 2 blocks west and 2 blocks north of his house? In how many ways can he avoid two dangerous intersections: one 2 blocks west and 2 blocks north and the other 3 blocks west and 2 blocks north? From problems like this, the student can model traffic flow in a major city or create the best design for a gas pipeline.

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Last updated: December 22, 2009

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