Projects will be selected by students, based on their interests and abilities, with approval of instructor.

** Six possible projects:**

(A) ** Mathematics of Elections and Voting **
E.g., we can use probability (MATH 135AB) to predicting elections.
Bayes' theorem gives us the ability to update the probability of a hypothesis (such as the prediction of a winner of an election) as new data becomes available (such as polls). Students will understand this statistical model and process, identify an appropriate application dataset and apply these techniques. Prerequisite: MAT 135A or equivalent.

(B) ** Constructing Cages and other symmetric graphs ** Constructing graphs
with high symmetric properties connects group theory and combinatorics. A $(k,g)$-cage is a graph where every vertex has degree $k$-regular and the smallest cycle has size $g$, having the minimum possible number of nodes. It is not known what are the possible cages for all values, e.g., is there a $(4,11)$-cage?
Students will investigate study the properties of such graphs.

(C) ** Random polynomials and their roots ** The study of random algebraic
polynomials is of theoretical interest, but they appear naturally
e.g., a random algebraic polynomial will arise if the coefficients of
an algebraic polynomial are subject to random error. This project will
investigate the zeros of a random algebraic polynomial and the
measurability of the number of real and complex zeros and its relation
to the zeros of its derivative

(D) ** Classification and clustering for prediction of restaurants **
Imagine you have access to all of the information on
Yelp about restaurants in Davis, and you are asked to predict the
health grade of (some of) these restaurants. You could use prior
health grade information for other restaurants to determine which
restaurant features (location, rating, etc.) best correlate to a
passing health grade and form a function that takes in these features
and outputs pass/not pass (this task is called classification) or you
could try to collect the restaurants together into similar groups that
will likely have the same health grade (this task is called clustering).

(E) ** Mechanical machines based on the geometry of lines and
circles ** There are a number of really cool machines and linkages
(e.g, pantographs) that were invented to make complicated mechanical
motions. These require the use of advance geometry to explain why and
how they work. The goal is to discuss the mathematical ideas that justify
the machines and possibly use them in a class context.
If you enjoyed MAT 141, 114, 116 and or you wish to teach
as a career, this project could be for you

(F) ** Fixed-points & combinatorial topology ** Topology is the study of properties
that remain under deformation and continuous change, what is preserved under
a continuous map? Topology can offer several interesting areas for research.
E.g.Often one can represent
topological spaces as gluing discrete objects like triangles. Students
could explore the notion of * homological group * to investigate
topological spaces. Also, when we have a continuous function from the disk into itself
it has point that is fixed. Fixed points of functions have many interesting
applications and properties. For instance, in Economics, a Nash equilibrium of a game is a fixed point of certain maps.

(a) A 20-30 pages document in LaTEX that contains the six required chapters: (1) Introduction presenting what the goal is, why do we care, relevant history, etc. (2) A summary of what the plan or method for solving will be (3) Technical details (4) In-depth analysis of one aspect of the project (5) References (6) Appendix containing relevant material (e.g., code or data with comments). An example of the way the document should be in LaTEX is available here .

(b) A poster presentation (up to 30 minutes long, no more than 15 slides) explaining the idea of the solution and the key math ideas used. Explanation should be to the level of a math 167 student (someone that knows the basics of linear algebra terminology, etc). Think of preparing a class about the project.

(b) In some cases, Write code that solves the problem in simple instances.

GRADING CRITERIA: Points will be decomposed as follows:

Content (10 pts) - the mathematics in the presentation should be correct. Your should explain what the questions are that you are answering. The code should run without bugs.

Clarity, Quality of presentation, Organization (10 pts). The reader can follow and understand the mathematics being described. The order of slides makes sense, correct spelling of terms. You should use pictures and diagrams when possible.

Creativity and Originality: ( 5 pts). You will get these last points for giving new spins to the challenge. E.g., if you create a youtube video instead of slides, or your MATLAB code can do more than just the minimum expected or can do it in many more cases. Another example is if you use LaTEX to create your transparencies instead of power point. But these are only suggestions to invite you to innovate!

NOTE on outside help: You may only discuss the project with your classmates and with Prof. De Loera or the TA. You are allowed to use any book available in the UC Davis library or online. All Sources must be cited!!