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\begin{center}
\Large \sc Math 2110: Introduction to Proof \& Logic:\\ Final Projects
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\vspace{0.1in}

Your last task this semester is a {\tiny small} project. 
You will pick a topic related to, but yet, a little beyond what was covered in class. Then 
you will write up your findings and record a short presentation.

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\noindent
\underline{\Large\bf Due Date:} The paper and recorded presentation are due by \underline{\bf Monday, May $4^\mathrm{th}$ at noon}.

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\noindent
Here is what I expect from you (and what you will be graded on):

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\begin{description}
   \item[Project Proposal] Pick out a topic and outline what you plan to study. Then talk to me about it.  I will give you some suggestions (possible examples, problems, theorems to consider). 

\underline{\bf Due:} No later than \underline{\bf Friday, April $24^{\mathrm{th}}$}.

   \item[The Paper] Your paper should be similar in structure to the one you did for your midterm mini-project. 
   \begin{itemize}
      \item The paper should be {\it at least} 5 pages long. As you try to fit in the other requirements, you may find that it grows to be a bit longer. The average paper length from previous classes was about 7-8 pages.
      \item This should be typed up nicely in \LaTeX.
      \item Your classmates are your audience. Make sure your paper is understandable from their current level of mathematical education [build on what we covered in class].
      \item The first page should include title, author, and abstract. 
      \item You should have a list of references at the end of your paper. You should have {\it at a minimum} 3 references. Make sure {\it at least} 1 reference is a {\sc solid} reference (i.e., something published and peer reviewed like a research article or textbook).
      \item Begin your paper with some background about your topic. Include some history. If you were studying Hilbert spaces, you should indicate who first defined such a space and maybe give a brief biography of that mathematician. If your topic doesn't lend itself to such a discussion, pick a prominent mathematician who made a big impact on your topic and give a brief biography.
      \item Your paper should include at least 1 ``big theorem'' and its proof. 
      \item You should give at least 3 ``concrete'' examples illustrating your topic.
      \item You should do at least 3 ``homework problems'' related to your topic. (These could be calculations or some easy-to-prove theorems -- just think of something I might assign if we covered this in class.) 
      \item Your paper should include a discussion of some application of your topic. This can be a scientific, ``real-world'', or mathematical application. 
      \item Ideally, you should do a inductive proof and proof by contradiction somewhere in the paper --- if you can't come up with a reason to use these techniques, come talk to me. I'll try to help you find a way to fit these things in somewhere. (Some topics just don't lend themselves to inductive proofs.)
   \end{itemize}

   \item[The Presentation] Everyone should record a 10 minute ({\it approximately}) presentation about their topic. Sum up what you learned. Give some examples. Describe your big theorem and at least ``sketch'' its proof. Create slides. You can either use \LaTeX or Powerpoint or whatever looks good.
\end{description}

\noindent {\bf Point breakdown:} Paper 75\%, Presentation 25\%.

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\begin{center}
\Large \sc Random Topic Suggestions!!!
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Here are some suggested topics. Your don't have to pick from this list, these are just random suggestions.
By the way, if you pick a topic closely related to someone else's topic, please feel free to
coordinate your papers and presentations --- you still have to write up your own stuff, but you can
build off each other's background.

\begin{itemize}
   \item Mersenne primes and perfect numbers.
   \item Error correcting codes, cryptography, and modular arithmetic.
   \item What is a Lie algebra?
   \item Difference equations.
   \item Cauchy Sequences. A metric space is called ``complete'' if every Cauchy sequence converges. It can be proved that
            every metric space can be embedded in a complete metric space. The real numbers are complete in this sense.
   \item Uniform continuity --- a powerful assumption.
   \item Normed spaces and inner-product spaces. 
   \item Path connected. Every path connected space is connected. But the converse does not hold. Path connectedness is
             more commonly used than regular connectedness.
   \item Product and Box topologies. Given topologies defined on spaces $X$ and $Y$ one can define a topology on $X \times Y$.
             This can be done for infinite cartesian products as well. The product of compact spaces is still compact.
   \item Symmetric groups and group actions. One can easily classify the platonic solids. Each of these has a group of symmetries.
             The rotational symmetries of the icosahedron form the smallest non-abelian simple group.
   \item Quotients of groups, vector spaces, topological spaces. Gluing topological spaces together.
   \item De Rham cohomology and the generalized Stoke's theorem.
   \item Dual spaces and bounded linear operators. What does bounding an operator's norm have to do with continuity?
   \item Sizes of infinity and transfinite induction.
   \item Permutations and determinants
   \item Change of basis in linear algebra and Jordan form. Matrix exponential.
   \item Complex variables. Better than Taylor series it's....Laurent series! How the coefficient of $x^{-1}$ computes integrals.
\end{itemize}

\dots Don't see anything that strikes your fancy? Come talk to me, maybe I help find a topic for you.

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