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\begin{tabular}{lclcl}
\textbf{MA 314}& \hspace{2in}& \textbf{Geometry} &\hspace{2in}& \textbf{Spring 2014}\\
&&&&\\
 && \textbf{MWF 10 - 10:50} && \textbf{Lovejoy 303 or Mudd 412A}
\end{tabular}
\end{center}
\end{minipage}
}

\begin{tabular}{lcl}
\textbf{Professor:} & \hspace{.2in} & Scott Taylor\\
\textbf{Office Hours:} &  &MWF 12 - 1, W 2-3, Tu 10-12 \\
&& \textit{and by appointment!}\\
\textbf{Office:} && Mudd 401\\
\textbf{Email:} && scott.taylor@colby.edu\\
\textbf{Webpage:} && {\tt http://www.colby.edu/personal/s/sataylor/}\\
\textbf{Course Blog:} && {\tt http://web.colby.edu/thegeometricviewpoint/}\\
\textbf{Prerequisites:} && MA 262 or MA 162, MA 253 or permission of instructor \\
\textbf{Text:} && Francis Bonahon, \textit{Low-Dimensional Geometry} \\
&&Richard Evan Schwartz, \textit{Mostly Surfaces}\\
\end{tabular}

\textbf{The Course:}

MA 314: Geometry of Surfaces, explores the notion of ``geometry'' by introducing metric spaces and the most important 2-dimensional geometries: Euclidean, Spherical, and Hyperbolic. These geometries will be investigated using calculus and linear algebra. The main result that will be proven in this course is that every compact 2-dimensional surface admits either a euclidean, spherical, or hyperbolic geometry. We will also investigate the relationship between tilings of  the euclidean plane, the sphere, and the hyperbolic plane and the geometries of surfaces. Throughout the semester, textbook readings will be augmented with journal articles exploring various applications of geometry.

\textbf{Objectives for increasing mathematical maturity:}

\begin{itemize}
\item Combine algebraic, analytic, and geometric modes of thought to systematically investigate different types of geometry.
\item Engage in significant self-teaching of mathematics.
\item Effectively communicate mathematics in both professional and informal styles.
\item Write proofs in conventional mathematical style
\end{itemize}

\textbf{Major Course Content Objectives}
\begin{itemize}
\item Understand the basic outline of the historical progression of understanding of ``geometry'' and its influence on culture.
\item Understand the axioms for euclidean, hyperbolic, spherical, and metric spaces.
\item Understand the relationship between notions of distance and path length.
\item Apply calculus in different geometric settings
\item Understand and prove major differences between euclidean, spherical, and hyperbolic geometry
\item Formalize intuitive geometric notions such as ``gluing'', ``area'', and ``geodesic'' and verify their properties in different geometric settings.
\item Understand the relationship between 2-dimensional geometries and linear algebra.
\item Prove that every closed surface supports a euclidean, hyperbolic, or spherical geometry.
\item Describe the relationship between tilings and 2-dimensional geometries. 
\item Understand the essence of various applications of hyperbolic geometry.
\end{itemize}

\textbf{Attendance:} I value your involvement in the class, therefore class attendance is mandatory. Absence for official Colby activities requires prior approval. Absence for religious reasons will be considered “excused” if the policy in the college catalogue is followed. I reserve the right to take attendance. More than 3 recorded unexcused absences will result in the reduction by 1/3 of the final course grade. Excessive tardiness or early departure may also result in such a reduction.

\textbf{Computing Resources:}	 You will want to use the software to complete some homework problems throughout the semester. You are allowed (nay, encouraged) to use software to solve any integral you encounter in this course.  \textit{Mathematica} is available on many computers at Colby, including the computers in Mudd 415. From Colby's fileserver you may also download and install \textit{Mathematica} on your personal computer for use while on campus. You may also wish to make use of \textit{Grapher}, software which comes bundled with every modern Macintosh. (It can be found under, ``Utilities''.) It is very easy to use and can draw almost every imaginable type of 2 and 3 dimensional graph (including solutions to differential equations and vector fields.) I highly recommend it. 

\textbf{Academic Honesty:} All work in this course must be your own and you should always be prepared to explain or defend it. You are encouraged to work with other students on the homework assignments, but you may not copy another student's work. On the take-home exams you may use your own texts and notes, but may not discuss the exam with others (except the instructor) and may not use other resources, in print or online. On the final exam you may not use any notes or books and may not discuss the exam with anyone except the instructor. Please read the section on Academic Honesty in the catalogue (pg 43). 

\textbf{Classtime:} Our class this semester has a very small number of students, so it will be run in a ``semi-seminar" style. This means that you should take special ownership of each class period and come to class prepared to explain, question, and discuss the material of the day. The amount of lecturing will be minimized and particular students may be assigned particular class periods for which they are to lead discussion.

\textbf{Evaluation:} The numerical course grade will be a weighted average of the cumulative grades with weightings as follows:

\begin{center}
\begin{tabular}{rlcrl}
25\% & Homework\\
5\% & Participation\\
5\% & Reading/Colloquium Responses\\
15\% & Blog Essay\\
15/20\%& Exam 1\\
15/20\%& Exam 2\\
20\% & Final Exam.\\
\end{tabular}
\end{center}

The maximum of your scores for Exams 1 and 2 will count at 20\% of your grade. The minimum of your scores on Exams 1 and 2 will count as 15\% of your grade.

\textit{Caveat:} earning less than 50\% on each of the three exams will result in a grade of ``F'' for the course.

Course letter grades will be assigned (subject to the above caveat) according to the following scale. Any curve will be determined at the end of the course according to the discretion of the instructor. An A+ may be awarded to students who do exceptionally well and who demonstrate an exceptional interest in the course.

\begin{center}
\begin{tabular}{|r@{\thinspace}c@{\thinspace}rl|r@{\thinspace}c@{\thinspace}rl|r@{\thinspace}c@{\thinspace}rl|r@{\thinspace}c@{\thinspace}rl|}
\hline
93 &--& 100 \%  & A    &  90 &--& 93 \% & A-   &  87 &--&  90 \%  & B+ & 83       &--& 87\% & B \\
80 &--& 83   \%  & B-   &  77 &--& 80 \% & C+ &  73  &--& 77 \%  &C     & 70       &--& 73\% & C-\\
67 &--& 70    \% & D+ &  63 &--& 67 \% & D    &  60 &--&  63 \% & D-    & \multicolumn{2}{r}{below}& 60\% & F\\
\hline
\end{tabular}
\end{center}

\textbf{Homework:} Homework is the most important part of this course. It is where you wrestle with new ideas, generate creative solutions, and convey your thoughts to others. You are expected to write complete, careful, and correct proofs.  You should start the homework early, some problems will require multiple attempts and careful thought. If you are having substantial difficulty with a particular problem or the entire homework set, you should email me or come to office hours. I am eager to help you! You are encouraged to work with a partner on the homework, but all work should be your own. In other words, you may discuss particular problems but you may not copy someone else's solution. Doing so is an act of academic dishonesty. 

\textit{Structure and Schedule:} Homework will be assigned and collected each class period. It will always be posted on the course webpage. You are responsible for checking the webpage. If no homework assignment is posted, you should refresh the webpage on your browser and, if that doesn't work, email me to let me know. In the special circumstance that there is no new homework, the webpage will make note of that. If you will not be in class on the day that homework is due you should arrange to turn it in at my office or to have a friend bring it to class. Late homework may be penalized. 

Homework assignments will typically have two parts: problems which can be solved simply by calculation (called ``Mortar'')  and problems which require careful thought and creative solutions (called ``Bricks''). The purpose of Mortar is to ensure technical mastery of basic concepts and the purpose of Bricks is to develop your ability to think geometrically and apply concepts from the reading and class to new situations. Together we use them to build our geometric knowledge and intuition.

Due to the small number of students and drastically different mathematical backgrounds, certain problems may be assigned only to certain students. If you are assigned a particular problem, you should take special ownership of it and be prepared to present the problem to your classmates the day it is due.

\textit{Grading and Rewrites:} You will receive simply ``credit'' or ``no-credit'' for doing the calculation-based problems. The other problems will be thoroughly graded for both mathematical correctness and completeness and clarity of the proof. Any Brick  for which you turned in an (attempted) solution on time may be rewritten within one week of its return to you. Rewrites are graded using the same criteria as the original and you have the potential to earn full credit on such problems.

Because there will be a lot of paper being passed back and forth between us, you must follow these guidelines for homework:
\begin{itemize}
\item Each assignment must have your full name, the date it is due, and the assignment number on each page of the assignment.
\item Problems must be turned in in the order they are assigned.
\item All solutions must be turned in on 3-hole punched paper and placed into the ``official'' homework binder.
\item Each Brick  must be written on its own 2-sided sheet of paper. You are encouraged not to turn in your first draft of any of these problems.
\item Rewrites must be turned in with the original solution. No credit will be given for rewrites without the original attached. It must be clear exactly which problems from which assignments are being rewritten.
\item Homework must be very neat. This means: no messy scratchwork, no cramped writing, no huge eraser marks. If these guidelines are not followed you may be penalized. If you are incapable of writing neatly, you should type your solutions. LaTex is the most popular mathematical typesetting software, but you may also use software like \textit{Scientific Word} or \textit{Mathematica}.
\end{itemize}

\textit{Proof Expectations:} It is expected that every graded proof be written in conventional mathematical style. Each statement should follow logically and naturally from the previous ones. In general, your work is your answer. It is possible for a proof to be substantially correct but not to receive full credit because the writing was not up to standard. The audience you should keep in mind is an imaginary classmate who is not quite as ``bright'' as yourself. The danger in communicating mathematics is always assuming your audience knows more than they do! This being geometry, good illustrations can help you communicate the important ideas in a proof. Be careful, however, that you do not rely on pictures to the detriment of logic. Always ask yourself if there are other possible pictures and if your argument relies too heavily on the particular picture you have drawn.

\textbf{Project:} Each class member will contribute an essay to the blog ``The Geometric Viewpoint''. 
The project will take a fair amount of time and will require independent investigation, exploration and writing. Completing the project will help improve your ability to convey mathematical concepts to non-mathematicians and will enable to encounter geometric ideas that cannot be fit into the regular content of the course.

\textbf{Participation:} I expect that you will volunteer answers to questions posed to the class as a whole and that you will be prepared to make a contribution to the discussion if specifically called upon. In the event that I call on you specifically, you do not necessarily need to be able to answer the question I ask, but you should be prepared to give a partial answer, an intelligent guess, or to ask a relevant question. I expect that you will have attempted each day's homework prior to class.

\textbf{Readings and Colloquium Responses:} Over the course of the semester, you will be assigned several non-textbook readings. These may include research articles and blog posts. For each you must turn in answers to the reading questions distributed with the assignment. \textit{Additionally}, you must attend three Mathematics/Statistics Colloquiua (or symposium presentations). For each you need to write a page summarizing and reacting to the lecture. 

\textbf{Exams:} There will be two take-home exams and a final exam. Each exam is cumulative, although the final exam is ``more cumulative''. Exams will be designed to test your understanding of the course material and your ability to devise original proofs. You must understand, and communicate your understanding of the material. You may not discuss the exam with other people and you may not use any resource other than your notes and textbooks. Here is the schedule of exams:
\begin{enumerate}
\item Exam 1: Distributed on March 5  and due by 12 PM on March 10
\item Exam 2: Distributed on April 9 and due by 12 PM on April 14
\item Final Exam: Exam period 1; 9 AM on Wed. May 14.
\end{enumerate}

\textbf{How to succeed in this course:}

Geometry is an intense 300-level mathematics course. The course, therefore, expects a level of sophistication and mathematical maturity beyond that expected by introductory calculus courses. In particular: you are responsible for your own learning and the professor is responsible for assisting you in your quest. You need to be willing to do computations and you need to be willing to wrestle with ideas. The following advice is meant to provide suggestions on get more out of the class and to improve your grades:

\begin{enumerate}
\spacing

\item	\textit{Do some math everyday.} Really. 

Learning mathematics is a lot like learning to play an instrument, play a sport, or learn a language. You must practice everyday. The daily homework problems are intended to help you in your daily practice. But you should spend additional time each day working on the weekly homework, reading the text, and studying previous material. On average, you should spend 2 - 3 hours studying for each hour spent in class. In our case, that's 6-9 hours per week of homework and studying.

\item \textit{Participate in class.} Yes, you.

Asking and answering questions is a great way to stay engaged with the material and verify for yourself that you know what's going on. The more people that participate, the more fun class is. I, and the rest of the class, value your questions and your answers, right or wrong. In fact, giving a wrong answer to a questions is a great way to learn the right answer. Make an effort to connect each class's activities to previous classes. Try to predict where the material will be going in the future. Take good notes and listen. If you can't do both, make a deal with a buddy: you take notes and they listen one day and the next day you switch.

\item \textit{Read the textbook.}

The lectures and the textbook will often present slightly different views on the same material. The examples in the text will (for the most part) differ from the example presented in class. As you read the text, relate what the text does to what happened in class – where is it the same, where is it different? If it differs, which approach do you like better – what are the pros and cons of each? 

\item \textit{Form a study group.}

Introduce yourself to other people in the class and meet up outside of class to study and work on homework. Be sure that you don't copy answers, but learn from each other and then write the answers on your own. Compare lecture notes to be sure you copied everything correctly. Ask each other questions and explain material to each other. Calculus-After-Hours is for 100-level calculus students, but it makes the math department a happening place -- it's a great place to meet up with your classmates.

\item \textit{Write your own problems.}

As you study for the exams, look back at all the examples done in class, in the text, or on homework. Try modifying them to make up your own problems. Try to solve your own problems – what makes your problems easier or more difficult than the ones you've seen before? Feel free to show me what you've done.

\item \textit{Be curious.}
We will be spending a lot of time, exploring the theoretical mathematical ideas. Be curious about why geometry (or a geometry) works the way it does. Ask ``why''. Try to find applications of what we're learning. Try to invent your own geometry. Schedule time for curiosity. Block off 2 hours every day for Geometry. Any time not spent working on homework, spend studying and playing with geometric concepts. Come up with your own research questions. Computers compute; humans ask creative questions.

\item \textit{Visit me in my office.}

I love working with students and I love to help you understand and appreciate the beautiful world of mathematics. Feel free to drop by, even when it's not my office hours. If I can’t chat, I’ll let you know. Ask crazy questions about the course. Ask questions about my research. Tell me about your past math experiences. Tell me about what subjects you love. Let me know when you have a concert or athletic event. 


\item \textit{Spread the studying out over the semester.}

If you do math everyday, as suggested above, you won’t have to cram for exams. You’ll be able to sleep and, consequently, to think. You'll be happier and more relaxed. You'll have time to write papers for your other classes. You'll have time to appreciate the snow and the spring. You don't need to pull all-nighters. 

\end{enumerate}
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