1. INSPIRATION AND DESCRIPTION OF THE PROJECT

Having taught mathematics for nearly ten years in the liberal arts environment, I have come to the conclusion that students perform best when:

• They feel that their professor cares about them and believes that they can be successful
• They are comfortable discussing their ideas or questions
• They are involved in active learning
• They work with each other in an environment that is supportive and responsive
• The material that they are studying is put into a context that is relevant to their lives

Motivated by these ideas, I proposed a weekend retreat that would  involve group student presentations on topics related to applications,  generalizations and extensions of the material in my Abstract Algebra I class.  In addition, I suggested we would hold informal discussions  about topics in Algebra I that would review the material in preparation  for Algebra II.  The idea was met with enthusiastic support by the students in the class and the planning process began.  An important element of the plan was to chose a location that was away from the New College campus so that students could immerse themselves in the project for the entire weekend.  Following a suggestion by Professor Sandra Gilchrist, we booked the facilities at the Keys Marine Laboratory in the town of Layton on Long Key, Florida.   Following a series of meetings, a group of fourteen students and I traveled to the Keys, the first weekend of Spring term 1999 (February 5-7,1999).  Participation in the retreat was voluntary and two of the sixteen people in the class chose not to participate.

All participants were asked to sign an honor code about their conduct through weekend  and I am happy to say that their behavior for the entire weekend was exemplary. We traveled to the retreat location in four cars that were driven by three students and myself. During our first evening, we scheduled our weekend.  It turned out that after several weeks of preparation we had to accommodate seven different student presentations. Four of the seven presentations involved two or more students and the other three were single presentations.  A detailed description of the program, as well as abstracts of the student presentations (prepared by the students) are included in this report. In addition to the presentations, we cooked some great dinners, watched movies, watched the sunset, played frisbee, and got to know each other very well.

The friendships, collaborations and excitement about mathematics and algebra in particular will remain with all of us for a long time to come.  As you will see from our evaluations at the end of this report, the main thing that we would do differently would be to make the retreat last longer and perhaps find a location that is a little closer to the New College campus.

[1] Alan C. Tucker, "Models that Work-Case studies in effective Undergraduate  Mathematics Programs", MAA Notes Number 38, Mathematical Association of America, (1995)
 
2. THE PARTICIPANTS

Standing (from left to right)

1. Mike  Cenzer   2. Gilliss Dyer  3. Joseph Corneli
4. Scott Moser 5. George Famiglio 6. Barry Liebowitz
7. Eirini Poimenidou (Prof) 8. Mike carlisle
 

Not  Standing (From left to right)

9. Adam Hollidge 10. Doug wahl 11. Dustin Soodak 12. Rachel Labes 13. Jake Byrnes 14. Rob Meyers 15. Andrea Saunders

February 5, 1999

 2:30 pm        Leave New College (One of the four cars left New College at 6pm)
 8                    Arrive Keys Marine Lab (One car arrived at 11:45pm)
 8-9:30            Dinner
 9:30-11        Discussion of goals of the retreat and program for Saturday and Sunday
 11-1 am       Movie  "Rashomon"

 8-10 am        Breakfast
 10-11:15       ImageUnderstanding, by Adam Hollidge and George Famiglio
 11:15-11:30   Coffee Break
 11:30-12:45   Rubik's Cube, by Scott Moser, Rachel Labes and Mike Carlisle

 12:45-2:00 LUNCH
 
 2:00-2:45      Boolean Algebras and Lattices, by Mike Cenzer
 2:45-3:30      Cayley Diagraphs, by Dustin Soodak
 3:30-4:00      Break-Group Photograph
 4:00-5:45       Fermat's Last Theorem, by Barry Liebowitz, Rob Meyers
                       and Joe Corneli
 5:45-7:00      Sunset Watch-Frisbee
 7:00-9:00      Dinner Preparation-Several different dishes prepared.
 9:00-10:00    Dinner
 10:00-10:30   Filling out of Evaluation Forms
 10:30-12:30   Movie "Raising Arizona"
 12:30-1:30     Movie "Algebra as a means of Understanding Mathematics"

 6:00-10:30       Brunch
 10:30-11:30     Wallpaper Groups, by Jake Byrnes, Andrea Saunders and Doug Wahl
 11:30-12:00     Break-Another Group Photograph (just in case)
 12:00-1:00       Calculating the Determinant using Tableaux, by Gilliss Dyer

 1:00-2:30         Clean and mop the dorm. Pack
 2:30                Leave the Dorm

4. ABSTRACTS - (provided by the students)

4.1 Image Understanding

    Image understanding is an attempt to extract knowledge about a 3D scene from 2D images. The relation between image parameters (data such as gray levels, surface texture, etc. obtained from the image) and object  parameters (parameters defining surfaces, edges, faces, etc. of the 3D  object) is expressed in the 3D recovery equations:

The object parameters are determined by solving the above equations after substituting the
observed values of c₁, . . .,c_m. Various methods exist  for solving these equations, including
brute force, table lookup, and  Newton iterations.  In our talk we present the group theoretical  approach using representation theory and Weyl's thesis (essentially, that the geometrical meaning of the parameters becomes very clear if  they are expressed in terms of invariants) to come to solutions that have an analytically closed form. Image understanding is FUN!!!

REFERENCES:
1. Kanatani, K,  "Group Theoretical Methods in Image Understanding", Springer-Verlag,
      Berlin: 1990.
2. Lidle & Pilz "Applied Abstract Algebra", Springer (1997)

 4.2  The Group Theory of Solving Rubik's Cube

The Rubik's cube has been a source of entertainment for millions, but what makes this puzzle an invention of mathematical genius?  We study the group theoretic structure underlying the Rubik-Ishinge Magic Cube and explain key concepts, such as permutation puzzles, conjugation, generators and orbits.  The group theoretic implications of the Rubik's cube are also addressed.  the mechanisms governing allowable moves on the Rubik's cube determine the number of achievable states.  It is interesting to note that this number is twelfth the size of the total permutation group.  We display two different notations used to describe the puzzle, and the benefits of each.  Finally, we discuss the limitations of the puzzle, and it's place as a subgroup of S48.

REFERENCES:
1. Gallian, Joseph.  "Contemporary Abstract Algebra", 4th edition.  Houghton Mifflin Co.
    (1998)  .p104.
2. GAP,  "Analyzing Rubik's Cube with GAP." GAP Group,
     http://wwww-gap.dcs.st-and.ac.uk/~gap/Intro/rubik.html
3. Hofstadter, Douglas.  "Magic Cubology-Metamagical Themas: Questing for the Essence of
    Mind and Pattern"  Basic Books Inc.  (1985)  p301.
4. Joyner, W.D.  "Mathematics of the Rubik's Cube", http://web.navy.mil/~wdj/rubik_nts.htm
   (1999)

My talk at the retreat dealt with lattices,boolean algebras and Karnaugh diagrams. A lattice is a sequence of chains with an order in each chain.  For example, the lattice of relations of sets A and B. The two chains in this lattice are defined as the higher ones contain the lower ones.

Boolean algebras are formed by a group of elements that have certain properties and operations that are well-defined and closed over the algebra. For example, (B, union, intersection,'(complementation), 0, 1).  The boolean algebras that have various variables in them with values that are 0 or 1 were the focus of my research.
    These are interesting because they can be made from logical strings of "and" and "or" statements. notational consideration: from now on P ?V=P+V and P ?V=PV.  A boolean polynomial (a series of variables equal to either zero or one) is equal to zero if all the atoms or minterms are equal to zero and  is equal to one if one or more of the terms is equal to one. We can use all the laws of multiplication and addition to reduce these as well as De Moivre's laws. (AB)'= A'+B' and (A+B)'= A'B'

    Finally, these polynomials and the problem of their reduction leads us to Karnaugh diagrams.  Karnaugh diagrams are simple charts that show, like a truth table, what happens to every possible combination of values for the miniterm.  Here is a simple example. Let P= XY+Y' (Diagram 1). Since every box under x is filled in and the box x‚y‚ is filled in, P is equivalent to X+X‚Y.

     The diagram can be expanded to as many dimensions as necessary, but after more than four it becomes unwieldy.  This diagram can be used to represent various sequences of coin flips. Say we have four coin flips, each with value one=heads and zero=tails. We can set up the polynomial as all values with three heads and one tail and all values with three tails and one head.Thus the polynomial P =(XYZW'+ XYZ'W+ XY'ZW + X'YZW) + X'Y'Z'W + X'Y'ZW' +X'YZ'W'+  XY'Z'W'. (See Diagram 2) This gives us a checkerboard. This interesting result carries into more dimensions with a few rearrangements.  The reason this occurs is that in moving from all miniterms with three ones and a zero to all those with three zeros and a one (and all moves within the group) is equivelent to shifting diagonially and because P contains 1/2 the possiblities, the diagram must look this way. This process holds up to n-dimensions for all polynomials in n-variables where the possibilities with 1 head, then 3 heads, then 5 all the way to n or n-1 heads, depending on whether n is odd or even.  Karnaugh Diagrams and Boolean algebras also relate to circuit and gate diagrams.  If you need to discuss this further please contact me at mcenzef@virtu.sar.usf.edu .

REFERENCES:
1. Lidle & Pilz "Applied Abstract Algebra", Springer (1997)
2. Humphreys &Prest "Numbers, Groups and Codes", Cambridge University Press (1989)
3. Elliot Mendelson "Boolean Algebra and Switching Circuits", McGraw-Hill, (1970).
 

4.4 -Caley Digraphs of Groups

Suppose G is a group and S is a set of generators for G. The Cayley digraph of G with generating set S, Cay(S:G), is constructed by assigning a vertex to each element of G then drawing an arrow from x to y if and only if xs = y for some s in S (because arrows are used as edges, this is called a directed graph). The arrows are drawn in different colors that depend on exactly which element of S is being represented.  This makes the graph a valuable tool for finding the product of a string of generators of a group.  Sometimes a group is actually defined in terms of a generating set.  In these cases, a Caley Digraph is much better than the traditional "multiplication table" method of  defining the group operation.

The property of Caley Digraphs which has probably recieved the most attention,  is Hamiltonicity.  A Hamiltonian path is a path along the arrows which passes each vertex exactly once before returning to the starting point. Hamiltonian paths on Caley Digraphs have found uses in both computer science and other areas in group theory.  One of the more
visually appealing applications is the duplication of some of Escher's"Circle Limit" pictures on a computer.

REFERENCES:
1. Gallian, Joseph.  "Contemporary Abstract Algebra" 4th edition. Houghton Mifflin Co.
    (1998) Ch. 30

 4.5 Fermat's Conjecture

 4.6 Wallpaper Groups

Our talk concerns the description of infinite tilings of the plane using wallpaper groups.  We begin by exploring rigid motions in the plane; translation (motion along a vector), rotation (about a point), reflection (by a mirror axis), and glide reflection (translation along a vector followed by a reflection in a mirror axis parallel to the vector).  We then present a geometric proof showing there exist only 4 rigid motions in the plane.  Using these motions, we begin our investigation of infinite tilings. Then we discussed the 7 Frieze groups, which restrict translations to one dimension. This naturally leads into the discussion of the wallpaper groups, where translations can occur in two dimensions.  We present a method for classifying any infinite tiling as one of the 17 wallpaper groups, and work through an example to demonstrate this process.  We also explain the relationship between lattice units and generating units.  Finally, we introduce the applet Kali (http://www.geom.umn.edu/java/Kali/), a template for tiling a drawing as one of the 17 wallpaper groups.

REFERENCES:
1. Crowe, D.W. "Symmetry, Rigid Motions, and Patterns."  The UMAP Journal. Pps 207-236.
    Volume 8.2
2. Schattschneider, D. "The Plane Symmetry Groups: their recognition and notation."
    The Plane Symmetry Groups. June/July 1978. pps 439-450.
3. Gallian, J.A. "Contemporary Abstract Algebra". Houghton-Mifflin Co.
    New York. 1998.

 4.7 Calculation of Determinants Using Tableaux

In my talk, I demonstrated an interesting alternative method for calculating the determinant of an nxn matrix. This method involves making (n-1)!/2 matrices from the original matrix by permuting the last n-1 columns. The first n-1 columns of a new matrix are then copied and appended to its end to form a (2n-1) x n "tableau". From each tableau, 2n terms of the determinant are acquired by taking the product of n forward diagonals- starting from the (1,1) diagonal and working across to the (1,n) diagonal- and n reverse diagonals- starting from the (1,n) position and working across to the (1, 2n-1) place (referred to henceforth as the forward and reverse diagonals). Each tableau is multiplied by the sign of the permutation which gave rise to it, and the signs of the diagonal terms follow one of three simple patterns determined by the value of n.

To see why this method works, we recall the definition of the determinant as a sum over every permutation 1 in symmetric group of size n (Sn). It turns out that the forward diagonal terms in a tableau correspond to the rotations which fix a regular n-gon, and the reverse diagonal terms correspond to reflections which fix the regular n-gon. Together, these permutations acting on a regular n-gon constitute the dihedral group of order n (Dn), which is a proper subgroup of Sn with 2n elements. A partition of Sn is formed by right cosets Dn1, where the permutation 1 representing a given coset can be chosen such that 1 remains unchanged. Each coset Dn1 corresponds to the tableau made by first permuting the columns of our original matrix by 1. We thus find a potentially useful method for solving the determinant of an nxn matrix using group theory. I also demonstrated that a third permutation of our elements could alleviate the need to extend beyond the nxn shape of our original matrix: an easy to perform permutation can allow us to read the forward diagonal terms of the tableau from the horizontal terms of an
nxn matrix, and to read the reverse diagonal terms from the vertical terms.

REFERENCES:
K.W. Johnson "The Calculation of Determinants Using Tableaux" (Penn. State), Private Correspondence.

5.  BUDGET

Lodging 15 participants $22/per night/per person        660.00
              for 2 nights

Food      Receipts  Publix                                               71.71
              SAM'S                                                              74.27
              No receipts Eirini Poimenidou                         20.00
              Rob Meyers                                                      12.00
              Doug Wahl                                                         8.00
              Andrea Saunders                                                8.00

Other    (Batteries+Video Tape) Mike Carlisle             10.00
             (T-shirt transfer sheets) Eirini Poimenidou       12.00

Gas & Toll    $25 +$8=$33 per car (4 cars)              $132.00

Contributed             Division of Natural Sciences      $300.00
                                                     Alumni Office        $300.00

TOTAL OUT OF POCKET EXPENSES                    $407.98
TOTAL EXPENSES (AVERAGE) PER PERSON    $  27.20
 
 

6. EVALUATIONS

6.1 Evaluation by Professor Poimenidou

My personal goals for the Abstract Algebra retreat were to:

Provide students with a multi-day, off-campus mathematical seminar that simulated the environment of a professional mathematics conference

Create an atmosphere that would promote interaction among the participating students .

Familiarize myself with each student so I can advise him/her better.

Highlight the relevance of Abstract Algebra in both real world applications and within mathematics.

Generate enthusiasm and reinvigorate interest in mathematics.

Provide a platform for an informal review of Abstract Algebra I in preparation for Abstract Algebra II.

Break the stereotypes of "who does mathematics", "where" and "why".

Give each student the opportunity and the responsibility to do her/his part for a group or individual project.

Celebrate the New College philosophy and create a new forum for out of classroom learning.

Have fun while "staying out of trouble".

Since the retreat was so recent we cannot yet evaluate its long term effects.  In the short term, every student appears to be more engaged and there is increased collaborative interaction among the students. The atmosphere of camaraderie and collegiality created at the retreat is permeating their interactions.  The students feel more comfortable asking questions both in and out of class and I feel that I have a better understanding of their needs and how to meet them..

The student evaluations contained in the following pages articulate the benefits they feel they have gained from this experience.  The students made a few suggestions that certainly should be taken into consideration if this activity is to be repeated in the future:

• The retreat should be longer by at least a day
• Students should only be involved in groups as opposed to individual presentations so as to minimize time spent in the classroom and maximize informal interactions
• Everyone should arrive at roughly the same time.
• There should be more discussions about the goals of the retreat before the event in order to better meet participant's expectations.

•  Every participant helped plan the retreat
• Every participant gave a presentation
• The retreat location was away from the New College campus and was conducive to interaction among participants.  In our case that meant a large kitchen, a large dinning-living room and a large enough dormitory so we could all stay under the same roof.
• We planned all the meals ahead of time, and cooked and ate together
• The Division of Natural Sciences and the New College Alumnae/i Association provided partial financial support to help defray the cost of the retreat
• We evaluated the event

Eirini Poimenidou        February 23, 1999
 

 6.2 Student Evaluations

Students were asked to evaluate the retreat in the evening of our first (and last) full day at the retreat. The following questions were suggested to assist in substantive feedback:

Q1. What did you enjoy the most?
Q2. What would do differently?
Q3. Would you do this again?
Q4. Do you feel differently about Mathematics?
Q5. Do you have any other comments?

Ok, now for the good things: This trip has really come together well. No one has starved, gotten lost, or set on fire.  From an idea that everyone thought was crazy, has come a great chance to spend time with my fellow math students.  It brings us all together a little bit more (if we don't kill each other first) and will definitely be a memorable experience.  It has been a lot of planning and work outside of everybody's normal course load. I'm impressed that Eirini has stuck to it and made it happen. None of the students had to be here, and their presence is testament to their excitement for mathematics, the program, and their professors. Maybe others will follow the example?
This trip will hopefully reappear in future years benefiting from the difficulties of this excursion".

"The proof is out there"