
RoseHulman Mathematics REU HistoryThe RoseHulman REU in Mathematics started with Gary Sherman and group theory. This research defined the new mathematical structure of cwatsets and started an investigation of their properties. The program enlarged to include more faculty to broaden the mathematical topics and has included since its inception in 1988 Hyperbolic Geometry and Tilings with Allen Broughton, Number Theory and Permutation with John Rickert, more Cwatsets with Tom Langley, and the present investigations of Inverse Problems, Geometric Analysis, Computational Number Theory, and Systems Biology.Cwatsets (Gary Sherman, 1998, 2000): A cwatset is a subset of binary nspace that is "nearly" a subgroup. For example, F = {000,110,101} is a cwatset in binary 3space. Notice that while the coset F + 110 = {110,000,011} is not F, it is nearly F in the sense that applying the transposition (1,2) to the components of the elements of F + 110 yields F. (Take a minute and find the appropriate permutations for the cosets F + 000 and F + 101. Are your permutations unique?) So, F is not closed under addition (i.e., is not a subgroup of binary 3space), but it is closed with a twist (cwat). Cwatsets have their roots in statistics, grow in groups and combinatorics, and were conceived by RoseHulman undergraduates, RoseHulman NSFREU participants, and Gary Sherman. Here are two references which should be readily available on your campus:
For more information see the Cwatset Research Home Page.
Hyperbolic Tilings of Riemann Surfaces (Allen Broughton, 1998, 1999, 2000, 2001, 2002, 2003): Three examples of tiled surfaces are the icosahedral tiling of the sphere and two different tilings of the torus by equilateral triangles and by isosceles triangles (click to view). Tilings of the hyperbolic plane induced by tilings of the higher genus surfaces may also be constructed: T245, T255, T355, and T433. However, the corresponding surfaces cannot be easily visualized as in the case of the sphere and torus, so group theoretic methods are needed to determine properties of the geometry. All tilings considered generate large symmetry groups of the surface, and the geometrical and combinatorial properties of the tiling are strongly reflected in the structure of the group and the group's action on the geometry. The geometric and combinatorial problems need to be solved by massive computations in the symmetry groups. Motivated by these problems, participants will perform group theoretic experiments, make discoveries and formulate conjectures by carrying out computer calculations using the software package Magma. The end goal will be to discover and prove theorems about the geometry and combinatorics of the surfaces and, of course, anything about groups or computations in groups discovered along the way. This will be the fourth summer of tiling research at RoseHulman and we expect to add to the ever growing knowledge of tilings of surfaces. A sample problem, on which progress was made last year, is to find all the tilings surfaces of low genus. Another is to determine the number of intersection points of two curves in the tiling. They are always exactly two on the sphere. Check it out in the icosahedral tiling of the sphere. A third project which has now been completed is to determine all the quadrilateral tilings of the hyperbolic plane which can be subdivided into a tiling by triangles. Although obviously helpful, no previous knowledge of the geometry of hyperbolic surfaces will be assumed. This will be learned on an as needed basis through the perspective of the group theory. The experimentation and testing of conjectures will be done by group theoretic calculations using Magma. The participants will learn how to use Magma on an as needed basis to solve problems. Therefore, prior knowledge of Magma is not assumed, though participants should have some programming experience so they will not be starting from scratch. Additionally, Maple will be used for geometric calculation. For more information see the Tilings Research Home Page.
Number Theory (John Rickert, 1999, 2001): Professor Rickert's REU project is on the partitions of integers. A partition of the integer n is an increasing sequence of integers whose sum is n. For example we write the 5 partitions of 4 as 4 = 1+1+1+1, 4 = 1+1+2, 4= 1+3, 4=2+2, 4 = 4. Partitions play an significant role in many parts of algebra, especially the study of permutations and the symmetric group. Indeed, the conjugacy classes of the symmetric group on n symbols are in 11 correspondence to the partitions of the integers. Especially interesting questions about partitions concern the number of partitions p_{n} of the integer n. Here is a table of the first few:
Looking at the fourth column should suggest a theorem. This is one of three congruence relations observed by Ramanujan. The sequence { p_{1} , p_{2} , p_{3} , . . . }, the associated qseries p_{1} q + p_{2} q^{2} + p_{3} q^{3} + . . . , and other related partition functions and qseries lead to many interesting number theoretic questions about partition numbers. Rediscovery of the work of Ramanujan, such as the theorem alluded to above, has led to an explosion of research on partition functions and qseries impacting areas as diverse as combinatorics and particle physics. We will be considering identities and related series, building on the results of researchers such as George Andrews. A good understanding of series is very useful here.
Cwatsets (Tom Langley, 2004,2005): A cwatset is a subset of binary space that is an additive subgroup wannabe. For example F = {000,110,101} is a subset of binary 3space which isn't an additive subgroup because, for example, 110 +101 = 011 is not an element of F. But F is nearly a subgroup in the sense that
Cwatsets were first discovered at RoseHulman by Gary Sherman and his students in the late 1980s and have their origins in statistics. The algebraic theory of cwatsets loosely parallels the theory of groups (cyclic cwatsets, subcwatsets, extensions, morphisms), supplying a rich inventory of questions. Just about any question on finite groups spawns a similar question on cwatsets. Connections to graph theory (each simple graph has an associated cwatset which completely describes the graph's isomorphism class) and algebraic coding theory (any cwatset is a nonlinear code) also provide fertile avenues of investigation and have sparked much interest within the mathematical community. Research topics for the summer of 2005 will explore the relations of cwatsets to graphs and coding theory, as well as the construction of "perfect" cwatsets and the study of isomorphism classes of cwatsets.
Geometric Analysis (David Finn, 2006, 2007, 2008, 2010, 2011, 2012) David Finn's research concerns the applications of partial differential equations and the calculus of variations to problems in differential geometry, i.e. geometric analysis. The goal of the project to be investigated is modelling the shape of a drop sugar cookie. The basic heurestic model to be investigated is that the shape of a sugar cookie (homogeneous cookie dough) is given as a surface with prescribed mean curvature. The reasoning behind the heurestic model is that the cookie dough becomes a liquid when it is heated and attains its equilibrium shape as a liquid before it solidifies later in the baking process.
After the cookie is baked, a natural question is: Can one recover the shape of the cookie from knowing the wetted domain (the region on the cookie sheet the cookie is in contract with) and properties of the cookie dough and the cookie drop (stiffness, density, volume)? Some questions to be investigated are:
A lot of the investigation will be done by numerical computation of solutions to generate conjuctures, No prior knowledge of partial differential equations, differential geometry, and numerical analysis is necessary. Some exposure to ordinary differential equations and/or basic analysis (advanced calculus/vector calculus) is extremely helpful. For more information see the Shape of a Cookie Page


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