|
|||
|
High School: Math Circle State Math Contest Summer High School Program Undergraduate: GRE Prep Course Undergraduate Colloquium Research Experience for Undergraduates Graduate: Mini-Courses Qualifying Exam Problem Sessions Graduate Fellowships Committees: Steering Internal Advisory External Advisory Outreach Advisory Other Information: People Master Calendar VIGRE Award Publications Undergraduates Graduates Postdocs 04-05 AR |
Summer 2001 Projects |
||
|
Rex Butler Viewmont High School, 1998 from Tooele, Utah University of Utah Faculty Mentor: Jim Carlson My project involved understanding the fundamental group of a knot through thee use of the Wirtinger presentation. I then showed that the property of n-coloration of a knot is also a basic property of the knot group, i.e. a know is n-colorable iff there exists a certain homomorphism from it's knot group to Dn. After trying (unsucessfully) to find another homomorphism to the symmetric group. I concluded other fundatmental knot concepts and properties might correspond to the existence of homomorphisms to various finite groups, thus suggesting we search for these homomorphisms. Aaron Cohen East High School, 2000 from Salt Lake City, Utah University of Utah Faculty Mentor: Jim Carlson My individual research project is on the spatial imbeddings of graphs, more specifically the existence of knots in graph embeddings. I have been using the text Topological Graph Theory by Dr. Gross, and The Knot Book, by Colin Adams. Various online journal articles and reports from Knot '96 have been a great help to me, and especially recently I have been trying to understand the work of Dr. Kouki Taniyama as of Knot '96, concerning spatial graphs. My main goal was originally to make more accessible to undergraduates and students with minimal background the remarkable properties of "knottedness" in spatial graphs, following from the work of Doctors Conway, Gordan, Kauffman, and others. In particular, I wasnted to show how the complete graph of seven vertices (K7) is 'intrinsically knotted,' that every spatial embedding of K7 contains a knotted Cayley Cycle, withouth the use of knot polynomials. To do this I have been studyin gthe ARf invariant as related to the 'pass-equivalence' or 'clasp-equivalence' of knots. Unfotunately, I have only been able to give a general idea of why it should be true that K7 is intrinsically knotted, using a vague gneralization I found for the Skein relation concerning knot crossings and the linking number, a relation normally proved using knot polynomials. I have not gotten as far as I wanted to in this direction, so I've started studying some of Dr. Taniyama's ideas on 'realizations' of spatial sub-graphs and Vassiliev invariants of graphs. This has been good for me, especially as it has forced me to learn and get a better understandin gof the homology of graphs in particular and spaces in general. Overall I think this project has been extrememly fruitful for me. Matthew Dalton Mountain View High School, 1998 from Orem, Utah University of Utah Faculty Mentor: Jim Carlson During the summer we read Colin Adams' "knot book". We covered many subjects in knot theory and hyperbolic geometry. My project was to study colorability of knots. During the summer I solved severel surprising questions:
Tara Henriksen Individual REU project Cimarron-Memorial High School, 1999 from Las Vegas, Nevada Faculty Mentor: Fred Adler There is a model that determines whether or not a patient with cystic fibrosis will become a candidate for a lung transplant or not. This model was written by Kerem et al, and was published by the New England Journal of Medicine (vol. 326, pp. 1187-1191). This model uses FEV1%, age, and gender as its sole variables to decide whether or not a patient is sick enough to be considered for a lung transplantation. Since then, Ted Liou and Fred Adler have been working to perfect this model by considering additional variables that affect a patient's predicted survivorship on a 5-year scale. These new variables are weight-for-age z-score, pancreatic sufficiency, diabetes mellitus, Staphylococcous aureus infection, Burkerholderia cepacia infection, and acute exacerbations. Their findings suggest that the old model is extremely inaccurate, and that the new model will prove to be a better predictor of the actual effects of lung transplantation. This new model also suggests that there is no survival benefit for patients who have transplantation. Over the summer I began my study of the relationship between disease and pulmonary function by identifying patterns in data collected from cystic fibrosis patients. This summer I learned to use the S-plus statistical software to begin the first task of identifying patternsIi data collected from cystic fibrosis patients. This summer I earned to use the S-plus statistical software to begin the first task of identifying patterns between individual patients and between patient grouus. My goal this summer was to decide the accuracy and usefulness of a one-year assessment of a CF patient. I began my study by making plots of Age vs. each variable that might have a pattern that would be clinically important. These variables were weight, acute exacerbations, predicted survival, and FEV1. It became clear that patterns in the data existed. Some observed patterns were that as weight decreases, predicted survival for most patients, regardless of age, decreased. It was also observed that as FEV1 decreases, most patients' predicted survival also decreases. I next set out to find patterns in the data, (specifically with the aforementioned variables) by looking at individual patients. In doing so, it as decided that FEV1 (which is a measurement taken of the percent of lung that a patient has) is quite an unpredictable variable. Stephen Jensen Individual REU project Viewmont High School, 1997 from Centerville, Utah Faculty Mentor: Stewart Ethier For my summer REU project, Professor Ethier and I are attempting to find a complete optimal drawing strategy for "Deuces Wild" video poker. "Deuces Wild" was chosen because of its relative popularity as well as the extra challenge provided by the presence of wild cards. We plan on finding the probabilities and expected values involved using combinatorical methods so we can present the answers in rational numbers with a common denominator. Although several analyses exist already, to our knowledge none have given rational answers which could be reproduced using basic probabilistic methods. These previous analyses were all computer based and the resulting numbers they generated were all decimals usually rounded to five significant figures. Also, previous analyses were published somewhat abridged since they were intended for use in actual play. So we seek to remedy the lack of both completion and exactness in our analysis.
The simplest way to do this analysis is to calculate the expected value for all 25 ways of playing each of the 52C5 possible hands. Since several computer analyses have already done this, we plan on using the most complete strategy as a starting point for our analysis. This strategy comes from the Macintosh program "Video Poker Strategy Master." We will also be using a DOS-based program called "Video Poker Tutor," which plays any given hand all 25 ways and gives their expected values, to check our work. (Click here to see project in .pdf format.)
Larsen Louder Twin Falls High School, 1997 from Twin Falls, Idaho University of Utah Faculty Mentor: Jim Carlson I am making a poster which will consist of a series of snapshots from a hypothetical movie showing the transformation figure 8 knot complement →hexahedral gluing diagram →figure 8 knot complement
It's a neat construction and I hope the project will help me gain some intuition into the mysteries of groups acting on spaces. It will be a good thing to understand if I ever get to study more advanced topics.
Robert Palmer Northridge High School, 1994 from Layton, Utah University of Utah Faculty Mentor: Jim Carlson My project was to write a program that allowed a user to draw a knot with a mouse or other pointing device, and determine from the drawing if that knot could be reduced to the "UN-knot." Ryan Rettberg Granger High School, 2000 from West Valley, Utah University of Utah Faculty Mentor: Jim Carlson My REU summer project is a java program (Knotster) that computes the colorabilities of knots/links along with my paper on the properties and effects of colorability(ies)/colorations of/on the crossing matrix of a knot. The paper describes how the colorability of a knot/link is determined via the crossing matrix of a knot, using special matrix forms in linear algebra; along with important consequences/properties of prime knots, composite knots, and links relating to nullity and multicolorability. The program allows the user to either draw the knot/link, type its dowker notation, type my notation for the link, and/or to type the crossing matrix for the link(in maple format) as input. Using this input the program can generate all of the other forms of input following the one which was inputted (i.e. drawing the knot outputs all the other forms) along with the colorabilities and a way of calculating the colorations. The program contains a number of options including: reduction over the integers modulo any positive integer; the matrix A such that A times the crossing matrix is the reduced matrix; conversion to maple format; the input of any rectangular matrix (this includes matrices which are not crossing matrices, for linear algebra purposes); color options for visualization and the distinguishing between link components; and a function that can select the crossings of a drawn link randomly, alternatingly, or trivially (for the ease of mouse clicks and the colorabilities of 'random' links). All in all my paper and program describe and calculate the colorabilities of links, which is a useful invariant for quickly distinguishing between two links, namely between a projection of some knot and the unknot.
The program can be found at the following location Allen Whitt Grissom High School, 1999 from Huntsville, Alabama University of Arizona Faculty Mentor: Hugo Rossi Over the course of the summer we have studied knot theory and hyperbolic geomtery. In our studies of each we encountered practical applications of our topics in the fields of chemistry and genetics. We read brief descriptions of how scientists in these other areas used the mathematics we were learning to advance knowledge in their respective fields. My project seeks to reverse this process: to use methods in chemistry and genetics to further knowledge in pure mathematics, specifically knot theory. I sought out the various techniques and methods used experimentally in the lab from both chemistry and molecular biology. I analyzed the techniques to deduce their properties and the characteristics they measure. Then, using the fact that both DNA and molecules from inorganic chemistry can knot themselves -- in fact, are found knotted in nature -- used these experiments to link knot invariants to molecular properties, so that we might derive a full proof machine to distinguish knots. |
|||
| VIGRE2 VIGRE Department of Mathematics University of Utah Content Disclaimer Comments | |||