Gröbner Basis

Probability and Statistics

Probability and Statistics

Rudy Guerra, Seminar Leader/Research Advisor

Topic Overview

Probability is an area of mathematics that deals with random variables and processes. Theoretical aspects of probability address finite and symptotic distributions, expected values, and limit theorems, among other things. Applied aspects of probability focus on probability models for random events, including queuing theory, renewal processes, and genetic mapping. Statistics is another area of mathematics that is a related, but yet distinct, from probability. It focuses on explaining and modeling variability of data. Ideas, principles and results from probability are used to model random phenomena. A lot of the research that theoretical statisticians conduct involves developing optimal methods for statistical inference (estimation and significance testing). Applied statisticians tend to work day-in and day-out with real data. They typically design studies, conduct exploratory data analysis, summarize data, develop models, estimate and evaluate parameter estimates. Applied statisticians are frequently in collaboration with scientists and intimately involved with scientific inference. 

The theory and practice of probability and statistics have been greatly affected by modern advances in computer technology. Today, statistical procedures that were ``nice in theory" years ago (e.g., permutation procedures), are now routinely conducted mainly because of advances in computation. Monte Carlo, bootstrapping, and randomization methods can all be used to investigate sampling distributions of estimators. Cross-validation is used to determine optimal classification trees. Markov chain Monte Carlo (MCMC) procedures are used to estimate posterior distributions. These, and other, computationally intensive methods have enormous implications for statistical practice. 

Seminar Description

The purpose of the seminar in probability and statistics is to introduce students to some of the modern methods in statistics that are based on intensive computation. In the ``real world" there are some very tough statistical problems that can only be addressed through statistical computing. By the end of the summer, students will have an appreciation for the nature of modern statistical practice, which is a wonderful combination of probability and probabilistic reasoning, statistics and statistical reasoning, data analysis (especially graphics), and computation.  

An outline of the course follows.

1. Basics of probability

2. Random variables and classical distributions

3. Expected values

4. The method of maximum likelihood Simulation

5. Randomization and bootstrap methods

There will be two types of projects, two of each kind for a total of four projects. Two projects will focus on theoretical questions about estimators. The other two projects will involve modeling of some real phenomena. All projects will require both analytical and computational work.

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Gröbner Bases

John B. Little, Seminar Leader/Research Advisor

Topic Overview

Systems of polynomial equations in several variables occur naturally in many parts of mathematics and its applications. The set of solutions of such a system forms a geometric object called a variety, and the corresponding algebraic object, the collection of polynomials that are zero at all points of the variety, is an example of what is called an ideal in the ring of polynomials. Starting in the mid-1960's, a powerful collection of computational tools based on Gröbner bases have been developed to answer questions about ideals and varieties, and these have been incorporated into many of the current generation of computer algebra systems such as Mathematica, Maple, REDUCE, and so forth. 

Seminar Description

We will begin the seminar by learning the basic algebra of polynomial rings and some elementary algebraic geometry of varieties. The next main focus will be the theory behind the Gröbner basis methods: the possible orderings on monomials in several variables, the division algorithm for polynomials in several variables, and Buchberger's Criterion and Algorithm for computing Gröbner bases. Using Mathematica, we will look at some first applications of Gröbner bases including elimination of variables, solving systems of equations, and "implicitization" of parametrically-defined curves and surfaces. In addition, we will learn a technique called "the FGLM algorithm" (FGLM = Faugere-Gianni-Lazard-Mora, the inventors!) that can be used to convert a Gröbner basis with respect to one monomial order into a Gröbner  basis with respect to another order if the set of solutions of the polynomial equations given by the basis is finite. The material for this part of the seminar will be taken from Chapters 1--3 of the text ``Ideals, Varieties, and Algorithms" that I wrote with David Cox and Don O'Shea, and some sections of a forthcoming ``sequel" to that book.  At this point, we will be ready to look at a very recent (1997) extension of the basis conversion idea called the "Gröbner Walk" (invented by Collart, Kalkbrener, and Mall) which will form the background for the research component of this seminar. Depending on the computing backgrounds of the students in the group, we might focus on implementing the ``Walk" and/or on applying it to polynomial equation-solving and problems from computational geometry and CAD including implicitization of surfaces such as the bisector surface of two space curves (the locus of points equidistant from the two curves).