Graduate Student Seminar 06-07

Abstract: Fermat's Last Theorem is the statement that for n > 2 the sum of two n-th powers of integers cannot itself be the n-th power of an integer. It was one of the most famous unsolved problems in mathematics until its proof in 1993/94 by Andrew Wiles and Richard Taylor. I will begin this talk with a brief historical account of some of the progress on the problem before 1993. Then I will give brief explanations of the tools and methods used in the eventual proof, but will restrict myself mainly to the very basics of elliptic curves and modular forms. It is my goal to explain the famous "Taniyama-Shimura-Weil Conjecture", the proof of which implied Fermat's Last Theorem..

Abstract: In 1931, Gershgorin (1901-1933) published his elegant theorem about estimating the eigenvalues of a square complex matrix. I will discuss the theorem, the simple proof as well as some other methods for estimating the eigenvalues of a matrix.

Abstract: Proofs in intuitionistic logic can be seen as programs. In this talk I will explain what I mean by this, and show how computing programs yield proof normalization. If time permits I will present as example the proof corresponding to the program "Office" from Microsoft.

Abstract: In algebra (algebraic geometry, combinatorics, number theory, ...) the resultant of two polynomials is defined up to a sign as an irreducible integer polynomial in their coefficients which vanishes whenever the polynomials have a common root. The resultant can be computed as the determinant of the so-called Sylvester matrix and is invariant under the action of the Moebius group. Drawing our inspiration from the quotation by G.-C. Rota chosen to be the title of this talk, we shall introduce by analogy (or, duality, if you wish) a new mathematical object via essentially replacing the word ``polynomials'' in the above. Surprisingly, this will ultimately lead us to a new mathematical characterization of the Kepler problem in classical mechanics.

Abstract: An introduction to combinatorial game theory through impartial games. Impartial games are games in which both players have the same options from every position. In particular, I'll talk about two of the more well-known such games, Nim and Wythoff's game

Abstract: In this lecture I will sketch out a few important ideas from General Relativity necessary to describe the Friedmann-Roberston-Walker (FRW) model of the universe. What is notable about this model is that it was the first model to predict that the universe started with a big bang. I will then show exactly how this was proven. If time allows I will discuss the history of the model and the reasons for my title.

Abstract: The concepts of null sets (sets with Lebesgue measure zero) and meagre sets (sets of first category) both give criterion for a given set of elements to be negligible in some sense. Meagre sets are most commonly used in general topology and descriptive set theory, whereas null sets are important for integration theory. What we are going to be interested in for this talk is the concept of duality; are there any similarities between these two classes of sets? any differences?

Abstract: Inverse semigroups first arose from the need to capture the nature of partial symmetries occurring in mathematics, a task in which groups have proven insufficient. An introduction to the algebraic theory of inverse semigroups will be given (focusing on the importance of idempotents), with motivation coming from the theory of partial bijections. Then, coming around full circle, the Wagner-Preston theorem will show that every semigroup can be faithfully represented by a so called symmetric inverse semigroup(a semigroup of partial bijections on a set). Time permitting, applications of category theory to inverse semigroups will be presented and some elementary applications of inverse semigroups to other areas of mathematics.

Abstract: The minimal free resolution of a monomial ideal is an invariant that gives algebraic and geometric insight into the ideal's structure. In this thesis, we study a number of techniques for computing resolutions and their Betti numbers from the combinatorial properties of the monomial ideal. Recursive and explicit formulas are given for the resolutions of certain classes of monomial ideals using techniques from graph theory, algebraic topology and the theory of convex polytopes.

Abstract: It is a well-known property of Pascal's triangle that the entries of the k-th row, without the initial and final entries 1, are all divisible by k if and only if k is prime. In this talk I will present a triangular array similar to Pascal's that characterizes twin prime pairs in a similar fashion. The proof involves generating function techniques. Connections with orthogonal polynomials, in particular Chebyshev polynomials, will also be discussed. If time allows, I will talk about another triangle, the so-called Stern-Brocot tree, and about some recent work on related number and polynomial sequences.

Abstract: I would like to devote the talk to an exploration of the geometric intuition behind the modern concept of a connection. The connection formalism is due to Elie Cartan, whose work provides a far reaching generalization to the usual idea of a connection as "derivative of the parallel transport operator". In this regards, Cartan's reformulation of differential geometry in terms of moving frames rather than local coordinates is essential to both the formal treatment of connections and also, as I hope to be able to show, to the intuitive understanding of the connexion (Cartan's spelling) concept.

Abstract: When we're talking about the Cantor set, of course. I'll describe the Cantor set, show that it has "length" 0, then show the surprising result that the Cantor set, when summed with itself, gives the entire interval [0,2].

Abstract: The relationships between the generators of an ideal encapsulate a great deal of information about the structure of the ideal. The study of these relations leads to the notion of a free resolution and the construction of free resolutions is an area of ongoing research. This talk will devote time to answer basic questions about the nature of these relations and provide many examples to illustrate the idea. Our focus will be on resolutions monomial ideals and we will provide a general method of supporting resolutions on cell complexes and a combinatorial description of the first few Betti numbers. For a detailed outline, including references, please see the pdf file linked to the right.(pdf)

Abstract: In mathematics, one of the most beautiful tools for producing mathematical papers is LaTeX . LaTeX has become essential to most undergraduate honours students, graduate students and professors. Since many students are tossed into the sea of LaTeX without help or guidance, I am giving a seminar (or tutorial, if you like) in the basics of LaTeX . I will be covering topics such as getting started, what programs and packages to use, creating mathematical formulas, making tables, the dalthesis class and other useful tidbits. Questions will be very much encouraged.

Abstract: This week, Pawel Pralat will be bringing in the movie titled "N is a Number - a portrait of Paul Erdos" for our viewing pleasure. Paul Erdos, a legendary mathematician who was so devoted to his subject that he lived as a mathematical pilgrim with no home and no job, died on the 20th of September, 1996 in Warsaw, Poland. He was 83. He was one of the century's greatest mathematicians, who posed and solved thorny problems in number theory and other areas and founded the field of discrete mathematics, which is the foundation of computer science. He was also one of the most prolific mathematicians in history, with more than 1,500 papers to his name. And, his friends say, he was also one of the most unusual.

Abstract: Several classical results in elementary number theory lend themselves quite naturally to solutions that are algebraic in nature. These problems provide excellent motivation for the development of very general algebraic theory. The goal of the talk is to develop the basic algebraic theory of numbers with the ultimate goal of providing a more conceptual proof of the law of quadratic reciprocity. This result forms the basis for class field theory by generalizing to the most general law of reciprocity. We will develop the relevant algebraic theory and then rephrase the law of quadratic reciprocity in terms of the Galois theory of quadratic and cyclotomic fields. For a detailed outline, please see the pdf file linked to the right. (pdf)

Abstract: Quantum computing is, as you would expect me to say, a fascinating subject. In this talk I will introduce you to the basics of the field, provide you with some insights, and give you a short tutorial on how to crack banking systems (provided you own a quantum computer). (pdf)