Mathematics and Computer Science
COLLOQUIUM
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SUMMER 2008 Wednesday, May 07 10:00 to 16:30 - Room C610 NUMBER THEORY DAY

SPRING 2008 Friday May 16 12:00 to 12:50 Room C620 Robert Benkoczi Assistant Professor, University of Lethbridge Ph.D. Simon Fraser University (2004) Research interests: Wireless networks, facility location, combinatorial optimization. On a covering problem with variable capacities. Facility location problems arise in many real life applications, in the management of cellular networks and of the supply chain of an enterprise to name a few. In cellular networks, for example, it is essential for the service provider to be able to use its resources efficiently, especially during peak periods when the number of mobile phones used in an area may exceed the service capacity of the provider's base station. Very recently, researchers have quantified the dependance of capacity on the covering range of a base station (or its transmission power). Intuitively, in an environment where all the wireless devices transmit at their maximum power setting, the amount of interference is significant and this decreases capacity. A simple analogy is that of talking to your neighbour in a room where everybody else shouts. The communication can be difficult. The task is now to find a way to effectively control the capacity of the base stations of a provider's entire network in order to maximize the number of mobile devices that can be served globally. In this talk, which targets a broad audience, I will present our recent results on this problem. The model we propose, that of a covering problem with variable capacities, is new in the literature and it is interesting from a theoretical point of view as well because it generalizes many classical combinatorial optimization problems. This is joint work with Daya Gaur and Shahadat Hossain. Thursday April 17 12:05 to 12:55 Room B650 Harald Helfgott Senior Lecturer, University of Bristol, UK Ph.D Princeton University (2003) Research interests: diophantine geometry, group theory, probability theory, combinatorics and analytic number theory. Currently: interest on the number, growth and distribution of discrete objects in algebraic structures. Growth in SL_3. Let $K$ be $\R$, $\C$ or a $\Z/p\Z$. Let $G = SL_2(K)$. Not long ago, I proved the following theorem: for every subset $A$ of $G$ that is not contained in a proper subgroup, the set $A A A$ is much larger than $A$. A generalisation to groups of higher rank was desired by many, but seemed hard to obtain.\\ I have obtained a generalisation to $SL_3(K)$. The role of both linearity and the group structure of $G$ should now be clearer than they were at first. Bourgain, Gamburd and Sarnak derived various consequences on expander graphs from my $SL_2$ result; analogous consequences should follow in the case of $SL_3$. Friday April 11 12:00 to 12:50 Room B650 Klaus Denecke Professor, University of Potsdam, Germany Research interests: General Algebra, Category Theory and Applications in Discrete Mathematics, Theoretical Computer Science and Mathematical Logic. What is General Algebra? I have frequently been asked "What is General Algebra?" I am unable to give a clear definition of this relatively new field of mathematics, but will describe some main streams and trends, influenced by my own interests and restricted knowledge. I will explain why and how classical parts of algebra like group theory and ring theory were generalized to semigroups, quasigroups, semirings and near-rings. Lattices and ordered algebraic structures and algebraic logic are also part of General Algebra, as are universal algebra and category theory. In the last ten years computer scientists have also used coalgebras to describe state-based systems, making coalgebras another important concept in General Algebra. Friday April 04 12:00 to 12:50 Room B650 Giuseppe Carenini Assistant Professor, UBC, Vancouver Ph.D University of Pittsburgh (2000) Research interests: Computational Linguistics - Natural Language Generation, Dialog, Statistical NLP HCI - Intelligent Interfaces, Information Visualization and Interactive Techniques Artificial Intelligence - User Modeling, Preference Elicitation, Decision Theory, Machine Learning Social Issues in Computing - Captology, Universal Access and Usability Interactive multimedia summaries of evaluative documents Many organizations are faced with the challenge of summarizing large corpora of text data. One important application is evaluative text, i.e. any document expressing an evaluation of an entity as either positive or negative. For example, many websites collect large quantities of online customer reviews of consumer electronics. Summaries of this literature could be of great strategic value to product designers, planners, manufacturers and consumers. In this seminar, I will first present and compare two approaches to the task of summarizing evaluative text. The first is a sentence extraction-based approach, while the second is a language generation-based approach. These approaches have been tested in a user study. In the second part of the seminar, I will describe an interactive multimedia interface which presents the knowledge extracted form a corpus of evaluative documents not only as a natural language summary but also in a hierarchical visualization mode. The interface is interactive in that it allows the user to explore the original dataset through intuitive visual and textual methods. Results of a formative evaluation of our interface show general satisfaction among users with our approach. Friday March 28 12:00 to 12:50 Room B650 Wendy Osborn Ph.D. University of Calgary Assistant Professor, University of Lethbridge Director of the SADL Research interests: Digital library technology, databases that handle non-standard data (spatial, multimedia, distributed databases). Southern Alberta Digital Library (SADL) The Southern Alberta Digital Library (SADL) at the University of Lethbridge is an extension of the New Zealand Digital Library (NZDL) project at the University of Waikato. The NZDL project manages the development of Greenstone, a software suite for creating online digital libraries. SADL has two primary goals: 1) to assist in the core development of the Greenstone digital library sofware, and 2) to host digital library collections that have ties to Southern Alberta. For this seminar, I will present some of the core development and applications that have taken place over the past few years. I will also present directions, in development and applications, that SADL will take in the near future. Friday March 14 12:00 to 12:50 Room B650 Matthew Greenberg Assistant Professor, University of Calgary Ph.D McGill University (2006) 2008 CMS Doctoral Prize Research interests: number theory, modular forms, elliptic curves, Heegner point computations. Analysis and Diophantine equations A Diophantine equation is a polynomial equation in finitely many variables with integer coefficients. A large part of number theory is concerned with the existence of rational (or, more generally, algebraic) solutions to particular (systems of) Diophantine equations. Since the field of rational numbers is not complete, analytic techniques involving successive approximation (e.g. Newton's method) cannot be brought to bear on Diophantine equations. Even so, analysis (both classical and $p$-adic) is an invaluable tool in the study of their properties and their solutions. Friday February 15 12:00 to 12:50 Room D511 John Irving Assistant Professor Saint Mary's University, Halifax Ph.D. University of Waterloo (2004) Research interests: algebraic combinatorics, particularly enumerative problems underlying questions in geometry and representation theory Minimal Transitive Factorizations of Permutations The problem at hand is to count decompositions of a given permutation into a product of a fixed number of transpositions of the form (1 i). We give a simple formula for the number of such decompositions into a minimal number of factors. Our derivation involves an interesting relationship between factorizations of permutations and maps on surfaces, a classical bijection between trees and Dyck sequences, and an enumeration of such sequences via the Cycle Lemma. Connections with algebraic geometry will be briefly discussed. Friday January 18 12:00 to 12:50 Room D511 Kerri Webb University of Lethbridge Research interests: Graph Theory, combinatorics. A major theorem about minors The monumental Graph Minors Project of Robertson and Seymour ranks among the deepest theorems in mathematics. Rota's Conjecture, regarding an extension of these results to matroids, is arguably the most important open problem in matroid theory. We survey the main results and applications from the Graph Minors Project, and describe progress towards Rota's Conjecture.

FALL 2007 Fridays - 12:00 to 12:50 - Room C610   December 07 Howard Cheng Associate Professor, University of Lethbridge Ph.D University of Waterloo (2003) Research interests: controlling intermediate expression growth in linear algebra problems, polynomial matrices and their generalization to Ore polynomial matrices ; compression and encryption algorithms for images and videos. Ore Polynomials: Applications and Algorithms Oystein Ore introduced the notion of noncommutative polynomials as a unified way of studying differential, difference (recurrence), and similar types of equations. These polynomials are used in modern computer algebra systems not only for solving these types of equations, but also for automatic theorem proving. Matrices of Ore polynomials can be used to study systems of linear differential and difference equations. Algorithms to reduce these matrices into a normal form can be applied to simplify the corresponding systems of equations.\\ In this talk, I will give a brief introduction on Ore polynomials and describe a few of the applications. I will also give an overview of the difficulties that arise when computing with these polynomials. It will be shown that many of these difficulties can be overcome using simple techniques from linear algebra and number theory, resulting in algorithms with lower complexity. November 23 Shafiq Joty Graduate student (Yllias Chali), University of Lethbridge Question Answering System Question Answering (QA) is retrieving answers to natural language questions from a collection of documents rather than retrieving relevant documents containing the keywords of the query which is performed by search engines. The Text Retrieval Conference (TREC) QA track is the major large-scale evaluation environment for open-domain question answering systems. We took the approach of designing a question answering system that is based on document tagging and question classification. Question classification extracts useful information (i.e. answer type) from the question about how to answer the question. Document tagging extracts useful information from the documents, which will be used in finding the answer to the question. Our system achieved one of the top scores in TREC-2007 QA track competition. November 09 Robert Benkoczi Assistant Professor, University of Lethbridge Ph.D. Simon Fraser University (2004) Research interests: Wireless networks, facility location, combinatorial optimization. Facility location problems with collection depots Facility location is an important problem in Economics. It has been around for almost 100 years now (98 to be more exact :)). It is a fascinating subject of study because so often the solutions to such problems rely on combining ideas from different fields such as computational geometry, graph theory, and linear and integer programming. This talk is an introduction to facility location and is targeted to a broad audience (some familiarity with the analysis of algorithms at the level of MATH 1xxx is the only prerequisite assumed for this talk). I will illustrate a few classical problems in facility location and I will conclude with some results obtained on a generalization of the classical problems, location with collection depots. Joint work with B. Bhattacharya, Q. Shi, J. Sember (SFU); S. Das (Indian Statistical Institute); A. Tamir (Tel-Aviv Univ.) October 26 Nathan Ng Assistant professor, University of Ottawa Ph.D UBC (2001) Doctoral Prize (2001) Research interests: analytic number theory, Riemann zeta function, prime numbers, and a variety of problems in multiplicative number theory. Chebyshev's Bias, Galois Groups, and L-functions Cheybshev observed the strange phenomenon that there appear to be more primes congruent to three modulo four than to one modulo four. This is counterintuitive since one expects that there are an equal number of primes congruent to three modulo four than to one modulo four. In this talk, I will explain this phenomenon known as "Chebyshev's bias" and I will discuss generalizations. For example, consider the polynomial $x^3-x-1$. If we consider this polynomial modulo p a prime, this polynomial is either irreducible, splits into a linear factor and a quadratic factor, or splits into three linear factors. I will explain which of these cases occurs the most frequently and I will explain why the "bias" arises. October 12 Steve Wismath Professor, University of Lethbridge Chair, dpt Math and CS Ph.D UBC (1989) Research interests: Analysis of Algorithms, Computational Geometry, Graph Drawing, Visibility Graphs. Universal Sets of n Points for 1-bend Drawings of Planar Graphs with n Vertices We show that any planar graph with n vertices can be point-set embedded with at most one bend per edge on a universal set of n points in the plane. An implication of this result is that any number of planar graphs admit a simultaneous embedding without mapping with at most one bend per edge. Joint work with: Hazel Everett, Sylvain Lazard and Giuseppe Liotta. September 21 Brandon Fodden PIMS post-doctoral fellow, University of Lethbridge Ph.D Queen's University (2007) Research interests: analytic and algebraic number theory, L-functions, Hilbert's 10th problem. Some unprovable statements in number theory We will discuss some number theoretic statements which are unprovable with respect to either Peano arithmetic or the ZFC axioms of set theory. We will cover Goodstein's theorem as well as the connection that Diophantine equations have with the consistency of formalized systems.