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Spring 2004 Tuesdays, 4:35 - 5:35pm in JWB 335 Math 6960-001 (1 - 6 credit hours) The goal of this Colloquium is to encourage interaction among graduate students, specifically between graduate students who are actively researching a problem and those who have not yet started their research. Speakers will discuss their research or a related introductory topic on a level which should be accessible to nonspecialists. The discussions will be geared toward graduate students in the beginning of their program, but all are invited to attend. This invitation explicitly includes undergraduate students. Talks will be held on Tuesdays at 4:35pm in JWB 335, unless otherwise
noted.
Speaker: Mike Woodbury Title: Elliptic Curves Abstract: Elliptic Curves are essentially just the set of solutions (x,y) to an equation of the form y^2 = x^3 + bx + c. Although simple, they are very powerful. Understanding elliptic curves is essential to understanding Wile's proof of Fermat's Last Theorem. They also are an important tool in Lenstra's factorization algorithm. Last summer I studied elliptic curves as part of the Summer REU program here at the University of Utah. My main focus was studying the group of rational points on elliptic curves. In my talk I will discuss this group structure, how projective geometry is used in the study of elliptic curves, and some of the major theorems about the group of rational points. Time permitting, I hope to also discuss why y^2 = x^3 - 432x + 8208 tells us something about \emph{all} curves whose rational solutions form a group of order 11. Speaker: Ken Chu Title: From Solutions of Systems of Polynomial Equations to Grobner Bases Abstract: I will start off with the story of how I encountered a system of nasty (non-linear) system of polynomial equations and proved that it admits only the zero solution. Grobner bases were used as a "simplification" tool to simplify that system. Then, I will move on to explain the "pathologies" of the Multivariate Division Algorithm --- the natural generalization to the multivariate case of the division algorithm for polynomials in one variable. Grobner bases were originally designed to fix these pathologies. Time permitting, I will actually tell you how Grobner bases are defined. Speaker: Aaron Bertram Title: The Weil Conjectures Abstract: Start with a homogeneous polynomial P (or a set of polynomials) with integer coefficients. E.g. take P(x,y,z) = x^n + y^n + z^n. Then you can ask several questions about the zeroes of P. Question 1. How many zeroes does P have "mod p" for a "good prime" p? More generally, how many zeroes does P have in the field with q elements, when q is a power of p? Question 2. What is the topology of the set of complex zeroes of P? More precisely, what are the betti numbers of this compact manifold? The Weil conjectures say that if you know the answer to Question 1 for a single prime p, then you know the answer to Question 2. This is very non-obvious! (and meant a Fields medal for Deligne when he proved it). I want to talk about this, the role of fixed-point theorems in the proof, and some applications. Speaker: Florian Enescu Title: Powers of Ideals in Polynomial Rings Abstract: Let I an ideal in a ring of polynomials and let f be a polynomial that does not belong to I. Although f does not belong to I, one can still have that f^p stays in I^q for some p and q positive integers. What can we say about such pairs (p,q)? Can the upper bound for q/p be an irrational number? We will discuss these questions along with classical results developed by Samuel, Nagata and Rees and will relate them to a recent question of R. Lazarsfeld. The talk is based on joint work with C. Ciuperca. Speaker: John Zobitz Title: Time Out for Delay Equations Abstract:TBA Delay differential equations (DDEs) can arise when adding more realism to a biological system (giving a population time to mature before reproducing) or when trying to reduce a complex biochemical scheme for easier analysis (as in the case of circadian rhythms). While on the surface DDEs look intimidating, a nice geometrical interpretation can be used to analyze their linear stability. This talk will describe DDEs, show a few examples, and demonstrate where I had to analyze a system with delays. Speaker: Dragan Milicic Title: Introduction to D-modules Abstract: D-modules are modules over rings of differential operators. They are a fundamental tool in modern algebraic geometry and representation theory. I'll discuss the origin of the theory in attempts to give a simple proof of the existence of fundamental solutions for PDE with constant coefficients. We are going to see how this analytic problem can be solved by algebraic methods. At the end, we will try to explain how these methods lead to a vast generalization of deRham theory and to the notion of "perverse sheaves" (which are neither perverse nor sheaves). Speaker: Matthew Clay Title: Splittings of Groups Abstract: Given two surfaces with boundary or one surface with two boundary components, we can make a new surface by gluing along these components. Likewise, given two groups with isomorphic subgroups or one group with two isomorphic subgroups we can make a new group by identifying these subgroups. Groups which can be formed in this way are said to split. We will looks at some examples and how this relates to actions of groups on trees. Speaker: Jingyi Zhu Title: Modeling Credit Risk With Diffusion Abstract: Credit risk pricing and risk measurement are central topics in today's financial world. I will discuss the basic mathematical methodologies used in two different schools of modeling and attempt to connect the two approaches. One approach is based on the first-exit time of a diffusion process, and the other is an application of the Poisson process, where the first arrival signals a credit event. This is sometimes called a jump diffusion process. We will show how these two types of diffusion can be unified to develop more powerful models. In our model, we deal with a free-boundary problem for a partial integro-differential equation. Numerical methods are developed and pricing/hedging of credit derivatives can be computed accordingly. Speaker: Mladen Bestvina Title: Thurston's Geometrization Conjecture Abstract: The Geometrization Conjecture is about the classification of all compact 3-manifolds. Roughly speaking, the conjecture states that every such manifold can be broken up into pieces in a certain natural way so that each piece admits one of eight 3-dimensional geometries. If true, the famous Poincare conjecture follows (it states that a simply connected closed 3-manifold is the 3-sphere). The solution of the Poincare conjecture is worth $1M. I will describe the Geometrization Conjecture and some of the geometries in more detail. A proof of the Geometrization Conjecture was recently announced by Grisha Perelman. SPRING BREAK Speaker: Elizabeth Doman Title: Multiple Metastatic Tumors Abstract: Consider a situation in which an invading organism expands its range not only by reproducing and random movement (diffusion), but also by generating long distance migrants who in turn create new colonies, and so on. Many species expand their ranges in this way. Some birds and insects can fly far distances, and the seeds of some plants can be carried away by the wind. This same idea can be applied to the spread of cancer through the body. Although it's not a large scale organism like a bird or plant, a cancer cell lives in a growing colony composed of other cancer cells, called a tumor. The tumor in turn produces long distance migrants that settle in other locations of the body. This process, known as metastasis, can be analyzed using the "scattered colony" model proposed by Iwata, Shigesada, and Kawasaki (2000). I will present their model, and show some results. Time: TBA Speakers: Aaron McDonald (Mathematical Biology) Title: Mathematical Epidemiology Abstract: There has been much work detailing the processes shaping ecological communities. One often reads about processes affecting the growth of a particular species, with focus placed on predator-prey dynamics, competition, reproduction, and dispersal dynamics. Many top predators are also limited by infectious diseases. Humans are of no exception. Throughout our documented history, human populations have been, and currently are, plagued by numerous infectious diseases. In this talk, I will discuss the human history of epidemiology and the subsequent development of disease modeling. Renzo Cavalieri (Algebraic Geometry) Title: TQFT's: bringing algebra into physics...or vice-versa Abstract: Quantum Field Theories (QFT) are the "hot" language in today's theoretical physics. Of course, physicists are very happy with them, whereas mathematicians are still struggling with giving these theories solid logical consistency and mathematical structure. I will introduce Topological Quantum Field Theories, a baby version of a QFT. In this case we are able to define exactly what it is, and to uncover some interesting mathematical relations. Josh Thompson (Topology) Title: "Building blocks" in 3-dimensions Abstract: People have long been interested in 3-dimensional space. As a result, we are naturally lead to the idea of a 3-manifold. Then one wonders what different kinds of 3-manifolds exist. Some time later, one asks whether there are some 3-manifolds which are the 'building blocks' out of which all others are built. If so, how do we recognize one when we see it? It turns out that there are different levels of 'building blocks'. We will meet one of the first levels of these 'building blocks' called Prime 3-Manifolds. Speaker: Jim Carlson Title: The Riemann Hypothesis: History and Influence Abstract: This will be a historical/expository talk with some comments on the Millennium Prize Problems. Speaker: Grady Wright Title: Connecting the Dots: The Role of Polynomial Interpolation in Numerical Analysis Abstract: Before the widespread use of calculators and computers, polynomial interpolation was taught regularly in high school algebra as a means to compute values of a function stored in tabular form (e.g. the natural logarithm). Nowadays, high school and even college students are rarely taught anything beyond the ancient Babylonian and Greek's linear interpolation. This is a bit ironic since polynomial interpolation has more of an impact on our everyday lives now than before the advent of the Digital Age. I make this argument because polynomial interpolation is one of the primary tools for developing the numerical methods used in, for example, forecasting the weather, flying airplanes, talking on cell phones, designing skis, etc. In this talk, I will review the basics of polynomial interpolation theory and show how to use it to develop numerical methods for finding roots of non-linear functions, evaluating definite integrals, computing solutions to ordinary differential equations (ODEs), and partial differential equations (PDEs). If time permits, I will try to connect this with my current research interests. Speaker: Meagan McNulty Title: Quorum Sensing in Pseudomonas aeruginosa Abstract: Quorum: n. 1. the number of members of a group required to be present to transact business or carry out an activity legally, usu. a majority. 2. a particularly chosen group. The bacteria Pseudomonas aeruginosa has the ability to sense its colony size, and change certain metabolic pathways to help ensure the survival of the colony. A 2003 model by Magnus G. Fagerlind et al. of P. aeruginosa quorum sensing will be derived and discussed. Speaker: Ryan Rettberg Title: Alexander Quandles Abstract: Given two knots it can be quite hard to prove that they are different knots or the same knot. Tricolorability is one of the easiest ways of distinguishing between some pairs of knots. I will discuss how this notion of knot coloring generalizes to invariants derived from Alexander quandles, and how these relate to other knot invariants. Speaker: None Title: Pizza
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