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| August 2007 | September 2007 | October 2007 | November 2007 |
| January 2008 | February 2008 | March 2008 | April 2008 |
| Speaker: | Valery Alexeev |
| Title: | Tropical geometry |
| Abstract: | Tropical geometry is a "hot", fairly new and popular field whose main attractions are: a very low starting point, almost no prerequisites needed; a wealth of applications in topology, algebraic geometry, combinatorics, physics, probability theory; wide-open areas for original research; and a large community of people interested. This combination makes it an attractive place for a student to start original research. Known applications in physics include an amazingly easy way to compute Gromov-Witten invariants and, in a perpendicular direction, a way to describe shapes of dimer crystals. |
| Speaker: | Dino Lorenzini |
| Title: | Algebraic Graph Theory |
| Abstract: | Associated to a graph G on n vertices are two (n x
n)-matrices, the adjacency matrix A and the Laplacian matrix L. Both A
and L have a set of eigenvalues, and a Smith normal form over the
integers. Much has been written on the relationships between the
eigenvalues and the combinatorics/topology of the graph. In this
seminar, we will investigate the information encoded in the Smith
normal form of the Laplacian of a graph. Links with recent
geometrically inspired theories could also be discussed, depending on
the interests of the participants. Prerequisites: a basic course in linear algebra, and a willingness to try things out. |
| Speaker: | Elham Izadi |
| Title: | Introduction to Algebraic Geometry |
| Abstract: | The VIGRE research group that I will run this year will concentrate on elementary problems in Algebraic Geometry. Most of the problems that we will work on will be concerned with explicitly desingularizing examples of algebraic varieties by blowing them up (an algebraic variety is the set of common zeros of some polynomials in several variables). For instance we will desingularize plane curves, some hypersurfaces in projective space, subvarieties of Grassmannians. We will start by covering some necessary background about what it means to be singular and nonsingular, what is a blow up etc. There are no prerequisites for the VIGRE group other than enthusiasm and hard work. You can start at any level and do as much as you can. We will split the interested students into groups of two or three (according to people's backgrounds and interests) who will work together on the same problem. |
| Speaker: | Brian Cook, University of Georgia |
| Title: | The Burr-Erdos Conjecture in Graph Theoretic Ramsey Theory |
| Abstract: | The graph-theoretic version of Ramsey's Theorem states, in its most simple form, that, for a given graph G, there exists a minimal integer r such that for any partition of the edges of the complete graph K_r = (V_r, E_r), say as E_1 and E_2, we shall have that either G is a subgraph of (V_r, E_1), or a subgraph of (V_r, E_2). For a given graph G, this minimal integer is denoted by r(G). In general, r(G) grows exponentially in |G|, where |G| denotes the number of vertices in G. However, for graphs that are sparse, we should expect that, asymptotically at least, we should be able to do much better. For a particular class of graphs in the 'more sparse' genre, a class known as the d-degenerate graphs, Burr and Erdos conjectured that we may improve this bound to a linear one. While the conjecture is still open, there has been much progress made. This talk focuses on some this work, particularly focusing on the partial results for graphs of bounded degree, p-arrangeable graphs, and subdivisions. |
| Speaker: | Will Kazez, University of Georgia |
| Title: | Contact Topology |
| Abstract: | The keywords for this talk are: families of 2-planes in 3-space, integrable vs. non-integrable distributions, singular foliations on surfaces, ODE's, knot theory, DLP television, and for good luck, fluid flows. Hopefully this ends up serving as a sort of coherent introduction to contact topology. |
| Speaker: | Alex Rice, University of Georgia |
| Title: | Density and Substance: An Investigation into the Size of Integer Subsets |
| Abstract: | What is the probability that an integer is prime? How about square? Squarefree? What portion of the integers can be written as the sum of three squares? What must be true of an integer subset in order for its reciprocal series to converge? How can we use these ideas from number theory to catch multinational corporations cooking their books and cheating on their taxes? These questions and more will be answered as we investigate two methods for measuring the size of a subset of positive integers, search for relationships between the two measures, and prove several intriguing results that both confirm and betray our intuition. The majority of the talk will be very accessible to anyone who is familiar with limits, infinite series, and set theory. A smaller portion of the talk will feature some more involved number theory, such as the three squares theorem and the Mobius function, but I will certainly give definitions and explanations of such things. Everything is pretty easy to wrap your head around. |
| Speaker: | Lev Konstantinovskiy, University of Georgia |
| Title: | The Hilbert Function |
| Abstract: | Given a polynomial ring S and an ideal I, there is a vector space of polynomials of degree n in the quotient ring S/I. Define h(n) to be the dimension of this vector space. It is a function of integers, called the Hilbert function of I. Given the ring S, which functions can be Hilbert functions? We will discuss MacAulay's result on the maximal growth of h. Another question is: Given a ring S and a function h, which ideals have this Hilbert function? |
| Speaker: | Jason Cantarella, University of Georgia |
| Title: | What can geometric measure theory do for you? |
| Abstract: | It's not uncommon to run into a situation in mathematics where you have a reasonable geometric functional defined on curves, or surfaces, or algebraic varieties and you want to prove that an object minimizing (or maximizing) your functional exists. One example is the classical isoperimetric problem: Find the solid of least boundary area with given volume in R^n. How do we know that such a solid exists? This talk will introduce the subject of geometric measure theory and show how it provides a general framework for proving existence theorems of this kind. We'll take as an example this natural challenge problem: Prove that planets should be round. More precisely, suppose you define the "gravitational potential energy" of a domain in R^3 by F(D) = \int_{x,y \in D} f(|x - y|) dVol_x dVol_y, where f(|x-y|) is an increasing function. Prove that the domain of volume V minimizing F is a round ball. The only absolutely essential prerequisite is some multivariable calculus, so the talk is (in principle) suitable for interested undergraduates. |
| Speaker: | Bobbe Cooper, University of Georgia |
| Title: | Tips for Surviving Graduate School |
| Abstract: | In this talk I will discuss strategies for managing your time and being productive in graduate school -- while keeping your sanity. We'll talk about balancing school and personal life, organizing office space, dealing with the multiple strains on your time, and getting motivation for long-term projects and day-to-day tasks. I'll also describe some nitty-gritty practical ways to start and continue mathematical research. This talk should be accessible to first-year graduate students. |
| Speaker: | Jim Stankewicz, University of Georgia |
| Title: | Introducing the p-adic Numbers |
| Abstract: | In the late 19th century and into the early 20th century, Kurt Hensel developed a novel way to extend the rational numbers for each prime integer p to the field of p-adic numbers. Since that time, the p-adic numbers have played a vital role in number theory, and in this talk I hope to introduce them in a friendly way. If time permits, we shall make mention of the Hasse-Minkowski theorem and the local-global principle. |
| Speaker: | Andrew Sornborger, University of Georgia |
| Title: | Multivariate Statistical Optimization Analysis of Ratiometric Imaging Data |
| Abstract: | In the biological imaging context, ratiometric fluorescent indicators have seen a recent surge in application due to the ability of fluorescence ratios to eliminate illumination variability and bleaching artifacts and to give quantitative estimates of variables of interest. A typical problem with ratiometric indicators is that taking ratios of noisy fluorescence measurements from two channels can result in poor ratio estimation. In this talk, I will discuss ratio estimates in imaging data. Our methods are generally applicable to a wide range of ratiometric indicators including voltage sensitive dyes, calcium dyes and genetically encoded FRET indicators such as cameleon. |
| Speaker: | Jesse Ratzkin, University of Georgia |
| Title: | The Maximum Principle |
| Abstract: | The maximum principle is one of the most important tools in differential equations. Unfortunately, most people never see it until taking a graduate course in partial differential equations. I will begin by explaining why, in a way, you already know the maximum principle. Then I will discuss some of its applications in PDE and geometry. |
| No meeting today. |
| Speaker: | Qianying Hong, University of Georgia |
| Title: | Partial Differential Equations in Image Analysis |
| Abstract: | Partial Differential Equations have been well studied in image analysis. In the talk, I will give an introduction to several famous PDE models in image analysis like: noise removing and edge enhancing. This time I will focus on the idea of level set and total variation in image analysis. |
| Speaker: | Benjamin Jones, University of Georgia |
| Title: | Schubert Calculus and Enumerative Geometry |
| Abstract: | We introduce some ideas from classical enumerative geometry and discuss the connection to what is known as Schubert Calculus. A question we will discuss is ``How many lines in 3-dimensional space meet 4 generic lines?'' The 20th century answer to this question lies in understand the cohomology ring of an important class of algebraic varieties called Grassmannians. |
| Speaker: | Caner Kazanci, University of Georgia |
| Title: | Energy Cycling in Ecological Networks |
| Abstract: | A common way to simulate ecosystems is to form a set of differential equations where the solution represents the state changing in time. Another way to analyze these systems is by formulating system-wide organizational properties. Ecological Network Analysis (ENA) formulates ecological network properties that quantify how the environmental inputs are shared among identities, or how much energy or matter cycling occurs within the system. Obviously, such analysis is essential in understanding how a specific ecological system functions, how it can be sustained or manipulated. However it is almost impossible to actually verify how well these algebraic definitions reflect their meanings. In this talk, we present a discrete stochastic method that provides an accurate computation of these ecological properties. |
| Speaker: | Charles Pooh, Wolfram Research |
| Title: | Infinitesimal and Discrete Calculus in Computer Algebra |
| Abstract: | We will give a comprehensive introduction to infinitesimal calculus and discrete calculus in computer algebra systems such as Maple and Mathematica. Theory, algorithms, concrete examples and applications will be discussed. |
| No meeting today |
| Speaker: | Maxim Arap, University of Georgia |
| Title: | An Introduction to Toric Varieties |
| Abstract: | After a brief historical and motivational remark for the theory of toric varieties there will be a brief review of basic definitions in algebraic geometry followed by examples. Subsequently, a definition of a toric variety will be given. The remaining part of the presentation will be devoted to the construction of (affine) toric varieties from certain combinatorial data. The core of the presentation will be based on some of the lectures that I attended at CRM in May of 2007. The presentation will be aimed at the audience with no background in algebraic geometry and many examples will be provided. |
| Speaker: | Edward Azoff, University of Georgia |
| Title: | Invariant Subspaces |
| Abstract: | Let T be a linear transformation acting on a complex vector space. A linear subspace S of the underlying vector space is invariant under T if T(S) is contained in S. The more invariant subspaces of T one can find, the more one learns about the structure of T. The best behaved linear transformations from this point of view are called reflexive. We will examine reflexive (collections of) linear transformations in both finite and infinite-dimensional settings. Applications of analytic function theory will be highlighted, and we will discuss progress on the invariant subspace problem -- whether every bounded Hilbert space operator leaves some non-trivial subspace invariant. Most of the talk should be accessible to those who have taken undergraduate courses in linear algebra and complex variables. |
| Speaker: | Ming-Jun Lai, University of Georgia |
| Title: | Introduction to Wavelets |
| Abstract: | We explain what kind of functions are called wavelets and why they are excellent for application. Then I will explain several methods to construct them. Finally we will propose some open problems. |
| Speaker: | Christof Meile, Marine Sciences, University of Georgia |
| Title: | Shades of Blue: Quantitative Methods in Marine Sciences |
| Abstract: | Oceanography is an interdisciplinary field of study, encompassing components from Biology, Chemistry, Physics and Geology. Their integration and quantitative analysis often necessitates or is facilitated by the use of numerical models. This presentation provides a brief overview on some research efforts in the Department of Marine Sciences using quantitative methods to unravel aspects related to ocean carbon cycling: The quantification of export of carbon from the photic surface waters to the deep ocean, modeling of carbon breakdown in sediments and nutrient cycling associated with seagrasses and numerical simulation of microbial dynamics in porous media at the grain scale. Finally, specific areas for potential collaborative endeavors are highlighted. |
| Speaker: | Jim Prestegard, Complex Carbohydrate Research Center, University of Georgia |
| No meeting today |
| Speaker: | Matt Mastin, University of Georgia |
| Title: | Introduction to Circle Packing |
| Abstract: | We will introduce circle packings on constant curvature spaces (k=1,0,-1). Several applications of circle packings will be mentioned, including their connection with conformal mappings on the complex plane and the Riemann Mapping Theorem. |
| Speaker: | Jim Stankewicz, University of Georgia |
| Title: | Report on Number Theory VIGRE Research Group |
| Abstract: | This talk is a review of the work done in the Number Theory VRG. We will discuss some basics of CM Elliptic Curves, the algorithm we developed for studying torsion, the computation we've done thus far and some theoretical results we have been working on. |
| Speaker: | Brandon Samples, University of Georgia |
| Title: | Report of Vigre Reasearch Group on Algebraic Graph Theory |
| Abstract: | A graph on n vertices has a matrix associated to it called the Laplacian matrix of a graph G, which we can call L(G). We considered the torsion subgroup of the group Z^n/Im(L(G)), which has interesting properties such as its order represents the number of spanning trees of the graph. I will discuss my exploration into a particular category of graphs called the Paley Graphs. Last year, the graph theory group completely understood the general structure of the group associated to the Paley graph, so I modified the question for powers of primes (particularly p^2 and p^3). I will give a general discussion about some basic computer computations involving matrices I did with Maple, and discuss what we had conjectured the structure of the group was by the close of the semester. |
| Speaker: | Irfan Bagci, University of Georgia |
| Title: | Lie Algebras, Lie Superalgebras and Cohomology |
| Abstract: | We will start with Lie algebras and their basic properties with lots of examples. After that we will discuss Lie superalgebras with examples and the classification of simple finite dimensional Lie superalgebras over the complex numbers given by V. Kac. Lastly I will briefly introduce relative cohomology for Lie superalgebras and present some of the results of research I am doing. The presentation will be aimed at the audience with no background in representation theory and as it can be understood from the abstract many examples will be provided. |
| Speaker: | Jie Yu, University of Georgia |
| Title: | Mean Reversion Model and Applications |
| Abstract: | Mean reversion is a mathematical methodology commonly used in the energy market and finance market. It describes the tendency for price to move towards an equilibrium level. The change of time method is used to solve this stochastic differentiation equation. Morevoer I will talk about its application in evaluating financial derivatives. |
| Speaker: | Maxim Arap, University of Georgia |
| Title: | Fano Varieties of Lines in Smooth Hypersurfaces in Projective Space |
| Abstract: | This presentation will be a report of some of the activity in Professor Izadi's VIGRE group on Algebraic Geometry. The presentation will begin with a brief overview of the history and applications of Fano varieties of lines. Then we shall discuss the construction of these varieties and see a concrete example of the Fano variety of lines on the Fermat cubic surface in projective space. The presentation will assume no background in Algebraic Geometry. |
| Speaker: | Bree Ettinger, University of Georgia |
| Title: | Ozone Prediction using Bivariate Splines to Approximate Functional Regression Models |
| Abstract: | Functional linear regression models are models where the explanatory variable is a random surface and the response is a real random variable. We will discuss the recent results for bounded or normal real random responses. We use bivariate splines to represent the random surfaces then we use this representation to construct least squares estimators of the regression function. We will discuss the two cases of the least squares estimators, one with a penalty term and one without. Finally we will explore the functional regression model’s application to predicting ground level ozone from 2006 EPA data over the continental United States. |
| Speaker: | Carrie Wright, University of Georgia |
| Title: | Lie Algebra Cohomology in Characteristic p |
| Abstract: | I am going to talk about computing Lie algebra cohomology over a field of characteristic p. I will also discuss some main results relating to the cohomology. The cohomology over a field of characteristic 0 or a field of characteristic p for large primes is known. In our VIGRE group we have looked at what happens when the prime is small. I will discuss the work of the VIGRE algebra group this past year and some future directions for new research. |
| Speaker: | Sybilla Beckmann, University of Georgia |
| Title: | Math Courses for Prospective Elementary and Middle Grades Teachers |
| Abstract: | We will discuss key features of the math courses for prospective elementary and middle grades teachers that are taught here and why more mathematicians (especially new mathematicians) should become interested in the preparation of teachers and in educational issues generally. |