|
MGO Colloquia Fridays, 12:45pm Upcoming Talks: Homological Methods in Commutative Algebra Andrew Crabbe, Syracuse University November 13, 2009 Abstract:
In this talk, we'll discuss the famous theorem of Auslander, Buchsbaum and Serre characterizing regular rings, a result which Kaplansky called a "resounding triumph of the homological method" that "marked a turning point in the subject of commutative rings." The talk will provide an overview of a number of fundamental topics in commutative ring theory, eventually working up towards the theorem. Some basic familiarity with rings is definitely desirable.
Previous talks:
Daniel Van Vliet, Syracuse University November 6, 2009 Abstract:
Linear methods of analysis have spawned very powerful technologies for approximation and signal processing, from the familiar fast Fourier transform (FFT) to wavelets, and others. However, recent developments have directed attention to nonlinear methods. While investigating nonlinear methods such as the Hilbert-Huang Transform, Blaschke products have emerged as an interesting set of functions to use as atoms for decomposition. Blaschke products have many properties familiar from linear methods (even inspiring special Blaschke products to be called “nonlinear Fourier atoms”). Developing methods based on Blaschke products presents a compelling challenge: many of the linear projection based methods can be generalized in some way to special classes of Blaschke products using classical tools. However using the complete set of Blaschke products gives the possibility of a truly nonlinear framework for doing approximation and signal processing.
Leonid Kovalev, Syracuse University October 23, 2009 Abstract:
In August 1994 the New York Times reported: "Mathematicians have found that every convex set of points has associated with it another set, which is called the polar set. Dr. Bourgain showed that if a convex set is small its polar set has to be big. Dr. Fefferman said that solution to this problem gave mathematicians insight into the geometry of infinite-dimensional spaces." The problem mentioned by NY Times concerns the product of volumes of a convex symmetric set and of its polar set. In 1939 Kurt Mahler conjectured that this product is minimal for the cube. This conjecture remains open to this day. I will describe the current state of the problem. (More details are available on Terence Tao's blog.
Graham Leuschke, Syracuse University October 9, 2009 Abstract:
A solid ball can be cut up into finitely many pieces, and those pieces rearranged by rigid motions (no stretching or squashing allowed) to form two solid balls the same size as the first. A pea can be cut up and reassembled into a ball the size of the sun. This is the Banach-Tarski Paradox. In this talk, I'll state the BTP carefully, give you an idea of how to prove it, explain why this geometric fact is interesting from both algebraic and analytical perspectives, and identify the elephant in the room that makes it possible: the Axiom of Choice.
Tony Perkins, Syracuse University October 2, 2009 Abstract:
The study of complex functions with two periods leads quite naturally to the theory of elliptic functions. Starting from this point of view I will define elliptic functions, establish some of their basic properties, and show some classic examples which draw connections between different branches of mathematics. Some basic knowledge of complex analysis is desirable but not essential.
Patrick Neary, Syracuse University April 24, 2009 Abstract:
Through the years, people have often considered shapes which minimize their surface area relative to volume. We now know the ideal shape to be a sphere. However, which shape, especially which polyhedron, minimizes edge-length relative to volume, remains an unanswered question today. We will discuss what progress has been made on this topic, especially in categorizing the solution space. This talk should be accessible to all graduate students and advanced undergraduates.
Treven Wall, Syracuse University April 3, 2009 Abstract:
The Bellman function technique has been used a lot in the past few years to answer longstanding open questions in harmonic analysis. It's a counter-intuitive approach, but it works remarkably well. Come and see what the fuss is all about! The talk should be accessible to anyone who has heard of an L^p space.
Dag Madsen, Syracuse University March 20, 2009
Yao Lu, Syracuse University March 6, 2009 Abstract:
Out-of-focus image restoration is a fundamental problem in image processing. When a detector is out of focus, we observe a blurred image. The aim of image restoration is to obtain an in-focus image from the out-of-focus observation. Most of the currently used practical models for image restoration are discrete. The discrete models impose bottleneck model errors, which cannot be compensated from the numerical methods developed based upon them. To overcome this drawback, we established the integral equation models for image processing by using their physics properites and developed multiscale approximation for the continuous models aiming at new discrete models having accuracy of a higher order. The talk is intended for students who have a basic knowledge of linear algebra and numerical analysis.
Aaron Luttman, Clarkson University February 27, 2009 Abstract:
Images captured by ground-based telescopes are necessarily contaminated both by noise and by atmospheric blur. The noise comes from three different sources, one of which follows a normal distribution and the other two of which follow a Poisson distribution. Given the contaminated, measured image, it is necessary to mathematically undo the blur and account for the noise, reconstructing the "true" image. Deblurring means to solve the linear operator equation Au=z, where A is the blurring operator, u is the true image, and z is the measured image. Like in linear algebra, this equation can have 0, 1, or infinitely many solutions, so we must instead solve a perturbed problem, using what's called regularization to pick out one solution. It is also necessary to ensure that the method used respects the Poisson nature of the noise. Using techniques from functional analysis, we must prove that this solution is truly an approximation to the function u we were trying to find. Several different kinds of regularization will be presented along with an explanation of what spaces of functions the resulting approximation lives in. The talk is intended for students who have a good knowledge of linear algebra and a basic knowledge of Lp spaces.
Derek Gustafson, Syracuse University February 20, 2009 Abstract:
The classical Poincarè Inequality allows us to locally control a function by its gradient, modulo constant functions. We shall answer the first question by explaining how this is generalized. Then we shall present some partial answers to the second question. This talk should be accessible to all grad students.
Tony Perkins, Syracuse University February 13, 2009 Abstract:
We will see how a question concerning some basic properties of irrational numbers can be answered by a direct application of some standard results from analysis. Everyone is welcome, including but not limited to interested undergraduate and beginning graduate students in mathematics.
Dayal Dharmasena, Syracuse University November 21, 2008 Abstract:
The Hartogs extension phenomenon is a prominent example of how the function theory of several complex variables differs from the theory of one complex variable. Every domain of the complex plane has a holomorphic function which can not be extended holomorphically beyond that domain. But in a space of dimension greater than one there exist domains on which any holomorphic function can be extended holomorphically into a larger domain. We will discuss some theorems, examples, generalizations and consequences related to the Hartogs Phenomenon. As time permits we will also discuss how to geometrically describe domains of holomorphy. This is an expository talk which should be accessible to anyone familiar with basic (similar to MAT 513) complex analysis.
Gouhui Song, Syracuse University November 14, 2008 Abstract:
We start with the definition and some properties of asymptotic equivalence of two sequences of numbers. A similar asymptotic definition can be derived for two sequences of matrices and some counterpart properties will also be shown. Applications will be mentioned if time allowed. The talk is accessible to all the graduate students with some background in linear algebra.
Cosmin Anitescu, Syracuse University November 7, 2008 Abstract:
We will give an overview of the Generalized Finite Element Method (GFEM) and its connection with the standard Finite Element Method (FEM) for solving partial differential equations. The main ideas of the method will be discussed along with some approximation results. The talk is directed towards all graduate students with an interest in applied mathematics.
Leonid Kovalev, Syracuse University October 31, 2008 Abstract:
Given a Hermitian matrix A with zeroes on the diagonal, is it possible to color the standard basis vectors using two colors so that A does not attain its norm on any monochromatic vector? The answer is known in dimensions up to 17, but in general the problem is open. It can be viewed as a non-quantitative version of Anderson's paving conjecture (1978), which has attracted much attention in operator theory. The talk is accessible to all graduate students and to undergraduates with solid knowledge of linear algebra.
Jarmo Jääskeläinen, University of Helsinki October 24, 2008 Abstract:
In the 1930s Ahlfors and Teichmüller realized the importance of quasiconformal mappings in complex analysis. Quasiconformal mappings occur naturally in various mathematical contexts and nowadays they are used everywhere in complex analysis. Quasiconformal mappings are principally mappings of "bounded distortion". We define quasiconformality and introduce some of the foundational properties of the mappings. The talk is accessible to all graduate students interested in analysis.
Marju Purin, Syracuse University October 10, 2008 ; Note special time & location : 11:40 in the Arents Room (312 Carnegie) Abstract:
We give a short introduction to almost split sequences (Auslander-Reiten sequences) starting with the basic notions of a module and short exact sequence. We prove the uniqueness of these sequences. We then proceed to introduce the Nakayama algebra. As our grand finale we produce all of the almost split sequences over a Nakayama algebra. This talk is aimed at hungry and/or beginning graduate students (it is completely accessible to second year graduate students).
|