California Polytechnic State University

Colloquium Schedule

• Fluid-structure interaction problems and upscaling of Stokes flow in elastic channels
  Peter Popov
  
  Abstract
  In this talk the problem of computing the effective permeability of a Stokes flow in a long channel with elastic walls is discussed. The problem is studied both numerically and by an asymptotic expansion method. First, the stationary, strongly coupled, fluid-structure interaction (FSI) problem is formulated. It is then solved numerically by an iterative procedure, which solves sequentially fluid and solid subproblems. Each of the subproblems is discretized by finite elements. The numerical method is then used to compute the permeability of a long elastic channel. The computational results are found to converge to the asymptotic one.

 

• Some results on the Buchsbaum Eisenbud Horrocks conjecture
  Melissa Kraus, Mathematics Undergraduate, Cal Poly
  
  Abstract

 

• Constructing non-positively curved spaces and groups
  Jon McCammond, University of California, Santa Barbara
  
  Abstract
  Geometric group theorists have been borrowing techniques from differential geometers for quite a while now. At this point, there is a well-developed theory of non-positively curved spaces (modeled after Riemannian manifolds with non-positive sectional curvature) and the groups which can act on them in reasonable ways. This theory has enormous consequences for the algebraic structure of the groups involved. After briefly reviewing the theory of non-positively curved spaces and groups, the remainder of the talk will focus on the construction of spaces to which the theory applies. To date, our ability to construct examples is much less developed than one would like. As I will explain, the key difficulty involves a seemingly elementary problem in Euclidean geometry.

 

• Introduction to geometric group theory
  Anton Kaul, Mathematics Department, Cal Poly
  
  Abstract

 

• New results on norms of composition operators
  Cliff Nelson, Mathematics Undergraduate, Cal Poly
  
  Abstract
  This talk will be about some theorems that provide a new answer to a question raised by Cowen and MacCluer in 1995. We will show that there is a class of symbols whose corresponding composition operator norm could not be determined by taking the supremum over all reproducing kernels but yet could be found by taking the supremum over all sums of two related reproducing kernels.
  
  This research was part of the Cal Poly Mathematics Research Program during the summer of 2003.

 

• Counting on determinants
  Art Benjamin, Department of Mathematics, Harvey Mudd College
  
  Abstract
  We demonstrate how determinants solve many interesting combinatorial problems. Determinants count non-intersecting lattice paths, spanning trees, and permutations with specified descent points. Elegant proofs of these results are based on the definition of the determinant and occasionally the principle of inclusion-exclusion. This talk is based on joint work with Naiomi Cameron of Occidental College.

 

• The nonlinear periodic steady state problem
  Al Jimenez, Mathematics Department, Cal Poly
  
  Abstract
  Flyer with Abstract

 

• Rings and algebras with polynomial identities:
  From classical theory to structure of T-ideals, combinatorial technique and invariants
  Angel Popov, American University in Bulgaria
  
  Flyer with Abstract

 

• Symmetries of differential equations
  Juha Pohjanpelto, Oregon State University
  
  Abstract
  A symmetry of a differential equation is a transformation that maps solutions of the equation to new solutions. Since their first systematic treatment by Sophus Lie over a century ago, symmetries have become an important tool in the study of differential equations. Classical applications of symmetries include the integration of ordinary differential equations, construction of particular solutions to partial differential equations and the classification of conservation laws, or first integrals, for such equations. Recently, symmetries have been employed in the study of both finite and infinite dimensional Hamiltonian systems and in the construction of separable coordinate systems for the field equations of mathematical physics. Moreover, there also seems to be a remarkable connection between symmetries and so-called completely integrable equations, which are nonlinear differential equations possessing soliton, or solitary wave, type solutions.
  
  Juha Pohjanpelto is a candidate for a tenure-track position in the Mathematics Department at Cal Poly.

 

• Conley decompositions for closed relations
  Tamas Wiandt, Rice University
  
  Abstract
  The talk presents a theory of dynamics of closed relations on compact Hausdorff spaces. It contains an investigation of set valued maps and establishes generalizations for some topological aspects of dynamical systems theory, including recurrence, attractor-repeller structure and the Conley Decomposition theorem. We also talk about possible notions of intensity of attraction and mention some applications of the theory.
  
  Tamas Wiandt is a candidate for a tenure-track position in the Mathematics Department at Cal Poly.

 

• Automorphisms of CR-Manifolds
  Gabriela Putinar
  
  Abstract
  This talk will contain a brief introduction to CR-manifolds, including some basic results and research topics. Our focus will then move to CR-automorphisms, and more specifically to the question of finite determination of CR-automorphisms by their jets at a point.

 

• Riemann sums approximating the Lebesgue and Bochner integrals
  Peter A. Loeb, University of Illinois at Urbana
  
  Abstract
  Flyer with Abstract

 

• Introduction to prehomogeneous vector spaces and the associated zeta functions
  Dr. T. Kogiso, Department of Mathematics, Josai University
  
  Abstract
  The theory of prehomogeneous vector spaces was originated by M. Sato around 1960. The motivation that he defined the space is the following. Let us quote a part of his words.
  
  "Now, as for the linear partial differential equations, what is particular for the classical Laplace equations and the wave equations is that their fundamental solutions can be explicitly described. Therefore, as a testing ground for a general theory of linear differential equations, I tried to find differential operators of higher order other than those of quadratic type. If possible, I wanted to find the general scheme of the differential operators with constant coefficients which are given by some homogeneous polynomials and whose fundamental solutions can be expressed as rational powers or log of some homogeneous polynomials."
  
  As originally expected, this theory worked as the "testing ground" for the general theory of linear differential equations, which is now called algebraic analysis. For example, M. Sato, M. Kashiwara, T. Kimura and T. Oshima gave the first application of algebraic analysis to prehomogeneous vector spaces. This space has applications to several fields.
  
  In this talk, I will begin with the definition of a prehomogeneous vector space, and then examples, its elementary properties and the applications to the number theory will be given. That is, the definition of the zeta function associated with the space and examples will be discussed. I will talk about the recent development of this field if time permits.

 

• Discretization of parabolic PDEs for Stefan problems and image segmentation
  Frederic Gibou, Department of Mathematics, Stanford
  
  Abstract
  Many phenomena in physical and life sciences can be modeled by partial differential equations that are parabolic in nature. Applications range from the conception of semi-conductors in Materials science (Stefan problem) to the treatment planning for cancer patients in radiation oncology (image segmentation). The modeling and numerical simulation of these equations share similar drawbacks, such as the computational burden imposed by a stringent time step restriction.
  
  In this talk we will discuss some new numerical algorithms that address some of these issues. First a fourth order accurate finite difference numerical discretization for the Laplace and heat equations with Dirichlet boundary conditions on irregular domains will be described. Then, we turn our focus to the Stefan problem and construct a third order accurate implicit discretization. Multidimensional computational results are presented to demonstrate the order of accuracy of these numerical methods. An adaptive grid refinement for the Poisson equation in the context of the incompressible Euler equations of fluid dynamics will be briefly presented, noting that the aim of this work is to construct an adaptive method for parabolic equations. Finally, a fast hybrid level Set/k-Means algorithm for image segmentation and its application to the segmentation of organs in the context of radiation oncology will be presented.
  
  Dr. Gibou is a candidate for a tenure-track position in the Mathematics Department.

 

• The inverse scattering problem in anisotropic media
  Borislava Gutarts, Department of Mathematics,University of California, Los Angeles
  
  Abstract
  Inverse problems have found many applications in the last 20 years: from medical diagnostics to oil exploration. In this talk two kinds of inverse problems will be defined: the Inverse Scattering Problem and the Inverse Boundary Value Problem. The interesting fact is that the 2-dimensional case is the hardest. I will state my uniqueness result for the Inverse Scattering Problem for the anisotropic wave equation in 2-D, at fixed energy. I will also outline some of the solved and open problems in the field of inverse scattering.
  
  Slava Gutarts is a candidate for a tenure-track position in the Mathematics Department at Cal Poly.

 

• Building generating functions brick by brick
  Anthony Mendes, Department of Mathematics, University of California, San Diego
  
  Abstract
  The relationship between the elementary and homogeneous symmetric functions can be interpreted as a signed sum of combinatorial objects called brick tabloids. We will show how manipulating these brick tabloids in a uniform way gives a new technique to finding generating functions.
  
  Anthony Mendes is a candidate for a tenure-track position in the Mathematics Department at Cal Poly.

 

• Geometric theory of hybrid dynamical systems
  Slobodan Simic, University of California, Berkeley
  
  Abstract
  Hybrid systems are dynamical systems which involve both continuous and discrete evolution. They arise naturally in a number of engineering applications. We will define a suitable geometric framework for their global analysis and focus on the Zeno phenomenon, in which a trajectory makes infinitely many discrete "jumps" in a finite amount of time. We will explain its underlying differential topological cause and give a local classification in dimension two. Finally, we will discuss structural stability of a class of hybrid systems on compact surfaces, and show that most such systems are structurally stable, generalizing a well known similar result for smooth systems.
  
  Dr. Simic is a candidate for a tenure-track position in the Mathematics Department at Cal Poly.

 

• An "Average" Approach to Fluid Dynamics
  James Peirce, Mathematics Department, U.C. Davis
  
  Abstract
  The study of fluid motion has developed into a major field of mathematical research. A fundamental problem is the following: if you start out with a nice smooth vector field describing the flow of a fluid, it will often get complicated as turbulence develops. In three dimensions, nobody knows whether the solution exists for all time, or whether it develops singularities and becomes undefined. In fact, numerical evidence hints at the latter! So one would like to know whether solutions exist for all time and remain smooth - or at least find conditions under which this is the case. One approach is to model the large scale motion of the fluid while averaging the small, computationally unresolvable scales, of the classical equations. The heart of my talk will be the discussion of one such averaged model. I will introduce a number of new results and give a broad description of the techniques used to prove the existence (sometimes global existence!) of solutions to these models.
  
  James Peirce is a candidate for a tenure-track position in the Mathematics Department at Cal Poly.

 

• The Asymptotic Behavior of Wiener-Hopf Determinants (Some Old and New Results)
  Levon Mikaelyan, Department of Informatics and Applied Mathematics, Yerevan State University, Yerevan, Armenia
  
  Abstract
  This talk will describe some results concerning the invertibility of convolution operators and the asymptotics of their Fredholm determinants. The class of convolution operators discussed ahve singular symbols and arise in certain applications.

 

• The Bispectral Problem and its Relations to Integrable Systems and Random
  Milen Yakimov, Mathematics Department, U.C. Santa Barbara
  
  Abstract
  The bispectral problem asks for finding all functions in two variables that are eigenfunctions of differential operators in each of them. It was posed in the early 80's by Duistermaat and Grunbaum in relation to problems in tomography and signal processing. In the last 20 years this problem attracted the attention of many researchers and it was connected to many (seemingly very distant) fields of pure mathematics such as integrable systems, representation theory, and noncommutative geometry. In this talk we will make an overview of the progress on the bispectral problem and its relation to integrable systems. We will also sketch parts of our recent work with Grunbaum, showing that most bispectral functions give rise to integral operators with the remarkable property that they posses a commuting differential operator. This greatly generalizes several examples that play an important role in the theory of random matrices.

 

• Axiom of Choice
  John De Pillis, University of California, Riverside
  
  Abstract
  It has been said (e.g., in a review by Lester Dubins in the MAA Monthly in the 1950's) that Alfred Tarski thought the Axiom of Choice was rubbish. He therefore developed the Banach-Tarski paradox (B-T) to prove that ridiculous results can arise using this "rubbish." We present a weak form of B-T where a full proof is accessible to undergraduate students with a strong calculus background.

 

• Archimedes, and a characteristic property of the circle
  Willem A. J. Luxemburg, California Institute of Technology, Pasadena
  
  Abstract
  We will discuss briefly the results and proof method of Archimedes dealing with his discovery of the volume and surface area of the 3-dimensional ball in relation to those of a cylinder and the right angled circular cone. Looking at these results from the calculus point of view we shall show that the surface area density function of the sphere has an interesting property that can be used to characterize the circle. The talk is accessible to graduate students and even to undergraduate students with a strong calculus background.
  
  Professor Willem Luxemburg is a prominent analyst. He is the author of more than 200 research articles in the field of analysis, functional analysis and non-standard analysis. He is also the author of several books including Operators and Semigroups on Banach Lattices, Riesz Spaces and Introduction to the Theory of Infinitesimals. He is the inventor the ultrapower approach to non-standard analysis.

 

• Physics/Math Colloquium
  Soliton and vortex lattices in nonlinear media: from isolated structures to molecular crystals
  Ricardo Carretero, Nonlinear Dynamical Systems Group, Department of Mathematics and Statistics, San Diego State University
  
  Abstract
  Nonlinear media host a wide variety of localized coherent structures (bright and dark solitons, vortices, aggregates, spirals, etc.) with complex intrinsic properties and interactions. In many situations such as optical communications, condensed matter waves and biochemical aggregates it is crucial to study the interaction dynamics of coherent structures arranged in periodic lattices. In this talk I will present results concerning chains and lattices of coherent structures in Bose-Einstein condensates. Particular attention will be given to:
  • spatially localized vibrations (intrinsic localized modes) in 1D chains of coupled bright solitons, and
  • vortex lattices in 2D condensates and its crystalline configurations.

   

• Symmetric spaces: What are they, what do they do, and why are they interesting?
  Joseph A. Wolf, Mathematics Department, U.C. Berkeley
  
  Abstract
  Many of the familiar spaces in geometry, or in analysis, or in number theory, are "symmetric spaces." This talk will be a quick introduction to symmetric spaces, more by examples than by theory, and an indication of their roles in geometry, analysis and number theory. As time allows there may be an indication of current directions of research involving symmetric spaces. The talk is intended for mathematicians who are not necessarily experts in the field and will be certainly accessible for the graduate students.
  
  Professor Joseph Wolf is known primarily as a geometer (associated in the minds of many with Chern's school) but he has also made considerable contributions in other areas of mathematics, such as representation group theory (in particular the analytic theory of Harish-Chandra modules). He is the author of more than 150 research articles and several books. His book "Space of Constant Curvature" has already been issued in 5 editions in several languages.

 

• A generalization of Pincus' formula
  Torsten Ehrhardt, Technical University of Chemnitz
  
  Abstract
  Let A and B be linear bounded operators on a Hilbert space such that the commutator AB - BA is a trace class operator. Then the operator determinant of e^A e^B e^-A-B is well defined and equals the exponential of 1/2 trace(AB - BA). This generalization of a formula due to Pincus can be applied to find explicit expressions for operator determinants which appear in the theory of Toeplitz plus Hankel determinants.

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Credits

Last Update: mm/dd/yyyy

Department of Mathematics
California Polytechnic State University
San Luis Obispo, CA 93407-0403
805.756.2206
math@calpoly.edu