Orthogonal Polynomials with respect to a measure are found by using the Grahm-Schmidt algorithm with respect to the induced inner product. These polynomials satisfy a recurrence relation. The recurrence relation has the form of a tridiagonal matrix. Which begs the question what would happen if we took this matrix as the initial condition of the Toda Equations.
It is a fairly rare phenomenon for two power series to commute under the operation of composition. This talk will be an introduction to the theory of commuting power series in the case where the coefficient ring is taken to be the integer ring of a finite algebraic extension of the p-adic numbers. In particular, a recent open conjecture from Jonathan Lubin attempts to classify when this can occur using the theory of formal groups and their endomorphism rings, and we will see why a proof is not too far off. Introductions to both p-adic numbers and formal groups will be given.
The naturality of quasiconformal mappings was first recognized in the 1930s, but an important result in this area had already been proven by Gauss the previous century. As their name suggests, quasiconformal mappings are almost conformal in a certain sense - namely the (infinitesimal) distortion which they produce is uniformly bounded, so that these mappings send infinitesimal circles to infinitesimal ellipses; whereas conformal (i.e. angle-preserving) mappings send such circles to other infinitesimal circles.
In this talk we present some equivalent definitions of quasiconformality as well as some (geometric and complex dynamical) applications.
Come hear a basic introduction to and see some examples of... what else? Formal groups of elliptic curves of course!
Gauss's Theorema Egregium is related to pattern formation on plants.
Precise knowledge of the eigenvalues of the Laplacian on domains in the plane is surprisingly scarce. We can only write them down explicitly in certain cases when symmetry allows us to throw open the lid to the cookie jar and make ourselves sick. This talk will focus on what could be termed "the dieter's solution" to eigenvalue problems - asymptotics. We'll discuss ways of sneaking out bits of cookie for a particular class of thin domains and the reasons why I've pulled a stool up next to this mathematical countertop in the first place.
Keywords: eigenvalue asymptotics, regular and singular perturbations, Dirichlet-to-Neumann operator
Proving mathematical theorems is an essential part of being a mathematician, however, most mathematics students are not exposed to mathematical proof or abstract mathematics until their sophomore or junior year in college. This transition from computational mathematics to theoretical mathematics tends to be a difficult one. In this talk, I will discuss the results from a study I conducted last Spring in which I explored the beliefs and misconceptions of students in a beginning proof-writing course.
In differential equations, geometry, and analysis one is eventually interested in evaluating functions of operators. I will introduce the class of operator algebras which satisfy the "C* identity" through a discussion of the spectral theorem from linear algebra. Via several examples we will explore how Connes's generalization of Grothendieck's theory of the structure of space brings C* algebras to the forefront as principal objects of study in Geometry.
Integrable Hamiltonian systems are quite an exceptional class of Hamiltonian systems. The integrals (constants of motion) of such systems can be analyzed using tools from symplectic geometry. In this talk we will discuss the applications of symplectic geometry to the study of integrable Hamiltonian systems. In particular, we will find the integrals and solutions of the Toda lattice.
This talk will cover some of the methods which artists through history have used to render 2-d images of 3-d objects and scenes.
This talk involves initial value problems for boundary layer flows.
To start, a parallel flow is considered and the governing equations for 2-D incompressible flow with small velocity perturbation are linearized and made dimensionless. These equations and the boundary conditions can be rewritten in a matrix operator form involving the disturbance vector. The problem is solved using Fourier and Laplace Transforms and it is shown that the Inverse Laplace Transform of the solution of the ODE is determined by the poles (discrete spectrum) and the branch cut(continuous spectrum corresponding to vorticity perturbations).
A solution of the initial value problem can also be presented as an expansion in a biorthogonal eigenfunctions system, where one component of the system is a solution of the direct problem, and the other component of the system is a solution of the adjoint problem. For this eigenfunction system, an orthogonality condition is found and it is then shown that the Inverse Laplace Transform can be expressed in the form of an expansion in the biorthogonal eigenfunction system.
This problem formulation and solution is then extended to hypersonic boundary layer flows. A parallel flow is still considered but the problem now involves compressible flow and the energy equation must be included in the analysis. It can be shown that the Inverse Laplace Transform can be expressed as a sum of residues associated with the poles as well as a sum of integrals along the sides of four branch cuts, two of which correspond to acoustic waves and one each for vorticity and entropy disturbances. In this compressible flow case, there are additional complications that arise from the coalescence of the poles with the vortcity/entropy branch cut. As in the incompressible case a solution of the problem can be presented as an expansion in a biorthogonal eigenfunctions system.
We will discuss the first purely number theoretic example of the incompleteness of Peano arithmetic, namely Goodstein's Theorem. All of the required logical notions will be introduced.
Penrose Tilings are a certain class of aperiodic tilings of the plane by isoceles triangles. This space of all such tilings admits a parametrization as space of sequences satisfying a grammar rule, modulo a countable equivalence relation. This quotient is a horrific space, it is not even Hausdorff! Connes and others have use methods of non-commutative geometry to analyze this space and explain certain remarkable features of Penrose Tilings. In this talk I will introduce Penrose Tilings and their basic properties. I will then derive the parameterization and introduce the non-commutative point of view.
In the 1950's, a young Greek architect and composer living in France named Iannis Xenakis began exploring a fairly original method of musical composition which he would term stochastic music. Its heavy reliance upon mathematics, and probability theory in particular, led to criticism and a lack of appreciation by both the music community and the general public. In this talk, we will discuss the composition scene as it was at the time of Xenakis (including all relevant musical terms) and examine some of the factors which led Xenakis to the use of mathematics as a foundation for composition. In particular, we will take a closer look at his piece, Pithoprakta, after which we will listen to it in its entirety.