Projects


Project Times

March 4: 12:30-12:55pm
  Jon Epstein
March 4: 1:05-1:30pm
 Emma Chiappetta  
March 6: 12:30-12:55pm
  Donato Cianci
March 6: 1:05-1:30pm
  Lin Zhao
March 8: 12:30-12:55pm
  Yuxiang Liu
March 8: 1:05-1:30pm
  Nicolas Petit

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Project Descriptions

Jon Epstein
Date: March 4
Title: An Introduction to Characteristics
Abstract: Nonlinear first-order PDE are in general difficult to handle. Nevertheless they arise many in different physical theories. The method of characteristics is a way of generating local solutions to an arbitrary first-order PDE by transforming it into a system of ODE. First, I will derive the associated characteristic ODE, and discuss how its initial conditions are related to boundary conditions for the original PDE. Secondly, I will attempt to provide some intuition through examples, including the Hamilton-Jacobi PDE.

Emma Chiappetta
Date: March 4
Title:
Abstract: Over the last decade, the Ricci flow equation has recently made it to the forefront of geometry. The new found popularity of this particular evolution equation is due in large to part to its central role in Perelman's proof of the Poincare conjecture. However, the Ricci flow has been a powerful tool for geometers, topologists and analysts since Richard Hamilton began to study it in the early '80s. In this talk I hope to demonstrate how the Ricci flow equation may be interpreted as a tensor version of the heat equation. What this means is that, in coordinates we may express each component function of the Riemannian metric(which we view as a (2,0) tensor.) In this coordinate system, each component function of the metric viewed satisfies a heat like-equation. Before proving the main result we will quickly review some concepts from both PDE theory and from Riemannian geometry.

Donato Cianci
Date: March 6
Title: Introduction to pseudodifferential operators
Abstract: In quantum mechanics we trade a "classical observable" (i.e. a smooth function on phase space) for a "quantum observable" (a self-adjoint operator on L^2). One purpose of pseudo-differential calculus is to answer the question: Given a classical observable, to what extent can I prescribe a corresponding quantum observable? This talk will introduce pseudo-differential operators. First we present the space of symbols and describe some of its properties. Then we will define the pseudo-differential operator of a certain symbol. I'll show in what sense a pseudo-differential operator is a generalization of a differential operator. If time permits, I'll define WKB solutions of a differential operator and determine the WKB solution of the Schrodinger equation.

Lin Zhao
Date: March 6
Title: Nystrom Method for Evaluation of Fredholm Determinants
Abstract: Fredholm determinants arise in physics and many mathematical settings. This paper investigates a simple and general numerical method for evaluating this quantity, the Nystrom method, which doesn't requires an eigenfunction expression of the interested operator, unlike many other existing numerical treatments. It also shows that essentially the approximation errors behaves like the underlying quadrature error.

Yuxiang Liu
Date: March 8
Title: Introduction to the Method of Fundamental Solutions
Abstract: The method of Fundamental Solutions is introduced in this article. It was proved that the solution to the Laplace equation with Dirichlet boundary condition is unique. We assume that the solution is a linear combination of the fundamental solutions to the Laplace equation. We tested the accuracy of the method of fundamental solutions in Laplace equation and also Helmholtz Equation in 2D. They both gave us a stable and accurate solution. It is found that as source points getting further away from the boundary points, the solution converges faster, in consistent with the result found by some other scholars. Also, Method of Fundamental Solutions is applied to solve the Laplace Equation and Helmholtz Equation in 3D, but failed. So further research need to be done about choosing the appropriate source points so that the application of the method of fundamental solution in 3D is possible.

Nicolas Petit
Date: March 8
Title: Dirac operators in the Real world and on manifolds
Abstract: While he was studying the quantum theory of the electron, Dirac asked himself a very simple question: is there an operator whose square is the laplacian? The answer, though, was all but trivial, and led to the study of Clifford algebras and Spin structures. In this talk I will study and solve the problem posed by Dirac for free space, then I'll give a flavor of the math required to simply speak about the problem in the more general setting of smooth manifolds.