## Projects

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 |

Here is a beamer template.

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.