| Team Directors: | Stuart Antman | (ssa@math.umd.edu, x5-5105, MTH 2309) |
| Manoussos Grillakis | (mng@math.umd.edu, x5-5173, MTH 2207) | |
| David Levermore | (lvrmr@math.umd.edu, x5-5127, MTH 3101) | |
| Jian-Guo Liu | (jliu@math.umd.edu, x5-5148, MTH 3313, on sabbatical this term) | |
| Konstantina Trivisa | (trivisa@math.umd.edu, x5-5067, MTH 4103) | |
| Peter Wolfe | (pnw@math.umd.edu, x5-5149, MTH 3314) |
| Student Participants: | David Bourne, Bin Cheng, Brian Davis, Gregory Crosswhite (UG), |
| J.T. Halbert, Cory Hauck, Ning Jiang, Weiran Sun, Linbao Zhang. |
Regular Meeting Time: 12:00pm-1:20pm Fridays in MTH 1311
The area of nonlinear partial differential equations and its applications is enormous, cultivated by many people on this campus. In addition to those mentioned above, applications might include dilute gases, semiconductor devices, plasmas, radiative transport, charged particles, neutronics, or mathematical finance. It is planned that the team will involve several other faculty members who will be invited by the directors to participate in response to the interests of the students.
Graduate Prerequisites: Analysis (MATH 630), or PDEs (MATH 673-674), or ODEs (MATH 670-671), or Numerics (AMSC 666 or 660), and some background in a field of application. Permission of an instructor.
Undergraduate Prerequisites: Differential Equations (MATH 246) or ODE (MATH 414). Advanced Calculus (MATH 410, plus either MATH 411 or MATH 412 taken before or concurrently with the RIT). Any of the following are useful but not required: complex variables (MATH 463), PDE (MATH 462), scientific computation (MATH 241 or AMSC 460), background in physics or engineering. Permission of an instructor.
Graduate Program: Team members will make presentations either on readings of fundamental papers, or on book chapters, or on their own investigations, or on interesting problems. When appropriate, these may be developed into more formal presentations for seminars, conferences, or publication. Team members will be asked to help mentor more junior team members.
Undergraduate Program: Same as above, as appropriate to the background of the student.
Work Schedule: Meetings will be held weekly for an hour (or possibly longer). These sometimes may be held jointly with related RITs.
Abstract: Mathematical models are being created to study systems of great complexity across a wide range of applications, many of which impact our daily lives. They are used to model the economy and set interest rates, to predict the weather and plan emergency responses, to manage traffic on our telecommunications networks, to design both semiconductor chips and their fabrication, to develop medical technology, to enforce environmental regulations, and many other things. A common feature of all these systems is that they are "high dimensional" while the information we care most about is relatively "low dimensional". This leads to the basic question, how much of the "high dimensional" details need to be modeled correctly to obtain correct "low dimensional" predictions. This minicourse will open a small window on how this question is approached.
The first lecture will introduce how "high dimensional" dynamical systems can be viewed as a linear evolution of densities on phase space. We will then discuss (in the setting of matrix algebra) how a high dimensional linear dynamical system can be reduced to a lower dimensional one. The second lecture will introduce a simple linear kinetic model for particles interacting with a background, and show how the ideas of the first lecture apply to it. The final lecture will be given by Cory Hauck (a graduate student) on developing models for semiconductor design.
Abstract of Lecture 1: Traffic Flow. The aim of this lecture is to give a mathematical formulation of traffic problems in terms of a nonlinear partial differential equation. We will begin our investigation of traffic problems by discussing the fundamental traffic variables: velocity, density, and flow. We will attempt to predict these quantities if they are intially known. Nearly uniform traffic flow is first discussed enabling the introduction of the concept of a traffic density wave. The method of characteristics is developed for nonuniform traffic problems. In particular we will discuss what happens to a line of stopped traffic after a light turns green. Difficulties in this theory occur when light traffic catches up to heavy traffic necessitating the analysis of traffic shocks, discontinuities in density. This lecture will give a special meaning to your daily commute!
Abstract of Lecture 2: The Heat Equation - Option Pricing. The first part of the lecture will be dedicated to the formulation of the equations of heat flow describing transfer of thermal energy. The second part will be an introduction to Option Pricing, the description of a model for the evolution of stock prices and the derivation of the Black-Scholes Option Pricing Formula. This well known formula for option pricing can be viewed as a solution to the heat equation.
Abstract of Lecture 3: Stochastic Models in Finance. We start the lecture with the formulation of a stochastic differential equation describing the evolution of stock prices. Using some ideas from portfolio theory we derive a deterministic differential equation, the well known Black-Scholes equation, whose solution yields the non-arbitrage price of options.
Abstract: Densities of particles making unbias independent random walks on a lattice are governed by the diffusion equation. This is one of the simplest examples of how ordered large-scale behavior emerges from small-scale random dynamics. It is a relationship that can be put on a firm mathematical foundation. It is easy to come up with examples of simple small-scale dynamics that exhibit coherent large-scale behavior for which the mathematical foundation of the relationship is either much harder to establish or unknown. For example, large-scale behavior governd Burgers equations arise by introducing an interaction-dependent bias into the random walks. Traveling waves and reaction-diffusion structures can be seen in other systems. One can see similar behaviors when the small-scale dynamics is chaotic rather than random, but establishing the coresponding mathematical results is much harder or impossible with our current understanding. Our course illustrates these ideas with a string of examples starting from the simple random walk.