|Team Directors:||Stuart Antman||(email@example.com, x5-5105, MTH 2309)|
|Manoussos Grillakis||(firstname.lastname@example.org, x5-5173, MTH 2207)|
|David Levermore||(email@example.com, x5-5127, MTH 3101)|
|Konstantina Trivisa||(firstname.lastname@example.org, x5-5067, MTH 4103)|
Regular Meeting Time: 1:00pm-2:50pm Fridays in MTH 0411
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.
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, or neutronics. 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 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. More senior 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.