Department of Mathematics of the Pennsylvania State University runs a yearly semester-long intensive program for undergraduate students seriously interested in pursuing a career in mathematical sciences. The *Mathematics Advanced Study Semesters* (MASS) program started in the Fall of 1996 and is held during the Fall semester of each year.

The principal part of the program consists of *three core courses* chosen from major areas in *Algebra/Number Theory*, *Analysis*, and *Geometry/Topology* respectively, specially designed and offered exclusively to MASS participants, and a weekly *MASS seminar*.

Additional features include a periodic listing of *challenging problems* from various fields of mathematics, *colloquium-type lectures* by visiting and resident mathematicians, and mathematical projects involving creative use of *computers*.

The following courses will be offered in the Fall of 2001:

## Schedule

*Combinatorics*

Instructor: George Andrews, Evan Pugh Professor of Mathematics

Teaching Assistant: Alexander Chindiapine

MWRF 1:25PM-2:15PM

103 MCALLISTER BUILDING*Geometry and Relativity: An Introduction*

Instructor: Nigel Higson, Distinguished Professor of Mathematics

Teaching Assistant: David Shoenthal

MWRF 10:10AM-11:00AM

103 MCALLISTER BUILDING*Mathematical Analysis of Fluid Flow*

Instructor: Andrew Belmonte, Assistant Professor of Mathematics

Teaching Assistant: Dima Matsnev

MWRF 11:15AM-12:05PM

103 MCALLISTER BUILDING*MASS Seminar*

Instructor: Sergei Tabachnikov, Associate Professor of Mathematics, Director of MASS Program

SEMINAR:

T 9:05AM- 11:00AM

LOCATION: 103 MCALLISTER BUILDING*MASS Colloquium*

Instructor: Multiple invited speakers

R 3:30PM- 4:30PM

LOCATION: 103 MCALLISTER BUILDING

## Course Outline

MATH 497A: Honors MASS Algebra

Combinatorics

INSTRUCTOR: George Andrews

TIME: MWRF 1:25 pm - 2:15 pm

We shall study combinatorial problems from the vantage point of generating functions. The text will be Herb Wilf's Generating Functionology. Generating functions allow the application of powerful analytic methods to combinatorial problems. We shall combine regular course work with consideration of new computer algebra packages that assist in exploring relevant analytic questions and in providing unexpected insights.

MATH 497B Honors MASS Geometry

Geometry and Relativity: An Introduction

INSTRUCTOR: Nigel Higson

TIME: MWRF 10:10 am - 11:00 am

This course will be two-thirds an introduction to differential geometry and one-third an introduction to relativistic mechanics. The two parts will be brought together at the end of the semester in a very brief account of general relativity. The differential geometric component of the course will present the extrinsic and intrinsic theory of curves and surfaces in Euclidean space. The relativistic component will present the mathematical foundations of special relativity, along with selected topics in particle mechanics and electromagnetism. The introduction to general relativity will reach as far as the Schwarzschild solution to Einstein's equations. The emphasis will be on mathematics, not physics.

MATH 497C: Honors MASS Analysis

Mathematical Analysis of Fluid Flow

INSTRUCTOR: Andrew Belmonte

TIME: MWRF 11:15 am - 12:05 pm

What can the flow of fluids teach us about mathematics? A hint is provided by considering some of the contributors to the development of fluid dynamics: Euler, d'Alembert, Poisson, Cauchy, etc. In this course we will discuss several different topics in mathematics, including partial differential equations and complex analysis, motivated by classical results in the dynamics of fluids. For example, the two dimensional flow around the cross-section of an airplane wing can be transformed into flow around a circle using mappings of functions in the complex plane—in fact many wings were designed using this transformation! Many other beautiful aspects of the mathematics of fluid dynamics come from ignoring viscosity, and we will discuss why this is a reasonable thing to do, physically and mathematically. This will lead us to the concept of a boundary layer, an idea which extends to other areas of mathematics (including its own area, asymptotics). Finally, we will look at some of the instabilities of boundary layers, and the connection to the flapping flight of birds and insects.

MATH 497D HONORS MASS SEMINAR

Interdisciplinary Seminar

INSTRUCTOR: Sergei Tabachnikov

TIME: T 9:05 am - 11:00 am

This seminar is designed to focus on selected interdisciplinary topics in algebra, geometry and analysis. These areas will be related to the other MASS courses. Seminar sessions may include presentations from student homework solutions.

## Calendar of Events

Arrival Day | 08/18 |

MASS Welcome Party & Orientation | 08/20 |

Classes Begin | 08/21 |

Labor Day Holiday | 09/03 |

Fall Break | 10/08 - 10/09 |

Midterm Exams | 10/01 - 10/05 |

Thanksgiving Holiday Classes end at 2:15 p.m. on Wednesday 11/21 |
11/23 - 11/24 |

Classes End | 12/07 |

Final Exams | 12/08 - 12/12 |

MASS Graduation Ceremony | 12/13 |