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 *colloquium-type lectures* by visiting and resident mathematicians and mathematical research projects.

The following courses will be offered in the Fall of 2009:

## Schedule

*Complex analysis from a fluid dynamics perspective*

Instructor: Andrew Belmonte, Associate Professor of Mathematics

Teaching Assistant: Andong He

113 McAllister Building, MWRF 10:10-11:00*Groups and their connections to geometry*

Instructor: Anatole Katok, Raymond N. Shibley Professor of Mathematics

Teaching Assistant: Vaughan Climenhaga

113 McAllister Building, MWRF 11:15-12:05pm*Explorations in convexity*

Instructor: Sergei Tabachnikov, Professor of Mathematics and MASS Director

Teaching Assistant: Pavlo Tsytsura

113 McAllister Building, MWRF 1:25-2:15*MASS Seminar*

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

113 McALLISTER BUILDING, Tuesday 10:10-12:05*MASS Colloquium*

Instructor: Multiple invited speakers

113 McALLISTER BUILDING, Thursday 2:30-3:30

## Course Outline

MATH 497A - Honors MASS Algebra

Groups and their connections to Geometry

Instructor: Anatole Katok, Raymond N. Shibley Professor of Mathematics

TA: Vaughn Climehaga

MWRF - 11:15-12:05pm

**Description**: In the introductory part of this course we will introduce principal classes of groups, both abstractly and as coming from various constructions in algebra, geometry and analysis, and develop basics of group theory. After that we will develop two principal themes:

- Groups related to geometric objects, and
- Geometric objects related to groups

Within the current mathematical landscape the course will provide introduction into several aspects of three major areas:

- Algebraic topology
- Theory of transformation groups
- Geometric group theory

**Readings**: There will be no single text for the course. The principal source will be lecture notes which will be developed and made available to students in real time. A variety of supplementary sources covering the background, various course topics, and directions for projects and future research, will be provided.

Math 497B - Honors MASS Analysis

Complex Analysis from a Fluid Dynamics Perspective

Instructor: Andrew Belmonte, Associate Professor of Mathematics

TA: Andong He

MWRF - 10:10-11:00am

**Description**: There is a deep connection between differentiable functions of complex variables and solutions to Laplace's equation. Due to this beautiful fact, there are many surprising connections between classical results in two-dimensional fluid flows and analytic functions in the complex plane. This course is an introduction to complex analysis from the perspective of these incompressible irrotational (ideal) fluid flows. We will additionally cover the opposite limit of very viscous fluids, which connects to a generalization of analytic functions known as polyanalytic functions. Topics will include: incompressible flows, vorticity, circulation, theorems of Kelvin and Helmholtz, streamfunctions, Euler's equation, Bernouilli's Theorems, analytic functions and the Cauchy-Riemann equations, conformal mapping, Kutta-Zhukovskii Theorem and lift on a swept wing, Milne-Thompson Circle Theorem, Blasius Theorem, d'Alembert's paradox, Cauchy's Theorem, Method of Residues.

**Readings**: T. Needham, Visual Complex Analysis (Oxford, 1999)

Math 497C - Honors MASS Geometry

Explorations in Convexity

Instructor: Sergei Tabachnikov, Director of MASS

TA: Pavlo Tsytsura

MWRF - 1:25-2:15pm

**Description**: The course is an introduction to the theory of convexity that plays an important role in analysis and geometry. Topics to be covered include Helly's theorem and its applications, geometry and combinatorics convex polyhedra, polar duality and its applications, lattice points in convex bodies.

**Readings**: A. Barvinok. A course in convexity (AMS, Providence, 2002)

## Calendar of Events

Arrival Day | August 22-23 |

MASS Welcome Party & Orientation | August 25 |

Classes Begin | August 24 |

Labor Day — No Classes | September 7 |

Midterm Exams | October 5 - 7 |

Thanksgiving Holiday — No Classes | November 23-29 |

Classes End | December 4 |

Final Exams | December 10 - 17 |

MASS Graduation Ceremony | December 18 |

## Enrollment

Participants are selected from applicants who will be juniors or seniors in the following academic year (sophomores may be admitted in some cases). All participants are expected to have demonstrated a sustained interest in mathematics and a high level of mathematical ability and to have mastered basic techniques of mathematical proof. The expected background includes a full calculus sequence, basic linear algebra, a transition course with proofs (such as discrete mathematics) and advanced calculus or basic real analysis. The search for participants is nationwide. International applications are invited as well. Each participant is selected based on academic record, two recommendation letters from faculty, and an essay (international applicants should demonstrate their mastery of English).

Candidates should submit:

- Application Form
- Transcript
- Record of Mathematics Courses
- A short essay describing their interest in mathematics
- Two letters of recommendation
- Financial disclosure form
- Transfer Protocol form

Application materials may be retrieved off the web, or requested by mail, fax, or e-mail.

Applications should be submitted through MathPrograms.org, ID: PSUMASS or sent by mail, fax, or e-mail to

107 McAllister Building

Department of Mathematics

Penn State University

University Park, PA 16802

(814) 863-8730 / Fax:(814) 865-3735

E-mail: mass@math.psu.edu

## Financial Arrangements

Successful applicants currently enrolled in U.S. colleges and universities will be awarded the Penn State MASS Fellowship which reduces the tuition to the in-state level. Best efforts will be made not to increase their out of pocked expenses. See the Financial Information for more details.

## Housing

All participants not enrolled at Penn State will be provided an opportunity to live in one of the residence halls on campus.

## Credits

The program elements total 16 credits, all of which are recognized by Penn State as honors credits and are transferable to participants' home universities. Students will also receive a certificate from the MASS Program at Penn State. Additional recognition may be provided through prizes for outstanding performance and for best projects.

## Administration

The overall supervision of the MASS program is provided by the Scientific Advisory Board which includes senior members of Penn State's mathematics faculty, and several outstanding mathematicians from other institutions.

The program is managed by the Director Sergei Tabachnikov.

Stephanie Zerby is the Administrative Assistant for the MASS program.

Participants are chosen by the *Selection Committee* headed by a member of the Scientific Advisory Board.