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.

## Courses

The following courses will be offered in the Fall of 2013:

*Number Theory in the Spirit of Ramanujan*

Instructor: George Andrews, Evan Pugh Professor of Mathematics

Teaching Assistant: Ayla Gafni

113 McAllister Building, MWF 10:10 - 11:00 a.m., T 11:15 a.m. - 12:05 p.m.*Elements of functional analysis*

Instructor: Boris Kalinin, Associate Professor of Mathematics

Teaching Assistant: Shilpak Banerjee

113 McAllister Building, MTWF 1:25 - 2:15 p.m.*Winding number in topology and geometry (and the rest of mathematics)*

Instructor: John Roe, Professor of Mathematics

Teaching Assistant: Dong Chen

113 McAllister Building, MWF 11:15 a.m. - 12:05 p.m., T 10:10 - 11:00 a.m.*MASS Seminar*

Instructor: Victoria Sadovskaya, Associate Professor of Mathematics, Interim MASS Director

113 McAllister Building, Thursday 10:10 am - 12:05 p.m.*MASS Colloquium*

Instructor: Multiple invited speakers

114 McAllister Building, Thursday 1:25 - 2:25 p.m.

## Course Outline

**Math 497A - Honors MASS Algebra**

**Number Theory in the Spirit of Ramanujan**

Instructor: George Andrews, Evan Pugh Professor of Mathematics

TA: Ayla Gafni

113 McAllister Building, MWF 10:10 - 11:00 a.m., T 11:15 a.m. - 12:05 p.m.

*Description*: The primary object of the course will be to understand those portions of elementary number theory that are closely related to the work of the Indian genius, Ramanujan. The honors objective will be to look at this mathematics from the broader perspective of the mathematical and societal influences surrounding Ramanujan's short, meteoric career. Perhaps this is explained best in the following description of Berndt's text prepared by the publisher:

"Ramanujan is recognized as one of the great number theorists of the twentieth century. Here now is the first book to provide an introduction to his work in number theory. Most of Ramanujan's work in number theory arose out of q-series and theta functions. This book provides an introduction to these two important subjects and to some of the topics in number theory that are inextricably intertwined with them, including the theory of partitions, sums of squares and triangular numbers, and the Ramanujan tau function. The majority of the results discussed here are originally due to Ramanujan or were rediscovered by him. Ramanujan did not leave us proofs of the thousands of theorems he recorded in his notebooks, and so it cannot be claimed that many of the proofs given in this book are those found by Ramanujan. However, they are all in the spirit of his mathematics. The subjects examined in this book have a rich history dating back to Euler and Jacobi, and they continue to be focal points of contemporary mathematical research. Therefore, at the end of each of the seven chapters, Berndt discusses the results established in the chapter and places them in both historical and contemporary contexts."

**Math 497B - Honors MASS Analysis**

**Elements of Functional Analysis**

Instructor: Boris Kalinin, Associate Professor of Mathematics

TA: Shilpak Banerjee

113 McAllister Building, MTWF 1:25 - 2:15 p.m

*Description*: The course will introduce students to various ideas and techniques of functional analysis. The topics will include spaces of functions, Banach and Hilbert spaces, functionals and operators, and applications.

**Math 497C - Honors MASS Geometry**

**Winding number in topology and geometry (and the rest of mathematics)**

Instructor: John Roe, Professor of Mathematics

TA: Dong Chen

113 McAllister Building, MWF 11:15 a.m. - 12:05 p.m., T 10:10 - 11:00 a.m.

*Description*: The winding number is one of the most basic invariants in topology: it is an integer that measures the number of times a moving point P goes around a fixed point Q, provided that P travels on a path that never goes through Q and that the final position of P is the same as its starting position. For example, the winding number of the tip of the minute hand of a clock (P), about the center of the clock (Q), between 11 a.m. and 4 p.m. the same day, is -5. This simple idea has far-reaching applications in almost every area of mathematics. For instance, in the course we’ll learn about how the winding number (and its generalizations)

Helps us show that every polynomial equation has a root (the fundamental theorem of algebra)

Guarantees a fair division of three objects in space by a single cut (the ham sandwich theorem)

Shows why every simple closed curve has an inside and an outside (the Jordan curve theorem)

Allows you to “renormalize” the difference of two infinities and get a finite answer (Toeplitz index theory)

Help explain why electrons fill successive “shells” around atomic nuclei (thereby giving rise to chemistry)

## Calendar of Events

Arrival Day | August 25 |

MASS Orientation | August 26, 9:30 a.m. |

Classes Begin | August 26 |

Labor Day — No Classes | September 2 |

Midterm Exams | October 7, 8, 9 |

Thanksgiving Holiday — No Classes | November 24-30 |

Classes End | December 6 |

Study Days | December 7-12 |

Final Exams | December 13, 16, 18 |

MASS Graduation Ceremony | December 19, 10 a.m. |

## 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

The deadline for MASS applications is **Friday, April 5, 2013.**

## 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.