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 2010:

## Schedule

*Differential equations from an algebraic perspective*

Instructor: Nigel Higson, Evan Pugh Professor of Mathematics

Teaching Assistant: Tyrone Crisp

113 McAllister Building, MWRF 10:10-11:00*Dynamics, mechanics and geometry*

Instructor: Mark Levi, Professor of Mathematics

Teaching Assistant: Pavlo Tsytsura

113 McAllister Building, MWRF 11:15-12:05pm*Function field arithmetic*

Instructor: Mihran Papikian, Assistant Professor of Mathematics

Teaching Assistant: Evgeny Mayanskiy

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

Function Field Arithmetic

Instructor: Mihran Papikian, Assistant Professor of Mathematics

TA: Evgeny Mayanskiy

MWRF - 1:25-2:15pm

**Description**: This course is a mixture of an abstract algebra course and a number theory course. Its aim is to explore the properties of F[T], the ring of polynomials over a finite field F, and to compare them to the properties of the ring of integers Z. From the point of view of Algebra, the rings Z and F[T] are quite similar. For example, both rings are principal ideal domains, both have the property that the residue class ring of any non-zero ideal is finite, both rings have infinitely many elements, and both rings have finitely many units. What we will see in the course is that the similarities of Z and F[T] extend beyond algebraic properties to the more subtle arithmetic properties, which are studied in Number Theory. For example, it is possible to define a zeta-function for F[T] having properties similar to the properties of the Riemann zeta-function. There are at least two different "zeta-functions" for F[T], and both will be discussed in the course. Especially intriguing is the zeta-function introduced by L. Carlitz in 1930s, which naturally leads to the analogue of Euler's formula for zeta(2), and to the theory of Drinfeld modules - an area on the frontier of current number theory. The only prerequisite for this course is a background in linear algebra. A prior exposure to abstract algebra is certainly helpful, although all the necessary background material will be covered at the beginning of the course.

**Readings**:

- Lidl and Niederreiter,
Finite fields

- M. Rosen,
Number theory in function fields

- D. Goss,
Basic structures of function field arithmetic

Math 497B - Honors MASS Analysis

Differential equations from an algebraic perspective

Instructor: Nigel Higson, Evan Pugh Professor of Mathematics

TA: Tyrone Crisp

MWRF - 10:10-11:00am

**Description**: The aim of this course is apply concepts from algebra and algebraic geometry to the study of differential equations. The main object of study will be the so-called Weyl algebra of differential operators. This is a noncommutative algebra (since for example differentiating then multiplying by `x` is not the same as multiplying by `x` then differentiating.) However it is nearly commutative, and this makes it possible to analyze it using a variety of geometric techniques. The Weyl algebra originally arose in quantum theory, and we'll take a little time to discuss how.

**Readings**:

- W. Fulton, algebraic curves (for a bit of algebraic geometry)
- S. Coutinho, A primer of algebraic D-modules (for the Weyl algebra)
- Other readings covering background material and supplementary topics will be provided during the course.

Math 497C - Honors MASS Geometry

Dynamics, mechanics, and geometry

Instructor: Mark Levi, Professor of Mathematics

TA: Pavlo Tsytsura

MWRF - 11:15-12:05pm

**Description**: The course will introduce students to the following topics: Geometry theory of ordinary differential equations; Linear and nonlinear systems in the plane; deterministic chaos and fractals as they arise in simplest dynamical systems; Variational principle of mechanics and optical-mechanical analogy. Many of the ideas developed in the course will be illustrated by mechanical demonstrations and/or applied to explain some interesting mechanical phenomena and paradoxes.

**Readings**:

- V. Arnold, Ordinary Differential equations
- A. Andronov, A.A. Vitt and S.E. Khaikin, Theory of oscillators, Dover, 1966.
- V. Arnold, Mathematical methods of Classical mechanics, Springer Verlag.
- S. Strogatz, Nonlinear Dynamics and Chaos. Addison Wesley.

## Calendar of Events

Arrival Day | August 22 |

MASS Welcome Party & Orientation | August 24 |

Classes Begin | August 23 |

Labor Day — No Classes | September 6 |

Midterm Exams | October 11-13 |

Thanksgiving Holiday — No Classes | November 22-28 |

Classes End | December 3 |

Final Exams | December 10-13-15 |

MASS Graduation Ceremony | December 16 |

## 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.