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 2012:

### Schedule

*Polynomials*

Instructor: Sergei Tabachnikov, Professor of Mathematics and MASS Director

Teaching Assistant: Nick Early

113 McAllister Building, MWRF 11:15 am-12:05 pm*Random walk and Brownian motion*

Instructor: Alexei Novikov, Associate Professor of Mathematics

Teaching Assistant: Brian Nowakowski

113 McAllister Building, MWRF 10:10 am-11:00 am*An introduction to geometric topology in dynamics*

Instructor: Federico Rodriguez-Hertz, Professor of Mathematics

Teaching Assistant: Kurt Vinhage

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

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

113 McAllister Building, Tuesday 10:10 am-12:05 pm*MASS Colloquium*

Instructor: Multiple invited speakers

113 McAllister Building, Thursday 2:30 pm-3:20 pm

## Course Outline

Math 497A - Honors MASS Algebra

Polynomials

Instructor: Sergei Tabachnikov, Professor of Mathematics and MASS Director

TA: Nick Early

MWRF 11:15 a.m. - 12:05 p.m.

**Description**: The course will cover various algebraic, analytic and geometrical properties of polynomials and their applications: discriminants and resultants, solving polynomial equations, estimating the number of roots, Fundamental Theorem of Algebra, interpolation, symmetric polynomials, trigonometric polynomials.

**Readings**: No textbook will be required. Possible reading: V. Prasolov, Polynomials, Springer.

Math 497B - Honors MASS Analysis

Random walk and Brownian motion

Instructor: Alexei Novikov, Associate Professor of Mathematics

TA: Brian Nowakowski

MWRF 10:10 - 11:00 a.m.

**Description**: The objective is to study properties of two basic examples in probability theory: random walk and Brownian motion.

**Readings**: Possibly – G. Lawler Random Walk and the Heat Equation, P. Doyle & J. Snell Random walks and electric networks

Math 497C - Honors MASS Geometry

An introduction to geometric topology in dynamics

Instructor: Federico Rodriguez-Hertz, Professor of Mathematics

TA: Kurt Vinhage

MWRF 1:25 - 2:15 p.m.

**Description**: The objective of the course is to build geometric tools that enables to give an accurate description of a dynamical system. We will start with circle diffeomorphisms and covering maps and their classification through topological invariants like rotation number and degree. Then we shall move to surface diffeomorphisms (particularly when the surface is the torus) and give some consequence of their homotopy type. In the mean time we shall introduce the notion of fundamental group, covering space, homotopy invariants, conjugacy invariants, suspension of a diffeomorphism, foliations, flows on surfaces.

**Readings**: The main reference we will use is the book “Introduction to the Modern Theory of Dynamical Systems” B. Hasselblatt and A. Katok, Cambridge University Press . We shall use some input from the books: A. Casson and S. Bleiler, “Automorphisms of Surfaces After Nielsen and Thurston”, Cambridge Univ. Press; A. Hatcher, “Algebraic topology” http://www.math.cornell.edu/~hatcher/#ATI; and/or “J. Milnor Topology from the “Differentiable Viewpoint”, Princeton University Press; and/or V. Guillemin and A. Pollack. “Differential Toplogy” Prentice Hall

## Calendar of Events

Arrival Day | August 26 |

MASS Orientation | August 27, 9:30 am |

Classes Begin | August 27 |

Labor Day — No Classes | September 3 |

Midterm Exams | October 8,9,10 |

Thanksgiving Holiday — No Classes | November 18-24 |

Classes End | December 7 |

Final Exams | December 12, 14, 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.