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 courses chosen from major areas in Algebra / Number Theory, Analysis, and Geometry / Topology, specially designed and offered exclusively to MASS participants. Each course features three lectures per week, a weekly recitation session conducted by a MASS teaching assistant, weekly homework assignments, a written midterm exam and an oral final exam. The program also includes a weekly interdisciplinary seminar that helps to unify all other activities and MASS Colloquium, a weekly lecture series by visiting and resident mathematicians.

Students with strong mathematical background may choose to work on a semester-long research project with a Penn State faculty member instead of taking one of the three courses. This option is subject to approval by the MASS Director.

## Courses

The following courses will be offered in the Fall of 2016:

*Hypercomplex numbers*

Instructor: Svetlana Katok, Professor of Mathematics

Teaching Assistant: Daren Wei*Approximation of functions and applications*

Instructor: Ludmil Zikatanov, Professor of Mathematics

Teaching Assistant: Nikita Lukyanov*An introduction to dynamics from a topological geometric viewpoint*

Instructor: Federico Rodriguez Hertz, Professor of Mathematics

Teaching Assistant: Scott Conrad*MASS Seminar*

Instructor: Sergei Tabachnikov, Professor of Mathematics, MASS Director*MASS Colloquium*

Instructor: Multiple invited speakers

## Course Outline

**Math 497A - Honors MASS Algebra**

**Hypercomplex numbers**

*Instructor:* Svetlana Katok, Professor of Mathematics*Teaching Assistant:*

113 McAllister Building, MWRF 11:15 am - 12:05 pm

*Description:* In this course we will study arithmetic, geometry and symmetry of hypercomplex numbers. The hypercomplex numbers are constructed by adding “imaginary units” to the real numbers. The complex numbers are a classical example of such a number system in dimension 2. It is easy to define addition, subtraction, and multiplication in each system of hypercomplex numbers, but the only dimensions in which there are hypercomplex numbers which allow division are 4 and 8. These hypercomplex numbers are called the Quaternions and the Cayley numbers (Octonions), respectively. With increases in dimension, some of the natural properties of the number systems cannot be maintained. In both the Quaternions and the Cayley numbers, multiplication is non-commutative, and in Cayley numbers multiplication is not even associative.

*Reading:* I.L. Cantor and A.S. Solodovnikov. “Hypercomplex numbers. An elementary introduction to algebras”. Springer-Verlag, New York,, 1989.

**Math 497B - Honors MASS Analysis**

**Approximation of functions and applications**

*Instructor:* Ludmil Zikatanov, Professor of Mathematics*Teaching Assistant:*

113 McAllister Building, 10:10 am - 11:00 am

*Description:* This is an introductory course on approximation of functions, a topic that has generated and still generates rich mathematical theory and is one of the driving forces in the modern scientific computing. Roughly, the problem of approximating an object (function or shape) consists of three main steps: (1) identify the possible approximants; (2) construct an approximation; (3) estimate the error. Two scenarios are often encountered in applications: first, when the function is explicitly known, and, second, when the function is a solution to another problem and cannot be found in explicit form. Classical examples for these scenarios include computing a best fit by a polynomial curve to given data points, or computing approximate solution to a differential equation modeling physical phenomena. The course covers classical results on approximation and interpolation with algebraic and trigonometric polynomials, numerical integration, and applications. The pool of applications touches upon: signal processing (the Fast Fourier Transform); iterative solution of linear systems (the method of Conjugate Gradients); numerical solution of models described by ordinary differential equations.

*Reading:* Most of the material can be found in the book T. Rivlin. "An introduction to the approximation of functions". Dover Publications, Inc., New York, 1981 and in lecture notes provided by the instructor during the course. Materials on applications are found in E. Isaacson and H. Keller. "Analysis of numerical methods". Dover Publications Inc., New York, 1994.

**Math 497C - Honors MASS Geometry**

**An introduction to dynamics from a topological geometric viewpoint**

*Instructor:* Federico Rodriguez Hertz, Professor of Mathematics*Teaching Assistant:*

113 McAllister Building, 1:25 pm - 2:15 pm

*Description:* We shall develop the basic theory of dynamics with an application to the study of homeomorphisms of surfaces. In the meantime, the basic properties of the hyperbolic plane and homotopy properties of surfaces will be discussed and applied.

*Reading:* A. Casson and S. Bleiler. "Automorphisms of Surfaces after Nielsen and Thurston". Cambridge University Press, Cambridge, 1988.

## Calendar of Events

Arrival Days | August 20, 21 |

MASS Orientation | August 22, 9:15 a.m. |

Classes Begin | August 22 |

Labor Day — No Classes | September 5 |

Midterm Exams | October 3,4,5 |

Thanksgiving Holiday — No Classes | November 20-26 |

Classes End | December 2 |

Study Days | December 5-10 |

Final Exams | December 9, 12, 14 |

MASS Graduation Ceremony | December 15, 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 **April 10, 2016.**

## 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.