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 2015:

*Lie Groups in Two, Three and Four Dimensions*

Instructor: Nigel Higson, Evan Pugh Professor of Mathematics

Teaching Assistant: Qijun Tan

113 McAllister Building, MWRF 10:10 - 11:00 a.m.*Classical Mechanics and Calculus of Variations*

Instructor: Mark Levi, Professor of Mathematics

Teaching Assistant: Oleg Rudenko

113 McAllister Building, MWRF 11:15 a.m. - 12:05 p.m.*Introduction to Applied Algebraic Geometry*

Instructor: Jason Morton, Assistant Professor of Mathematics and Statistics

Teaching Assistant: Sara Jamshidi

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

Instructor: Sergei Tabachnikov, Professor of Mathematics, MASS Director

113 McAllister Building, Tuesday 10:10 a.m. - 12:05 p.m.*MASS Colloquium*

Instructor: Multiple invited speakers

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

## Course Outline

**Math 497C - Honors MASS Geometry**

**Lie Groups in Two, Three and Four Dimensions**

*Instructor:* Nigel Higson, Evan Pugh Professor of Mathematics*Teaching Assistant:* Qijun Tan

113 McAllister Building, MWRF 10:10 - 11:00 a.m.

*Description:* Lie groups arise from the continuous symmetries that are seen in nature and mathematics. Sometimes the symmetries are easy to appreciate (such as the continuous rotational symmetries of a circle, as opposed to the discrete, 3-fold rotational symmetries of an equilateral triangle) and sometimes they are not. This course will given an introduction to the theory of Lie groups, with an emphasis on examples in low dimensions, where many of the most interesting applications to mathematics and physics are to be found. Topics will include the basic properties of matrix Lie groups, Lie algebras and the exponential map; examples from real Euclidean space, complex Hermitian space, the quaternions and the octonions; the mechanics of rotating bodies; complexification; representations; applications to spin, the eightfold way, Lorentz transformations, and other things.

*Objectives:* The first objective of the course to will to teach the basic theory of Lie groups and Lie algebras. Beyond that, the course will present many examples where mathematical techniques from geometry, algebra, analysis and combinatorics interact with on another, and contribute to our understanding of the laws of nature.

*Reading:* No textbook will be used. Readings will be suggested by the instructor, and lecture notes will be developed and distributed as the course progresses.

**Math 497B - Honors MASS Analysis**

**Classical Mechanics and Calculus of Variations**

*Instructor:* Mark Levi, Professor of Mathematics*Teaching Assistant:* Adam Zydney

113 McAllister Building, MWRF 11:15 a.m. - 12:05 p.m

*Description:* Classical mechanics and calculus of variations lie at the foundation of the modern theory of dynamical systems. This is the field where geometry, differential equations and number theory interact with each other and with physics. I will describe examples of this interaction, along with the fundamental concepts and ideas of the subject, with numerous special examples and with many entertaining problems.

*Objectives:* I would like to give a clear intuitive understanding of the subject of classical mechanics and of calculus of variations, and to show how many ideas that appear mysterious or arbitrary are actually very natural and simple, if looked at in the right way. We will also solve many intuition-enhancing problems.

*Reading:* Mark Levi. Classical Mechanics with Calculus of Variations and Optimal Control: An Intuitive Introduction. ISBN-10: 0-8218-9138-3; ISBN-13: 978-0-8218-9138-4.

**Math 497A - Honors MASS Algebra**

**Introduction to Applied Algebraic Geometry**

*Instructor:* Jason Morton, Assistant Professor of Mathematics and Statistics*Teaching Assistant:* Sara Jamshidi

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

*Description:* Fundamentals of algebraic geometry. Polynomial rings, ideals, varieties. Affine and projective space, Segre varieties, secant varieties, and tensor rank. Group actions on varieties. Techniques for application including identifying algebraic varieties in nature, computational methods, and solving systems of polynomial equations. Applications to statistics, quantum information, and computational complexity.

*Objectives:*

-Learn basic concepts of algebraic geometry.

-Learn how to recognize when algebraic geometry could be used to study a problem and take the first steps in that direction.

-Acquire computational tools.

-Learn about current applications.

*Reading:* Ideals, Varieties, and Algorithms by Cox, Little, and O'Shea will be used in the first part of the course to supplement lectures so that students can acquire the needed definitions and basic techniques in algebraic geometry.

## Calendar of Events

Arrival Days | August 23, 24 |

MASS Orientation | August 24, 9:30 a.m. |

Classes Begin | August 24 |

Labor Day — No Classes | September 7 |

Midterm Exams | October 5, 6, 7 |

Thanksgiving Holiday — No Classes | November 22-28 |

Classes End | December 4 |

Study Days | December 5-10 |

Final Exams | December 11, 14, 16 |

MASS Graduation Ceremony | December 17, 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 4, 2015.**

## 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.