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.

## Courses

The following courses will be offered in the Fall of 2017:

*Elliptic functions and elliptic curves*

Instructor: Yuriy Zarkhin, Professor of Mathematics

Teaching Assistant: Dmitrii Pedchenko*Geometry of infinite dimensional spaces (functional analysis and its applications)*

Instructor: Moisey Guysinsky, Senior Lecturer

Teaching Assistant: Agnieszka Zelerowicz*Knot theory*

Instructor: Sergei Tabachnikov, Professor of Mathematics and MASS Director

Teaching Assistant: Caleb Springer*MASS Seminar*

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

Instructor: Multiple invited speakers

## Course Outline

**Math 497A - Honors MASS Algebra**

**Elliptic functions and elliptic curves**

*Instructor:* Yuriy Zarkhin, Professor of Mathematics*Teaching Assistant:* Dmitrii Pedchenko

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

*Description:* In this course we will study analytic, geometric and arithmetic properties of elliptic curves. From the analytic point of view elliptic curve is the special doubly-periodic (aka elliptic) meromorphic functions in one complex variable (with given lattice of periods) that may be viewed as counterparts of (periodic) trigonometric functions (like cotangent). From the algebraic point of view an elliptic curve is a smooth degree 3 curve on the projective plane. The arithmetic approach deals with the existence and construction of rational points on such curves and/or with solutions of congruences of degree 3. An interaction of different approaches makes the very subject of elliptic curves extremely deep and beautiful and opens the way for unexpected applications. Elliptic functions and elliptic curves play an important role in complex analysis, mathematical physics, algebraic geometry, number theory, mathematical logic and mathematical cryptography. A delicate analysis of elliptic curves played a crucial role twenty years ago in the proof of Fermat Last Theorem by Andrew Wiles.

*Reading:* L.C. Washington ``Elliptic curves: Number Theory and Cryptography", 2nd edition. Chapman & Hall, CRC Press, 2008.

**Math 497B - Honors MASS Analysis**

**Geometry of infinite dimensional spaces (functional analysis and its applications)**

*Instructor:* Moisey Guysinsky, Senior Lecturer*Teaching Assistant:* Agnieszka Zelerowicz

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

*Description:* In this course, we study how geometric methods could be used to understand properties of functions. Spaces of functions could be thought as infinite dimensional linear spaces. Geometry of those spaces will be studied using both analogies and insights coming from the familiar finite-dimensional situations (such as convexity) and new features associated with infinite dimension, e.g. reflexivity or its absence. We also see the applications to problems from analysis and differential equations.

**Math 497C - Honors MASS Geometry**

**Knot Theory**

*Instructor:* Sergei Tabachnikov, Professor of Mathematics and MASS Director*Teaching Assistant:* Caleb Springer

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

*Description:* In this course, we shall study geometry, topology, and combinatorics of knots and links. We shall start with the classical period (late 19th - first half of 20th century), and discuss tabulating of knots, unknotting numbers, linking numbers, knots and surfaces, operations on knots. Then we shall study quantum knot invariants, including knot polynomials and their applications, and relations with models of statistical mechanics. We shall also study knot invariants of finite type and their relation with Lie algebras. We shall present some applications of knot theory in biology and chemistry, including the structure of DNA.

*Reading:* C. Adams "The Knot Book", Amer. Math. Soc., 2004.

## Calendar of Events

Arrival Days | August 19, 20 |

MASS Orientation | August 21, 9:15 a.m. |

Classes Begin | August 21 |

Labor Day — No Classes | September 4 |

Midterm Exams | October 2 - 4 |

Thanksgiving Holiday — No Classes | November 19-25 |

Classes End | December 1 |

Study Days | December 2 - 7 |

Final Exams | December 8, 11, 14 |

MASS Graduation Ceremony | December 15 |

## 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 11, 2017.**

## 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.