# MASS 2018

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 2018:

• Polynomial interpolation: an introduction to algebraic geometry
Instructor:  Jack Huizenga, Assistant Professor of mathematics
Teaching Assistant:  Daniel Levine
• A mathematical journey with Fourier series
Instructor: Nigel Higson, Evan Pugh Professor of Mathematics
Teaching Assistant: Shintaro Nishikawa
• Comparison geometry
Instructor: Anton Pertunin , Professor of Mathematics
Teaching Assistant:  Sergio Zamora
• MASS Seminar
• Instructor:  Moisey Guysinsky, Teaching Professor
• MASS Colloquium
• Instructor:  Multiple invited speakers

## Course Outline

Math 497A - Honors MASS Algebra

Polynomial interpolation: an introduction to algebraic geometry

Instructor:  Jack Huizenga, Assistant Professor of mathematics
Teaching Assistant: TBA

113 McAllister Building, MWRF TBA

Description:  Classical Lagrangian interpolation describes when it is possible to assign the values of a single variable polynomial at fixed points.  We will study the aspects of linear algebra which are relevant to give a streamlined treatment of Lagrangian interpolation.  The proper generalization of Lagrangian interpolation to several variables is a very interesting question which is best studied by using the tools of multilinear algebra and algebraic geometry.  In contrast to the single variable case, the geometric positions of the points where values are to be assigned is highly relevant to the solution of the problem.  If the points are in "special" position--for example if some of the points are collinear--then it can become impossible to find a polynomial of the appropriate degree with the desired values at the points.  We will give an introduction to algebraic geometry focused on the aspects of the subject relevant to the interpolation problem.  Along the way we will study linear algebra, multilinear algebra (especially exterior algebras), algebraic varieties, Grassmannians, secant varieties, and specialization methods in algebraic geometry.  Additional related topics suitable for student projects include algorithms for fast matrix multiplication, tensor rank, topics in applied algebraic geometry, and the Alexander-Hirschowitz theorem.

Reading:  We will primarily use the course lecture notes as a resource, but several sources for additional reading will be suggested.

Math 497B - Honors MASS Analysis

A mathematical journey with Fourier series

Instructor:  Nigel Higson, Evan Pugh Professor of Mathematics
Teaching Assistant: TBA

113 McAllister Building, MWRF TBA

Description:  The theory of Fourier series is about writing possibly complicated periodic functions as infinite linear combinations of simple ones.  The simple ones are sin(x) and cos(x), as well as sin(2x) and cos(2x), and sin(3x) and cos(3x), and so on.  What makes these functions simple is that their derivatives are easy to understand.  What makes Fourier theory useful is that by expressing possibly complicated periodic functions as combinations of simple ones, it becomes possible to say something about their derivatives.  As a result, Fourier theory is useful nearly everywhere derivatives are useful, which is to say nearly everywhere.  In this course we shall develop the mathematical theory of Fourier series and study a diverse range of applications in mathematics and beyond.

Reading:  T.W. Körner, ``Fourier Analysis,“ Cambridge University Press, 1988.

Math 497C - Honors MASS Geometry

Comparison geometry

Instructor:  Anton Petrunin, Professor of Mathematics
Teaching Assistant: TBA

113 McAllister Building, MWRF TBA

Description:  This course is an invitation to differential geometry which is focused on problem-solving. We will study smooth curves in the plane and surfaces in space by comparing them with model curves and surfaces, for example, with straight line or circle, plane or sphere.

Reading:  We will primarily use the course lecture notes as a resource, but several sources for additional reading will be suggested.

## Calendar of Events

 Arrival Days August 18-19 MASS Orientation TBA Classes Begin August 20 Labor Day — No Classes September 3 Midterm Exams TBA Thanksgiving Holiday — No Classes November 18-24 Classes End TBA Study Days TBA Final Exams TBA MASS Graduation Ceremony TBA

## 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

Applications should be sent by mail, fax, or e-mail to

MASS Program
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 15, 2018.

## 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 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.