# Polynomial Interpolation: an Introduction to Algebraic Geometry

## Math 497A: Honors MASS Algebra -- Fall 2018

 Instructor: Jack Huizenga Office: 324 McAllister email: huizenga@psu.edu URL: http://www.personal.psu.edu/jwh6013/
 Teaching Assistant: Daniel Levine Office: 010 McAllister email: dul190@psu.edu URL: http://personal.psu.edu/dul190/
 Meeting times: MWF 11:15-12:05, 113 McAllister. Problem session/recitation: Th 11:15-12:05, 113 McAllister. First meeting: Monday, August 20 Jack's office hours: Monday 4-5, Tuesday 4-5, or by appointment. Daniel's office hours: Friday 3-4, or by appointment. Textbook: We will primarily use the course lecture notes as a resource, which I will write up as the semester goes on. Several additional sources for reading will also be suggested.
 Brief course 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.
 Prerequisites: MASS enrollment or consent of the instructor. The course will be fairly self-contained, but move at a rapid pace. Familiarity with proofs will be expected. It will be helpful to be comfortable with basic notions from theoretical linear algebra, but this will also be quickly reviewed.
 Homework: Weekly homework assignments will be posted on the webpage most Wednesdays, and they are due in class the following Wednesday at the beginning of class. Please write your solutions clearly and carefully. Homework is by far the most important part of the course. The only way to learn advanced mathematics is to do advanced mathematics.
 Collaboration: You are encouraged to discuss homework problems with your fellow students. However, you have to write up your solutions by yourselves and show originality. Please write the names of any students you collaborated with on your assignment.
 Attendance: Attendance in class and recitation is expected.
 Midterm: There will be a midterm exam on Monday, October1. The exact time and format of the exam will be announced later.
 Final Exam: A final oral exam will be scheduled for each student on December 10, 11 or 12. The exam will follow the standard MASS final exam format. More about that later.
 Final Project: Each student will be required to complete a project and present it during the final exam (this is part of the standard MASS exam format). A list of possible project titles will be distributed at about the time of the midterm, along with further details about the project requirement.
 Grading and Expectations: The homework and exams will count for the following portions of your grade. Homework: 50% Midterm: 20% Final exam and project: 30% Letter grades will be assigned based on the cumulative score. Grade ranges will be discussed after the midterm.
 Academic Integrity Statement: All Penn State policies regarding ethics and honorable behavior apply to this course.
 Disability Statement: Penn State welcomes students with disabilities into the University's educational programs. The Office for Disability Services Web site provides contact information for every Penn State campus: http://equity.psu.edu/ods/dcl . For further information, please visit the Office for Disability Services Web site: http://equity.psu.edu/ods .
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