Philadelphia Undergraduate Mathematics Conference Series

Highlights

• A plenary talk by William Dunham, Truman Koehler Professor Emeritus of Mathematics at Muhlenberg College and Research Associate in Mathematics at Bryn Mawr College
• A mini-course on coding theory by Kathryn Haymaker BMC '07, Assistant Professor of Mathematics at Villanova;
• A poster session for undergraduate and graduate students (any mathematics topic welcome!;
• Sessions for undergraduate research presentations;
• A professional development session.

Schedule

• 9 a.m. Registration and Coffee
• 9:20 a.m. Welcome
• 9:30 to 10:20 a.m. Plenary Lecture
• 10:30 to 11:20 a.m.  Mini-Course Lecture I
• 11:30 a.m. to 12:30 p.m. Poster Session
• 12:30 to 1:30 p.m. Lunch
• 1:30 to 2:45 p.m. Undergraduate 15-Minute Talks
• 3 to 3:50 p.m. Mini-Course Lecture II
• 4 to 4:30 p.m. Professional Development Session
• 4:45 p.m. Reception

Faculty Organizers

• Janet Fierson (La Salle University), Maria Lorenz, Irina Mitrea, and
• Ellen Panofsky (Temple University) and Bryn Mawr College math faculty

Euler in Two Acts

William Dunham

Research Associate in Mathematics

Bryn Mawr College

Leonhard Euler (1707 – 1783) is one of the towering figures from the history of mathematics. Here we look at two results that show how he acquired his lofty reputation. In the first, Euler considers the infinite series 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + … – i.e., the sum of reciprocals of the primes – and establishes that the sum “is infinite.” The proof from 1737 rests upon his famous product-sum formula and requires a host of analytic manipulations so typical of Euler’s work. The other result involves 1 + 1/4 + 1/9 + 1/16 + … – i.e., the sum of reciprocals of the squares. Euler first evaluated this in 1734, but here we examine his 1755 argument that uses l’Hospital’s rule, not once, not twice, but thrice! Euler has been described as “analysis incarnate.” These two theorems, it is hoped, will leave no doubt that such a characterization is apt. NOTE: This talk should be accessible to any mathematics major or minor.

Abstract: Mini-course in Coding Theory

Kathryn Haymaker

Assistant Professor of Mathematics

Villanova University

Whenever data (documents, pictures, or other files) are transmitted or stored, errors may occur. Error-correcting codes are used to strategically add redundancy to information so that the original message can be recovered. The study of these codes and their properties is called coding theory, and historically the field has capitalized on several areas of mathematics to design and study good codes. In the first part of this mini-course, we will start with the foundations of coding theory that were established by Claude Shannon and Richard Hamming. We will also see how more recent applications such as distributed storage and cloud computing are revolutionizing the subject of coding theory. In the second session, we will see how linear subspaces defined by matrices can provide a useful description of a code, and how to detect errors and decode using linear algebra techniques. We will study several algebraic constructions of codes and discuss their practical applications. Note: The first session will be accessible to a general undergraduate audience, and the second session will be accessible to those who have studied linear algebra.