Sample Colloquium Topics
"Mathematical Aspects of the Music of Bach"
Connect to "Mathematics/Bach"
"Comprehending Beauty in Mathematics and in Music"
- Please see under "Mathematics/Bach" for details.
Connect to "Comprehending Beauty"
"The Burnside Counting Theorem and Group Characters"
- Please see under "Comprehending Beauty" for details.
"Nearly So and Nearly Not"
- In 1911, W. F. Burnside published (in a form that is only scarcely recognizable when compared to modern notation) a remarkable theorem relating two abstract algebraic concepts that were then still in their youth: what we now refer to as orbits and group characters. Polya in 1957 and Liu in 1968, among others, showed how these concepts can be applied to more recent problems in mathematics. In this talk, Dr. Hill shows how the Burnside Counting Theorem can begin with a simple problem relating to the planning of a set of children's blocks and can extend to basic questions in group character theory, with applications to spectroscopy in chemistry. This talk assumes a basic background in linear algebra, but does not require a background in group theory; it is accessible to undergraduates in mathematics and the physical sciences.
"Rigor and Pitfall in the Work of Regiomontanus"
- The supposition that "true" and "false" do not exhaust the possible truth values that can be assigned to statements gives rise to many-valued logics. This talk explores some systems of logics with more than two truth values. No technical background is assumed, but some mathematical frame of mind is helpful.
"Story Problems: their History, Purposes, and Solutions"
- Johannes Mueller of Koenigsberg (1436-1476), who adopted the pen name of Regiomontanus, was probably the most significant and influential mathematician of the 15th century. The printing press and observatory that he set up at Nuremberg were intended to advance the interest of both science and literature. His major work, "De triangulis omnimodis" (1464, pub. 1533), may be regarded as the first mathematical treatise in Western Europe to rise above the elementary and imprecise writings of the preceding centuries. Still, this book curiously combines Greek deductive reasoning with arguments that are often at best not rigorous and at worst simply incorrect. This talk compares those elements in selected proofs from the work of Regiomontanus.
"Zero and the Null Set: a Mathematical Talk on Nothing"
- "Story problems" (or "word problems") have often been feared or hated by generations of students. Yet these problems have a fascinating history, sometimes progressing over centuries from the practical to the absurd, and often presenting additional problems of interpretation, assumptions, or cultural context. In this talk (which assumes only a bit of high school algebra as background), Dr. Hill categorizes these problems from PRACTICE to PREPOSTEROUS and surveys, in particular, the possibility that a given problem may, upon inspection, turn out to have many legitimate solutions.
- The concept of nothing, zero, came curiously late into the history of mathematics, and it gained intellectual acceptance against much resistance. Still, this idea turns out to be an intriguing thread through the long development of mathematical thought. The relationship between zero and the empty set leads to the remarkable construction in the 20th century of a full set theory out of the mathematical concept of nothing. In this talk, Dr. Hill traces the history of "nothing" in mathematics from prehistory to the present, with literary references from Homer to Hemingway.
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